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The effect of the risk-free rate is not straightforward. At a fixed

stock price, the rising risk-free rate increases the value of the call

option. Indeed, the option holder may defer paying for shares and

invest this payment into the risk-free assets until the option matures.

On the contrary, the value of the put option decreases with the risk-

free rate since the option holder defers receiving payment from selling

shares and therefore cannot invest them into the risk-free assets.

However, rising interest rates often lead to falling stock prices,

which may change the resulting effect of the risk-free rate.

Dividends effectively reduce the stock prices. Therefore, dividends

decrease value of call options and increase value of put options.

Now, let us consider the payoffs at maturity for four possible

European option positions. The long call option means that the in-

vestor buys the right to buy an underlying asset. Obviously, it makes

sense to exercise the option only if S > K. Therefore, its payoff is

PLC Â¼ max [S Ã€ K, 0] (9:2:1)

The short call option means that the investor sells the right to buy an

underlying asset. This option is exercised if S > K, and its payoff is

PSC Â¼ min [K Ã€ S, 0] (9:2:2)

The long put option means that the investor buys the right to sell an

underlying asset. This option is exercised when K > S, and its payoff

is

PLP Â¼ max [K Ã€ S, 0] (9:2:3)

The short put option means that the investor sells the right to sell an

underlying asset. This option is exercised when K > S, and its payoff is

96 Option Pricing

PSP Â¼ min [S Ã€ K, 0] (9:2:4)

Note that the option payoff by definition does not account for the

option price (also named option premium). In fact, option writers sell

options at a premium while option buyers pay this premium. There-

fore, the option sellerâ€™s profit is the option payoff plus the option

price, while the option buyerâ€™s profit is the option payoff minus the

option price (see examples in Figure 9.1).

The European call and put options with the same strike price

satisfy the relation called put-call parity. Consider two portfolios.

Portfolio I has one European call option at price c with the strike

price K and amount of cash (or zero-coupon bond) with the present

value Kexp[Ã€r(T Ã€ t)]. Portfolio II has one European put option at

price p and one share at price S. First, let us assume that share does

not pay dividends. Both portfolios at maturity have the same value:

max (ST , K). Hence,

c Ã¾ Kexp[Ã€r(T Ã€ t)] Â¼ p Ã¾ S (9:2:5)

Dividends affect the put-call parity. Namely, the dividends D being

paid during the option lifetime have the same effect as the cash future

value. Thus,

c Ã¾ D Ã¾ K exp [Ã€r(T Ã€ t)] Â¼ p Ã¾ S (9:2:6)

Because the American options may be exercised before maturity, the

relations between the American put and call prices can be derived

only in the form of inequalities [1].

Options are widely used for both speculation and risk hedging.

Consider two examples with the IBM stock options. At market

closing on 7-Jul-03, the IBM stock price was $83.95. The (American)

call option price at maturity on 3-Aug-03 was $2.55 for the strike

price of $85. Hence, the buyer of this option at market closing on 7-

Jul-03 assumed that the IBM stock price would exceed $(85 Ã¾ 2.55) Â¼

$87.55 before or on 3-Aug-03. If the IBM share price would reach say

$90, the option buyer will exercise the call option to buy the share for

$85 and immediately sell it for $90. The resulting profit4 is

$(90Ã€87.55) Â¼ $2.45. Thus, the return on exercising this option equals

2:45=2:55Ãƒ 100% Â¼ 96%. Note that the return on buying an IBM

share in this case would only be (90 Ã€ 83:95)=83:95Ãƒ 100% Â¼ 7:2%.

97

Option Pricing

(a) 20

Profit

15

10

Short Call

5

Stock price

0

0 5 10 15 20 25 30 35 40

âˆ’5

Long Call

âˆ’10

âˆ’15

âˆ’20

(b) 20

Profit

15

Long Put

10

5

Stock price

0

0 5 10 15 20 25 30 35 40

âˆ’5

âˆ’10

Short Put

âˆ’15

âˆ’20

Figure 9.1 The option profits for the strike price of $25 and the option

premium of $5: (a) calls, (b) puts.

98 Option Pricing

If, however, the IBM share price stays put through 3-Aug-03, an

option buyer incurs losses of $2.45 (i.e., 100%). In the mean time, a

share buyer has no losses and may continue to hold shares, hoping

that their price will grow in future.

