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Chapter 6 : Options Markets and Option Chapter 6 : Options Markets and Option PricingPricing

• Options contracts are a form of derivative securities, or simply derivatives. These are securities whose prices are determined by the prices of other securities.

• Options give its holder the right to either purchase or sell an asset for a specified price on or before some expiration date.

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• The option is called a call option if it gives the right to purchase an asset. On the other hand, a put option gives the right to sell an asset in the future.

• Price set for calling (buying) an asset or putting (selling) an asset is called the exercise or strike price.

• For example, a July call option on Microsoft stock with exercise price $80 entitles its owner to purchase Microsoft stock for a price of $80 at any time up to the expiration date in July.

• The holder of the call is not required to exercise the option. The holder will choose to exercise only if the market value of the asset exceeds the exercise price.

• If the call option is left unexercised after expiry, its value will reduce to zero.

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• The purchase price of the option is called the premium. It represents the compensation the purchaser of the call option must pay for the ability to exercise the option if exercise becomes profitable.

• Sellers of call options (who are said to write calls) receive premium as a payment against the possibility they will be required in the future to deliver the asset in return for an exercise price lower than the asset’s market value.

• The writer of the call option clears a profit equals to the premium if the option is unexercised.

• If the call is exercised, the profit to the option writer is equal to the premium derived minus the difference the value of the stock that must be delivered and the exercise price that is paid for those shares. The writer will incur a loss if the difference is larger than the initial premium.

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Example 1

Consider a June 1999 maturity call option on a share of Microsoft

Stock with an exercise price of $80 per share selling on May 3, 1999

for , to expire on June 18, 1999. Until then, the purchaser of the

call is entitled to buy shares at $80.8

15$

i) Scenario 1: Microsoft stock falls below $80 at expiry date. The call is left to expire worthless.

ii) Scenario 2: Microsoft stock sells for $83 at expiry date. The investor should still leave the call unexercised because final profit = $83 $80 $5.125 = $2.125

iii) Scenario 3: Microsoft stock sells for over $85.125. Exercising the call will clear a profit.

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Example 2

Consider a June 1999 maturity put option on Microsoft with an exercise price of $80 selling on May 3, 1999 for $4.75.

Scenario 1: Microsoft stock price falls to $72 at expiration. The profit of the put would be $80 $72 $4.75 = $3.25

Scenario 2: Microsoft stock closes above $80. The put option is left unexercised.

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• An option is said to be “in the money” when its exercise would produce a positive payoff. It is “out of the money” when exercise would be unprofitable.

• American option: can be exercised on or before its expiration.

• European option: can be exercised only at expiration.

• Bullish strategy: purchasing call options, writing put options.

• Bearish strategy: purchasing put options, writing call options.

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Why might an option strategy be preferable to direct investment?

Consider the following situation: Current stock price of Microsoft is $80 and you believe that its value will increase in future. But you also know your analysis could be incorrect. Suppose you have $8000. A call option with a six month maturity at exercise price of $80 sells for $10, and the semi-annual interest rate is 2%.

Consider the following three strategies:

A: Purchase 100 shares of MicrosoftB: Purchase 800 call options on MicrosoftC: Purchase 100 call options for $1000. Invest $7000 in a

bank to earn 2% interest.

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Let us trace the possible values of these three portfolios when the options expire in six months:

Microsoft Price

Portfolio $55 $80 $85 $90 $95 $100

A$5500

(31.25%)$8000(0%)

$8500(6.25%)

$9000(12.5%)

$9500(18.75%

)

$10000(25%)

B$0

(100%)$0

(100%)$4000(50%)

$8000(0%)

$12000(50%)

$16000(100%)

C$7140

(10.75%)$7140

(10.75%)$7640

(4.5%)$8140

(1.75%)$8140(8%)

$9140(14.25%)

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Rate of return to three strategies

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Note that

• Options offer “leverage”. Modest increases in the rate of return on the stock result in disproportionate increase in the option rate of return.

• Combining options with a risk free asset can also offer insurance.

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Option Strategies

• Protective Put: Suppose you would like to invest in a stock, but you are unwilling to bear potential losses beyond some level. One strategy to consider is to invest in the stock and purchase a put option on the stock one feature of the protective put is that whatever happens, you are guaranteed a payoff equal to the strike price of the option.

