Tienoven, Thornas-Jelle Van - The Influence of Pricing Differences on the Application of European-style Asion Options, A Liturature Study

7/29/2019 Tienoven, Thornas-Jelle Van - The Influence of Pricing Differences on the Application of European-style Asion Optio

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Bachelor Thesis Finance, June 2007

THE INFLUENCE OF PRICING DIFFERENCES ON THE

APPLICATION OF EUROPEAN-STYLE ASIAN

OPTIONS: A LITERATURE STUDY

Thomas-Jelle van Tienoven,

272411

[email protected]

Supervisor:

Jiajia Cui

Finance Department, K9.34

Tilburg University


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Abstract

In this thesis I investigate how the complexity of pricing Asian options could make a

difference in applying the options in line with determined trading objectives. In thecomprehensive literature on pricing methods of European-style Asian options I

simplify the Monte Carlo Simulation Approach to a basic concept and compare this to

the Black-Scholes Option Pricing Model for European vanilla options. Subsequently I

discuss practical examples on the application of European-style Asian options and

make a distinction between stand-alone trades and embedment in financial contracts.

Accordingly, I find that in certain situations institutional and individual investors

prefer the complex structure of Asian like payoff diagrams and conclude that in these

cases the pricing differences between European vanilla options and European-style

Asian options play a significant role in setting preferences for the application of these

options to achieve determined trading objectives.


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Table of contents

1. Introduction............................................................................................................03

2. Background ............................................................................................................05

* 2.1 Trading objectives..............................................................................................05

* 2.2 Derivatives markets ...........................................................................................06

* 2.3 Options markets .................................................................................................06

* 2.4 European vanilla stock options ..........................................................................07

* 2.5 European-style Asian options ............................................................................10

3. Pricing methods......................................................................................................13

* 3.1 The Black-Scholes Option Pricing Model .........................................................13

* 3.2 The Monte Carlo Simulation Approach.............................................................14

4. The application of European-style Asian options ...............................................18

* 4.1 Stand-alone trades ..............................................................................................18

* 4.2 Embedment in financial contracts......................................................................19

5. Conclusion and recommendation .........................................................................21

* 5.1 Conclusion .........................................................................................................21

* 5.2 Recommendation ...............................................................................................22

* 5.3 Acknowledgement .............................................................................................22

References ...................................................................................................................23


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1. Introduction

Asian options are commonly traded on currencies and commodity products

with low trading volume. End-users of commodities or energies tend to be exposed toaverage price over time, so Asian options are attractive for them. Asian options are

also popular with corporations, such as exporters, who have ongoing currency

exposures (Contingency Analysis, 1996). Asian options are also attractive because

they tend to be less expensive than comparable vanilla options. This is because the

volatility in the average value of an underlying asset is lower than the volatility of the

current value of the underlying asset at maturity.

In situations where the underlying asset is thinly traded or there is the potential

for its price to be manipulated, an Asian option offers protection. It is more difficult to

manipulate the average value over an extended period of time than it is to manipulate

it just at expiration. The longer the holding period of the option in which the average

price of the underlying asset is influenced, the more the investor is protected for price

manipulation.

Asian options are often used as they more closely replicate the requirements of

firms exposed to price movements on the underlying asset. For example, an airline

might purchase a one year Asian call option on fuel to hedge its fuel costs

(vDerivatives, 2005). In order to minimize the volatility on the market price for fuel,

the airline would prefer to hold an Asian option based on the average market rate of

fuel rather than an option that is based on as single strike price.

The lower prices of Asian options relative to European options make them

useful for situations in which the user wants to hedge or speculate on the price series

but is unwilling to spend the full price required by European options. So Asian

options are cash flow reducing. Moreover, this type of option is mainly used in

markets where prices are unusually volatile commodities like crude oil, natural gas,

copper, aluminum (Overseas Development Institute, 1995) and may be susceptible

to distortion or manipulation. For example, Denali Consulting (2005) outlines the

urge for managers to maintain focus on substantial price swings related to increase in

demand or shortage in supply.

