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Valuation of Structured Products
Pricing of Commodity Linked Notes
Shahid Jamil, Stud nr: SJ80094
Academic Advisor: Jochen Dorn
Department of Business Studies
Aarhus School of Business, University of Aarhus
February 2011
1
Abstract
Structured products including commodity linked structured products have complex
composition. These are suitable for those investors who want a complete or partial
protection of their investment. A typical structured product is a combination of a risk
free bond and an option. The bond part guarantees capital protection while the option
part provides the possibility of payoff. The option pricing is the tricky part of these
products and a wide range of theories are available to price them. In this thesis the well
known Black & Scholes option pricing frame work is applied and the theoretical
estimated price of the selected commodity linked structured notes are compared with
their issue price to evaluate if these products are offered to the investors at fair price.
2
Table of Contents
1. Introduction .......................................................................................................................... 5
2. Structured Products .................................................................................................................. 8
2.1 Structured products are suitable for investors who .......................................................... 9
2.2 Disadvantage of structured Products ................................................................................ 10
2.3 Difference between a Conventional Bond and a Commodity Linked Bond ...................... 10
2.4 Commodity –Linked Bonds, a brief history ....................................................................... 11
2.5 Classification ...................................................................................................................... 12
2.5.1 Classic Products .......................................................................................................... 12
2.5.2 Corridor Products ....................................................................................................... 13
2.5.3 Guarantee Products ................................................................................................... 13
2.5.4 Turbo Products ........................................................................................................... 13
2.6 Products with exotic option components ......................................................................... 13
2.6.1 Barrier Products ......................................................................................................... 13
2.6.2 Rainbow Products ...................................................................................................... 14
2.7 Structure of structured products ...................................................................................... 14
2.7.1 The Bond Component ................................................................................................ 15
2.7.2 The Option Component .............................................................................................. 17
2.7.3 Swaps.......................................................................................................................... 17
2.7.4 Participation Rate ....................................................................................................... 18
3 Understanding Options ............................................................................................................ 19
3.1 Exotic Options ................................................................................................................... 19
3.2 Path dependent options .................................................................................................... 19
3.2.1 Asian options .............................................................................................................. 19
3.2.2 Lookback options ....................................................................................................... 21
3.2.3 Ladder options............................................................................................................ 21
3.2.4 Barrier options............................................................................................................ 21
3.3 Time dependent options ................................................................................................... 22
3.4 Multifactor options ........................................................................................................... 23
3.5 Payoff modified options .................................................................................................... 23
4 Option Pricing Theory ............................................................................................................... 25
4.1 Assumptions ...................................................................................................................... 25
4.2 Stochastic Process ............................................................................................................. 26
3
4.2.1 Properties of a stochastic process.................................................................................. 26
4.2.2 The Markov Property ................................................................................................. 26
4.2.3 Wiener Process ........................................................................................................... 26
4.2.4 Generalized Wiener Process ...................................................................................... 27
4.2.4 Geometric Brownian motion ...................................................................................... 28
4.2.6 Ito’s Lemma ................................................................................................................ 29
4.2.7 Risk Neutral Valuation ................................................................................................ 30
5 The Black- Scholes Equation (BS) ............................................................................................. 31
5.1 Options on dividend paying stock ..................................................................................... 35
5.2 Commodity Options .......................................................................................................... 35
5.3 Options on many underlying ............................................................................................. 36
5.4 Black- Scholes Pricing Formulas ........................................................................................ 37
5.5 Upper and Lower bounds for the call option .................................................................... 38
5.6 Forward Contract .............................................................................................................. 39
5.7 Futures contracts .............................................................................................................. 40
5.8 Futures Options ................................................................................................................. 42
5.9 Pricing of European futures options ................................................................................. 43
6 An overview of the selected products ..................................................................................... 45
6.1 DB Råvarer 2013 Basel (the “Notes”) ................................................................................ 46
6.1.1 Payoff Structure ......................................................................................................... 46
6.1.2 Risk Factors ................................................................................................................. 47
6.1.3 Issuance costs ............................................................................................................. 47
6.1.4 Embedded option ....................................................................................................... 47
6.1.5 The underlying asset ........................................................................................... 48
6.2 Analysis of Råvarer Basis 2010 .......................................................................................... 48
6.2.1. Payoff Structure ........................................................................................................ 48
6.2.2 Risk factors ................................................................................................................. 49
6.2.3 Issuance costs ............................................................................................................. 50
6.2.4 Embedded option ....................................................................................................... 50
6.2.5 Underlying Asset ........................................................................................................ 50
7 Pricing of the selected products ............................................................................................... 51
7.1 Pricing of DB Råvarer 2013 Basel ...................................................................................... 52
7.1.1 Deriving the Zero Coupons ......................................................................................... 52
7.2 Term Structure for Råvare.r Basis 2010 ............................................................................ 53
4
7.3 Option Pricing .................................................................................................................... 55
7.4 Estimating Volatility .......................................................................................................... 55
7.5 Monte Carlo Simulation .................................................................................................... 56
7.6 Variance Reduction Technique ......................................................................................... 57
7.7 Generating Random numbers ........................................................................................... 57
7.8 DB Råvarer Basel 2013 ...................................................................................................... 58
7.9 Pricing of Råvarer Basel 2010 ............................................................................................ 59
8 Evaluation of the Model ........................................................................................................... 61
8.1 Possible extensions to the thesis ...................................................................................... 62
9 Conclusion ................................................................................................................................ 65
References ................................................................................................................................... 67
5
1. Introduction
Structured products are drawing more and more investors now a day. Retail and
institutional investors alike are piling into these products. Structured products are
suitable for defensive or conservative investors because investments in structured
products assure a complete or partial protection of their invested capital but at the same
time, they can take advantage of the economic exposure to the growth potential of the
selected underlying. The underlying asset can be a single or a basket of the underlying
assets.
The popular structured products offer exposure to the equities, foreign exchange,
indices, volatility indices, commodities and commodity indexes such as S&P
commodity index, the Dow Jones-AIG commodity index or the Rogers International
commodities index. Commodity indices differ from the equity indices. The underlying
investments are not shares, bonds or the commodities themselves but “futures” contracts
on a single or a basket of commodities (a contract to buy or sell an asset at a given
future date for a set price). Futures contracts generally expire after three months;
therefore the so called rolling principal is applied for futures index where the index
sponsor replaces near to expiry contracts with the longer maturity contracts.
Structured products have become very advanced too in their structure. The complex
mechanisms within their structures are sometimes difficult to understand by the
investors and even sometimes by the financial managers too.
So, the theme of this thesis is to present an in depth analysis of the selected structured
products. The following will describe exactly what this thesis will be answering
• What are structured products and their composition?
• How the individual components of a structured product are valued. That is, how
are bonds priced (coupon and zero coupon bonds), and how the underlying
embedded options are priced (plain vanilla and exotic options)?
• What is the theoretical fair value of the selected structured products and whether
these products are offered to the investors at fair value, overvalued or
undervalued?
6
Structured products can be found in a wide variety of underlying assets. The underlying
assets can be from equities to equity indices, foreign exchange, interest rates,
commodities or commodity indices. This thesis will mainly focus on valuing
commodity linked structured products.
The contents of the thesis are split into two parts. The first section mainly focuses on the
theory behind the structured products. This section consist of chapters 2, 3, 4 and 5.
Section two consists of the valuation of the selected products and comparison of the
theoretical price with the issuing price of the selected products. Chapters 6, 7, 8 and 9
will be part of the second part of this thesis.
Chapter 2 begins with the introduction and definition of structured products and then
defining the commodity linked structured products. It will also describe in general how
structured products are engineered. This chapter will also discuss the advantages and
disadvantages of commodity linked products, their brief history, the difference between
a conventional bond and a structured bond and explanation of different concepts within
structured products.
Structured products normally consist of two components i.e. the bond and an embedded
option. The option component is generally the most tricky and complex part of these
products. Chapter three, four and five consist of the option theory including their
classification and the underlying concepts involved in option pricing in particular
stochastic process, geometric Brownian motion and generalized Wienner process. Black
and Scholes option pricing framework along with underlying assumptions and the
concept of risk neutral world will be discussed in chapter five.
Chapter six and seven deal with the pricing issues of the selected products. Chapter six
begins with the analysis of prospectus of the selected products. While chapter seven
starts with pricing of the bond and option part of these structured products. Valuation of
the option components start with an overview of the Monte Carlo simulation. The
qualitative and quantitative comparison of theoretical and issue price of the products is
performed. Chapter eight starts with a critical evaluation of the underlying assumptions
and their effect on valuation model. This thesis will give an understanding of how the
structured products are priced and therefore will trigger readers interest to find
improvements in the pricing model and possibly to use other pricing models too. So, a
7
brief description of the alternative option pricing models is also discussed in chapter 8.
Finally, the thesis ends up with a brief conclusion in chapter 9.
Although, structured products are available with a wide variety of underlying assets, but
this thesis will focus on commodity linked products. The products are selected from the
Danish market, therefore, interest rates will also be considered from the Danish market.
The valuation will be performed by applying the well known Black & Scholes option
pricing theory because the main theme of this thesis is the valuation of the selected
products and not to evaluate the performance of different option pricing models.
Therefore, other advanced option pricing models are not considered in this thesis. In
order to price the option component in Black & Scholes frame work, Monte Carlo
simulations are applied which follows the principal of risk neutral random walk. Tax
issues will not be considered in the valuation process. Default risk of the issuing firm
will be also disregarded because the selected products in this thesis are from the issuers
with very good credit ratings.
The data for this thesis has been downloaded directly from the Dow Jones UBS
commodity index web site, while the data for deriving zero coupon term structure was
down loaded from data stream.
Finally, I would like to thank my advisor Jochen Dorn for his use full guidance and
patience during the thesis.
8
2. Structured Products
Structured products have emerged as an important instrument in financial markets. A
structured product can be defined as a security that combines the features of a fixed
income security with the characteristics of a derivative transaction. Generally, a
structured product contains two components i.e. a fixed income security (a zero coupon
bond that guarantees full or part of the invested capital) and an option or forward – like
instrument which has a specific class of the asset as an underlying. The underlying
assets can be equity, interest rate, an index, inflation, foreign exchange, commodities or
credit. The underlying can be a single asset or a basket of multiple assets. The additional
payoff of a structured product depends on the performance of the underlying asset.
Structured products are also said to be a ‘marriage of a fixed income security and an
option like instrument1
When the underlying in a structured product is a commodity or a basket of commodities
or a commodity index then they are called commodity- linked structured products. The
underlying commodities can be for example crude oil, gas oil, metal (gold, silver,
copper, and precious metals), energy etc.
’.
Commodity indices are different from the other indices. The underlying investments are
not bonds or shares or the commodities themselves but it is ‘futures’ contracts on the
commodities. A futures contract is defined as the contract which gives its holder the
right but not the obligation to buy or sell an asset (commodities, equity, foreign
exchange etc) at a given future date at an agreed price. Futures have normally three
months expiry date. These expiry dates are normally standardized. The index sponsor is
therefore required to replace the expiring futures contracts with the new ones traded on
the futures market (every three months). This is called the ‘rolling’ principle. When a
futures contract expires, the index will treat it as sold and the proceeds are reinvested in
the new futures contract that will again expire after the next three months.
1 BNP Paribas equities & Derivatives handbook
9
The index level takes care, the price movements in the underlying commodities and
takes into account the price difference between the old and the new futures contract that
are rolled.2
Commodity linked structured products are available in a wide variety of range. One of
them is Commodity- Linked bonds/ Notes, which is also the topic of this thesis.
Commodity linked notes or bonds are classified into two classes, i.e. The Forward type
and the option type. In a forward type bond, the coupon and or principle payment to the
bearer are linearly related to the price of the stated reference commodity i.e. it allows
the holder to receive either the nominal face value or the designated commodity amount
at maturity . While, an option type bond, the coupon payments are similar to that of a
conventional bond but at maturity, the bearer receives the face value plus an option to
buy or sell a predetermined quantity of the commodity at specified price
3
2.1 Structured products are suitable for investors who
. In literature
both the terms (bonds and note) are used interchangeably.
4
1. Want protection of their invested capital by hedging the risk of existing
investments.
2. Want to enhance the return from their investment while controlling risks,
whereby the structure is designed to enhance equity return with leverage.
3. Want to diversify with the adjustable risk/ return profiles and market cycle
optimization capabilities of structured products.
4. Want to exploit their market view with more freedom and flexibility.
5. Want a growth by capitalizing on the market upside while protecting the
downside.
6. Want to benefit from periodic returns with limited risks. This income type of
structure is built to deliver coupons while protecting capital
2 Barclays Wealth, Light Energy Commodity Plan 2,3 Joseph Atta- Mensah. Commodity- Linked Bonds 4 BNP Paribas equities & Derivatives handbook
10
2.2 Disadvantage of structured Products
Despite the fact, that structured products including commodity linked products provide
capital protection and a possible payoff from the option component, the investor still
loses the payoff associated with a traditional risk free bond. In a structured product, the
investor receives only the invested capital if the option expires out of the money, but in
a traditional bond, the investor receives a risk free interest of his invested capital along
with the invested capital. This lost risk free interest or profit is called the Opportunity
cost and can be defined as the “forgone risk-free rate of return that could have been
achieved if the principal would have been invested in the safe fixed- income securities
such as Treasury bills”5
For example, if an investor invests 100 DKK in treasury bonds for one year with a 5%
interest rate. He will receive DKK105 at maturity while if he invests in a structured
product, he will only get DKK 100 at maturity plus a possible payoff from the option
embedded in it, because he will actually invest 100*1,05^-1 = DKK 95 in the risk free
investment and DKK 5 will go to buy a call option plus administration fee . This DKK 5
from investment in risk free bond will be the opportunity cost that he will miss if he
would invest in a structured product. The payoff of a traditional bond will exceed as
long as the option component of a structured product is out of the money, at the money
or if it is in the money but still below DKK105.
