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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

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