At market closing on 7-Jul-03, the put option for the IBM share

with the strike price of $80 at maturity on 3-Aug-03 was $1.50. Hence,

buyers of this put option bet on price falling below $(80Ã€1.50) Â¼

$78.50. If, say the IBM stock price falls to $75, the buyer of the put

option has a gain of $(78:50 Ã€ 75) Â¼ $3.50.

Now, consider hedging in which the investor buys simultaneously

one share for $83.95 and a put option with the strike price of $80 for

$1.50. The investor has gains only if the stock price rises above

$(83:95 Ã¾ 1:50) Â¼$85:45. However, if the stock price falls to say $75,

the investorâ€™s loss is $(80 Ã€ 85:45) Â¼ Ã€$5:45 rather than the loss of

$(75 Ã€ 83:95) Â¼ Ã€$8:95 incurred without hedging with the put

option. Hence, in the given example, the hedging expense of $1.50

allows the investor to save $(Ã€5:45 Ã¾ 8:95) Â¼$3:40.

9.3 BINOMIAL TREES

Let us consider a simple yet instructive method for option pricing

that employs a discrete model called the binomial tree. This model is

based on the assumption that the current stock price S can change at

the next moment only to either the higher value Su or the lower value

Sd (where u > 1 and d < 1). Let us start with the first step of the

binomial tree (see Figure 9.2). Let the current option price be equal to

F and denote it with Fu or Fd at the next moment when the stock price

moves up or down, respectively. Consider now a portfolio that con-

sists of D long shares and one short option. This portfolio is risk-free

if its value does not depend on whether the stock price moves up or

down, that is,

SuD Ã€ Fu Â¼ SdD Ã€ Fd (9:3:1)

Then the number of shares in this portfolio equals

D Â¼ (Fu Ã€ Fd )=(Su Ã€ Sd) (9:3:2)

The risk-free portfolio with the current value (SD Ã€ F) has the future

value (SuD Ã€ Fu ) Â¼ (SdD Ã€ Fd ). If the time interval is t and the risk-

99

Option Pricing

Su2

Fuu

Su

Fu

Sud

S

F

Fud

Sd

Fd

Sd2

Fdd

Figure 9.2 Two-step binomial pricing tree.

free interest rate is r, the relation between the portfolioâ€™s present value

and future value is

(SD Ã€ F) exp(rt) Â¼ SuD Ã€ Fu (9:3:3)

Combining (9.3.2) and (9.3.3) yields

F Â¼ exp(Ã€rt)[pFu Ã¾ (1 Ã€ p)Fd ] (9:3:4)

where

p Â¼ [ exp (rt) Ã€ d]=(u Ã€ d) (9:3:5)

The factors p and (1 Ã€ p) in (9.3.4) have the sense of the probabilities

for the stock price to move up and down, respectively. Then, the

expectation of the stock price at time t is

E[S(t)] Â¼ E[pSu Ã¾ (1 Ã€ p)Sd] Â¼ S exp (rt) (9:3:6)

This means that the stock price grows on average with the risk-free

rate. The framework within which the assets grow with the risk-free

rate is called risk-neutral valuation. It can be discussed also in terms of

the arbitrage theorem [4]. Indeed, violation of the equality (9.3.3)

100 Option Pricing

implies that the arbitrage opportunity exists for the portfolio. For

example, if the left-hand side of (9.3.3) is greater than its right-hand

side, one can immediately make a profit by selling the portfolio and

buying the risk-free asset.

Let us proceed to the second step of the binomial tree. Using

equation (9.3.4), we receive the following relations between the option

prices on the first and second steps

Fu Â¼ exp (Ã€rt)[pFuu Ã¾ (1 Ã€ p)Fud ] (9:3:7)

Fd Â¼ exp (Ã€rt)[pFud Ã¾ (1 Ã€ p)Fdd ] (9:3:8)

The combination of (9.3.4) with (9.3.7) and (9.3.8) yields the current

option price in terms of the option prices at the next step

F Â¼ exp (Ã€2rt)[p2 Fuu Ã¾ 2p(1 Ã€ p)Fud Ã¾ (1 Ã€ p)2 Fdd ] (9:3:9)

This approach can be generalized for a tree with an arbitrary number

of steps. Namely, first the stock prices at every node are calculated

by going forward from the first node to the final nodes. When the

stock prices at the final nodes are known, we can determine the

option prices at the final nodes by using the relevant payoff relation

(e.g., (9.2.1) for the long call option). Then we calculate the option

prices at all other nodes by going backward from the final nodes to

the first node and using the recurrent relations similar to (9.3.7) and

(9.3.8).

