Afm - Derivatives

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ADVANCED FINANCIAL MANAGEMENT - DERIVATIVES

Tel. No.: 25394777 / 67120221 E-mail: [email protected] Website: www.quoinacademy.com(1)

DERIVATIVES: MANAGING FINANCIAL RISK

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INTRODUCTION

Derivative instruments are defined by the Securities Contracts (Regulation) Act to include (1) a security derived from a d

instrument, share, secured/unsecured loan, risk instrument or contract for differences, or any other form of security and (2

contract that derives its value from the prices/ index of prices of underlying securities. Derivative contracts have several varia

The most common variants are forwards, futures and options. Three broad categories of participants —hedgers, speculators

arbitrageurs—trade in the derivatives market.

The derivatives market performs a number of economic functions. First, prices in an organised derivatives market reflect

perception of the market participants about the future and lead the prices of underlying to the perceived future level. The pric

of derivatives converge with the prices of the underlying at the expiration of the derivatives contract. Thus, derivatives help in

discovery of the future as well as current prices. Second, the derivatives market helps to transfer risks from those who ha

them but may not like them those who have an appetite for them. Third, derivatives, due to their inherent nature, are linked

underlying cash markets. With the introduction of derivatives, the underlying market witnesses1 higher trading volumes beca

of participation by more players who would not otherwise participate for lack of an arrangement to transfer risk. Fou

speculative trades shift to a more controlled environment of derivatives market In the absence of an organised derivatimarket, speculators trade in the underlying cash markets. Finally, derivatives markets help increase savings and investmen

the long run.

FORWARD CONTRACTS

A forward contract is an agreement to buy or sell an asset on a specified date for a specified price. One of the parties to

contract assumes a long position and agrees to buy the underlying asset on a certain specified future date, for a cert

specified price. The other party assumes a short position and -agrees to sell the asset on the same date for the same pr

Other contract details like delivery date, price and quantity arc negotiated bilaterally by the parties to contact.

Forward contracts are normally traded outside stock exchanges. They are popular on the Over the Counter (OTC) market.

The salient features of forward contracts are as follows:

(i) They are bilateral c an asset/security tracts and, hence, exposed to counterparty risk;

(ii) Each contract is customer designed, and, hence, is unique in terms of contract size, expiration date and the a. d

for a type and quality;

(iii) The contract price is generally not available in public domain.

(iv) On the expiration date, the contract has to be settled by delivery of the asset and

(v) If a party wishes to reverse the contract, it has to compulsorily go to the same counterparty, which often results

high price being charged.

However, forward contracts in certain markets have become very standardized, as in the case of foreign exchange, ther

reducing transaction costs and increasing transaction volume.

Forward contracts are very useful in hedging and speculation. A classic hedging application would be that of an exporter w

expects to receive payment in dollars, three months later. He is exposed to the risk of exchange rate fluctuations. By using

currency forward market to dollars forward, he can lock-on a rate today and reduce his certainty. Similarly, an importer w

required to make a payment in dollars two months hence can reduce his exposure to exchange rate fluctuations by buy

dollars forward. If a speculator has information or analysis, which forecasts an upturn in a price, he can go along on the forw

market instead of the cash market. The speculator would go long on the forward, wait for the price to rise and then take

reversing transaction to book profits. Speculators may well be required to deposit a margin up-front. However, this is general

relatively small proportion of the value of the assets underlying the forward contract. The use of forward markets here supp





leverage to the speculator.

Limitations:

Forward markets are afflicted by several problems:

(i) Lack of centralization of trading,

(ii) (ii) Liquidity and

(iii) (iii) Counterparty risk.

The basic problem in the first two is that they have too much flexibility and generality. The forward market is like a real est

market in that any two consenting adults can form contracts against each other. This often makes them design terms of the d

that are very convenient in that specific situation, but makes the contracts non-tradable. Counterparty risk arises from

possibility of default by any one party to the transaction. When one of the two sides to the transaction declares bankruptcy,

other suffers. Even when forward markets trade standarised contracts and, hence, avoid the problem of illiquidity,

counterparty risk remains a very serious issue.