.
2.3 Difference between a Conventional Bond and a Commodity Linked Bond6
Commodity linked bonds are different from conventional bonds in many aspects. Some
of the key differences between are
1. In conventional bonds, the investor receives fixed coupon payments i.e. interest
payments during the life cycle of the bond (annually or semiannually etc) and
the face value at the maturity of the bond. While the holder of a commodity
linked bond receives the physical units of the underlying commodity or
equivalent of its monetary value. Similarly the coupon payments may or not be
5 Lehman Brothers, A guide to Equity_Linked Notes
11
in units of the underlying commodity (it depends upon the performance of the
underlying).
2. The nominal return of a conventional bond held to maturity is known while the
real return is not known because of inflation. On the other hand both the real and
the nominal return of a commodity linked bond are unknown.
3. The results of Atta- Mensah study also show that the coupon rate for a
conventional bonds are greater than that of the commodity linked bonds whose
terminal payoff is greater of the face value and monetary value of a pre-specified
unit of a commodity.
4. The coupon rate of a conventional bond is less than that of a commodity linked
bond that pays its holders on maturity the minimum of the face value and the
monetary value of a pre-specified unit of a commodity.
2.4 Commodity –Linked Bonds, a brief history
The concept of structured note is considered to be relatively new in the financial
markets. In reality these products have been in existence for a considerable time. For
example callable notes and equity linked securities i.e. convertibles and debt with equity
warrants are the precursors to the structured note products that are common place today.
Commodity linked bonds were introduced during 1970’s when the oil backed bonds
were issued by the Mexican Government in the financial market. These bonds were
called Petrobonds. Each 1000 Peso bond was linked to 1.95353 barrels of oil with a
coupon payment of 12. 658% annually and had three years to maturity. Later on the
French Government issued gold backed bonds during 1973. They were known as
‘Giscard’ in the financial markets. These bonds have 7% coupon rate and redemption
value was indexed to the one kilogram bar of gold. In 1981 Eco Bay Mines Company of
Canada also issued gold warrants. Commodity linked bond with sliver as an underlying
was issued during 1983 and again in 1985 by the Sunshine Mining Company in USA.
The objective was to hedge against the fluctuations in the price of silver.
Later on, bonds indexed to other commodities like nickel, copper, silver, cobalt and
platinum were issued during 1984 by Inco. (a leading producer of metals). Now a days,
commodity linked bond are issued by many investment banks around the world. These
bonds are linked to the performance of a basket of specific commodities or commodity
12
index for example Goldman Sachs Commodity Index (GSCI), London Metal Exchange
(LME), S & P commodity index or Dow Jones UBS commodity index.
2.5 Classification
Structured products including commodity linked structured products are available with
a wide variety of product characteristics and heterogeneous characteristics in the
market. However, Pavel A Stoimenov and Sascha Wilkens, in their article about the
equity linked structured products in the German market, “Are Structured Products
Fairly Priced”? have classified them according to the underlying option components
embedded in the product. As is shown in the figure 1, structured products can be
divided into two categories i.e. Plain vanilla Option- Component and exotic Option-
component. Plain vanilla products are further classified into Classic, Corridor,
Guarantee and Turbo while Exotic components products are further classified into
Barrier and rainbow. Barrier structured products are further divided into Knock- in,
Partial Knock- in and knock – out products. The brief definitions of these products are
discussed below.
Figure 1
2.5.1 Classic Products
A classic structured product has the basic characteristics of a bond except that the issuer
has the right to redeem it at maturity by repayment of its nominal value or delivery of
previously fixed number of specified shares. In general, structured products can be
13
categorized into with and without coupon payments. Products with coupon payments
are called as ‘reverse convertibles’, while those without coupon payments are named as
‘discount certificates’.
2.5.2 Corridor Products
The payout of a corridor structured product depends on whether the underlying at
maturity is quoted within a certain range.
2.5.3 Guarantee Products
A guarantee product is similar to that of a corridor product. The only difference is that
in a guarantee product, fixed minimum repayments are guaranteed to the investor. So, if
price of the underlying falls below the reference value, then the investor will always get
the guaranteed amount.
2.5.4 Turbo Products
The payout of a turbo product is doubled if the underlying is quoted within a certain
price range at maturity. This is called turbo effect. But there are three possibilities at
maturity. If, for example, L and K are lower and upper reference prices, then at
maturity, if
1. St fixing ≤ L, the product is redeemed in shares;
2. L < St fixing < K, a cash settlement with s(2 St fixing - L) occurs;
3. K ≤ St fixing, the maximum amount s (2K - L) will be paid.
2.6 Products with exotic option components
2.6.1 Barrier Products
Barrier products are the most common type of structured products, where the embedded
option is a barrier one. The redemption of a barrier product depends upon whether the
underlying reaches a certain fixed price barrier during its lifetime. In a knock in product,
if the underlying reaches or crosses a fixed pre specified lower price barrier, then the
stocks are delivered at maturity. In such a case the product behaves like a classic
product. A knock in pays the maximum amount if the underlying is always above this
barrier regardless of the St Fixing . In the case of a knock out product, if the underlying
14
reaches or crosses the pre specified upper price barrier, then the issuer loses his choice
of redemption and the products behaves like a regular bond in this case. In a partial time
knock in product, the barrier criterion is tested only within a certain time interval,
generally a few months immediately before maturity.
2.6.2 Rainbow Products
The rainbow products have more than one underlying. In a rainbow product, the issuer
has the right to choose between the specified underlying on redemption.
2.7 Structure of structured products
Commodity linked notes like other types of structured products provide partial or 100%
capital protection depending upon the investor’s specific needs. A typical commodity
linked bond provides 100% capital protection at maturity independent of the
performance of the underlying commodity or commodity index. The structure of a
simple commodity linked note can be sketched as
This figure shows that when an investor buys a structured product (equity, commodity
or any other asset as an underlying), he/ she actually has bought a package which
consists of a bond and an option or a swap (forward or future) and the fee on top of it.
The payoff of a structured product (note) is equal to the par amount of the note plus a
commodity / equity etc linked coupon. The payoff is either
1. Zero, if the underlying has depreciated from the initial agreed upon strike level
Bond Commodity linked Note Principal
Payoff Option/ Swap
15
2. Or the participation rate times the percentage change in the underlying
commodity/ equity times the par amount of the note7
In order to understand how a structured note works, we consider a simple example,
where an investor wants to invest 100 DKK over five years with full capital protection
and exposure to the S & P Commodity index. This means that the investor will get at
least his/ her 100 DKK at the end of five years no matter if the index depreciates or
appreciates. The investment bank will apply the appropriate interest rate (treasury
bonds, LIBOR or REPO rate) and find out the present value of 100 DKK (future value).
For example, if the five years interest rate was 5 %. Then the present value of this five
years zero coupon bond will be 78.35 DKK (100 * 1.05-5) today. It means that the
structure provider (investment bank) will have 100-78.35 = 21.65 DKK to purchase an
option or a futures/ forwards contract. Now let’s consider that a five – years S & P call
option costs 23.65 and 2 is the administration and margin costs, then the investor will
benefit from 83. 87 % ((21.65 – 2) / 23.65) participation in the S & P index’s upside.
.
2.7.1 The Bond Component
The bond component of a structured product is the most important part of it. It is also
the major part of any structured product. The bond component ensures that the investor
will receive the agreed amount of his/ her investment at maturity. The agreed amount
can be a 100 % of the invested capital or it can also be partial protection depending
upon the product. Structured products in general have the characteristics of a zero
coupon bond but it can also have coupon payments (annual or a semi- annual). The
main advantage of a zero coupon bond is that the investor gets all his investment back at
the same time instead of coupon payments at the end each period (annual or semi-
annual). The risk free interest rate applied to a zero coupon bond ensures that the
present value of the investment will grow continuously until maturity. The risk free
interest rate is normally taken from the government bonds (the rate at which the state
borrows money).
If an amount A is invested for n years at an interest rate R per annum and if R is
compounded once per annum i.e. m=1, then the terminal value of the investment will be
7 Lehman Brothers ( equity – Linked Notes)
16
𝐴(1 + 𝑅𝑚
)𝑛𝑚
If 𝑚 → +∞, we compound more and more frequently, then we obtain the well known
compounding frequency interest rates and the future value or the terminal value will be given as
𝐴𝑒𝑅𝑛
In the same way the present value of a future amount can be written as
𝐴𝑒−𝑅𝑛
The pricing of a zero coupon bond or any other fixed income security can be derived if
we know the zero coupon (ZC) yield curve. The term structure of ZC rates (also known
as ZC yield curve) is the curve relating maturities t (time horizons) with the
corresponding ZC interest rate R(t).
𝑃𝑟𝑖𝑐𝑒 = ∑ 𝐹𝑡(1+𝑅(𝑡))𝑡
𝑇𝑡=1 = ∑ 𝐹𝑡𝑇
𝑡=1 𝐵(𝑡) (1)
Here
• B(t) means the discount factor at time t ( the prices of zero coupon rates with face
value of 1)
• R(t) is the zero coupon rate derived from B(t) and
• F (t) is the known cash flow or also called the principal amount.
When a structure note/ bond have the features of a coupon bond, then it can be
considered as the portfolio of zero coupon bonds. The price of such a bond can be
written as the present value of the sum of all cash flows (coupon payments) for each
period plus the principal amount and can by the following expression
𝑃𝑟𝑖𝑐𝑒 = ∑ 𝐶𝐹𝑡𝑛 .𝐵𝑡 + 𝐹𝑡𝐵𝑛 = ∑ 𝐶𝐹𝑡𝑛 . (1 + 𝑅𝑡)−𝑡 + 𝐹𝑡 (1 + 𝑅𝑡)−𝑛 (2)
Where,
• CFt is the cash flow in time t
17
2.7.2 The Option Component
The option component is the 2nd part of a structured product. Option component
provides the chances of payoff. Options are of two types i.e. a call and a put. A call
option provides its holder the right to buy an asset at a certain date on a pre specified
price while the put option gives its holder the right to sell an asset at a pre specified
price on a certain date. Call options are normally embedded in structured products
because it is easy to earn on something whose price is increasing rather than decreasing
as in case of a put option. Option component is also the risky part of any structured,
because the payoff depends on the performance of the underlying. If the option
embedded in a structured product expires out of the money (i.e. the strike price of the
call option is higher than the corresponding price of the underlying) than it will not be
exercised and the holder will get no profit but instead loose the money to buy that
option but he will still receive his invested capital. If on the other hand, an option
expires in the money, then it will be exercised and the holder will earn profit along with
the guaranteed capital. The expression at the money means that when the strike price of
the call option is equal to the price of the underlying.
There are two possibilities to exercise an option. In a European style option, the holder
can exercise his right to buy or sell an asset only at the maturity of the option. In an
American style, the holder can exercise his right to buy or sell the underlying before the
maturity of the option too.
2.7.3 Swaps
Commodity linked structured products can also be found with swaps in their structures.
Forwards are an example of swap and commodity swaps are in fact a series of forward
contracts on a commodity with different maturity dates and the same delivery prices.8
The commodity linked products as mentioned by Schwartz9
8 John C Hull: P-173 Option, Futures and Other derivatives
issued for example by
Sunshine company during 80’s or by the Mexican Government during 1979 backed by
silver and oil respectively are examples of the products with forward type component in
the structured product, where the company promised to pay either the face value or
market value of the underlying commodity.
9 Eduardo S. Schawartz: The Pricing of Commodity Linked bonds
18
2.7.4 Participation Rate
Participation rate determines that how much the product will participate in the
performance of the underlying. It can be defined as the exposure of a product to
movements in the price of its underlying. A participation rate of 100 % means that the
investor would receive the return that will be exactly equal to the rise in the price of the
underlying. For example if the underlying has increased by 25% at maturity, then the
investor will also receive 25% return. But if it is low as mentioned in the example on
page 9 i.e. 87.83 % then the investor will get DKK 21(87.83% * 25).
Participation rate depends on the value of the option embedded in the structured
product, the administration and other issuing costs and the present value of the bond
component of the product. The participation rate depends on many factors. For example
if the issuing costs of the product are low then the participation rate can be higher.
Similarly, if the value of embedded option is high/ low then the participation rate can
lower/ high. Participation rate is generally not set prior to the expiry of the issuance
period and it appears as estimate in the prospectus. The participation rate can be
calculated by the following relation
𝑃𝑎𝑟𝑡𝑖𝑐𝑖𝑝𝑎𝑡𝑖𝑜𝑛 𝑟𝑎𝑡𝑒 =𝑖𝑠𝑠𝑢𝑒 𝑝𝑟𝑖𝑐𝑒 − 𝑐𝑜𝑠𝑡𝑠 − 𝑃𝑉 𝑜𝑓 𝑏𝑜𝑛𝑑
𝑃𝑟𝑖𝑐𝑒 𝑜𝑓 𝑜𝑝𝑡𝑖𝑜𝑛 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡∗ 100 (3)
Participation rate is also named as Gearing. The above equation also shows that there
other factor which determines the participation rate. For example, the interest rate used
to calculate the present value of the bond component, the life time of the product and
volatility of the underlying asset. For example a low interest rate will result in high
present value of the bond component and can reduce the participation rate and vice
versa. Similarly volatility of the underlying asset can also affect participation rate. If the
volatility of the underlying asset is lower, consequently the option will have lower value
and ultimately a higher participation rate. Cheaper options embedded in the structured
products also result in high participation rate. For example, exotic options are generally
cheaper than plain vanilla options. Therefore, now a day exotic options are generally
embedded in the structured product which increase the participation rate and can result
in higher payoffs at the end.
