So far, we have focused on the pricing of call options. In many important cases, put prices can

be derived simply from the prices of calls. This is because prices of European put and call

546

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Fifth Edition

547

15 Option Valuation

F I G U R E 15.5

Payoff

The payoff pattern of

a long call“short

put position

Long call

ST

X

Payoff

+ Short put

ST

Payoff

= Leveraged equity

ST

X

options are linked together in an equation known as the put-call parity relationship. Therefore,

once you know the value of a call, put pricing is easy.

To derive the parity relationship, suppose you buy a call option and write a put option, each

with the same exercise price, X, and the same expiration date, T. At expiration, the payoff on

your investment will equal the payoff to the call, minus the payoff that must be made on the

put. The payoff for each option will depend on whether the ultimate stock price, ST, exceeds

the exercise price at contract expiration.

ST X ST X

ST X

Payoff of call held 0

ST)

Payoff of put written (X 0

ST X ST X

Total

Figure 15.5 illustrates this payoff pattern. Compare the payoff to that of a portfolio made

up of the stock plus a borrowing position, where the money to be paid back will grow, with

interest, to X dollars at the maturity of the loan. Such a position is a levered equity position in

Bodie’Kane’Marcus: V. Derivative Markets 15. Option Valuation © The McGraw’Hill

Essentials of Investments, Companies, 2003

Fifth Edition

548 Part FIVE Derivative Markets

which X/(1 rf)T dollars is borrowed today (so that X will be repaid at maturity), and S0 dol-

lars is invested in the stock. The total payoff of the levered equity position is ST X, the same

as that of the option strategy. Thus, the long call“short put position replicates the levered

equity position. Again, we see that option trading provides leverage.

Because the option portfolio has a payoff identical to that of the levered equity position, the

costs of establishing them must be equal. The net cash outlay necessary to establish the option

position is C P: The call is purchased for C, while the written put generates income of P.

Likewise, the levered equity position requires a net cash outlay of S0 X/(1 rf)T, the cost of

put-call parity

the stock less the proceeds from borrowing. Equating these costs, we conclude

relationship

rf )T

C P S0 X/(1 (15.2)

An equation

representing the

Equation 15.2 is called the put-call parity relationship because it represents the proper re-

proper relationship

lationship between put and call prices. If the parity relationship is ever violated, an arbitrage

between put and call

opportunity arises.

prices.

Suppose you observe the following data for a certain stock.

15.3 EXAMPLE

Stock price $110

Put-Call

Call price (six-month maturity, X $105) 17

Parity

Put price (six-month maturity, X $105) 5

Risk-free interest rate 10.25% effective annual

yield (5% per 6 months)

We use these data in the put-call parity relationship to see if parity is violated.

C P S0 X/(1 rf)T

17 5 110 105/1.05

12 10

This result, a violation of parity (12 does not equal 10) indicates mispricing and leads to an

arbitrage opportunity. You can buy the relatively cheap portfolio (the stock plus borrowing

position represented on the right-hand side of the equation) and sell the relatively expensive

portfolio (the long call“short put position corresponding to the left-hand side, that is, write a

call and buy a put).

Let™s examine the payoff to this strategy. In six months, the stock will be worth ST. The

$100 borrowed will be paid back with interest, resulting in a cash outflow of $105. The writ-

ten call will result in a cash outflow of ST $105 if ST exceeds $105. The purchased put

pays off $105 ST if the stock price is below $105.

Table 15.3 summarizes the outcome. The immediate cash inflow is $2. In six months, the

various positions provide exactly offsetting cash flows: The $2 inflow is realized risklessly with-

out any offsetting outflows. This is an arbitrage opportunity that investors will pursue on a

large scale until buying and selling pressure restores the parity condition expressed in Equa-

tion 15.2.

Equation 15.2 actually applies only to options on stocks that pay no dividends before the

maturity date of the option. It also applies only to European options, as the cash flow streams

from the two portfolios represented by the two sides of Equation 15.2 will match only if each

position is held until maturity. If a call and a put may be optimally exercised at different times

Bodie’Kane’Marcus: V. Derivative Markets 15. Option Valuation © The McGraw’Hill

Essentials of Investments, Companies, 2003

Fifth Edition

549

15 Option Valuation

Cash Flow in Six Months

TA B L E 15.3

Position Immediate Cash Flow ST 105 ST 105

Arbitrage strategy

ST ST

Buy stock 110

Borrow X/(1 rf )T

$100 100 105 105

Sell call 17 0 (ST 105)

105 ST

Buy put 5 0

Total 2 0 0

before their common expiration date, then the equality of payoffs cannot be assured, or even

expected, and the portfolios will have different values.