Furthermore, next to the abovementioned examples on stand-alone Asian

options, which are traded in the OTC market, Asian options also exist as parts of


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financial contracts. Life insurance contracts for example may be in need to guarantee

a minimum rate of return being the average of the past returns. For further reading

hereupon, see Jiajia Cui, Pricing Interest Rate Guarantees embedded in Life

Insurance Contracts by Monte Carlo Simulation, MSc thesis, University of

Amsterdam, Amsterdam, The Netherlands, August 2003.

These abovementioned examples underline the need for guidance in trading

objectives on whether to use European-style Asian options compared to European

vanilla options. The aim of this thesis is to provide an overview on these two types of

options, how they are generally priced and valuated and the differences herein,

leading two the research question:How can valuation and pricing differences between

European vanilla options and European-style Asian options play a role in applying

these options in line with determined trading objectives?

First, I will briefly outline the two major trading objectives and introduce the

derivatives market and options market respectively. Subsequently I summarize both

types of options focused upon, that is European vanilla options and European-style

Asian options. Next, I will go deeper into pricing and valuation methods on the two

option types and on the basis of this overview I discuss the application of European-

style Asian options in contrast to European vanilla options.


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2. Background

2.1 Trading objectives

Any investor whether institutional or an individual needs to set objectivesbefore he or she actually starts with making investment decisions. There is no perfect

investment process, but the Association for Investment Management and Research

(AIMR) suggests following its basic framework. The first step in this framework is to

set objectives. Such objectives center on the risk-return trade-off between return

requirements and risk tolerance.

On the one hand an investor can require a high rate of return on his or her

investment, but logically this is partially involved with a higher level of risk compared

to an investor that requires a less high rate of return. For instance, individual investors

that have a high level of risk tolerance can invest in a way of arbitraging.

Arbitrage opportunities arise when two or more securities are mispriced

relative to each other. If such a scenario occurs, an investor can take a long position in

one derivative and sell short another related derivative. For instance, if the price of a

European put option exceeds the value predicted by put-call parity, an investor can

make use of this arbitrage opportunity by selling short the overpriced put, take a long

position in the related call, sell short the underlying asset and take a long position in

risk-free securities, like bonds. The payoff diagram for the arbitrageur will look like

this:

S(T) < X X S(T)

short put - (X - S(T)) 0

long call 0 S(T) - X

short stock - S(T) - S(T)

long bond X X

0 0

In both states of outcome, that is whether the price of the underlying asset at

maturity is lower or higher than the exercise price, the payoff will be zero. But since

the put is overpriced, it will gain more money than the replication with a long call.

Creating the profit from this mispricing is called an arbitrage opportunity.

On the other hand an investor can focus on risk reduction by investing in a

financial position contrary on the position that he or she currently is in. This is

commonly done by institutional investors, like companies. Reducing risk by taking an


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opposite position is moreover known as hedging. It is an extreme case of

diversification, in which negative correlation between two risks is involved. Financial

risk can be reduced by buying and selling derivative securities. If an investor holds a

position in 1,000 Microsoft shares for instance, he or she can hedge this by buying put

options on Microsoft. In case the stock prices increase over the time, the gain

hereupon is relatively lowered by the put premium paid. But more strongly, in case

the stock price decreases over the time, the loss hereupon is reduced by the payoff on

the put options. In other words, the financial risk is reduced and the position is so-

called hedged.

2.2 Derivatives markets

Derivatives exist to transfer risks from those who do not want to bear it to

those who do (Marthinsen, 2005). These are financial instruments whose returns are

derived from those of other financial instruments. Derivatives are contractual

instruments whose performance is determined by how an underlying asset or

instrument performs. While derivatives can be powerful speculative instruments,

businesses most often use them to hedge in order to protect themselves against shocks

in currency values, commodity prices and interest rates for example. At the end of

June 2004 the Bank for International Settlements (2007) recorded a total notional

amount of all the outstanding positions in the derivatives market of $220 trillion. This

trading volume underlines the importance of financial derivatives.