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3 Understanding Options
Options are classified into plain vanilla and exotic options. Plain vanilla options are
standard options while the exotic options are complex in nature. The complex options
have low prices as compared to the standard options. Therefore, exotic options are
generally embedded in structured products, which also make the products interesting for
investor’s point of view. Some examples of exotic options are barrier, chooser, look
back, Asian, Himalayan and basket options. Their detail will be discussed later on. If
the price of an embedded option/ options is lower, then the participation rate will be
higher and more payoff for the investor.
3.1 Exotic Options
An option whose characteristics, including strike price calculations/ determinations,
payoff characteristics, premium payment terms or activation/expiration mechanisms
vary from standard call and put options or where the underlying asset involves
combined or multiple underlying assets are called exotic options (Das2001, p718).
Exotic options are also called thirds generation risk management products. Although it
is hard to classify all the options, but they can be roughly divided into five to six
categories.10
3.2 Path dependent options
In path dependent options the final payoff depends on particular path that asset prices
follow over their life rather than asset’s value at expiration. The path of the underlying
determines payoff and structure of the options. Path dependent options are further
divided into weak and strong path dependent options. In strong path dependency, the
payoff depends on some property of the asset price path along with the value of the
underlying at present moment of time and some other extra variable (Wilmot 2007,
p252). Examples of strong path dependent options are
3.2.1 Asian options
Asian options are examples of strong path dependent options. In Asian options the
payoff is determined by comparing the strike/ spot price of the underlying with the
10 Das divided exotic options into five classes while Wilmott into six categories.
20
average value of strike/ spot price during a specific period of time. They are strongly
path dependent because their value prior to expiry depends on the path taken and not
just where they have reached. Asian options were originated from Tokyo office of the
bankers trust in 1987. Asian options are normally cheaper than the plain vanilla options
because averages are less volatile and therefore less risky. Average can be calculated by
means of arithmetic, or geometric average of the prices. In Asian options
• There is a specific period over which the prices are taken. End of the averaging
interval can be shorter than or equal to the options expiration date, the starting
value can be any time before. In particular, after an average option is traded, the
beginning of the averaging period typically lies in the past, so that parts of the
values contributing to the average are already known.
• The market generally uses discrete sampling, like daily fixing.
• Weighting different weights may be assigned to the prices to account for a non-
linear, i.e. skewed, price distribution
• The wide range of variations covers also the possible right for early execution.
Asian options are popular in risk management for currencies and commodities because
they provide protection against rapid price movements or manipulation in thinly traded
underlying at maturity, i.e. reduction of significance of the closing price through
averaging. These options reduce hedging costs because they are cheaper than standard
options. Average Price Options can be used to hedge a stream of (received) payments
(e.g. a USD average call can be bought to hedge the ongoing EUR revenues of a US
based company). Different amounts of the payments can be reflected in flexible
weights, i.e. the prices related to higher payments are assigned a higher weight than
those related to smaller cash flows when calculating the average. With Average Strike
Options the strike price can be set at the average of the underlying price which is a
helpful structure in volatile or hardly predictable markets.
An average price call pays (AT – K) +, where AT denotes the geometric or arithmetic
average price of the underlying𝑆𝑡𝑖. The geometric average of the underlying can be
calculated as
𝐺𝑒𝑜𝑚𝑒𝑡𝑟𝑖𝑐 𝑀𝑒𝑎𝑛 = �∏ 𝑆𝑡𝑖𝑛𝑖=0
𝑛 (4)
21
And the arithmetic or the simple average can be calculated as
𝐴𝑇 = 1𝑛
∑ 𝑆𝑡𝑖𝑛𝑖=0 (5)
Week path dependency means that the option depends only on the underlying price and
the time. Barrier options are examples of week path dependent options. The payoff in
these options depends on if the underlying hits a pre specified value at some time before
expiry.
3.2.2 Lookback options
In these options the purchaser has the right at expiration to set the strike price of the
options at the most favorable price for the asset that has occurred during a specified
time. In a lookback call option, the buyer can choose to buy the underlying asset at the
lowest price that has occurred over a specified period, typically the life of the option.
Details about lookback options can be found in Fx options and structured products by
Uwe Wystrup.
3.2.3 Ladder options
The strike price in these options is periodically reset based on the underlying evolution
of the asset price. A ladder option can be identical to lookback when the amount of reset
is set to infinity.
3.2.4 Barrier options
Barrier options are weekly path dependent options. Das also classified them as limit
dependent options because their payoff depends on the realized asset path via its level.
Certain aspects of the contract are triggered if the asset price becomes too high or too
low. For example, an up- and – out call option pays off the usual max (S-K, 0) at expiry
unless at any time previously the underlying asset has traded at a value Su or higher. It
means if the asset reaches this level then it is said to ‘knock out’ and become worthless.
The option can also be “knocked in” instead of “Knock out”, where the payoff is
received only if the level is reached (Wilmott 2007, P288). Barrier options can be
divided into two types (out option and in option) i.e. up- and – out, down- and- out, up-
and- in and down- and- in.
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• The ‘out option’ pays off only if a specified level is not reached. Otherwise the
option is said to have knocked out and becomes worthless.
• The ‘in option’ pays off as long as the level is reached before expiry. If the
barrier is reached then it is said to have knocked in. In options contracts starts
worthless and only become active when the predetermined barrier is reached.
If the barrier is set above the initial asset value then it is said to have an ‘up option’ and
if the barrier is set below the initial asset value then it is said to have ‘down option’
Barrier options generally are of American style. It means that the barrier level is active
during the entire duration of the option: any time between today and maturity the spot
hits the barrier, the option becomes worthless. If the barrier level is only active at
maturity the barrier option is of European style and can in fact be replicated by a
vertical spread and a digital option.
Apart from a lower or an upper barrier, double barrier options are also available. Double
barrier options have both upper and lower barrier. In double ‘out’ option the contract
becomes worthless if either of the barriers is reached. In a double ‘in’ option one of the
barriers must be reached before expiry, otherwise the option expires worthless.
In some cases a so called rebate is paid if for example in an ‘out’ option the barrier level
is reached. The rebate may be paid as soon as the barrier is triggered or not until expiry.
The above mentioned barrier options are standard in nature. The barrier options can also
have exotic type features for example resetting of barrier, outside barrier options, soft
barriers and Parisian options. Detail discussion can be found in Wilmott 2007, p300.
3.3 Time dependent options
In time dependent options the buyer has the right to nominate a specific characteristic of
the option as a function of time for example the expiration of the option. Preference or
chooser option is an example of time dependent option. In a chooser option, at a
predetermined date (normally after commencement and before expiry) the buyer can
choose if the contract should be a call or a put option. Bermudan options are also
example of time dependent options, where early exercise of the option is possible on
certain dates or periods.
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3.4 Multifactor options
In multi factor options, the payoff depends on the relationship between multiple assets.
It means there is second source of randomness such as a second underlying asset.
Compound, basket, exchange, quanto, rainbow are the examples of multifactor options.
In Compound options (options on options), the holder has the right but not an obligation
to buy or sell another predetermined options at a pre agreed time. The compound
options can be a call on a call, a put on a put, a call on a put and put on a call.
Compound options have two strike prices and two exercise dates. For examples in a call
on a call option, on the first exercise date T1, the holder of the compound option is
entitled to pay the first strike price K1, and receive a call option. This call option gives
him the right to buy the underlying asset for the second strike price K2 on the second
exercise date. The compound option will be exercised on the first exercise date only if
the value of the option on the date is greater than the first strike price.
In basket options the payoff is based on the cumulative performance of the underlying
assets and in exchange options the holder has the right to exchange one asset for
another. The underlying assets can be individual stocks or stock indices, currencies or
commodities etc. If the payoff is determined on performance of maximum or minimum
of two more underlying assets, then these option are named as Rainbow option. A
quanto option can be any cash-settled option, whose payoff is converted into another
currency at maturity than that of the underlying asset at a pre-specified rate, called the
quanto factor. There can be quanto plain vanilla, quanto barriers, quanto forward starts,
quanto corridors, etc.
3.5 Payoff modified options
These options entail adjustment to the linear and smooth payoffs that are associated
with conventional options (Das 2001, p723). Examples include
• Digital options: Digital options have discontinuous payouts irrespective of the
normal options whose payoffs are smooth. In a normal option, if it is further in
the money the higher the payout to the purchaser. While in digital option the
payout is normally fixed provided if certain conditioned are met. For examples,
in the typical structure of digital option, if the strike price is reached the payouts
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are fixed predetermined amounts no matter how much the option is in the
money.
• Contingent premium: A contingent premium option is basically a European
option. The premium will be paid to the writer if the contingent premium option
finishes "in the money". Otherwise, if the option expires "at the money" then, no
premium will be paid. It means no premium is paid in the beginning of the
contract and is due at expiration of the options only if it expires in the money. In
other words, the contingent premium structure is a combination of a
conventional option and a digital option.
• Power options: A power option is a derivative with payoff depending on the asset
price at expiry raised to some power α, where α is higher than 1.
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4 Option Pricing Theory
This section will discuss the underlying concepts in Black & Scholes option pricing
theory. For example, the assumption of model, stochastic process, Markove property,
generalized Wienner process, geometric Brownian motion, Ito’s lemma, risk neutral
evaluation and finally the Black & Scholes option pricing formulae are discussed.
4.1 Assumptions
The famous Black and Scholes (B&S) model has several underlying assumptions like
other option pricing models. The understanding of these assumptions will help to
analyze the advantages and the drawbacks of the model. The underlying assumptions
are discussed below
1. The markets are efficient i.e. the markets are assumed to be liquid. There is price
continuity. The markets are fair and provide all information to all the players. It
means no transaction costs in buying or selling stock or options.
2. The underlying is perfectly divisible and short selling is allowed. A seller who
does not own a security will simply accept the price of the security from a buyer
and will agree to settle with buyer on the same future date by paying him an
amount equal to the price of the security on that date
3. There are no costs of carrying to the commodity (evaporation, obsolescence,
insurance etc) and that the commodity is held for speculative purposes like a
stock.
4. The commodity price, firm value and the interest rate follow continuous time
diffusion processes. It means that the interest rate is known and it is constant
(risk free) through time (Schwartz 1982). In other words there exists a risk free
security that returns $1 at time T when 1e-r(T-t) is invested at time T.
5. The stock/commodity price follows a random walk in continuous time
(geometric Brownian motion) with a variance rate proportional to the square of
the price. Thus the distribution of the possible stock/ commodity price at the end
of any infinite interval is log normal and the variance rate of the return on the
stock/ commodity is constant
6. The model deals with European style options only that can be exercised at
maturity only.
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4.2 Stochastic Process
We know that the prices of commodities, stocks and interest rates etc change over time
in the financial markets. If the change in value is uncertain over time i.e. if the change in
price of a commodity, equity or currency exchange is unpredictable over time then this
kind of price behavior is called a stochastic process. In other words any variable whose
value changes over time in an uncertain way is said to follow a stochastic process (Hull
2008, p 259) and it can be discrete time and continuous time stochastic process. In a
discrete time process the value of a variable is assumed to change at fixed time intervals
of time, while changes can take place at any time in a continuous time stochastic
process.
4.2.1 Properties of a stochastic process
4.2.2 The Markov Property
A stochastic process is said to have the Markov property, when only the present value
of a variable is relevant to predict its future value (Hull 2008, p 259) i.e. the process has
no memory beyond where it is now. It means that the past history of that variable and
pattern of changes in value would be irrelevant to predict future prices. So it means that
to predict the future price of a commodity bundle, the only relevant price will be the
today’s price and it will be independent of its price during the last week or year.
4.2.3 Wiener Process
Wiener process is a particular type of Markov process which has a mean change of zero
and a variance rate of 1.0 per year. Wiener process is also called Brownian motion
(named after a Scottish botanist Robert Brown). Brownian motion has been used in physics
to describe the motion of the particle that is subject to a large number of small molecular
shocks. It is among the simplest type of continuous stochastic process. In mathematical finance,
this concept was first used by Louis Bachelier during the 1900 in his PHD thesis, where he
presented the stochastic analysis of stock and option markets
A variable Z follows a Wiener process if it has the following two properties (Hull 2008,
p261)
1. The change ∆Z during a small period of time ∆t will be
27
∆𝑍 = 𝜀 √∆𝑡 (6)
Here, ε has a standardized normal distribution with mean zero and standard deviation of
1, that is; N (0, 1).
2. The values of ∆ Z for any two different short interval of time ∆t are independent.
It means that Z has independent increments and ∆Z 1 is independent of ∆Z2 if ∆t1
does not overlap with ∆t2.
The first property shows that ∆ Z has a normal distribution with i.e.
Mean of ∆ Z = Z (t) – Z (0) = 0
Standard deviation of ∆ Z = √∆𝑡
And variance of ∆ Z = ∆ t
The Wiener process is both the Markov and Martingale process (zero drift stochastic
process). By martingale process, it means that the expected value at any future time is
equal to its value today. Martingale property is an important part of the risk neutral
evaluation.
4.2.4 Generalized Wiener Process
It is clear from the Wiener process that if we choose it as a model then the stock/
commodity price can take negative values at any point in time with a probability of 0.5
and it will have a constant zero mean and it is not an ideal model to price stock prices.