FUTURES/FUTURE CONTRACTS

Futures markets are designed to solve the problems that exist in forward markets. A futures contract is an agreement betwe

two parties to buy or sell an asset at a certain time in the future, at a certain price. But unlike forward contracts, futures contraare standardized and stock ex-change traded. To facilitate liquidity in the futures contracts, the exchange specifies cert

standard features for the contract. It parties to buy/sell is a standardized contract with a standard underlying instrumen

standard quantity an asset /security and quality of the underlying instrument that can be delivered, and a standard timing

such settlement. A futures contract may be offset prior to maturity by entering into an equal and føfb’s daily opposite transact

The standardised items in a futures contract are:

(i) Quantity of the underlying,

(ii) (ii) Quality of the underlying,

(iii) (iii) The date/month of delivery, (iv)

Future contracts are a significant improvement over forward contracts as they eliminate counterparty risk and offer mliquidity.

Futures Terminology

Important terms associated with futures contracts are as follows:

Spot Price: The price at which an instrument/asset trades in the spot market.

Future Price: The price at which the futures contract trade in the future market.

Contract Cycle: The period over which a contract trades. For instance, the index futures contracts typically have one mon

two months and three months expiry cycles that expire on the last Thursday of the month. Thus, a January expiration contr

expires on the last Thursday of January and a February expiration contract ceases trading on the last Thursday of February. the Friday following the last Thursday, a new contract having three month expiry is introduced for trading.

Expiry Date: It is the date specified in the futures contract. This is the last day on which the contract will be traded, at the end

which it will cease to exist.

Contract Size: The amount of asset that has to be delivered under one contract. For instance, the contract size of the N

future market is 200 Nifties.

Basis Basis: is defined as the futures price minus the spot price. There will be a different basis for each delivery month for e





contract. In a normal market, basis will be positive. This reflects that futures prices normally exceed spot prices.

Cost of Carry: The relationship between futures prices and spot prices can be summarised in terms of the cost of carry. T

measures the storage cost plus the interest that is paid to finance the asset, less the income earned on the asset.

Initial Margin: The amount that must be deposited in the margin account at the time a futures contract is first entered into is

initial margin.

Marking to Market: In the futures market, at the end of each trading day, the margin account is adjusted to reflect the investgain or loss depending upon the futures closing price. This is called marking to market.

Maintenance Margin: This is somewhat lower than the initial margin. This is set to ensure that the balance in the mar

account never becomes negative. If the balance in the margin account falls below the maintenance margin, the inves

receives a margin call and is expected to top up the margin account to the initial margin level before trading commences on

next day.

Payoffs Payoff for Futures

A pay off is the likely profit/loss that would accrue to a market participant with change in the price of the underlying as

Futures contracts have linear payoffs. In simple words, it means that the losses as well as profits, for the buyer and the selle

futures contracts, are unlimited. The pay off for futures, that is, for buyers (long futures) and sellers (short futures) is discussbelow.

Pay off for Buyer of Futures: Long Futures

The pay offs for a person who buys a futures contract is similar to the pay off for a person who holds an asset. He has

potentially unlimited upside as well as downside. Take the case of a speculator who buys a two month Nifty index futu

contract when the Nifty stands at 1220. The underlying asset in this case is the Nifty portfolio. When the index moves up,

long futures position starts making profits and when the index move down it starts making losses.

Pay off for Seller of Futures: Short Futures

The pay off for a person who sells a futures contract is similar to the pay off for a person who shorts an asset. He ha

potentially unlimited upside as well as downside. Take the case of a speculator who sells a two month Nifty index futu

contact when the Nifty stands at 1220. The underlying asset in this case is the Nifty portfolio. When the index moves down,

short futures position starts making profits and when the index moves up, it starts making losses. The pay off for futures

illustrated below.

Example: On January. 15, X bought a January Nifty futures contract that cost him Rs 5,38,000. For this he had to pay an in

margin of Rs 43,040 to his broker. Each Nifty futures contract is for the delivery of 200 Nifties. On January 25, the index clos

at 2,720. How much profit/loss did he make?

Solution: X bought one futures contract costing him Rs 5,38,000. At a market lot of 200, this means he paid Ps 2,690 per N

future. On the futures expiration day, the futures price converges to the spot price. If the index closed at 2,720 this must be

futures close price as well. Hence, he would have made of profit of (Rs 2,720 — Ps 2,690) x 200 = Rs 6,000.