Let ST = Stock price at expiryX = option’s strike price

ST X ST >X

Stock ST ST

Put X ST 0

Total X ST

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Value of a protective put position at expiration

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Comparing profit on the protective put strategy with stock investment. For simplicity, consider an at-the-money protective put scenario.

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• Covered calls: A covered call position is the purchase of a share of stock with the simultaneous sale of a call on that stock. The position is covered because the potential obligation to deliver the stock is covered by the stock held in the portfolio.

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Value of a covered call position at expiration

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• Intrinsic option value: the value S0 X (where S0 is the current stock price) is often called the intrinsic value of an option. It gives the payoff that can be obtained by immediate exercise.

• If the option is surely to be exercised (e.g., if the stock price increases substantially), then it makes more sense to consider the “adjusted” intrinsic value of the stock: S0 PV(X), which is the stock price minus the present value of X. If we assume continuous compounding, we can write PV(X) = XerT, where r is the prevailing interest rate and T is the period to maturity.

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Binomial models for option pricing

Assumptions:

• The underlying variable (the price of the security underlying the option) has a binomial distribution.

• It is feasible to create a risk-free portfolio by hedging a long (buy) position in the underlying asset with a short (sell) positions in a number of fairly priced call options on that asset.

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The binomial lattice approach

• The cost of establishing the risk-free portfolio is the cost of buying the underlying asset minus the premiums from the written (sold) options.

• Consider the following example. Let

S = 35X = 35r = 10% or 0.01 in decimalsR = 1 + r

Time to expiry of the option is one year.

At the end of the one year period the asset price will rise by 25% to 43.75 or fall by 25% to 26.25.

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• u is the multiplicative upward movement in the asset price (1 + percentage rise in asset price) = 1.25

• d is the multiplicative downward movement in the asset price (1 + percentage fall in asset price) = 0.75.

• We require d < (1 + r) < u, otherwise the risk-free asset will show a higher return than the risky asset, which is contrary to financial theory.

S = 35Su = 43.75

Sd = 26.25

C

cu = Max[0, 43.75 35] = 8.75

cd = Max[0, 26.25 35] = 0

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• To create a fully hedged portfolio, consider a covered-call strategy by buying one stock and selling H call options,

• Being fully hedged, the portfolio value will be the same whether the price of the asset rises or falls. So,

S Hc

Su Hcu = 26.25

Sd Hcd = 26.25

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There are two problems to solve:

1) Find out how many options to sell that makes the portfolio riskless (i.e. what is the magnitude of H?)

2) Determine the fair price at which these options should be sold.

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• To answer Question 1, we consider the hedge ratio, which shows the number of options that have to be written over each unit of stock in order to achieve a perfectly hedged portfolio.

2

75.8

5.17

075.8

75.025.135

cdcu

duSH

So, if asset rises to 43.75:Su Hcu = 1.25(35) 2(8.75) = 26.25Sd Hcd = 0.75(35) 2(0) = 26.25

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• To answer Question 2, note that the perfect hedging strategy is riskless, so it should earn only the risk free rate of return.

• Now, the end of year payoff is $26.25, so its present value is $26.25/1.1 = 23.86. i.e. 35 2c = 23.86, so c = $5.57.

• In general, note that

R(S Hc) = (Su Hcu)

where R = 1 + r

So

HR

HcuuRSc

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but noting that

we have

,

cdcu

duSH

57.51.1

75.025.11.125.1

075.025.1

75.01.175.8

R

duRu

cddudR

cu

c

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The multi-period binomial model• The binomial approach can be generalized so that the life of the option can be divided

into any number of periods, i.e., binomial trials.• As an example, let

S = X = 35;r = 10% = 20%

and the one year period is divided into four quarterly sub-periods (i.e., number of binomial trials = 4).

• Note that

0.25

0.2 0.25

0.2 0.25

1 1.024

1.10517

0.904837

0.59539

1 0.40461

T t n

T t n

R r

u e e

d e e

R dP

u d

u RP

u d

where T t is the life of the option in years (or fractions there of) and n is the number of binomial trials.