So we have to consider a better model called Generalized Wiener Process. The basic
Wiener Process also states that ∆Z has a zero drift rate and a variance rate of 1.0. Zero
drift means that the expected value of Z at any future time is equal to its current value
and the variance rate of 1.0 means that the change in a time interval of length T equals
T. Here we consider a discrete time random walk
X0 = x, Xi = Xi-1 + a ∆t + b √∆t εi where ε i ~ N (0, 1)
And the increments are given by ∆ Xi = a ∆t + b √∆t εi
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Here ‘a’ is the constant drift rate and ‘b’ is the volatility rate. When we take smaller and
smaller time steps ∆t, then the above equation can be written as
𝑋 (𝑡) = 𝑥 + 𝑎 𝑡 + 𝑏 𝑍 (𝑡) (7)
The above equation is called the Generalized Wiener Process. The above equation can
be written in the differential form as follows
𝑑𝑋 = 𝑎 𝑑𝑡 + 𝑏 𝑑𝑍 (8)
So for a stock price we can conclude that
• Its expected percentage change in a short period of time remains constant, not
it’s expected absolute change in a short period of time.
• The size of the future stock price movements is proportional to the level of the
stock price
4.2.4 Geometric Brownian motion
Generalized wiener process fails to capture a key aspect of the stock / commodity prices
i.e. the percentage return required by the investors is independent of the stock price. It
means that the investor will demand the same return, does not matter if the stock price is
DKK 10 or DKK 100. So, the assumption of constant expected drift rate needs to be
replaced by the assumption that expected return (i.e. expected drift divided by the stock
price) is constant. So, if P is the price of a commodity bundle at time t, then the
expected drift rate in P should be assumed to be µP for some constant parameter µ. It
means that the expected increase in P (in a short period of time ∆t) is µP ∆t. The
parameter µ is the expected rate of return on the price, expressed in decimal form.
If the volatility of the commodity price is zero, then the model implies that
∆P = µ P∆t And as ∆t approaches to zero then
dP = µP dt or 𝑑𝑃𝑃
= 𝜇 𝑑𝑡 (9)
Integrating the equation (7) between 0 and time T, we get
29
𝑃𝑇 = 𝑃0 𝑒𝜇𝑇 (10)
PT and P0 are the commodity prices at time T and time 0 respectively. This equation
shows that when variance rate is zero, the commodity price grows at a continuously
compounded rate of µ per unit of time. But in practice, commodity prices or stock prices
exhibit volatility. Here it can be assumed that the variation in the percentage return in
short period of time ∆t is the same regardless of the commodity price. It means that the
investor is just as uncertain of the percentage return when the price is DKK 100 as when
it is DKK 10. It means that the standard deviation of the change in a short period of time
∆t should be proportional to the commodity price and it can be written as follows
𝑑𝑃 = 𝜇 𝑃𝑑𝑡 + 𝜎𝑃 𝑑𝑧
Or 𝑑𝑃𝑃
= 𝜇 𝑑𝑡 + 𝜎 𝑑𝑧 P > 0 (11)
This equation is called the Geometric Brownian motion. The important feature of this
equation is that the commodity or stock price will never become negative.
4.2.6 Ito’s Lemma
Ito’s lemma is the most important rule of stochastic calculus. It was discovered by the
mathematician K. Ito in 1951. According to Ito’s lemma the ordinary rules of calculus
do not apply to the stochastic processes For example, consider a function F(Z) = Z2
where Z is a Brownian motion. Then according to ordinary calculus, dF (Z) = 2ZdZ.
But this is not true for stochastic processes. In order to drive rules for stochastic
calculus, we have to apply Taylor expansion i.e.
𝑑𝐹 = 𝑑𝐹𝑑𝑍𝑑𝑍 + 1
2𝑑2𝐹𝑑𝑍2
𝑑𝑡…….. (12)
So from the F (Z) = Z2 we have 𝑑𝐹𝑑𝑍
= 2Z and 𝑑2𝐹𝑑𝑍2
= 2 substituting these values in the
above Taylor expression we get 𝑑𝐹 = 2𝑍𝑑𝑍 + 𝑑𝑡
Now we consider the Geometric Brownian motion equation i.e.
𝑑𝑃 = 𝜇 𝑃𝑑𝑡 + 𝜎𝑃 𝑑𝑧 (13)
30
Here 𝑎 (𝑃, 𝑡) = 𝜇𝑃 𝑎𝑛𝑑 𝑏(𝑃, 𝑡) = 𝜎𝑃
Now consider the function 𝐹(𝑃) = log𝑃
Then 𝑑(𝑙𝑜𝑔𝑃)𝑑𝑃
= 1𝑃
𝑎𝑛𝑑 𝑑2(logP)𝑑𝑃2 = − 1
𝑃2
And by Ito’s lemma 𝑑𝐹 = 𝑑𝐹𝑑𝑃𝑑𝑃 + 1
2𝑏2(𝑡,𝑃) 𝑑2𝑃
𝑑𝑝2𝑑𝑡 and substituting the values we
obtain 𝑑𝐹 = 1𝑃𝑑𝑃 − 1
2𝜎2𝑃2 ∗ 1
𝑃2 𝑑𝑡 (14)
And substituting equation (13) in equation (14), we get
𝑑𝐹 = 𝜇 𝑑𝑡 + 𝜎 𝑑𝑧 − 12𝜎2 𝑑𝑡
Or 𝑑𝐹 = �𝜇 − 12𝜎2� 𝑑𝑡 + 𝜎 𝑑𝑧 (15)
Since 𝜎 and 𝜇 constant, this equation shows that F= log P follows a generalized Wiener
process. It has a constant drift rate of �𝜇 − 12𝜎2� and constant variance rate of 𝜎2. The
change in ln P between time 0 and some future time T is therefore normally distributed
with mean �𝜇 − 12𝜎2� and variance 𝜎2𝑇. It means that
ln𝑃𝑡 ~∅ �𝑙𝑛𝑃0 + �𝜇 − 𝜎2
2� 𝑇,𝜎2𝑇� (16)
Equation (16) shows that 𝑃𝑡 is normally distributed. A variable has a lognormal
distribution if the natural logarithm of variable is normally distributed. This model
implies that that the price of a commodity bundle/ stock price at time T, given the price
today, are log normally distributed. The standard deviation of the stock price is𝜎√𝑇. It
is proportional to the square root of how far ahead we are looking (Hull 2007, p 271).
4.2.7 Risk Neutral Valuation
Risk- neutral valuation is an important concept in option pricing and particularly while
deriving the famous Black- Scholes option pricing equation. According to this principal
we can assume that the world is risk neutral when pricing options. It means that present
value of any cash flow can be obtained by discounting its expected future value at risk
31
free interest rate. So, in order to calculate the payoff of a derivative at particular time by
risk neutral valuation we assume that expected future return from the underlying asset is
the risk free interest rate “r”, that is, 𝜇 = 𝑟. Then we have to calculate the expected
payoff from the derivative and finally discount this expected payoff at the risk free
interest rate. In a risk- neutral world all individual investors are indifferent to risk and
they require no compensation (premium) for risk and the expected return on all
securities is the risk free interest rate. The solutions obtained in the risk- neutral world
also hold in the real world (risk- averse world) because in the risk-averse assumption
the expected growth rate in the stock price changes and discount rate that must be used
for any payoffs from the derivative changes. It happens that these two changes always
offset each other exactly (Hull 2007, p290).
5 The Black- Scholes Equation (BS)
The Black- Scholes model was derived by Fischer Black and Myron Scholes in the
early 1970s for the pricing of stock options. Robert Merton also participated in the
creation of this novel model. Therefore, sometimes, this model is also named as the
Black- Scholes- Merton model. This model was perhaps the biggest breakthrough in the
field of option pricing and rapidly got its acceptance among the financial engineers. The
success of financial engineering in the last 30 years is highly because of this model. In
1997, Myron Scholes and Robert Merton because of creating this model were also
awarded the Nobel Prize for economics.
The BS model shows that the value of an option depends on two factors i.e. the stock
price and the time to expiry/ maturity provided that the ideal market conditions
discussed earlier hold. Therefore, it is possible to create a hedge position by having a
long position in one option and a short position by some amount ∆ in the underlying ,
whose value will not depend on the stock price. If this portfolio is hedged continuously
then the portfolio of these two will be risk free and the expected return will be the risk
free interest rate. The riskless portfolio can be created because stock price and the
derivative price are both affected by the same underlying source of uncertainty i.e.
stock/ commodity price movements. In any short period of time, the price of the
derivative is perfectly correlated with the price of the underlying (Hull 2007, p285).
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In order to find the value “f” of a derivative written on an asset by BS model, the stock/
commodity price P follows geometric Brownian motion given in equation (11). In
general, the value f will be a function of many parameters in the contract i.e.
f (P, t, σ, μ,K,T, r )
Here
• P and T are variables
• σ and μ are associated with the underlying asset’s price
• Strike price K and time to maturity T depends on the specific details of the
contract.
• And interest rate r depends on the currency in which the asset is quoted.
We assume here that value of an option is a function of time t and current price P of the
underlying and drop the other parameters. Therefore we can write as follows
f (P, t )
To begin with we assume that we know the value f of the option and
• Form a portfolio, which we hold for a period of length dt, by taking a long position
in the option and a short position of a quantity Δ in the underlying.
• Determine the quantity Δ so that our portfolio is risk-free over a time period of
length dt.
• By a no arbitrage argument, the rate of return of this risk-free portfolio must be
equal to the risk-free interest rate r. What comes out of this restriction is an equation
whose solution is the option price f.
Suppose we know the value of the option f (P, t) at time t and Π denote the value of a
portfolio consisting of a long position in the option and a short position in a quantity Δ,
delta, of the underlying
Π(𝑡) = 𝑓 (𝑃, 𝑡) − Δ · 𝑃 (17)
In equation (12), the term f (P, t) is the option part of the portfolio and Δ · P is the short
asset position (negative sign).
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During the next time period of length dt the change in the portfolio value is given by
𝑑Π = 𝑑𝑓 (𝑃, 𝑡) − Δ · 𝑑𝑃 (18)
(Δ is fixed during the period t, t + dt). The portfolio Π is self financing, replicating, and
hedging strategy. It replicates a risk free investment. There is no stochastic term,
therefore it is hedged.
From Ito’s lemma we can compute df and we get
𝑑𝑓(𝑃, 𝑡) = 𝜕𝑓𝜕𝑡𝑑𝑡 + 𝜕𝑓
𝜕𝑃𝑑𝑃 + 1
2𝜎2𝑃2 𝜕
2𝑓𝜕𝑃2
𝑑𝑡 (19)
Substituting equation (19) into equation (18), we get
d Π = 𝜕𝑓𝜕𝑡𝑑𝑡 + 𝜕𝑓
𝜕𝑃𝑑𝑃 + 1
2𝜎2𝑃2 𝜕
2𝑓𝜕𝑃2
𝑑𝑡 - Δ · dP
Re-arranging the above equation,
𝑑 Π = �𝜕𝑓𝜕𝑡
+ 12𝜎2𝑃2 𝜕
2𝑓𝜕𝑃2
�𝑑𝑡 + �𝜕𝑓𝜕𝑃− ∆�𝑑𝑃 (20)
In the right hand side of equation (20), the expression containing the term dt is the
deterministic term while the with dP is random. The random term is the risk factor in
the portfolio. The risk in the portfolio can be removed if
�𝜕𝑓𝜕𝑃− ∆� = 0 (21)
I.e. if, for the small period of time from t to t + dt, we choose the quantity Δ as
𝜕𝑓𝜕𝑃
= ∆ (22)
Therefore, equation (17) becomes
Π(𝑡) = 𝑓 (𝑃, 𝑡) − 𝜕𝑓𝜕𝑃
𝑃 (23)
Then the randomness reduces to zero. The reduction in randomness is called hedging.
The perfect elimination of risk by exploiting correlation between two instruments
34
(option and the underlying in our portfolio) is called Delta hedging (Wilmott, p142).
Delta hedging is an example of perfect hedging. It means if we are allowed at any point
in time t to continuously re-balance our portfolio by choosing the quantity Δ in the
underlying (i.e. if we are entitled to continuous trading), then we have constructed a
portfolio which is risk-less and with dynamics given by
𝑑 Π = �𝜕𝑓𝜕𝑡
+ 12𝜎2𝑃2 𝜕
2𝑓𝜕𝑃2
�𝑑𝑡 (24)
If there are no arbitrage opportunities in our market, then it must hold that our risk-less
portfolio should yield the risk-free interest rate r, i.e.
𝑑 Π = 𝑟 Π 𝑑𝑡 (25)
Substituting the values of d Π and Π from equation 24, and 23 in equation 25 we get
�𝜕𝑓𝜕𝑡
+ 12𝜎2𝑃2 𝜕
2𝑓𝜕𝑃2
�𝑑𝑡 = 𝑟 �𝑓 − 𝜕𝑓𝜕𝑃𝑃� 𝑑𝑡
After re-arranging the above equation we get
𝜕𝑓𝜕𝑡
+ 12𝜎2𝑃2 𝜕
2𝑓𝜕𝑃2
+ 𝑟 𝜕𝑓𝜕𝑃𝑃 − 𝑟𝑓 = 0 (26)
Equation (26) is the famous BS equation. The price of any option which depends on P
and t must satisfy the BS equation otherwise it cannot be price of the derivative without
creating arbitrage opportunities for the traders. BS equation is a linear parabolic partial
differential equation. By the term linear means, if we have two solutions of the equation
then the sum of the two solutions is itself a solution. Parabolic means it has a second
order derivative with respect to one variable P and a first order derivative with respect
to the variable t. Therefore equations of this type are called heat or diffusion equations
of mechanics (Wilmott 2007, p158).