The extension of the parity condition for European call options on dividend-paying stocks

is, however, straightforward. Problem 22 at the end of the chapter leads you through the

extension of the parity relationship. The more general formulation of the put-call parity con-

dition is

P C S0 PV(X) PV(dividends) (15.3)

where PV(dividends) is the present value of the dividends that will be paid by the stock dur-

ing the life of the option. If the stock does not pay dividends, Equation 15.3 becomes identi-

cal to Equation 15.2.

Notice that this generalization would apply as well to European options on assets other than

stocks. Instead of using dividend income in Equation 15.3, we would let any income paid out

by the underlying asset play the role of the stock dividends. For example, European put and

call options on bonds would satisfy the same parity relationship, except that the bond™s coupon

income would replace the stock™s dividend payments in the parity formula.

Let™s see how well parity works using real data on the Microsoft options in Figure 14.1

from the previous chapter. The April maturity call with exercise price $70 and time to expira-

tion of 105 days cost $4.60 while the corresponding put option cost $5.40. Microsoft was sell-

ing for $68.90, and the annualized 105-day interest rate on this date was 1.6%. Microsoft was

paying no dividends at this time. According to parity, we should find that

P C PV(X) S0 PV(dividends)

70

5.40 4.60 68.90 0

(1.016)105/365

5.40 4.60 69.68 68.90

5.40 5.38

So, parity is violated by about $0.02 per share. Is this a big enough difference to exploit? Prob-

ably not. You have to weigh the potential profit against the trading costs of the call, put, and

stock. More important, given the fact that options trade relatively infrequently, this deviation

from parity might not be “real” but may instead be attributable to “stale” (i.e., out-of-date)

price quotes at which you cannot actually trade.

Put Option Valuation

As we saw in Equation 15.3, we can use the put-call parity relationship to value put options

once we know the call option value. Sometimes, however, it is easier to work with a put option

Bodie’Kane’Marcus: V. Derivative Markets 15. Option Valuation © The McGraw’Hill

Essentials of Investments, Companies, 2003

Fifth Edition

550 Part FIVE Derivative Markets

valuation formula directly. The Black-Scholes formula for the value of a European put op-

tion is3

rT T

P Xe [1 N(d2)] S0e [1 N(d1)] (15.4)

Using data from the Black-Scholes call option in Example 15.2 we find that a European put

option on that stock with identical exercise price and time to maturity is worth

15.4 EXAMPLE .10 .25

$95e (1 .5714) $100(1 .6664) $6.35

Black-Scholes Notice that this value is consistent with put-call parity:

Put Option

P C S0 .10 .25

PV(X) 13.70 95e 100 6.35

Valuation

As we noted traders can do, we might then compare this formula value to the actual put

price as one step in formulating a trading strategy.

Equation 15.4 is valid for European puts. Listed put options are American options that offer

the opportunity of early exercise, however. Because an American option allows its owner to

exercise at any time before the expiration date, it must be worth at least as much as the corre-

sponding European option. However, while Equation 15.4 describes only the lower bound on

the true value of the American put, in many applications the approximation is very accurate.

15.4 USING THE BL ACK-SCHOLES FORMUL A

Hedge Ratios and the Black-Scholes Formula

In the last chapter, we considered two investments in Microsoft: 100 shares of Microsoft stock

or 700 call options on Microsoft. We saw that the call option position was more sensitive to

swings in Microsoft™s stock price than the all-stock position. To analyze the overall exposure

to a stock price more precisely, however, it is necessary to quantify these relative sensitivities.

A tool that enables us to summarize the overall exposure of portfolios of options with various

exercise prices and times to maturity is the hedge ratio. An option™s hedge ratio is the change

hedge ratio

in the price of an option for a $1 increase in the stock price. A call option, therefore, has a pos-

or delta

itive hedge ratio, and a put option has a negative hedge ratio. The hedge ratio is commonly

The number of shares

called the option™s delta.

of stock required to

If you were to graph the option value as a function of the stock value as we have done for

hedge the price risk

of holding one option. a call option in Figure 15.6, the hedge ratio is simply the slope of the value function evaluated

at the current stock price. For example, suppose the slope of the curve at S0 $120 equals

0.60. As the stock increases in value by $1, the option increases by approximately $0.60, as

the figure shows.

For every call option written, 0.60 shares of stock would be needed to hedge the investor™s

portfolio. For example, if one writes 10 options and holds six shares of stock, according to the

hedge ratio of 0.6, a $1 increase in stock price will result in a gain of $6 on the stock holdings,

3

This formula is consistent with the put-call parity relationship, and in fact can be derived from it. If you want to try to

do so, remember to take present values using continuous compounding, and note that when a stock pays a continuous

T

flow of income in the form of a constant dividend yield, , the present value of that dividend flow is S0(1 e ).