Derivatives come roughly in two basic categories: option-type contracts and

forward-type contracts. These may be exchange-listed, such as futures and stock

options, or they may be privately traded. The major difference between the two basic

categories is explained by the right to exercise an option or the obligation to execute

the contract at delivery date. This thesis deals with financial instruments in the options

market.

2.3 Options markets

The options markets have been tremendously successful, reflected by the

heavy trading volume in these markets for example the Chicago Board Options

Exchange (2007) announced a volume of 674,735,348 option contracts over 2006.

Options are derivative instruments that entail a contract between two parties

a buyer and a seller. It gives the buyer the right to purchase or sell something at a later


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date agreed at a todays settled price. Options exist on financial assets, like stocks and

bonds, but also on commodities, futures contracts and foreign currencies. Most of

these types of options are traded on exchange-listed basis. However, the over-the-

counter market, a market in which option trading is conducted privately between two

parties who prefer rather contracting with each other than to a public transaction, has

been revived and is now very large and widely used by corporations and financial

institutions.

A wide variety of options exist, but generally they can be divided into two

main categories. The payoff for path-independent options is not affected by the

sequence of prices of the underlying asset as it moves towards the expiration of the

option. In other words, the average price over time of the underlying asset does not

influence the payoff of the option. In contrast, path-dependent options are affected by

the price fluctuation of the underlying asset over the holding period of the option

(Chance, 2004).

Recently, customized options have been made available to investors. Now

innovation still occurs in the market for customized options that are not exchange-

listed (Bodie, Kane and Marcus, 2005). The path-dependent exotic option I am going

to deal with in this thesis is the Asian option.

In order to be able to answer the conducted research question I will elaborate

on differences between this European-style Asian stock option and the European

vanilla stock option. Hereby I assume the options to be written on non-dividend

paying stocks. Differences in these two stock options can be found in both the payoff

construction and the valuation method.

2.4 European vanilla stock options

European vanilla stock options give its holders the right to sell or buy the

underlying asset for a specified price, called the strike price, on some specified

expiration date.

The term European does not refer to geographic regions of these options or

their underlying assets. It is confined to the possibility to exercise an option only at

maturity, in contrast to so-called American options, which can be exercised prior to

the expiration date.

In general, this type of option can be divided into call options and put options.

A call gives the right to buy the underlying; a put option gives the right to sell. The


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holder of a European call is not required to exercise the right to buy. He or she will

only choose to do so if the market value of the asset to be purchased exceeds the

strike price at the end of the holding period. If the call is left unexercised, it will be

worthless. Since the holder cannot be forced to exercise the option, the minimum

value of a European call therefore is:

0eC

Logically, no one would pay more for the right to buy a stock than for the stock itself.

Therefore the maximum value of a European call is:

0SCe

In case the call is exercised, the value will be the difference between the stock price at

expiration and the strike price. The mathematical way of presenting the value of a

European call at expiration is as follows:

),0( XSMAXCTe=

But more strongly, what is important is the value of the European call prior to

expiration. The price of a European call must at least equal the greater of zero or the

stock price minus the present value of the exercise price. If this is not the case,

arbitrage opportunities arise, like aforementioned. An investor could then buy the call

and invest in risk-free bonds and take a short position in the stock. Since the

combination of buying a call and risk-free bonds is less then the stock sold short, one

has a positive initial cash flow. At maturity of the call, again, the exercise price can be

higher than the stock price, of which the difference would lead to another positive

cash flow. If the exercise price turns out to be lower, then there would be no cash flow

at maturity. In other words, if the following lower bound for pricing a European call

would not hold, investors would have the opportunity the gain money from a risk-free

position:

))1(,0( 0T

e rXSMAXC

+


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Similar equations holds for European put options. A holder of a European put

has the right to sell the underlying asset for a specified price, called the strike price,

on some specified expiration date. Since this is not obligatory the put holder will only

do so if exercising provides a positive cash flow. Therefore the minimum value of a

put is:

0eP

An investor that holds European puts expects the price of the underlying asset to

decrease. If the stock price is lower than the exercise price the put holder will have a

positive cash flow, because he or she can short sell the stock for the exercise price and

buy back the stock for the current price. So the gain from a European put is the

difference between the strike price and the stock price at maturity. The mathematical

way of presenting the value of a European put at expiration is as follows:

),0(Te

SXMAXP =

The best case scenario for a put holder is when the company that has written the put

goes bankrupt. Then the stock at maturity is worth zero and the put holder earns the

strike price (minus the premium paid to buy the right to sell). The maximum value of

a European put therefore is the present value, discounted for interest rate r, of the

strike price:

T

erXP + )1(

Subsequently, the value of a European put must always at least equal the greater of

zero or the present value of the strike price minus the stock price. Otherwise arbitrage

opportunities arise. An investor could then sell short the put, borrow money the

present value of the strike price X, and buy the stock. This generates an initial positive

cash flow as buying the stock is less than selling short the put and borrow money

equivalent to the discounted strike price. At maturity of the put, the stock price could

be lower than the exercise price, what leads to another positive cash flow. If the stock


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price turns out to be higher than the exercise price, there would be no cash flow at

maturity. In conclusion, if the following lower bound for pricing a European put

would not hold, investors would have the opportunity the gain money from a risk-free

position:

))1(,0( 0SrXMAXPT

e +

For further reading on European vanilla stock options, see John C. Hull, Options,

Futures and other Derivatives, 6th edition, 2006, chapter 9.

2.5 European-style Asian stock options

Asian options are options where the payoff depends on the average price of the

underlying asset during at least some part of the life of the option (Hull, 2006).

Because of this fact, Asian options have a lower volatility and hence rendering them

cheaper relative to their European counterparts. They are commonly traded on

currencies and commodity products which have low trading volumes. Asian options

were originally used in 1987 when Banker's Trust Tokyo office used them for pricing

average options on crude oil contracts (Global Derivatives, 2007). This explains the

term Asian, which is not related to any geographic positioning.

Asian options, also referred to as average options, are related to the category

of exotic options. The payoff of these non-vanilla types of options is path-dependent.

Other examples are Russian options, Israeli options, chooser options and barrier

options. In this thesis I am dealing with the Asian option type and since I am going to

compare this type with European vanilla options I will deal with European-style Asian

options only.

Generally, Asian options are broadly segregated into three categories;

arithmetic average Asians, geometric average Asians and both these forms can be

averaged on a weighted average basis, whereby a given weight is applied to each

stock being averaged. This can be useful for attaining an average on a sample with a

highly skewed sample population (Global Derivatives, 2007). A further breakdown of

these options implies that they can either be based on the average price of the

underlying asset, or on the average strike price. In other words, the average price can


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replace either the stock price at expiration, or the strike price at expiration. In sum,

four main types of European-style Asian options with accompanying payoff diagrams

exist:

Average price call ),0( XSMAX avg =

Average price put ),0(avg

SXMAX =

Average strike call ),0( avgT SSMAX =

Average strike put ),0( Tavg SSMAX =

Now assume a three-period case for all types of European-style Asian options

mentioned above. There are eight possibility outcomes; the average price of the

underlying asset can go up three times but also down three times. All the possibilities

are stated in the second column of the table below. The probability of each outcome

can be found in the third column. In column four the current price at maturity is given,

taken into account an increase (up) in S(t) of 25% and a decrease (d) in S(t) of 20%.

Column four gives the average price of the asset over all three periods. The last four

columns represent the payoffs in each outcome for the discussed types of European-

style Asian options.

From the bottom line in the table above one can find the values for each type of Asian

option. This value is calculated as the sum of the product of the payoffs and the

respective probability of that payoff, discounted at the risk-free rate of 5% (Chance,

2004).