The BS equation shows that value of a stock option when expressed in terms of the
value of the underlying, does not depend on drift rate or expected return µ. This is
dropped out when the dP term is eliminated from the portfolio. It is also clear from this
35
equation that the variables appear in this equation (stock price, time, volatility and risk-
free interest rate) are all independent of risk preferences.
Another point to be noted from equation (26) is that we can not see whether this
equation is valuing a call or a put option. It means we will have to specify the final
conditions and specify the option value f as a function of the underlying at expiry date
T. For example, if we have a call option, then we know that f(P,T) = max( P- K, 0) and
for a put option the final condition will be f(P,T) = max(K- S, 0)
5.1 Options on dividend paying stock
Let q be the amount of continuous and constant dividend yield. Then equation (26) can
be re-written as
𝜕𝑓𝜕𝑡
+ 12𝜎2𝑃2 𝜕
2𝑓𝜕𝑃2
+ (𝑟 − 𝑞) 𝜕𝑓𝜕𝑃𝑃 − 𝑟𝑓 = 0 (27)
5.2 Commodity Options
Options are called commodity options when the underlying is a commodity or
commodities. Commodity options differ from the security options in the way that it
cannot be exercised before the future fixed (expiry) date. Therefore, in a European
option rather than an American style option11
𝜕𝑓𝜕𝑡
+ 12𝜎2𝑃2 𝜕
2𝑓𝜕𝑃2
+ (𝑟 + 𝑞) 𝜕𝑓𝜕𝑃𝑃 − 𝑟𝑓 = 0 (28)
, the holder of the commodity option can
choose whether or not he wants to buy the commodity at the specified price. Then we
have to adjust the general BS equation (26). Commodities have the cost of carry. That
is, the commodities cannot be held without storage cost. So, if we assume that q is the
storage cost associated with the commodity, which means that if we simply hold the
commodity, it will lose value even if the price remains fixed. It means that for each unit
of commodity held an amount qP dt will be required during the short period of time dt
to finance the holding (like a negative dividend). Therefore, equation (26) can be
modified as follows (Wilmott, p148)
11 F. Black, The Pricing of Commodity Contracts, Journal of financial Economics, 3 (March 1976)
36
The BS model can also be derived in many other ways instead of the classical risk-
neutral valuation. For example the Martingale approach, the binomial model and capital
asset pricing model (CAPM)
5.3 Options on many underlying
Options with many underlying assets are called basket options, options on baskets or
rainbow options. The basket can be any weighted sum of the underlying assets as long
as the weights are all positive. The value of these options depends on price, time to
maturity and additional variable i.e. the correlation between assets in the basket. The
payoff from the basket depend on the degree of correlation among the underlying assets
i.e. if the underlying assets are highly correlated with each other, then the option’s
payoff will be high and vice versa. The basic option pricing with one underlying can be
extended to more than one underlying too. First of all the idea of lognormal random
walk needed to be extended for multiple assets. I.e. The geometric Brownian motion of
an asset price (equation 11) can be easily extended to multiple assets. Therefore,
equation 11 can be written as
𝑑𝑃𝑖 = 𝜇𝑖𝑃 𝑖𝑑𝑡 + 𝜎𝑖𝑃𝑖𝑑𝑍𝑖 (29)
Here 𝑃𝑖 is the price of ith asset, i= 1, 2, 3…..,d and 𝜇𝑖 and 𝜎𝑖 are the drift and volatility
of the assets and dZ is the Wiener process for the respective asset. dZi can be still
considered to be as a random number drawn from normal distribution with a mean of
zero and a standard deviation of dt0,05 i.e .
𝐸[𝑑𝑍𝑖] = 0 𝑎𝑛𝑑 𝐸 �𝑑𝑍𝑖2 � = 𝑑𝑡
In baskets options, the assets are also correlated with each other, therefore the log
normal random walks are also considered to be correlated with each other. That is
𝐸[𝑑𝑍𝑖𝑑𝑍𝑗] = 𝜌𝑖𝑗𝑑𝑡
Here 𝜌𝑖𝑗 is the correlation coefficient between ith and jth random walks. The symmetric
matrix with ρij as the entry in the ith row and jth column is called the correlation matrix.
For example, if we have a basket option with three underlying assets i.e. n = 3 and the
correlation matrix can be written as
37
1 𝜌12 𝜌13𝜌21 1 𝜌23𝜌31 𝜌32 1
Here, 𝜌𝑖𝑖 = 1 𝑎𝑛𝑑 𝜌𝑖𝑗 = 𝜌𝑗𝑖. The correlation matrix is positive definite.
We can apply multi dimensional ito’s lemma to manipulate functions of many random
variables. So, if we a have a function of variables Pn where n= 1, 2, 3 ….. And time t,
then 𝑑𝑓 = � 𝛿𝑓𝛿𝑡
+ 12∑𝑑𝑖=1 ∑ 𝜎𝑖𝑑
𝑗=1 𝜎𝑗𝜌𝑖𝑗𝑃𝑖𝑃𝑗𝜕2𝑓𝜕𝑃𝑖𝑃𝑗
� 𝑑𝑡 + ∑ 𝜕𝑉𝜕𝑃𝑖
𝑑𝑖=1 𝑑𝑃𝑖 (30)
The pricing model for basket options following BS model can be derived in the same
way as for the single asset, i.e. by setting up a portfolio consisting of one basket option
and short a number ∆ o f each o f th e asset p rice P in th e basket. Employ the
multidimensional Ito’s Lemma, take Δi = ∂V/∂Pi to eliminate the risk, and set the
return of the portfolio equal to the risk-free rate. We are able to arrive at the multi
dimensional version of the Black and Scholes equation (Wilmott 2007, p277)
�𝛿𝑓𝛿𝑡
+ 12∑𝑑𝑖=1 ∑ 𝜎𝑖𝑑
𝑗=1 𝜎𝑗𝜌𝑖𝑗𝑃𝑖𝑃𝑗𝜕2𝑓𝜕𝑃𝑖𝑃𝑗
� + 𝑟 ∑ 𝜕𝑉𝜕𝑃𝑖
𝑑𝑖=1 𝑃𝑖 − 𝑟𝑓 = 0 (31)
5.4 Black- Scholes Pricing Formulas
Consider now the case of a call option with maturity T and strike price K written on a
stock / commodity paying no dividends. Assume that we stand at time t, that the current
price of the underlying is P0, that the interest rate is r and that the volatility of the stock
is σ. Here P follows geometric Brownian motion described by equation (11).
Then we know that
𝐶 = 𝑒−𝑟(𝑇−𝑡)𝐸[max(𝑃 − 𝐾) , 0] (32)
Recalling, we know that P follows a lognormal distribution with mean
�𝜇 − 12𝜎2� 𝑎𝑛𝑑 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑜𝑓 𝜎2𝑇 and
𝐸 [𝑃] = 𝑃𝑒𝑟(𝑇−𝑡) (33)
38
Then value of the European call option can written as
𝐶 = 𝑃0 𝑁(𝑑1) − 𝐾𝑒−𝑟(𝑇−𝑡)𝑁(𝑑2) (34)
Equation (34) can also be written as
𝐶 = 𝑒−𝑟(𝑇−𝑡)[𝑃0 𝑁(𝑑1)𝑒𝑟(𝑇−𝑡) − 𝐾𝑁(𝑑2)] (35)
Where N(.) is the cumulative probability distribution function for standardized normal
distribution and
𝑑1 =ln�P0K �+�r+
σ2
2 �(T−t)
σ√T−t
And 𝑑2 = 𝑑1 − 𝜎√𝑇 − 𝑡
N(d2) is the probability that the option will be exercised (P > K) in a risk- neutral world
so that KN(d2) i.e. strike price times the probability that the strike price will be paid.
The expression 𝑃0 𝑁(𝑑1)𝑒𝑟(𝑇−𝑡) is the expected value in risk- neutral of a variable
which is equal to PT if PT > K and to zero otherwise. We can also say that the expected
value of the call option at maturity will be 𝑃0 𝑁(𝑑1)𝑒𝑟(𝑇−𝑡) − 𝐾𝑁(𝑑2).
The value of a European put option P on a non dividend paying underlying can be
written as
𝑃𝑢𝑡 𝑜𝑝𝑡𝑖𝑜𝑛 𝑣𝑎𝑙𝑢𝑒 = −𝑃0 𝑁(−𝑑1) + 𝐾𝑒−𝑟(𝑇−𝑡)𝑁(−𝑑2) (36)
The value of European call and put option on dividend paying stocks can be written as
𝐶 = 𝑃0 𝑒−𝑞(𝑇−𝑡)𝑁(𝑑1) − 𝐾𝑒−𝑟(𝑇−𝑡)𝑁(𝑑2) (37)
𝑃𝑢𝑡 𝑜𝑝𝑡𝑖𝑜𝑛 𝑣𝑎𝑙𝑢𝑒 = −𝑃0 𝑒−𝑞(𝑇−𝑡)𝑁(−𝑑1) + 𝐾𝑒−𝑟(𝑇−𝑡)𝑁(−𝑑2) (38)
5.5 Upper and Lower bounds for the call option
Structured products have in general embedded call options and therefore it is important
to evaluate whether the prices of call option follows the no-arbitrage argument. We
know that call price has boundary conditions (upper and lower bounds). Boundary
39
conditions tell us how the solution must behave for all time at certain values of the asset
(Wilmott 2007, P159). If an option is above the upper bound or below the lower bound,
then there exist arbitrage opportunities for investors. So from equation (37), the lower
bound for a call option can be written as
𝐶 + 𝐾𝑒−𝑟(𝑇−𝑡)𝑁(𝑑2) ≥ 𝑃0 𝑒−𝑞(𝑇−𝑡)𝑁(𝑑1) Or
𝐶 ≥ 𝑃0 𝑒−𝑞(𝑇−𝑡)𝑁(𝑑1) − 𝐾𝑒−𝑟(𝑇−𝑡)𝑁(𝑑2)
And the upper bound for call option is given by
𝐶 ≤ 𝑃0
It means that the option value is always less than the corresponding underlying asset or
the underlying asset is always worth more than the corresponding call option.
Therefore, stock price is an upper bound to the option price. Similarly, for a put option,
the option can never be worth more than the strike price i.e. it can not be worth more
than the present value of K today (Hull 2008, p206). If it is not true, then the arbitrageur
could make riskless profit by writing the option and investing the proceeds of the sale at
the risk free interest rate.
𝑝 ≤ 𝐾 𝑜𝑟 𝑝 ≤ 𝐾 exp (−𝑟𝑇)
5.6 Forward Contract
A forward contract is an agreement to buy or sell the asset P at a future date T (delivery
date) for a fixed price K (delivery price). The payoff from a long position of forward
contract is therefore
𝑉(𝑃,𝑇) = 𝑃 − 𝐾 (39)
The delivery price K is typically delivered so that at initiation the contract has value
zero. By no arbitrage argument we have
𝐾 = 𝑒𝑟𝑇𝑃0 (40)
40
A forward contact sometimes also called a cash forward sale. There are some
disadvantages associated with the forward contracts i.e.
• Default risk, particularly if the prices are either high or low by the delivery date,
which negate the main value of a forward contract- price certainty
• The only way to legally terminate a contract was by mutual agreement, which would
be unlikely when the market price was significantly different from the delivery
price;
• There was no easy way to resell the contract, because it had customized terms that
specifically suited the seller and buyer—hence, forward contracts were highly
illiquid.
Eventually, organized exchanges developed that solved these basic problems. To lower
the risk of default, the exchanges required that money to be deposited with a 3rd party to
ensure the performance of the contract. The exchanges also standardized the contracts
by stipulating the types of contracts that they would sell, including its terms.
Standardized contracts were easier to sell or to offset with another contract that
eliminated the liability of the original contract. Standard specifications include the
amount of the commodity, the grade, and delivery dates. These standard forward
contracts were called futures, and the exchanges developed listings for these contracts
that greatly increased their liquidity.
5.7 Futures contracts
A futures contract is an agreement between two parties to buy or sell an asset at a
certain time in the future for a certain price, the current futures price of the asset.
Futures contracts are similar to forward contracts, but the principal difference lies in the
way payments are being made. In a forward contract, the whole gain or loss is realized
at the end of the life of the contract. But in case of the futures contract, the gain or loss
is realized day by day through a mechanism known as marking to market.
In mark to market mechanism, the contract is settled every day and simultaneously a
new contract, with the same maturity, is written according to the current futures price of
the underlying. Any profit or loss during the day is recorded in the account of the
contract holder. At any point in time, the value of the futures contract itself is therefore
41
zero (i.e. it is a zero sum game, which means that short side’s loss or gain is the long
side’s gain or loss) , but we have also a capital gain which comes from the previous
days cash settlements. The futures price of an asset, for a given maturity, varies from
day to day, but at maturity it must be the same as the price of the underlying.
In general, at any point in time t the value of the futures contract is zero, but we have a
capital gain given by
Capital gain = change in futures prices = Ft − F0
At the expiry of the contract, the total cash flows will amount to
(ST − FT−1) + · · · + (F2 − F1) + (F1 − F0) = ST − F0
In order to find out the futures price F(P,T) for a futures contract with expiry T on an
asset P, we assume that the price follows Geometric Brownian motion given in equation
(8). We create a portfolio Π consisting of one long futures contract and short Δ of the
underlying. The portfolio value is given by
Π = −ΔP (41)
We know that the value of the futures contract is zero. But, when computing the
portfolio change during the next time period we need to take into account the cash
settlement which is given by the change in the futures price, i.e.