The factors that determine the price change, u and d, can be

estimated from the known stock price volatility [1]. In particular, it

is generally assumed that prices follow the geometric Brownian

motion

dS Â¼ mSdt Ã¾ sSdW (9:3:10)

where m and s are the drift and diffusion parameters, respectively, and

dW is the standard Wiener process (see Section 4.2). Hence, the price

changes within the time interval [0, t] are described with the lognor-

mal distribution

pï¬ƒï¬ƒ

ln S(t) Â¼ N( ln S0 Ã¾ (m Ã€ s2 =2)t, s t) (9:3:11)

In (9.3.11), S0 Â¼ S(0), N(m, s) is the normal distribution with mean

m and standard deviation s. It follows from equation (9.3.11) that the

expectation of the stock price and its variance at time t equal

101

Option Pricing

E[S(t)] Â¼ S0 exp (mt) (9:3:12)

Var[S(t)] Â¼ S0 2 exp (2mt)[ exp (s2 t) Ã€ 1] (9:3:13)

In addition, equation (9.3.6) yields

exp (rt) Â¼ pu Ã¾ (1 Ã€ p)d (9:3:14)

Using (9.3.13) and (9.3.14) in the equality (y) Â¼ E[y2 ] Ã€ E[y]2 , we

obtain the relation

exp (2rt Ã¾ s2 t) Â¼ pu2 Ã¾ (1 Ã€ p)d2 (9:3:15)

The equations (9.3.14) and (9.3.15) do not suffice to define the three

parameters d, p, and u. Usually, the additional condition

u Â¼ 1=d (9:3:16)

is employed. When the time interval Dt is small, the linear approxi-

mation to the system of equations (9.3.14) through (9.3.16) yields

p Â¼ [ exp (rDt) Ã€ d]=(u Ã€ d), u Â¼ 1=d Â¼ exp [s(Dt)1=2 ] (9:3:17)

The binomial tree model can be generalized in several ways [1]. In

particular, dividends and variable interest rates can be included. The

trinomial tree model can also be considered. In the latter model, the

stock price may move upward or downward, or it may stay the same.

The drawback of the discrete tree models is that they allow only for

predetermined innovations of the stock price. Moreover, as it was

described above, the continuous model of the stock price dynamics

(9.3.10) is used to estimate these innovations. It seems natural then to

derive the option pricing theory completely within the continuous

framework.

9.4 BLACK-SCHOLES THEORY

The basic assumptions of the classical option pricing theory are

that the option price F(t) at time t is a continuous function of time

and its underlying assetâ€™s price S(t)

F Â¼ F(S(t), t) (9:4:1)

and that price S(t) follows the geometric Brownian motion (9.3.10) [5,

6]. Several other assumptions are made to simplify the derivation of

the final results. In particular,

102 Option Pricing

. There are no market imperfections, such as price discreteness,

transaction costs, taxes, and trading restrictions including those

on short selling.

. Unlimited risk-free borrowing is available at a constant rate, r.

. There are no arbitrage opportunities.

. There are no dividend payments during the life of the option.

Now, let us derive the classical Black-Scholes equation. Since it is

assumed that the option price F(t) is described with equation (9.4.1)

and price of the underlying asset follows equation (9.3.10), we can use

the Itoâ€™s expression (4.3.5)

@F @F s2 2 @ 2 F @F

dF(S, t) Â¼ mS Ã¾ Ã¾S dt Ã¾ sS dW(t) (9:4:2)

@S2

2

@S @t @S

Furthermore, we build a portfolio P with eliminated random contri-

@F

bution dW. Namely, we choose Ã€1 (short) option and shares of

@S

5

the underlying asset,

@F

P Â¼ Ã€F Ã¾ S (9:4:3)

@S

The change of the value of this portfolio within the time interval dt

equals

@F

dP Â¼ Ã€dF Ã¾ dS (9:4:4)

@S

Since there are no arbitrage opportunities, this change must be equal to

the interest earned by the portfolio value invested in the risk-free asset

dP Â¼ rP dt (9:4:5)

The combination of equations (9.4.2)â€“(9.4.5) yields the Black-Scholes

equation

@F s2 2 @ 2 F

@F

Ã¾ rS Ã¾S Ã€ rF Â¼ 0 (9:4:6)

@S2

2

@t @S

Note that this equation does not depend on the stock price drift

parameter m, which is the manifestation of the risk-neutral valuation.