This way of pricing Asian options makes use of the binomial model. For

European vanilla options this binomial model requires a large number of time steps to

give an accurate approximation to the price for standard European vanilla options,priced with the Black-Scholes model. Unfortunately there is no similar pricing

Outcome Path Probability S(T) S(avg) Avg price call Avg price put Avg strike call Avg strike put

1 uuu 0,171 97,66 72,07 22,07 0,00 25,59 0,00

2 uud 0,137 62,50 63,28 13,28 0,00 0,00 0,78

3 udu 0,137 62,50 56,25 6,25 0,00 6,25 0,00

4 duu 0,137 62,50 50,63 0,63 0,00 11,87 0,00

5 udd 0,110 40,00 50,63 0,63 0,00 0,00 10,63

6 dud 0,110 40,00 45,00 0,00 5,00 0,00 5,00

7 ddu 0,110 40,00 40,50 0,00 9,50 0,00 0,50

8 ddd 0,088 25,60 36,90 0,00 13,10 0,00 11,30

Option value = 5,71 2,37 5,92 2,48

Payoff


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formula for Asian options that are arithmetic. There exists a formula for Asian options

that have a geometric average, but these types are rarely traded in the market or

embedded in financial contracts.

Furthermore, this binomial model can be used in an illustrative way, but is

impractical in realistic scenarios. In the example describe above, I dealt with a

hypothetical three-period case. If for example the average of an Asian option is being

hold for n periods, the binomial model needs to calculate 2 paths either up or down.

For a 60-period case this would lead to 1,152,921,504,606,850,000 paths.

Some formulas to price Asian options exist and are used in practice, but lead

only to an approximated price. I will go over some of these approximation formulas

later in my thesis. For further reading on pricing European-style Asian options based

on the binomial model, see Don M. Chance, An Introduction to Derivatives & Risk

Management, 6th edition, 2004, chapter 14.


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3. Pricing methods

3.1 The Black-Scholes Option Pricing Model

As the binomial tree model is a basic method for conceptualizing theapproximation of valuating calls and puts, it is not straightforward for practical use.

The commonly used model for accurately pricing European vanilla stock options is

the Black-Scholes model. Based on the Capital Asset Pricing Theory and stochastic

calculus an accurate pricing formula was developed (Black and Scholes, 1973). The

core concepts of this model are the use of natural logarithm and the exponential

function. Furthermore, the model is based on a specific set of assumptions. First,

stock prices behave randomly and evolve according to a lognormal distribution with

the natural logarithm, e, of the return on the stock. Second, the risk-free rate and

volatility of the log return on the stock are constant throughout the options life, to

simplify the model. Next, the model assumes no dividend payments, taxation or

transaction costs and more strongly, it ignores the possibility of early exercise, such

that it automatically deals with European options. The end formula for pricing a

European call is obtained through complex mathematics, but the result is as follows:

)(*)(* 210 dNXedNSCTr

ec= ,

where

T

TrXSd c

)2()ln( 201

++= ,

and

Tdd=

12 .

From this pricing method derives that the price of a European call equals the expected

value of the stock price at expiration, given the fact that the stock price exceeds the

exercise price at expiration ( XST > ), discounted at the continuously compounded

interest rate, Trce . The second part is the expected payout of the entire exercise price

at expiration. Since this is a future value, it should be multiplied by Trce , but the


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present value of the call price is needed, so it can be cancelled out. So both )( 1dN

and )( 2dN are the cumulative normal probabilities of these expressions of the model.

Inserting the put-call parity into the pricing formula for European calls as

explained above provides the pricing formula in the Black-Scholes model for

European put options:

Tr

eecXeSCP

+= 0 (put-call parity),

rewriting into the Black-Scholes model gives

)](1[)](1[ 102 dNSdNXePTr

ec =

This is the commonly used formula for pricing European puts on stock options under

the aforementioned assumptions. Klemkosky and Resnick (1979) derive from their

study that inefficiency in their empirical model results to test put-call parity

consistency is a result of option overpricing or underpricing. This influences the

accuracy of the Black-Scholes model. For further reading upon the Black-Scholes

model and put-call parity, see Stoll (1969), MacBeth and Merville (1979), and Chriss

(1997).