Example: X sold a January Nifty futures contract for Rs 5,38,000,. on January 15. For this he had to pay an initial margin of

43,040 to his broker. Each Nifty futures contract is for the delivery of 200 Nifties. On January 25, the index closed at 2,520. H

much profit/loss did me make?

Solution: X sold one futures contract costing in Rs 5,38,000. At a market lot of 200, this works out to be Ps 2,690 per N

future. On the futures expiration day, the futures price converges to the spot price. If the index closed at 2,520 this must be

futures close price as well. Hence, he would have made profit of (Rs 2,690 — Rs 2,520) x 200 = Rs 34,000.





Example: On January 15, X bought one January Nifty futures contract that cost him Rs 2,69,000. For this he had to pay

initial margin of Rs 21,520 to his broker. Each Nifty contract is for the delivery of 200 Nifties. On January 25, the index closed

1,280. How much profit/loss did he make?

Solution: X bought one futures contract for Rs 2,69,000. At a market lot of 200, this means he paid Rs 1,345 per Nifty futu

On the futures expiration day, the futures price converges to the spot price. If the index closed at 1,280, this must be the futu

close price as well. Hence, he made of loss of (Rs 1,345 — Rs 1,280) x 200 = Rs 13,000.

Example: X sold one January Nifty futures contract for Ps 2,69,000, on January 15. For this he had to pay an initial margin

Rs 21,520 to his broker. Each Nifty futures contract is for the delivery of 200 Nifties. On January 25, the index closed at 1,3

How much profit/loss did he make?

Solution: X sold one futures contract for Rs 2,69,000. In a market lot of 200, this works out to be Rs 1,345 per Nifty future.

the futures expiration day, the futures price converges to the spot price. If the index closed at 1,390, this must be the futu

close price as well. Hence, he made of loss of (Rs 1,390 — Rs 1,345) x 200 = Rs. 9,000.

Pricing Futures

The pricing of futures is illustrated below with. reference to

(1) The Cost.-of-Carry Cost of carry Model,

(2) Pricing equity index futures and

(3) Pricing stock futures.

The Cost-of-Carry Model

The cost-of-carry model explains the dynamics of pricing that constitute the estimation of the fair value of futures. The fair va

calculation of futures is used to decide the no arbitrage limits on the price of a future contract. According to this model, us

discrete compounding, where interest rates are compounded at discrete intervals, (for example, annually/semi-annually)

price of the contract is defined as:

F= S+ C

where F= Futures price,

S= Spot price, and

C = Holdings costs or carry posts

This can also be expressed as:

F= S (1 + r)T

where r = Cost of financing and T = Time till expiration

If F< S(1 + r)T

or F>S(1+r)T, arbitrage opportunities would exist, that is whenever the futures price moves away from the

value, there would be chances for arbitrage. The components of holding cost vary with contracts on different assets. At tim

the holding cost may even be negative. In the case of commodity futures, the holding cost is the cost of financing plus cost

storage and insurance purchased and so on. In the case of equity futures, the holding cost is the cost of financing minus

dividends returns.

Using continuous compounding, the Equation would be expressed as





F= SerT

where r = Cost of financing (using continuously compounded interest rate),

T= Time till expiration, and

e= 2.71828

To illustrate cost of carry, let us take an example of a futures contract on a commodity and work out the cost of contract. Tspot price January 1, Year 1, of silver is assumed to be Rs 7,000/kg. Assuming an annual cost of financing of 15 per cent a

no storage cost, the fair value of the future price of 100 gms of silver one month hence (January 30, Year 1) would be

follows:

F= S (1 + r)T+ C= Rs 700 (Rs 7,000 ÷ 10) [1.15] x 30/365 = Rs 708

If the contract is for a three month period expiring on March 30, Year 1, the cost of financing would increase the future price, t

is, F= Rs 700 (1.15) x 90/365 Rs 724.5. If, however, the one month contract was for 10,000 kgs, it would involve storage c

and the price of the future contract would be Rs 708 plus the cost of storage.

Pricing Equity Index Futures

A futures contract on the stock market gives its owner the r ight and obligation to buy or sell the portfolio of stocks characteri

by the index. Stock index futures are cash settled; there is no delivery of the underlying stocks.

