E[S(t)] ¼ S

0

exp (mt) (9:3:12)

Var[S(t)] ¼ S

0

2

exp (2mt)[ exp (s

2

t) 1] (9:3:13)

In addition, equation (9.3.6) yields

exp (rt) ¼ pu þ (1 p)d (9:3:14)

Using (9.3.13) and (9.3.14) in the equality (y) ¼ E[y

2

] E[y]

2

,we

obtain the relation

exp (2rt þ s

2

t) ¼ pu

2

þ (1 p)d

2

(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 ¼ 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 ¼ [ exp (rDt) d]=(u d), u ¼ 1=d ¼ 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 asset’s price S(t)

F ¼ 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,

Option Pricing 101

. 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 Ito’s expression (4.3.5)

dF(S, t) ¼ mS

@F

@S

þ

@F

@t

þ

s

2

2

S

2

@

2

F

@S

2

dt þ sS

@F

@S

dW(t) (9:4:2)

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

bution dW. Namely, we choose 1 (short) option and

@F

@S

shares of

the underlying asset,

5

P ¼F þ

@F

@S

S(9:4:3)

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

equals

dP ¼dF þ

@F

@S

dS (9:4:4)

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 ¼ rP dt (9:4:5)

The combination of equations (9.4.2)–(9.4.5) yields the Black-Scholes

equation

@F

@t

þ rS

@F

@S

þ

s

2

2

S

2

@

2

F

@S

2

rF ¼ 0(9:4:6)

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.

102 Option Pricing

Get *Quantitative Finance for Physicists* now with the O’Reilly learning platform.

O’Reilly members experience live online training, plus books, videos, and digital content from nearly 200 publishers.