3.2 The Monte Carlo Simulation Approach

A fast and very accurate model for pricing path-dependent options such as

Asian options has yet not been found. Still several researchers have devised

approximation algorithm for Asian options. Chalsani, Jha and Varikooty (1997)

conclude that pricing path-dependent options in the binomial tree model of Cox, Ross

and Rubinstein (1979) is notoriously hard and that the Black-Scholes differential

equation for the price of such options has no known closed form solution.

Furthermore they summarize the researches on the approximation algorithm into tree

categories.

Geman and Yor (1993) considered an approach by deriving formulas for the

Laplace transform of an Asian option. The second method, as being investigated by

Levy and Turnbull (1992), Levy (1992) and Turnbull and Wakeman (1991), consists


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of approximations on the distribution of arithmetic average on which the payoff of an

Asian option is based. The pricing method most often used in practice has been

studied by Kemna and Vorst (1990) and Carverhill and Clewlow (1990). The latter

have explored Fourier transform techniques, while the first have used Monte Carlo

simulation to price options based on average value assets.

By presenting a dynamic hedging strategy from which the value of an average

option can be derived, Kemna and Vorst (1990) explain mathematically why no

explicit formula for an average option like the European-style Asian option can be

derived. According to them the combined process ),( tt AS , that is the price of the

underlying asset S at time t and the average value A at time t, is not Gaussian in

character which means it is not a function in the form of

22 )2/()(

)(cbx

aexf

= . For

mathematical proof of this, see Kemna and Vorst (1990).

Therefore they are obliged to use numerical computations, for which they take

a Monte Carlo simulation approach in finding a valuation model for Asian options.

This simulation approach generates random numbers according to probabilities that

are associated with uncertainty (Chance, 2004). This method does not take into

account American-style Asian options since these are exposed to price manipulation

as they can be exercised early. Therefore only European-style Asian options are takeninto account.

The generated numbers from the simulation represent possible changes in

stock price S , making up the average of the European-style Asian option at a certain

time interval from a certain point in the future until the maturity date. Mathematically

this is written out as t . Also, the numbers are based on the continuously

compounded interest rate Trce and how sensitive the stock price is for changes, i.e. the

volatility of the stock

. The variable

is a random number generated from astandard normal probability distribution. This leads to the general formula:

tStSrS c +=

As Kemna and Vorst use their theoretical numerical approximation of the value of an

average option in line with the Monte Carlo simulation method, they use the normal


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distribution of

1

logi

i

St

St, with mean

n

TTr ))(2

1( 0

2

and variancen

TT )( 02

in

the formula to generate a random sequence of future stock prices at nT . For

mathematical proof of this, see Kemna and Vorst (1990). Independent drawings,

nxx ,...,1 from the standard normal distribution are generated to simulate this random

sequence:

i

TiTi

nx

TT

n

TTr

SS)(

))(2

1(

loglog 00

2

)1()(

+

+=

For each series the change in stock price in the final time interval is calculated and

then inserted into the formula for the price of an Asian option at maturity. This leads

for each series to an expectation of the price of on Asian option. Discounting the

average of all generated expected option price to obtain the present value of the option

is done using the continuously compounded interest rate, )( 0TTrce . So the expectation

of the price of an Asian option, ~C, is:

( )=

n

i

TTr

Tc

ieY

1

)( 0* ,

where

}0,max XSYavgT

= .

More complex options require changes in the Monte Carlo simulation

procedure, but with European-style Asian options the speed and simplicity of this

procedure is desirable. Kemna and Vorst (1990) and Grant, Vora and Weeks (1997)

for example have researched on modifications of the procedure, but in this thesis I

keep the European-style Asian option pricing method very simplified. Researches,

however, have shown that with the efficient pricing method using modified and

extended versions of the Monte Carlo simulation procedure result in sufficient

analytical approximations on the price of European-style Asian options. For further


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reading hereupon, see Chalsani, Jha and Varikooty (1997), Vanmaele, Deelstra,

Liinev, Dhaene and Goovaerts (2006), and Reynaerts, Vanmaele, Dhaene, and

Deelstra (2006).