In other words, investors do not expect a portfolio return exceeding

the risk-free interest.

103

Option Pricing

The Black-Scholes equation is the partial differential equation with

the first-order derivative in respect to time and the second-order de-

rivative in respect to price. Hence, three boundary conditions deter-

mine the Black-Scholes solution. The condition for the time variable is

defined with the payoff at maturity. The other two conditions for the

price variable are determined with the asymptotic values for the zero

and infinite stock prices. For example, price of the put option equals

the strike price when the stock price is zero. On the other hand, the put

option price tends to be zero if the stock price approaches infinity.

The Black-Scholes equation has an analytic solution in some

simple cases. In particular, for the European call option, the Black-

Scholes solution is

c(S, t) Â¼ N(d1 )S(t) Ã€ KN(d2 ) exp[Ã€r(T Ã€ t)] (9:4:7)

In (9.4.7), N(x) is the standard Gaussian cumulative probability

distribution

d1 Â¼ [ ln (S=K) Ã¾ (r Ã¾ s2 =2)(T Ã€ t)]=[s(T Ã€ t)1=2 ],

(9:4:8)

1=2

d2 Â¼ d1 Ã€ (T Ã€ t)

The Black-Scholes solution for the European put option is

p(S, t) Â¼ K exp[Ã€r(T Ã€ t)] N(Ã€d2 ) Ã€ S(t)N(Ã€d1 ) (9:4:9)

The value of the American call option equals the value of the Euro-

pean call option. However, no analytical expression has been found

for the American put option. Numerical methods are widely used for

solving the Black-Scholes equation when analytic solution is not

available [1â€“3].

Implied volatility is an important notion related to BST. Usually,

the stock volatility used in the Block-Scholes expressions for the

option prices, such as (9.4.7), is calculated with the historical stock

price data. However, formulation of the inverse problem is also

possible. Namely, the market data for the option prices can be used

in the left-hand side of (9.4.7) to recover the parameter s. This

parameter is named the implied volatility. Note that there is no

analytic expression for implied volatility. Therefore, numerical

methods must be employed for its calculation. Several other functions

related to the option price, such as Delta, Gamma, and Theta (so-

called Greeks), are widely used in the risk management:

104 Option Pricing

@2F

@F @F

DÂ¼ ,GÂ¼ 2,QÂ¼ (9:4:10)

@S @S @t

The Black-Scholes equation (9.4.6) can be rewritten in terms of

Greeks

s2 2

Q Ã¾ rSD Ã¾ S G Ã€ rF Â¼ 0 (9:4:11)

2

Similarly, Greeks can be defined for the entire portfolio. For example,

@P @S

the portfolioâ€™s Delta is . Since the shareâ€™s Delta equals unity,

@S @S

Delta of the portfolio (9.4.3) is zero. Portfolios with zero Delta are

called delta-neutral. Since Delta depends on both price and time,

maintenance of delta-neutral portfolios requires periodic rebalancing,

which is also known as dynamic hedging. For the European call and

put options, Delta equals, respectively

Dc Â¼ N(d1 ), Dp Â¼ N(d1 ) Ã€ 1 (9:4:12)

Gamma characterizes the Deltaâ€™s sensitivity to price variation. If

Gamma is small, rebalancing can be performed less frequently.

Adding options to the portfolio can change its Gamma. In particular,

delta-neutral portfolio with Gamma G can be made gamma-neutral if

it is supplemented with n Â¼ Ã€G=GF options having Gamma GF .

Theta characterizes the time decay of the portfolio price. In add-

ition, two other Greeks, Vega and Rho, are used to measure the

portfolio sensitivity to its volatility and risk-free rate, respectively

@P @P

yÂ¼ ,rÂ¼ (9:4:13)

@s @r

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