𝑑Π = 𝑑𝐹 – Δ𝑑𝑃 (42)
Applying Ito’s Lemma, we finally get
d Π = 𝜕𝐹𝜕𝑡𝑑𝑡 + 𝜕𝐹
𝜕𝑃𝑑𝑃 + 1
2𝜎2𝑃2 𝜕
2𝐹𝜕𝑃2
𝑑𝑡 - Δ · dP (43)
Assuming that Δ = 𝜕𝐹𝜕𝑃 and to eliminate the risk, the following condition must hold
𝑑Π = 𝑟Π𝑑𝑡 (44)
Substituting equations 41and 43 into 44, we get
42
𝜕𝐹𝜕𝑡
𝑑𝑡 + 𝜕𝐹𝜕𝑃
𝑑𝑃 + 12𝜎2𝑃2
𝜕2𝐹𝜕𝑃2
𝑑𝑡 − Δ · 𝑑𝑃 = r(−∆P)dt
�𝜕𝐹𝜕𝑡
+ 12𝜎2𝑃2 𝜕
2𝐹𝜕𝑃2
+ r∆P�dt − Δ 𝑑𝑃 + 𝜕𝐹𝜕𝑃𝑑𝑃 = 0
𝜕𝐹𝜕𝑡
+ 12𝜎2𝑃2 𝜕
2𝐹𝜕𝑃2
+ 𝑟𝑃 𝜕𝐹𝜕𝑃
= 0 (45)
The important point to note in equation (45) is that there are only three terms instead of four as
in BS equation. The final condition is
𝐹(𝑃,𝑇) = 𝑃
i.e. The futures price and the underlying must have the same value at maturity.
It can also be shown that 𝐹(𝑡,𝑃) = P𝑒𝑟(𝑇−𝑡)
Since 𝜕𝐹𝜕𝑡
= −𝑟 𝑃𝑒𝑟(𝑇−𝑡) , 𝜕𝐹𝜕𝑃
= 𝑒𝑟(𝑇−𝑡) 𝑎𝑛𝑑 𝜕2𝐹
𝜕𝑃2= 0 and substituting these values in
equation (45) we get zero. It should also be noted here that futures price is equal to the
forward contract if interest rate is considered to be constant.
When we also assume that P follows Geometric Brownian motion (11) and again substituting in
equation 34, we get
𝑑𝐹 = −𝑟𝑃𝑒𝑟(𝑇−𝑡) + 𝑒𝑟(𝑇−𝑡) (𝜇𝑃𝑑𝑡 + 𝜎𝑃𝑑𝑍) Or
𝑑𝐹 = −𝑟𝐹𝑑𝑡 + µ𝐹𝑑𝑡 + 𝜎𝐹𝑑𝑍
𝑑𝐹 = ( 𝜇 − 𝑟)𝐹𝑑𝑡 + 𝜎𝐹𝑑𝑍 (46)
It means that futures price follow Geometric Brownian motion with drift (µ-r). In a risk neutral
world i.e. µ= r, so the above equation becomes
𝑑𝐹 = 𝜎𝐹𝑑𝑍 (47)
5.8 Futures Options
When the underlying of an option is a futures contract, then it is called a futures options
or options on futures. A futures option is the right, but not the obligation, to enter into a
futures contract at a certain futures price by a certain date. If a call option is exercised,
43
the holder acquires a long position in the underlying futures contract plus a cash amount
equal to the current futures price minus the strike price. If a put option is exercised, the
holder acquires a short position in the underlying futures contract plus a cash amount
equal to the strike price minus the current futures price.
Permanent trading of futures options was approved in 1987, and since then the
popularity of these contracts has grown very fast (Hull 2007, p341). Futures options are
getting popularity because
• The futures contracts are more liquid and easy to trade than the underlying asset.
• A futures price is known immediately from trading on the futures exchange,
while the spot price of the underlying asset may not be so readily available.
• Futures on commodities are easier to trade than the commodities themselves.
• There is lower transaction costs associated with futures options than spot
options.
• Futures options facilitate hedging, arbitrage and speculation, because future
options are usually traded side by side in the same exchange
Futures options are usually settled in cash, because exercising a futures contract does
not lead to delivery of the underlying asset, as in most circumstances the underlying
futures contract is closed out prior to delivery. This is good for investor with limited
capital who may find it difficult to come up with the funds to buy the underlying asset
when an option is exercised Hull 2007, p344).
5.9 Pricing of European futures options
For a European call option with maturity date To, strike price K and written on a futures
contract with expiry Tf > To, the payoff function is given by
𝑚𝑎𝑥[(𝐹(𝑇0,𝑃) − 𝐾), 0]
It can also be written as 𝑚𝑎𝑥 �𝑃((𝑇0)𝑒𝑟�𝑇𝑓−𝑇0� − 𝐾),0� (48)
And for European put option the payoff function is given by
max [�𝐾 − 𝐹(𝑇0,𝑃)�, 0]
44
Or max �(𝐾 − 𝑃(𝑇0)𝑒𝑟�𝑇𝑓−𝑇0�� , 0]
Here F(T0) is the futures price at maturity. If the futures contract mature at the same
time as the options contract, then F(T0)= P(T0) and the two options are equivalent.
In a risk neutral world, the European futures options can be evaluated by considering
future prices as assets paying dividends at a continuous yield q = r. The price of a
European Call option C and put option P is then given by
𝑐 = 𝐹𝑒−𝑟𝑇𝑁(𝑑1) − 𝐾𝑒−𝑟𝑇𝑁(𝑑2) (49)
𝑝 = 𝑒−𝑟𝑇[𝐾𝑁(−𝑑2) − 𝐹𝑁(−𝑑1)] (50)
Where 𝑁(𝑑1) and 𝑁(𝑑2) are cumulative normal distribution function given by
𝑑1 = ln�𝐹𝑒
−𝑟𝑇
𝐾 �+�𝑟+ 𝜎2
2 �𝑇
𝜎√𝑇=
ln�𝐹𝐾� + 𝜎2𝑇/2
𝜎√𝑇 (51)
𝑑2 = 𝑑1 − 𝜎√𝑇 (52)
The interest rate factor has been dropped out of the formula because the investment in a
futures contract is zero. The formula derived above is the same as the value of a security
option that pays continuous dividend at a rate equal to the stock price times the interest
rate, when the option can only be exercised at maturity. Equations (49) and (50) are also
named as Black’s Model for valuing futures options. This model does not require the
options contract and the futures contract to mature at the same time.
45
6 An overview of the selected products
Structured products to be analyzed in this thesis are chosen from the Danish market.
The chosen structured products are DB Råvarer 2013 Basel issued by Danske Bank and
Råvarer Basis 2010 issued by Nordea Bank. Both banks are the leading financial
institutions in Denmark. The prospects for structured products from different issuers
contain more or less identical information about the products. The typical information
included in the prospectus is
• Name of the product
• Name of the issuer and the guarantor
• ISIN code for the issuing product
• Offer date (period) of the product
• Initial date
• Expiry date
• Issue price
• Risk factors associated with the product
• Minimum and or maximum redemption amount
• The underlying asset/assets, index/ indices and their category
• Number of underlying assets and their contributing weights
• Brief description of the underlying assets/ indices
• Initial price of the underlying
• Credit rating of the issuer
• Specified currency of the notes
• Formula for the final payoff
• Coupon type
• Coupon payment dates if applicable
• Initial or final valuation period or date.
• Annual costs related to the notes e.g. commission, listing costs etc
• Participation rate
46
6.1 DB Råvarer 2013 Basel (the “Notes”)
DB Råvarer 2013 Basel is issued and guaranteed by Danske bank. The offer period of
the product was from 12 April 2010 to and including 29 April 2010. The product has an
ISIN code of DK0030240247 and it will be listed on Copenhagen stock exchange
(OMX.com). The investor has to keep this product for three years (i.e. A zero coupon
commodity index linked note maturing in three years), because investment period of the
product is three years i.e. from 10 May 2010 to 13 May 2013. DB Råvarer 2013 Basel is
commodity linked note. The note consists of a bond with maturity of three years and a
call option on Dow Jones UBS commodity index. The note was issued at a price of
105% of the aggregate nominal amount. The minimum amount of the issuing notes
would be DKK 25,000,000. The issuer has the right to cancel the issue if the
subscription of the notes will be less the DKK25, 000,000.
6.1.1 Payoff Structure
The product will pay no coupon payments until expiry. The final redemption amount
will depend on the performance of the underlying index but it will not be less than 95%
and it will not be more that 35%. It means if the index performance is between 95% to
100%, then the redemption amount will be calculated accordingly but it will not be less
than 95%, and if the index performance is 100% or more, then the investor will get
100% or a maximum of 135% of nominal amount at maturity depending upon index
performance. The percentage of maximum payoff (Cap) will be set no later than 6 May
2010. The final commodity linked redemption amount will be according to following
formula
𝐶𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑎𝑚𝑜𝑢𝑛𝑡 ∗ (100% + 𝑀𝑎𝑥 �−5%;𝑀𝑖𝑛 �𝐶𝑎𝑝; 𝐼𝑛𝑑𝑒𝑥 𝐹𝑖𝑛𝑎𝑙𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝐼𝑛𝑑𝑒𝑥
− 100%��) (53)
Here
• Initial index means the initial price of the underlying at the initial date which is
10 May 2010. This initial date is also the strike date and therefore the index
price on this date will be the strike price too. The initial index value set by the
bank was 130.2348.
• Final index means the value of the index at maturity which is 13 May 2013. The
valuation date of the option will be 26 April 2010, which will be considered as
47
the initial spot price of the index. So it means that the major amount of the
redemption amount will come from the zero coupon bond and the rest from the
option part.
6.1.2 Risk Factors
The issuance price of the bond is 105% of the nominal amount. It means that the
investor pays 5% premium or issuing costs which will be gone no matter if the option
component expires worthless or in the money. If the underlying commodity index
performs below 100 %, then the investor can lose only 5% of the nominal amount. But
from 100% and onward performance of the underlying, the investor will get at least
100% of the nominal amount. So the only loss to the investor in this case will be the 5%
of its initial invested plus an opportunity cost that he would have earned if he would
have invested in the risk free investment i.e. bonds.
The other risk associated with the bond can be the default risk of the issuer. In this case,
Danske bank (issuer) has a good credit rating. Credit rating of Danske at the time of
issuance of the bond was ‘Aa3 ‘according to Moody’s and ‘A’ according to Standard &
Poors and Fitch. It means that, the bank has the strong capacity to meet its financial
commitments and obligations, are judged to be of very high quality and are subject to
very low credit risk. So, we can say that the risk associated with commodity linked note
is very low.
6.1.3 Issuance costs
The issuer also charges costs associated with the issuance of the bond. The issuance
costs included are arranger fee of 0,967% annually, estimated listing fee of NASDAQ
OMX Copenhagen A/S of 0.017% annually and a fee to VP securities which is about
0.17% annually. So the total annual issuance cost will be about 1% of aggregate
nominal amount of DKK 50,000,000.00
6.1.4 Embedded option
The embedded option is the most important part of the structured product. The option
embedded in this product is a capped call option with three years of expiry. The payoff
is set to the level of 35% maximum and if the index performance is between 95% -
48
100% than the final redemption amount will be calculated accordingly. The final payoff
can be calculated from the equation 53.
6.1.5 The underlying asset
The underlying asset is the Dow Jones UBS Commodity index composed of futures
contracts on physical commodities. Unlike equities, which typically entitles the holder
to a continuing stake in a corporation, commodity contracts normally specify a certain
date for the delivery of the underlying physical commodity. In order to avoid the
delivery process and maintain a longer futures position, nearby contracts must be sold
and contracts that have not yet reached the delivery period must be purchased. This is
called “rolling” a futures position.
The Dow Jones UBS Commodity index is composed of commodities traded on US
exchanges except aluminum, nickel and zinc which trade on the London Metal
exchange.
6.2 Analysis of Råvarer Basis 2010
Råvarer Basis 2010 was issued by KommunalBanken A/S on 16 June 2006. The ISIN
code for the product is DK0030030861. The note will mature in four years and the
maturity date was 16June 2010. The subscription period of the product was from 22
May 2006 to and including 9 June 2006. The product was available at a minimum
amount of DKK 10,000 for the investors while the total subscription will be a minimum
of DKK 25,000,000.00. The product is a combination of a zero coupon bond and a call
option. The underlying asset is the DJAIJ commodity index. According to Reuters
(May7, 2009), DJAIG commodity index was acquired by Dow Jones UBS and therefore
renamed as Dow Jones UBS commodity index12
6.2.1. Payoff Structure
. The product was issued at 104% of
the aggregate principal amount (Nominal amount). The product offered no coupon
payments until expiry. This product was also listed on the Copenhagen Stock exchange.
Final payoff of the product depends on the performance of the underlying index but the
minimum redemption amount will be at par. It means this product guarantees the
12 http://uk.reuters.com/article/idUKN0731630220090507
49
investor 100% of the invested amount no matter what the index performance will be
negative. This product does not have any limit for the up side payoff unlike DB Råvarer
Basel. The final redemption amount will be calculated according to the following
formula
𝑃𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙 ∗ [1 + 𝑃𝑎𝑟𝑡𝑖𝑐𝑖𝑝𝑎𝑡𝑖𝑜𝑛 𝑟𝑎𝑡𝑒 𝐷𝐽𝐴𝐼𝐺2010 ∗ {max �0; 𝐷𝐽𝐴𝐼𝐺𝑒𝑛𝑑−𝐷𝐽𝐴𝐼𝐺 𝑠𝑡𝑎𝑟𝑡 𝐷𝐽𝐴𝐼𝐺𝑠𝑡𝑎𝑟𝑡
�}]
Here
• 𝐷𝐽𝐴𝐼𝐺𝑠𝑡𝑎𝑟𝑡 , is the official closing price of the DJAIG on 16 June 2006 as
determined by the calculation agent.