The difference in complexity of pricing European-style Asian options

compared to European vanilla options is substantial. For the pricing of the vanilla

options a well-defined pricing formula is available and can be considered as exact.

This enhanced the ability to sweeping standardization of the European puts and calls.

Next to the use in OTC markets these types of options are therefore also traded on

exchange-list basis. In the literature the use of these options are extensively described.

Institutional investors play this options market by continuously profiting from

arbitrage opportunities due to the over- and underpricing of vanilla options. Also

these financial instruments are being used for several hedging strategies. For basic

ideas, see Hull (2006).

For the European-style Asian options, however, no ultimate exact pricing

method has been found, leaving the method described above only to be an accurate

analytical approximation. This brings along pricing errors, making the application of

this type of options more complex. Still, the Asian option is preferable in useful in

practice. I discuss the application in the next section.


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4. The application of European-style Asian options

4.1 Stand-alone trades

In practice a rough distinction can be made between ways of trading withEuropean-style Asian options. On the one hand they are traded as stand-alone

products in the over-the-counter market, as European-style Asian options are not

exchange list-based standardized options. The basic findings upon the pricing

methods used to value European-style Asian options can be a factor that withholds the

financial markets to trade this type of options based on exchange-list. Milevsky and

Posner (1998) indicate that the entire volume of Asian options is traded on OTC and

lies somewhere between five to ten billion US dollars.

This financial instrument as a stand-alone can be supportive and useful for

both companies and individual investors or fund managers. For one reason, fund

managers that hold a portfolio of stocks can face losses from price movements of one

of these stocks. The fund manager therefore needs to rehedge this price fluctuation

continuously in order to meet the minimum rate of return. By making use of a

European-style Asian option on the underlying stock, one does not need to do so, as

the option is based on the average stock price over the holding period. Hence, the

fund manager does not need to rehedge the portfolio continuously. Here, the Asian

option is being used as a hedging instrument.

Secondly, applying European-style Asian options might be useful for

companies that need to buy commodities like crude oil, aluminum or gold at a specific

time of the year. If such companies sell their product regularly throughout the year,

they are likely to face profit margins as the price of the commodities is volatile.

Taking a position in Asian options on the underlying commodity allows managers to

smoothen their profit margin on sales. This is a cost efficient way of hedging cash

flows over extended periods. Also, in thinly traded asset markets the use of European-

style Asian options is preferable as in these markets price manipulation near maturity

is possible (Vanmaele et al., 2006) and furthermore for underlying assets of Asian

options exchange rates and interest rates are often used (Reynaerts et al., 2006). So

the Asian option functions as a hedging instrument here.


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4.2 Embedment in financial contracts

The characteristics of Asian options make them also useful in complex

financial contracts and strategies. Nielsen and Sandmann (2003) explain that Asian

options can also be part of retirement plans, life insurance or corporate financing, by

giving an example. They consider an investment plan where periodic payments are

spread over a certain period invested in a specified risky asset. At maturity the value

of this periodic investment is composed of the price of the asset and the price on each

of the investment days. In order to guarantee a minimum rate of return on this

investment plan an Asian option on the average return can be used as a risk reduction

instrument. This is especially of relevance when the periodic payments are part of a

retirement scheme.

Also, Schrager and Pelsser (2004) underline the use of Asian options by

elaborating on the existing literature on insurance contracts. They extend the way of

viewing a single premium Unit Linked contract as a stock along with a put option on

that stock, by explaining that in practice payments of multiple premiums are

straightforward. It is shown that for a generic structure of the cost and mortality

deductions the structure of the payoff remains Asian like (Schrager and Pelsser, 2004).

They also prove the analogy of the guarantee with Asian options by finding equality

between both the insurance contract and its equivalent.