The main differences between commodity and equity index futures are that:

(i) There are no costs of storage involved in holding equity and

(ii) Equity comes with a dividend stream, which is a negative cost if you are long the stock and a positive cost if you

short the stock. Therefore cost of carry = financing cost — dividends. Thus, a crucial aspect of dealing with eq

futures, as opposed to commodity futures, is an accurate forecasting of dividends. The better the forecast

dividend offered by a security, the better is the estimate of the futures price. The pricing of equity index future

illustrated below with reference to (i) expected dividend amount and (ii) expected dividend yield.

Pricing Index Futures Given Expected Dividend Amount

The pricing of index futures is also based on the cost-of-carry model, where the carrying cost is the cost of financing

purchase of the portfolio underlying the index, minus the present value of dividends obtained from the stocks in the in

portfolio.

Example: Nifty futures trades on a stock exchange (NSE) as one, two and three-month contracts. Money can be borrowed a

rate of 15 per cent per annum. Compute the price of a new two month futures contract on Nifty of X Ltd (XL).

Solution: Let us assume that XL will be declaring a dividend of Rs 10 per share after 15 days of purchasing the contract. T

current value of Nifty is 1,200 and Nifty trades with a multiplier of 200. The value of the contract is 200 x Rs 1200 = Rs 2,40,0

If XL has a weight of 7 per cent in Nifty, its value in Nifty is Rs 16,800 (Rs 2,40,000 x 0.07). If the market price of XL is Rs 140

traded unit of Nifty involves 120 shares (Rs 16,8O0/14€. To calculate the futures price, we need to reduce the cost -of-carrythe extent of the dividend received. The amount of dividend received is Rs 1,200 (120 x Rs 10). The dividend is received

days later and, hence, compounded only for the reminder of the 45 days. To calculate the futures price we need to compute

amount of dividend received per unit of Nifty. Hence, we divide the compounded dividend figure by 200. Thus, the futures pr

is





Pricing Index Futures Gwen Expected Dividend Yield If the dividend flow throughout the year is generally uniform, that is, th

are few historical cases of clustering of dividends in any particular month, it is useful to calculate the annual dividend yield.

F= S (1 + r — q)T (28.3)

Where, F= futures price,

S= spot index value,

r = cost of financing,

q = expected dividend yield, and

T= holding period

Example: A two month futures contract trades on the NSE. The cost of financing is 15 per cent and the dividend yield on Nift

2 per cent annualised. The spot value of Nifty is Rs 1,200. What is the fair value of the futures contract?

Solution:

Fair value = Rs 1,200 (1 + 0.15 — 0.02) x 60.365 = Rs 1,224.35

The cost-of-carry model explicitly defines the relationship between the futures price and the related spot price. The differen

between the spot price and the futures price is called the basis. As the date of expiration comes near, the basis reduces: there

a convergence of the futures price towards the spot price. On the date of expiration, the basis is zero. If it is not, then there is

arbitrage opportunity. Arbitrage opportunities can also arise when the basis (difference between spot and futures price) or

spreads (difference between prices of two futures contracts) during the life of a contract are incorrect. How these arbitra

opportunities can be exploited is discussed subsequently. There is nothing but cost-of-carry related arbitrage that drives

behaviour of the futures price. Moreover, transactions costs are very important in the business of arbitrage. However, th

pricing models give an approximate idea about the true future price. The price observed in the market is the outcome of

price discovery mechanism (demand-supply principle) and may differ from the so called true price.

Pricing Stock Futures

A futures contract on a stock gives its owner the right and obligation to buy or sell the stocks. Like index futures, stock futu

are also cash settled; there is no delivery of the underlying stocks. Just as in the case of index futures, the main difference Sto

between commodity and stock futures are that: (i) There are no costs of storage involved in holding stock, and (ii) Stocks co

with a dividend stream, which is a contract that negative cost if you are long) the stock and a positive cost if you are short

stock. Therefore, cost of carry = financing cost — dividends. Thus, a crucial aspect of dealing with stock futures, as opposed

commodity futures, is an accurate forecasting of dividends. The better the forecast of dividend offered by a security, the bette

the estimate of the futures price. The pricing of stock futures is discussed below (shares). when (i) no dividend is expected,

when dividend is expected.

Pricing Stock Futures When No Dividend Expected

The pricing of stock futures is also based on the cost-of-carry model, where the carrying cost is the cost of financing purchase of the stock, minus the present value of dividends obtained from the stock. If no dividends are expected during the

of the contract, pricing futures on that stock is very simple. It simply involves multiplying the spot price by the cost of carry.