• 𝐷𝐽𝐴𝐼𝐺𝑒𝑛𝑑 , is the arithmetic means of the official closing price of DJAIG on
each of the determination dates. The determinatioon dates are
• 01 December 2009, 04 January 2010, 01 February 2010, 01 March 2010, 06
April 2010, 04 May 2010 and 01 June 2010.
• Participation rate will be fixed by the calculation agent on 13 June 2006 and it
will not be less than 70%. The expected participation rate is about 87%
6.2.2 Risk factors
The prospectus does not mention any credit rating information of the issuing bank
(Kommunal Bank) and also the distributor of the notes which is Nordea Bank. It
includes in the prospectus that the status of instruments is senior. It means that in case
of default, the note holders will have the priority for getting their money back. Apart
from the default risk
• The investor will lose 4% of the invested amount because the notes are issued at
104% of the nominal amount.
• The investor will miss the opportunity of earning the profit from investing in a
risk free bond.
• The good news for the investor here is that he will get at least 100% of the
invested amount if the underlying index performs negative or zero plus an
opportunity to earn unlimited profit if the underlying performs more than zero.
50
6.2.3 Issuance costs
The issuing costs include commission and concession which is about 3% of the
aggregate principal amount. The total estimated expenses based on an issue of DKK
200,000,000 are DKK 200,000 which include a fee of DKK 40,000 for listing on stock
exchange, VP: DKK 10,000, service charges to Nordea bank about DKK 5,000 and a
license fee of DKK 100,000
6.2.4 Embedded option
The embedded option is call option and the strike price is the arithmetic average of the
seven determination dates mentioned above. Therefore, it is considered to be an Asian
style option, because the payoff depends on the average performance of the underlying
commodity index.
6.2.5 Underlying Asset
DJAIG was the underlying index in this option but after acquisition by Dow Jones
commodity index, the underlying index is Dow Jones UBS commodity index. The
description of the index is already mentioned above during analyzing DB Råvarer basel
2013.
51
7 Pricing of the selected products
This section will deal with the valuation of the structured products. All structured
products face the risk of trading at prices that differ substantially from their fair values.
Given that there would otherwise be no liquidity in the secondary market, issuers
generally act as market makers for their own products during the exchange trade.
Alternatively, structured products can be bought and sold over-the-counter with the
issuing firm. As a consequence, almost any transaction involves the product's issuer as
one counter-party, and there is a possibility that unfavourable prices will be quoted. As
a rule of thumb, the more complex the product, the higher the margins entailed in
quotes13
The price of a commodity linked bond can be calculated as
𝐵(𝑃, 𝑡) = 𝐶𝑟
(1 − 𝑒−𝑟𝑇) + 𝐹𝑒−𝑟𝑇 + 𝑊(𝑃,𝑇)14 (54)
Where 𝑊(𝑃,𝑇) is the Black and Scholes solution (as derived earlier in last section) to
the value of a call option with exercise price K. The above equation shows that the
value of a commodity linked bond is equal to the discounted value of future coupon
payments (if any) and face value of the bond plus a call option to buy the reference
commodity bundle at agreed exercise price. The above relation holds if we assume that
• The firm issuing the bond has no default risk
• Interest rate is constant
• There is no cost of carrying to the commodity (evaporation, obsolescence,
insurance etc) and that the commodity is held for speculative purpose like a
stock15
.
13 The pricing of structured products in Germany; By Wilkens; Journal of Derivatives Vol.11 14 Option pricing theory and its application; by Eduardo S. Schwartz, The Journal of Finance. May 1982 15 Details can be found in the above mentioned reference 14.
52
7.1 Pricing of DB Råvarer 2013 Basel
This section will consist of pricing the bond and the option components of the DB
Råvarer 2013.
7.1.1 Deriving the Zero Coupons
In order to calculate the present value of the bond component of DB Råvarer Basel we
need to find out the discount rates or the interest rate. Since the note will mature in three
years therefore we need to find the set of zero coupon bonds whose maturity is from one
to three years. The table below shows a set of zero coupon bonds issued by Real Kredit
Denmark with maturity from one to years. These bonds were issued around the same
period when DB Råvarer was issued. Therefore at that time these bonds were also not
listed on the stock exchange and consequently no market prices were available in the
beginning. Therefore, we assume that the price of the bond would be equal to the
market price (i.e. the par value is the same as the principal value). The term structure is
as follows
Table 1
Bond maturity Coupon rate Price Principal
1 Year 2% 100 100
2 Years 2% 100 100
3 Years 2% 100 100
So, discount factors and the interest rates calculated on the basis of continuous
compounding (equation 2) are given in the following table. The discount factor and the
interest rate are actually nearly the same because of the fact that we have assumed that
the par value and principal value of the bonds are equal. The detailed calculations can
be found in the excel spread sheet on CD Rom
Table 2
Time to Maturity Disc. Factor r(0,T) 1 0,980198673 0,020201 2 0,960789439 0,020201 3 0,941764534 0,020201
53
The graph below shows a decreasing trend in discount factor with the passage of time
while the interest rate is constant, because all bonds have the same coupon rates and
market price of the bonds is assumed to be equal to the principal amount.
So, from table 2 we can find out the present value of the bond component as given
below
PV = 100* 0, 94176 = 94,176 (55)
7.2 Term Structure for Råvare.r Basis 2010
In order to derive the zero coupon term structure we need to find the set of bonds which
were issued during the same period as the issuance of the Råvarere Basis 2010. The
maturity of the structured note is four years therefore we need a set of bonds with
maturity from one to four years. BRF kredit issued bonds with maturity from one to
four years. These bonds were issued during June 2006 and were listed on Copenhagen
the stock exchange. The term structure is given in the following table
Table 3
Time to Maturity Coupon Price Principal 1 3% 100 100 2 3% 100 100 3 3% 100 100 4 3% 100 100
0
0,2
0,4
0,6
0,8
1
1,2
1 2 3
Discount factor
r(0,T)
54
The market prices of the bonds were not available during the June 2006 because they
started to trade during July 2006. Therefore, here we make the same assumption again
that the market price of the bonds is equal the principal amount. The term structure can
be derived by assuming the continuous compounding again and is given in table 4
below.
Table 4
Time Discount factor R(0,t) 1 0,970445534 0,030454534 2 0,941764534 0,030454534 3 0,913931185 0,030454534 4 0,886920437 0,030454534
The graph below shows that the discount factor is decreasing as the time to maturity is
increasing. On the other hands the interest rate is unchanged during the four years
because we assumed principal and market price of the bonds to be equal.
After calculating the discount factor and interest rates we can calculate the present value
of the Råvaere 2010. Therefore,
PV= 100* 0.88692
= 88. 692 (56)
0
0,2
0,4
0,6
0,8
1
1,2
0 1 2 3 4 5
Discount factor
R(0,T)
55
7.3 Option Pricing
This section will deal with the pricing of the embedded options within the selected
structured products. The theoretical price of the option will be calculated by the Black
& Scholes formula and also by the Monte Carlo simulations. The Black–Scholes option-
pricing framework is applied because of its wide acceptance, its simplicity and
elegance, and its mathematical tractability. All conclusions derived in the thesis can be
used as benchmarks for other more sophisticated option-pricing models.
Before we can calculate option price, we need to calculate one of the most important
parameter named as volatility.
7.4 Estimating Volatility
Volatility “σ” is an important parameter in option pricing. It is the degree of uncertainty
about the stock returns and is also known as standard deviation. To estimate volatility,
the stock, commodity etc prices are usually observed at fixed interval of time (e.g. daily
closing price, weekly or monthly etc). Volatility can be estimated by the following
formula
𝜎 = � 1(𝑀−1)
∑ (log𝑃𝑡 −𝑀𝑖=1 log𝑃𝑡−1)2 (57)
Here
• log𝑃𝑡 𝑎𝑛𝑑 log𝑃𝑡−1,are log of stock/commodity prices at time t and t-1
respectively, M is the number of observations in the series
The selection of appropriate number of observation M is also tricky. More data
generally lead to accuracy but choosing too old data can also be irrelevant to predict the
future volatility. Choosing data over the most recent 90 to 180 days can estimate
reasonably well estimate of volatility. In option pricing however, it is good idea to set M
equal to the number of days to which the volatility is to be applied. Therefore, if an
option matures in two years, then it is good idea to take the daily data for the recent two
years.
56
Equation (57) calculates the so called historical volatility on daily basis. In option
pricing, we normally use annual volatility. So the annualized volatility σ (yearly) can be
estimated by multiplying the equation (57) with the square root of number of trading
days in a year (assume 252 trading days/ year). That is,
𝜎𝑦𝑒𝑎𝑟𝑙𝑦 = 𝜎𝑑𝑎𝑖𝑙𝑦 ∗ √252 (58)
There are other ways to estimate volatility which are not considered in this thesis.
7.5 Monte Carlo Simulation
Monte Carlo simulation is being named after the city of Monte Carlo which is famous
for its casinos. In Monte Carlo simulations random numbers are generated according to
the probabilities assumed to be connected with the source of uncertainty for example
stock, commodities, interest rates or option prices etc. Monte Carlo simulations has
been using since 1977 for option pricing. The method is proved to be accurate for
option pricing. The method is particularly good for the pricing of Asian options and
other path dependent options. Monte Carlo simulation is based on the principal of risk
neutral random walks of the stock, commodities etc.
Monte Carlo simulation for option pricing has many advantages. For example, simple
mathematics is involved to perform the simulation, correlations can be easily modeled
and simulation can be performed on excel spreadsheet too. We can generate more
simulations to get accurate results (Wilmott 2007, p586). As we know in a risk neutral
world the present value of any cash flow can be obtained by discounting its expected
future value at risk free interest rate i.e.
𝑂𝑝𝑡𝑖𝑜𝑛 𝑣𝑎𝑙𝑢𝑒 = 𝑒−𝑟(𝑇−𝑡)𝐸[𝑝𝑎𝑦𝑜𝑓𝑓(𝑃)]
The above relation holds if the expectation is with respect risk neutral random walk i.e.
the geometric Brownian motion in equation (11) mentioned in the previous section.
Precisely the method of pricing the option with Monte Carlo simulation has the
following steps
• First of all produce random numbers from standardized normal distribution or
some other approximation.
57
• Start with the today’s value of the asset 𝑃0 over the required period of time
(from today up to maturity). The asset price up to maturity can be found by the
so called Euler method which is a discrete way of simulating the time series for
𝑃0 and is given by
𝛿𝑃 = 𝑟 𝑃 𝛿𝑡 + 𝜎𝑃√𝛿𝑡 𝜑
Here 𝜑 is drawn from the standardized normal distribution. Applying Ito’s
lemma to the above equation we can simulate the asset price as follows
𝑃(𝑡 + 𝛿𝑡) = 𝑃𝑡𝑒�𝑟−𝜎
2
2 �𝛿𝑡+ 𝜎√𝛿𝑡 𝜑
• Perform many such realizations over the time horizon and then calculate the
arithmetic average of corresponding payoffs
• Finally take the present value of this average payoff according to risk free
interest rate. This will be the option value (Wilmott 2007, p582).
7.6 Variance Reduction Technique
One of the drawbacks of Monte Carlo simulation is that it generates high variances
which lead to computational inefficiency and therefore can produce biased estimates of
the option price16
7.7 Generating Random numbers
. But according to Wilmott, we can apply variance reduction
techniques to overcome this drawback. One of the variance reduction techniques is
called Antithetic variable. In this technique first of all we calculate the option price by
generating random variables in a usual way. Secondly, we find the option price by
producing negative random variables. The option value will be the average of these
values. This technique works well because of the symmetry in the Normal distribution
and it also easy to implement.
There are many ways to generate random numbers. The most common methods are
NormSinv(RND()) in excel, Box- Muller method and Marsaglia- Bray. For the pricing
of DB Råvarer basel 2013, random numbers are generated by Marsaglia – Bray method.
Marsagalia- Bray method is used because it is the most efficient way to generate 16 Option pricing and Monte Carlo simulations. George M. Jabbour. Journal of economics and business research Sept.2005
58
random numbers and we can get better convergence rate than other random number
generating methods17
7.8 DB Råvarer Basel 2013
.
The embedded in DB Råvarer Basel 2013 is considered to be plain vanilla option. The
only thing to consider is the minimum and maximum payoff from the investment. The
option is valued considering the fact that we are standing on the valuation date which is
26- April-2010. So we consider that that the official closing price on 26-April-2006 is
the initial value i.e. P0 and the strike price, as mentioned in the prospectus, will be the
official closing price on the index on 10- May-2010. Since the maturity of the option
component is considered to be the same as that of the structured bond which is three
years. Therefore, historical volatility is estimated from the last three years data on Dow
Jones UBS commodity index. Since the time period between the initial date and the
strike date is about 11 trading days, therefore, it is assumed that the time for the option
is equal to 0.04 (11/252). The raw data and calculations can be found in excel file on the
CD Rom. The E-view output of log return and the volatility is calculated and the daily
historical volatility estimate is 0.014991. The annual volatility estimate according to
equation (45) is
𝜎𝐴𝑛𝑛𝑢𝑎𝑙 = 0.014991 ∗ √252 = 23.797% (59)
Table5
Initial index P0 Final index KT Interest rate r Maturity T Volatility
135.75 130.23 2% 0,04 23.79
The value of the option is estimated from the standard Black and Scholes formula for
pricing the European call option and the Monte Carlo simulations. The simulations are
performed following the risk neutral random walk up to maturity of the contract. The
results are given as
Black and Scholes price of the Call option: 6.37
From Monte Carlo simulations
17 Lecture slides
59
Table6
Simulations 5000 20000 40000 90000
Call Price 6.28 6.39 6.37 6.37
The idea to compare Call value from Black and Scholes formula and Monte Carlo
simulations is that as long we increase the number of simulations, the Monte Carlo
estimate of the call price converges to the Black and Scholes call price. It means that the
increase in number of simulations can lead to increase in accuracy by Monte Carlo
simulations.