A third example of the embedment of Asian options is an Oranje-Nassau bond

contract. This type of contract consists of a Dutch government bond in combination

with an option on the average asset value of a commodity. The settlement price of

such a contract was defined as the average Brent Blend oil price over the last year of

the contract. The redeemer of the contract was to receive the face value of the bond

plus the difference between this face value and the settlement price, but had to forfeit

a percentage of the coupon rate. Similar structures are briefly described by Kemna

and Vorst (1990) and include examples of Mexican Petrobonds, Delaware Gold

indexed bonds, Petrolewis oil indexed notes and BT Gold Notes Limited notes. In

case where the settlement price is lower than the face value, the average option leads

to the profit of an arbitrage opportunity.

Finally, Asian options included in bond contracts protect the issuer of any

possible price manipulation of the underlying asset at maturity. Next to that it gives

the bond holder also the ability to gain profit from a firm of which its profits mainly

depends on the price of an underlying asset. For example, if an asset has a low price


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during the entire holding period of an option, but increase tremendously near maturity,

then the firm would not be able to pay out the option premiums to the holders in case

the option would be a European vanilla call option. The use here of a European-style

Asian option would be preferable in order to smoothen premiums to be paid out by

this firm and hedges the firms cash flows.


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5. Conclusion and recommendation

5.1 Conclusion

The aim of this thesis is to provide clear answers to the research question:How can valuation and pricing differences between European vanilla options and

European-style Asian options play a role in applying these options in line with

determined trading objectives? From my discussion on the application of European-

style Asian options and the simplified outline on differences in pricing methods

between these options I have drawn my conclusions.

The available literature on the various approaches for pricing European-style

Asian options was too comprehensive to discuss entirely. Since the emphasis of my

thesis is not on how to price the mentioned options I simplified this to a basic concept.

I summarized an overview on findings that no accurate exact formula for pricing an

Asian option is available and that the complexity substantiates its use in practice from

the standardized European vanilla option. Next to that I mentioned the use of

European vanilla options briefly and discussed the use of European-style Asian

options. For each application I indicated for what purposes it was being used.

In line with the research question stated above I wanted to investigate how the

complexity of pricing Asian options could make a difference in applying the options

in line with determined trading objectives. For European vanilla options it is obvious

to conclude that these derivatives are used to profit from arbitrage opportunities in

case they are mispriced. Aggressive derivatives traders are constantly looking for such

opportunities in the market. For European-style Asian options these relative simple

arbitrage opportunities are not readily available due to the complexity of pricing them.

So European-style Asian options are not used to gain from arbitrage opportunities in

stand-alone traded options. In case these options are embedded in financial contracts

like Oranje Nassau bond contracts, however, they do lead to arbitraging.

Furthermore, European options are commonly used in hedging strategies and

extensive literature hereupon is available. The differences in pricing method does not

influence the possibility to use Asian options in hedging strategies as from the

discussion on the application can be found that these type of options is commonly

used to hedge positions as well. The examples I used provide evidence that in certain

situations institutional and individual investors prefer the complex structure of Asian


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like payoff diagrams, for example a guarantee of a minimum rate of return on a

investment plan, equality with multiple premium Unit Linked insurance contracts,

protection against price manipulation near or at maturity and price smoothening of

underlying assets in order to hedge cash flows. I conclude that in these cases the

pricing differences between European vanilla options and European-style Asian

options play a significant role in setting preferences for the application of these

options to achieve determined trading objectives as the structure of European-style

Asian options is preferred.

5.2 Recommendation

In this literature study I have investigated the differences in pricing methods

between European-style Asian options and European plain vanilla options.

Subsequently I looked for practical examples on the application of Asian options and

how in practice is determined when European-style Asian options are preferred rather

than vanilla options considering the pricing differences.

Further research on the application of Asian options could show how in

practice is dealt with the complexity of pricing. This could be extended to various

path-dependent options, as they exist in several variants.

5.3 Acknowledgement

I would like to thank my supervisor Jiajia Cui for her comments and

suggestions.


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Tienoven, Thornas-Jelle Van - The Influence of Pricing Differences on the Application of European-style Asion Options, A Liturature Study