Example: SBI futures trade on NSE as one, two and three-month contracts. Money can be borrowed a 15 per cent per annu

What will the price of a unit of new two month futures contract on the SBI be if no dividends are expected during the two mo

period, assuming spot price of the SBI is Rs 228?

Solution: Futures price, F = Rs 228 x (1.15) x 60/365 = Rs 233.30





Pricing Stock Futures When Dividends Are Expected

When dividends are expected during the life of the futures contract, pricing involves reducing the cost of carry to the exten

the dividends. The net carrying cost is the cost of financing the purchase of the stock, minus the present value of dividen

obtained from the stock.

Example: XL futures trade on NSE as one, two and three month contracts. What will the price of a unit of new two-mo

futures contract on XL be if dividends are expected during the two month period? Assume that XL will be declaring a dividend

Rs 10 per share after 15 days of purchasing the contract. The market price of XL may be assumed as Rs 140.

Solution: To calculate the futures price, we need to reduce the cost-of-carry to the extent of dividend received. The amoun

dividend received is Rs 10. The dividend is received 15 days later and, hence, compounded only for the remainder of 45 da

Thus, the futures price, F = Rs 140 x (1.15) x 60/365 — [10 x (1.15) x 45/3651 = Rs 133.08.

OPTIONS/OPTIONS CONTRACTS

Options are fundamentally different from forward and futures contracts. An option gives the holder of the holder of the option

right to do something. The holder does not have to necessarily exercise this right. In contrast, in a forward or futures contr

the two parties have committed themselves to doing something. Whereas it costs nothing (except margin requirements) to en

into a futures contract, the purchase of an option requires an upfront payment. This section discusses and illustrates options

a derivative contract, with reference to (i) Option terminology, (ii) Comparison of options and futures, (iii) Option payoffs,

Pricing options and (v) Using stock options.

Option Terminology

Index Options

These options have the index as the underlying. Some options are European while others are American. American options

be exercised at any time upto the expiration date. Most exchange traded options are American. European options can

exercised only on the expiration date itself. European options are easier to analyse than American options, and properties of

American option are frequently deduced from those of its European counterpart. Like index futures contracts, index opti

contracts are also cash settled.

Stock Options

Stock options are options on individual stocks. A contract gives the holder the right to buy or sell shares at the specified price.

Buyer of an Option

The buyer of an option is the one who by paying the option premium buys the right but not the obligation to exercise his op

on the seller/writer.

Writer of an Option

The writer of a call/put option is the one who receives the option premium and is thereby obliged to sell/buy the asset if

buyer exercises the option on him. There are two basic types of options, call options and put options.

Call Option

A call option gives the holder the right but not the obligation to buy an asset by a certain date for a certain price.

Put Option

A put option gives the holder the right but not the obligation to sell an asset by a certain date for a certain price.





Option Price/Premium

Option price is the price that the option buyer pays to the option seller. It is also referred to as the option premium.

Expiration Date

The date specified in the options contract is known as the expiration date, the exercise date, the strike date or the maturity.

Strike Price

The price specified in the options contract is known as the strike price or the exercise price.

In-the-Money Option

An in-the-money (ITM) option is an option that would lead to a positive cashflow to the holder if it were exercised immediately

call option on the index is said to be in-the-money when the current index stands at a level higher than the strike price (that

spot price > strike price). If the index is much higher than the strike price, the call is said to be deep ITM. In the case of a p

the put is ITM if the index is below the strike price.

At-the-Money Option

An at-the-money (ATM) option is an option that would lead to zero cashflow if it were exercised immediately. An option on

index is at-the-money when the current index equals the strike price (that is, spot price = strike price).

Out-of-the-Money Option

An out-of-the-money (OTM) option is an option that would lead to a negative cashflow if it were exercised immediately. A

option on the index is out-of-the- money when the current index stands at .a level that is less than the strike price (that is, s

price < strike price). If the index is much lower than the strike price, the call is said to be deep OTM. In the case of a put, the p

is OTM if the index is above the strike price.

Intrinsic Value of an Option

The option premium can be broken down into two components

(i) intrinsic value and (ii) time value. The intrinsic value of a call is the amount the option is ITM, if it is ITM. If the call is OTM,

intrinsic value is zero. Putting it another way, the intrinsic value of a call is Max[O,(S t — K] which means the intrinsic value o

call is the greater of 0 or (St — K). Similarly, the intrinsic value of a put is Max[0, K— St] , that is, the greater of 0 or (K- St ).

the strike price and St is the spot price.

