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 ¼F þ
@F
@S
S(9:4:3)
The change of the value of this portfolio within the time interval dt
equals
dP ¼dF 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

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