So adding the PV of the bond and the option components according to equation (54),
the theoretical value of the DB Råvarer Basel 2013 is
𝑃𝑉 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑒𝑑 𝑛𝑜𝑡𝑒 = 94.17 + 6.37 = 100.54 (60)
The product was issued at 105 of the nominal amount. So, it means that the theoretical
price estimated in Black & Scoles world is less than the issue price. Therefore, we can
say that the product was offered at higher price than its theoretical fair price.
7.9 Pricing of Råvarer Basel 2010
The embedded option in Nordea Bank’s structured product is assumed to be Asian type
because the payoff depends on the arithmetic average of the strike price at some fixed
interval of time (fixing dates) during the life time of the product. The product had four
years maturity time which is from 16-June-2006 to 16-June -2010. So we assume that
we are standing in past time which is 16-June 2006, which will be the initial price on
the Dow Jones UBS commodity index. The historical volatility is estimated taking the
last four years index prices from the date of option valuation (16-June-2002 to 16- June-
2006). The daily volatility estimate is 0.0100. So the annualized volatility estimate will
be
𝜎𝐴𝑛𝑛𝑢𝑎𝑙 = 0.01 ∗ √252 = 15.8% (61)
The final value of the index is estimated by Monte Carlo simulation following risk
neutral assumption. Each valuation date is estimated by performing 5000 simulations
and by assuming that there are 21 trading days (one month) among all the fixing dates.
60
The simulated index values are then continuously compounded to maturity date. Table
below shows each fixing date.
Table 7
Fixing Date Simulated Index value
01-Dec-2009 200.40
04- Jan-2010 200.52
01-Feb-2010 200.75
01-Mar-2010 200.82
06-Apr-2010 200.91
04-May-2010 201.16
01-Jun-2010 201.24
The final price of the index is equal to the arithmetic average of the fixing dates, which
is equal to 200.83. So the parameters required to estimate option are
Table 8
Initial index value P0 170.51
Final Index KT 200.83
Interest rate r 3%
Time to Maturity T 4 years
Volatility 15.8%
So, the estimate of the Call option with the above mentioned parameters performed with
5000 simulations is
Price of the Call Option = 21.68
The theoretical price of the Råvarer Basel 2010 can now be calculated by adding the PV
of the bond and the option components according to equation (54). Therefore,
𝑃𝑉 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑒𝑑 𝑛𝑜𝑡𝑒 = 88.69 + 21.68 = 110.37 (62)
Equation (62) shows the calculated theoretical price for Råvarer Basel 2010 in Black &
Scholes world, with the assumptions of a perfect world discussed earlier. The issuing
price was 104% of the nominal amount. It means that the theoretical price is higher than
the issue price of the note. These results show that Black & Scholes option pricing
61
method under price the option when it is in the money and over price the option when it
is out of money. We can not make a concrete conclusion on the basis of only two
products evaluation that these observations are true but James D. Macbeth and Larry
also in their analysis showed that Balck & Scholes model tends to under price the option
that are in the money and overprice them is they are out of money18
8 Evaluation of the Model
.
The model misprices the options because of the set of assumption we took into
considerations.
• We assumed that the volatility of the returns remains constant throughout the
maturity of the product, which is not true in the real world. As a matter of fact
volatility of returns is not constant instead it changes over time. It can also be
observed in our data. It is also found from studies that higher value of volatility
may lead to higher theoretical value of the option and vice versa. Although in
both products the underlying is the same index but the time period is different.
Even then the annual volatility for the data from April 2010 to April 2013 is
different than the data from June 2006 to June 2010. The volatility estimates for
both the time periods were 23.79% and 15.8% respectively. We also assumed
historical volatility to calculate the options but according to Markove property,
only the present value of a variable is relevant to predict its future behavior. It
means historical volatility may not be good predictor of the future volatility.
• We assumed that the log of returns follow a normal distribution. This
assumption is not true always. The descriptive statistics for both the time series
data are given below. The data from these two tables show, that the returns in
both the time series are not normally distributed because Jarque-Bera test for
normality for both the data series is well above the probability values.
• Kurtosis for both the time series data is also nearly four in the first table while it
is more than four in the second table, which is also a violation for normality to
18 An Empirical Examination of the Black-Scholes Call Option Pricing Model, James Macbeth and Larry
Journal of finance Dec 1979
62
hold. High values of kurtosis means, that the returns have heavy tails which is
not a characteristic of the normal distribution. Returns from both the time series
data also have skewness (also called volatility smile). It means that the returns
are not distributed symmetrically around the mean. Positive and negative
skewness in table 9 and 10 show that data in both time series is right and left
skewed respectively.
Table9: Råvarer Basel 2010
Mean 0,000554 Median 0,000682 Maximum 0,048236 Minimum -0,31199 Std. Dev. 0,010067 Skewness 0,084407 Kurtosis 3,685963 Jarque-Bera 20,79348 Probability 0,000031
Table 10: Råvarer Basel 2013
Mean -0,00031 Median -0,000028 Maximum 0,056475 Minimum -0,064023 Std. Dev. 0,014991 Skewness -0,264433 Kurtosis 4,517711 Jarque-Bera 81,26153 Probability 0,000
• In BS model for option pricing, we assumed that the interest rate is known and
constant. In reality interest rate is not constant all the time but it also changes
over time. Therefore, the assumption of constant interest rate can also lead to
misprice both the bond and option components with the structured products.
8.1 Possible extensions to the thesis
The objective of this thesis was to see how the structured products are designed and the
theory involved for their estimation. The option component embedded was estimated by
using well known Black and Scholes option pricing formula. The model is applied
because of its wide acceptance, its simplicity and elegance, and its mathematical
tractability. The results obtained in this thesis can be used as a bench mark for further
63
readings. There are a number of possible extensions to this thesis. Some of them could
be
• The pricing of the structured products could also be performed by considering
the default risk of the firm issuing these products and by relaxing the assumption
of constant risk free interest rate.
• Option pricing can be performed in many ways. For example BS model assumes
that the asset’s price change continuously over time producing a log normal
distribution (geometric Brownian motion). Another possibility is that the prices
do not follow geometric Brownian motion but instead it can be assumed that
process of price changes follows jumps. There are further three possibilities in
jump assumption i.e. price changes continuously (diffusion model), or the
continuous changes are overlaid with jumps (jump – diffusion model) or the
price changes are only jumps (pure jumps). These processes are collectively
called Levy process.
The Constant Elasticity of Variance Model is an example of the diffusion model where
the risk neutral process for a stock price is given by
𝑑𝑆 = (𝑟 − 𝑞)𝑆𝑑𝑡 + 𝜎 𝑆𝛼 𝑑𝑍 (63)
Here α is a positive constant. If it is equal to one, then the equation becomes Geometric
Brownian motion.
Another useful option pricing model was suggested by Merton and is called Merton’s
mixed jump-diffusion model. In this model the continuous price changes are associated
with jumps. The risk neutral process of asset’s price behavior can be written as
𝑑𝑆𝑆
= (𝑟 − 𝑞 − 𝛾𝑘)𝑑𝑡 + 𝜎𝑑𝑍 + 𝑑𝑝 (64)
Here 𝛾 is average number of jumps per year and 𝑘 is the average size of the jump and is
measured as a percentage of the asset price. The percentage jump size is assumed to be
drawn from a probability distribution in the model. dZ is Wienner process and dp is the
poisson process generating the jumps.
64
Variance- Gamma model is an example of pure jump model. A gamma process is pure
jump process where small jumps occur very frequently and large jumps occur only
occasionally. Details about the model can found in Hull, p594.
Another possibility to price the option components embedded in structured products is
to relax the assumption of constant volatility. Hull and White, and the well known
Heston model to price the options in stochastic volatility frame are examples of the
models that consider stochastic volatility.
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9 Conclusion
Structured products are getting popularity among the investors now a day. Therefore,
the main objective of this thesis was to see how these products in general and
commodity linked products in particular are valued and how they are engineered. These
products can be divided into different categories according to their payoff structure.
Structured products generally consist of two components. The major component is a
zero coupon bond which ensures complete or partial protection of the invested capital
and an option component, which provides the opportunity of payoff. The issuing firms
discount the future payment into present value according to risk free interest rate and
then subtract it form the future amount. The difference is used to buy the option. While
deriving the zero coupon term structure, it was noted the discount factor decreases as
the time to maturity increases. The options are classified into plain vanilla and exotic
options. Exotic options are further subdivided into five to six groups according to their
payoff profile. The valuation of the embedded options is the tricky part of the valuation
process because the options embedded in these products are generally complex and
some time difficult to understand.
Option pricing theories involve set of assumptions and concepts that needed to be
understood before we can estimate them. Plain vanilla options are comparatively easy to
estimate while exotic option calculation is challenging. In this thesis the famous Black
& Scholes option pricing frame work is applied which is based on ideal market
assumptions like no transaction costs, constant volatility and interest rate, log returns
are normally distributed and the assumption that the underlying follows Geometric
Brownian motion. First, Brownian motion was introduced, but it can allow price to
become negative was therefore, replaced by Geometric Brownian motion because here
the prices never become negative. The Black Scholes frame work is based on delta
hedging principal and therefore was also explained. The ability to completely hedge the
option means that the expected returns must be equal the risk free rate of return.
Exotic options are difficult to price in particular when payoff structure become advance.
Therefore, Monte Carlo simulations were introduced in the pricing section of the option
component and discussed how it can be used to estimate the complex options.
66
Two commodity index linked notes were analyzed in the thesis. The embedded option
in DB Råvarer Basel 20013 is assumed to be standard European style call option with
the exception that the maximum payoff was limited up to a level of 35%. The embedded
option in Nordea Bank’s commodity linked note Råvarer Basel 2010 was Asian type,
because the final payoff was calculated on the basis of arithmetic average of each fixing
date (from December 2009 to June 2010). In this thesis we estimated the annualized
historical volatility and disregard other techniques to estimate volatility.
The theoretical fair price of DB Råvarer Basel 2013 was 100.54 while its issuance price
was 105. It means that the theoretical price is lower than the issuance price. In case of
Nordea Bank’s Råvarer Basel 2010, the estimated fair price was 110.37 while the issue
price was 104. It means the model underpriced Råvarer Basel 2013 and over priced
Råvarer 2010. Another important point to note here was that Black & Scholes frame
work underpriced the option which was in the money and overpriced the option which
was out of money. The model mispriced the embedded options because the underlying
assumptions in the model are not realistic. It is shown in several studies that volatility is
not constant but it changes over time. Similarly, interest rate also changes over time.
The assumption of log normal returns is also not true always. The data set for the two
selected products also showed that returns are not normally distributed but they are
skewed and have heavy tails.
In the end, suggestions are discussed to improve the option pricing model, for example
levy models and stochastic volatility models. The assumptions in these models are more
realistic than Black & Scholes model.
Studies also show that traders and market makers still resist using the so called cutting
edge option pricing formulas and the most widely used option pricing model is still
Black & Scholes with some so called ad hoc changes and frequent updating of
parameters. So we can say that the results obtained from this model can be used as a
bench mark for further studies.
67
References
Barclays Wealth, Light Energy Commodity Plan
Black, F, & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal
of Political Economy, 81(3), 637
BNP Paribas equities & Derivatives handbook
Bruce Tuckman, Fixed income securities. Wiley & sons
Das 2001, structured products and hybrid securities. John Wiley & sons
Douglas R. Emery A closer look at Black–Scholes option thetas. J Econ Finan (2008)
Eduardo S Schawartz 1982, “Option pricing theory and its application. The pricing of
commodity- linked bonds” The Journal of Finance Vol. XXXV11 NO. 2.
John Crosby, January 2007, A multi- factor jump- diffusion model for commodities.
Quantitative Finance, Vol. 8. No. 2. March 2008. PP. 181-200
John Crosby, June 2007. Pricing a class of exotic commodity options in a multi- factor
jump- diffusion model. Quantitative Finance, Vol. 8, No. 5, PP. 471-483
John C Hull 2008. Options, Futures and other derivatives. Pearson Prentice Hall
JAMES D. MACBETH, An Empirical Examination of the Black-Scholes Call Option
Pricing Model, Journal of finance Dec 1979
Joseph Atta- Mensah (1992). The Valuation of Commodity- Linked Bonds. University
of New Brunswick.
68
Lehman Brothers, A guide to Equity_Linked Notes
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Finance VOL. XLII, NO. 4. September 1987. PP. 1071-1076
Sergei Mikhailov .Heston’s Stochastic Volatility Model Implementation, Calibration
and Some Extensions
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Web addresses for the selected products
http://www.danskebank.dk/da-dk/Privat/Opsparing-og-
investering/Investering/Produkter/strukturerede-
produkter/Documents/2010/DBRÅVARER2013/Preliminary_Final_Terms_Final_Basal
http://www.nordea.dk/sitemod/upload/Root/main_dk/Privat/Raadgivning/Opsparing/Str
ukturerede_produkter/Oevrige_obligationer/prospekt_raavarerBasis2010.pdf