Time Value of an Option

The time value of an option is the difference between its premium and its intrinsic value. Both calls and puts have time value.

option that is OTM or ATM only has time value. Usually, the maximum time value exists when the option is ATM. The longer

time to expiration, the greater is an option’s time value, other things being equal. At expiration, an option would have no tvalue.

Futures and Options

Options are different from futures in several respects. At a practical level, the option buyer pays for the option in full at the tim

is purchased. After this, he only has an upside. There is no possibility of the options position generating any further loss to

(other than the funds already paid for the option). In contrast, futures are free to enter into hut can generate very large loss

This characteristic makes options attractive to many occasional market participants who cannot put in the time to closely mon

their futures positions.





Buying put options is buying insurance. To buy a put option on the Nifty is to buy insurance that reimburses the full exten

which the Nifty drops below the strike price of the put option. This is attractive to many people and to mutual funds creat

“guaranteed return products”. The Nifty index fund industry will find it very useful to make a bundle of a Nifty index fund a n

Nifty put option to create a new kind of Nifty index fund, which gives the investor protection against extreme drops in the N

Selling put options is selling insurance. Anyone who feels like earning revenues by selling insurance can set himself up to do

on the index options market.

More generally, options offer “non-linear payoffs”, whereas futures only have “linear payoffs”. By combining futures and optioa wide variety of innovative and useful payoff structures can be created. The distinction between futures and option

summarised in Table

Distinction Between Futures and Options

Options Payoffs

A pay off for derivative contacts is the likely profit/loss that would accrue to the market participant with change in the price of

underlying asset. The optionality characteristic of options results in a non-linear pay off for options. In simple words, it mea

that the losses for the buyer of an option are limited. However, the profits are potentially unlimited. For a writer, the payof

exactly the opposite. His profits are limited to the option premium. However, his losses are pote-ntially unlimited. These n

linear pay offs are fascinating as they lend themselves to be used to generate various pay offs by using combinations of opti

and the underlying. We illustrate below six basic pay offs.

Pay off Profile for Seller of Asset: Short Asset In this basic position, an investor shorts the und-erlying asset, the Nifty

instance, for 1,220 and buys it back at a future date at an unknown price, S,. Once it is sold, the investor is said to be “short”asset. The investor sold the index at 1,220. If the index falls, he profits. If the index rises, he loses.

Pay off Profile of Buyer of Asset: Long Asset In this basic position, an investor buys the under-lying asset, the Nifty

instance, for 1,220 and sells it at a future date at an unknown price, S. Once it is purchased, the investor is aid to be “long” asset. The investor would make profit if the index goes up. If the index falls he would lose.

Pay off Profile for Buyer of Call Options: Long Call A call option gives the buyer the right to pay the underlying asset at

strike price specified in the option. The profit/loss that the buyer makes on the option depends on the spot price of

underlying. If upon expiration, the spot price exceeds the strike price, he makes a profit. The higher the spot price, the m

profit he makes. If the spot price of the underlying is less than the strike price, he lets his option expire unexercised. His loss

this case is the premium he paid for buying the option.

Pay off Profile for Writer to Call Options: Short Call A call option gives the buyer the right to buy the underlying asset at

strike price specified in the option. For selling the option, the writer of the option charges a premium. The profit/loss that

buyer makes on the option depends on the spot price of the underlying. Whatever is the buyer’s profit is the seller’s loss. If upexpiration, the spot price exceeds the strike price, the buyer will exercise the option on the writer. Hence, as the spot p

increases, the writer of the option starts making losses. The higher the spot price, the more is the loss he makes. If up

expiration the spot price of the underlying is less than the strike price, the buyer lets his option expire unexercised and the wr

gets to keep the premium.





Pay off Profile for Buyer of Put Options: Long Put A put option gives the buyer the right to sell the underlying asset at

strike price specified in the option. The profit/loss that the buyer makes on the option depends on the spot price of

underlying. If upon expiration the spot price is below the strike price, he makes a profit. The lower the spot price, the more is

profit he makes. If the spot price of the underlying is higher than the strike price, he lets his option expire unexercised. His l

in the case is the premium he paid for buying the option.