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S

Su

Sd

Su2

Sud

Sdu

Sd2

Su3

Su2d

Su2d

Sud2

Su2d

Sud2

Sud2

Sd3

Su4

Su3d

Su3d

Su2d2

Su3d

Su2d2

Su2d2

Sud3

Su3d

Su2d2

Sud3

Su2d2

Su2d2

Sud3

Sud3

Sd4which may be simplified as

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S = 35

Su = 38.68

Sd = 31.67

Su3

= 47.25

Su2d= 36.68

Sud2

= 31.67

Sd3

= 25.93

Su2 = 42.75

Sud = 35

Sd2

= 28.65

Su4

= 52.21

Su3d= 42.75

Su2d2

= 35

Sud3

= 28.66

Sd4

= 23.46

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C= 4.37

6.48

1.52

13.07

4.51

0

0

9.38

2.62

0

17.21 = max [0, Su4 X]= max [0, 52.21 35]

7.75= max [0, Su3d X]= max [0, 42.75 35]

0= max [0, Su2d2 X]= max [0, 35 35]

0= max [0, Sud3 X]= max [0, 28.66 35]

0= max [0, Sd4 X]= max [0, 23.46 35]

The binomial tree of the options follows the same pattern

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How do we work out the option value at the end of the 3rd, 2nd and 1st period?

Without invoking the binomial tree, one can work out the initial option price using the formula.

0

1 !1 MAX 0,

! !1

nn jj j n j

nj

nC P P u d S X

n j jr

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• These examples are unrealistic because the life of option is divided only into 4 discrete time periods. If the time to expiry is divided into infinitestimally small periods then the binomial pricing formula has a more practical application. In fact, when trading is in effect continuous, the binomial model converges to the famous Black-Scholes model.

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• The Black-Scholes pricing for call option was introduced by Black and Scholes (1973) and Merton (1973) and is given by

where

21 dNXedNSeC tTrtT

dZPdN

tTdd

tT

tTrXSnd

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2

1

2

= annual dividend yield of underlying stock

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• The B-S formula may look intimidating at first glance, but it can be explained at an intuitive level for certain special cases.

• Consider the special case of = 0. Then Se(T t) = S. Now N(d) may be loosely interpreted as the probability that the call option will expire in the money.

• If N(d) → 1, then C = S Xer(T t) which is just the adjusted intrinsic value. If N(d) → 0 then c = 0 which means the option is worthless.

• Notice also that N(d1) and N(d2) also increase with higher stock price.

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Example. Work out the value of a call option under the following circumstances:

5714.018.0

6664.043.0

18.025.05.043.0

43.025.05.0

25.025.001.095100

2

2

1

N

N

d

nd

S = 100 ; = 0X = 95 ; T t = 0.25r = 0.10 ; = 0.50

Now,

Therefore,

70.13

94.5264.66

5714.0956664.0100 25.01.0

eC

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How applicable is the Black-Scholes formula in practice?

The Black-Scholes formula is in fact based on the following assumptions:

1) , the dividend paid, is a constant until the option expires.

2) r and 2 are both constant

3) Sudden plunge or jump in stock prices are ruled out.

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• Stock standard deviation is often a variable, rather than a constant.

• Implied volatility of an option refers the volatility level that the B-S option price implies.

• What about put options?

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Put-Call Parity Relationship

• Suppose you buy a call option and write a put option, each with the same exercise price, X and the same expiry date. At expiry, the payoff is as follows:

ST X ST > X

Payoff of call held 0 ST X

Payoff of put written (X ST) 0

Total ST X ST X

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The payoff pattern of a long call – short put position

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• Consider another portfolio made up of stock (S0) plus borrowing X/(1 + r)T s.t. X dollars will be repaid at maturity. The total payoff of this levered quality position is

ST X = long call – short put pay off

• So the cost of establishing the two portfolios must be the same too. i.e.

C P = S0 X/(1 + r)T

• If the put-call parity relationship is violated, then arbitrage opportunity will arise. Discussion in class.

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• The put-call parity relationship applies only to European options and options on stocks that pay no dividends. For stocks that pay dividends, the put-call parity relationship may be modified as

C P = S0 X/(1 + r)T PV(dividends)

= S0 PV(X) PV(dividends)

• The Black-Scholes formula also provides a direct formula for the value of a European put option P = XerT[1 N(d2)] S0eT[1 N(d1)], which agrees with put-call parity.

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Example. Use the data from the previous example to find the put option value using the put-call parity relationship, we have

P = C + PV(X) S0

= 13.70 + 95e0.10 0.25 100 = 6.35

Now, using the Black-Scholes formula,

P = 95e0.10 0.25(1 0.5714) 100(1 0. 6664) = 6.35