Pay off Profile for Writer of Put Options: Short Put A put option gives the buyer the right to sell the underlying asset at

strike price specified in the option. For selling the option, the writer of the option charges a premium. The profit/loss that buyer makes on the option depends on the spot price of the underlying. The buyer’s profit is the seller’s loss. If upon expir athe spot price happens to be below the strike price, the buyer will exercise the option on the writer. If upon expiration the s

price of the underlying is more than the strike price, the buyer gets his option expire unexercised and the writer gets to keep

premium.

Pricing Options

An option buyer has the right but not the obligation to exercise on the seller. The worst that can happen to a buyer is the l os

the premium paid by him. His downside is limited to this premium, but his upside is potentially unlimited. This optionality ha

value expressed in terms of the option price. Just like in other free markets, it is the supply and demand in the secondary ma

that drives the price of an option. Ther@- are various models that help us get close to the true price of an option. Most of the

are variants of the celebrated Black-Scholes Model for pricing European options.

Black-Scholes Option Pricing Model/Formulae

Black and Scholes start by specifying a simple and well known equation that models the way in which stock prices fluctua

This equation, called Geometric Brownian Motion, implies that stock returns will have a lognormal distribution, meaning that

logarithm of the stock’s return will follow the normal (bell shaped) distribution. They then propose that the option’s p rice

determined by only two variables that are allowed to change: time and the underlying stock price. The other factors, namely,

volatility, the exercise price, and the risk free rate do affect the option’s price but they are not allowed to change. By

forming a portfolio consisting of a long position in stock and a short position in calls, the nsk associated with the stock

eliminated. This hedged portfolio is obtained by setting the number of shares of stock equal to the approximate change in

call price for a change in the stock price. This mix of stock and calls must be revised continuously. This process is known

delta hedging. They then turn to a little known result in a specialised field of probability known as stochastic calculus. This re

defines how the option price changes in terms of the change in the stock price and time to expiration. They then reason that t

hedged combination of options and stock should grow in value at the risk free rate. The result then is a partial differen

equation. The solution is found by forcing a condition called a boundary condition on the model that requires the option price

converge to the exercise value at expiration. The end result is the Black and Scholes Model.

The Black-Scholes formulas for the prices of European calls and puts on a non-dividend paying stock are:

• The Black-Scholes equation is done in continuous time. This requires continuous compounding. The r that figures in this

n(1 ÷ r). Example, if the interest rate per annum is 12 per cent, you need to use I n 1.12 or 0.1133, which is the continuou

compounded equivalent of 12 per cent per annum.





• N 0 is the cumulative normal distribution. N (d1) is called the delta of the option, which is a measure of change in optio n p

with respect to change in the price of the underlying asset.

• σ a measure of volatility, is the annualised standard deviation of continuously compounded returns on the underlying. W

daily sigma are given, they need to be converted into annualised sigma.

• sigmaannual = sigmadaily X √Number of trading days per year. On a average there are 250 trading days in a year.

• X is the exercise price, S the spot price and T the time to expiration measured in years.

Pricing Index Options Under the assumption of the Black-Scholes Options Pricing Model, index options should be valued

the same way as ordinary options on common stock, the assumption being that investors can purchase, without cost,

underlying stocks in the exact amount necessary to replicate the index, that is, stocks are infinitely divisible and the ind

follows a diffusion process such that the continuously compounded returns distribution of the index is normally distributed.

use the Black-Scholes formula for index options we must, however, make adjustments for the dividend payments, replacing

current index value S in the model with q is the annual dividend yield and T is the time to expiration in years. Consider Exam

28.9

Example : A three-month call option on the Nifty with a strike of 1,180 is available for trading. The Nifty stands at Rs 1,150, a

it has a volatility of 30 per cent per annum. The annual risk free rate is 12 per cent. We can calculate the price of the 1,1

option using the Black-Scholes option pricing formula. We take T 0.25, S= 1,150, X= 1,180, r=ln(1.12), and 0= 0.3. Substitut

these values in the formula, we get the call price as Rs 70.15. The put price on an option with the same strike works out to

Rs 67.19.

Pricing Stock Options Much of what was discussed about index options also applies to stock options. The factors that aff

option prices are listed below.

The Stock Price The payoff from a call option will be the amount by which the stock prices exceeds the strike price. Call opti

therefore, becomes more valuable as the stock price increases and less valuable as the stock prices decreases. The pay

from a put option will be the amount by which the strike price exceeds the stock price. Put options, therefore, become m

valuable as the stock price decreases and less valuable as the stock price increases.

The Strike Price In the case of a call, as the strike price increases, the stock price has to make a larger upward move for option to go in-the-money. Therefore, for a call option, as the strike price increases, options become less valuable and as

strike price decreases they become more valuable. Put options behave exactly in the opposite way to call options.

Time to Expiration: Both put and call American options become more valuable as the time to expiration increases. Cons

the case of two options that differ only as far as their expiration date is concerned. The owner of the long-life option has all

exercise opportunities open to the owner of the short-life option, and more. The long-life option must, therefore, always be wo

at least as much as the short life option.

Volatility:The volatility of a stock price is a measure of how uncertain we are about future stock price movements. As volat

increases, the chance that the stock will do very well or very poorly increases. The value of both calls and puts, therefo

increases as volatility increases.

Risk Free Interest Rate: The affect of the risk free interest rate is less clear cut. It is found that the put option prices decline

the risk free rate increases, whereas the prices of calls always increase as the risk free interest rate increases.

Dividends: Dividends have the effect of reducing the stock price on the ex-dividend date. This has a negative affect on

value of call options and a positive affect on the value of put options.

Application of Black-Scholes Option Pricing Formula to Stock Options

The Black-Scholes option pricing formula, with some adjustment, can be used to price American calls and puts options





stocks. Pricing American options becomes a little difficult because unlike European options, American options can be exerci

any time prior to expiration. However, it is never optimal to exercise a call option on a non-dividend paying stock bef

expiration. When no dividends are expected during the life of the option, the option can be valued simply by substituting

values of the stock price, strike price, stock volatility, risk free rate and time-to-expiration in the BlackScholes formula. Howev

when dividends are expected during the life of the option, it is sometimes optimal to exercise the option just before t

underlying stock goes ex-dividend. Hence, when valuing options on dividend paying stock, we should consider exerc

possibilities at to times: (i) just before the underlying stock goes ex-dividend and (ii) at the expiration of the options contract.

Therefore, owning an option on a dividend paying stock today is like owning two options: one being a long maturity option wit

time-to-maturity from the starting date till the expiration day, and the other being a short maturity option with a time-to-matu

from the. starting date till just before the stock goes ex-dividend.

Some adjustment needs to be made before using the Black-Scholes formula. The first step is to value the option on

assumption that it will be exercised on expiry. Thus, the present value of the dividends is deducted from the stoik price and

adjusted value, Sd, is used in the BlackScholes Model. The second step is to assume that the option will be exercised j

before the exdividend date. The unadjusted stock price is used. In addition, the time to expiry is shortened to be the period up

the ex-dividend date. Following these adjustments, the Black-Scholes model can be applied. The actual value of the option

be the highest of the two valuations. Consider Example 28.10.

Example: Assume that the price of a stock is Rs 50, the exercise price is Rs 45, the risk free rate of interest is 6 per cent annum and that the cx dividend adjustment of 2.5 will occur 0.1644 years hence. The volatility of the stock is 20 per cent. T

discount rate on dividend is also taken to be 6 per cent. We have now two call options, a long maturity call option with a matu

of 0.25 years, which can be exercised on the expiration date, and a short maturity call option with a maturity of 0.166 yea

which can be exercised just before the ex-dividend date. We will now value both these options.

• The details of the long option are: T— 0.25, r= 0.06, D= 2.5, S= P.s 50, X= Rs 45, and S d= [S—D/ (1 = r)T] = Rs 47.52. T

stock price to be used in the Black-Scholes option pricing formula is S d, the adjusted price of the stock after deducing

present value of the dividends. Using these values, we get the price of the long option as Rs 3.84.

• The details of the short option are: T= 0.166, r— 0.06, D 2.5, S= Rs 50 and X= Rs 45. Since the option is exercised just bef

the stock goes ex-dividend, the unadjusted stock price of Ps 50 is used. Using these values, we get the price of the short opt

as Rs 5.56.

Thus, using the above approximation, the American option on the dividend paying stock would be valued at the higher of

two options, that is, at Rs 5.58.

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