186 Mathematics for Finance
treated in detail in more advanced texts. In place of this, we shall exploit an
analogy with the discrete time case.
As a starting point we take the continuous time model of stock prices de-
veloped in Chapter 3 as a limit of suitably scaled binomial models with time
steps going to zero. In the resulting continuous time model the stock price is
given by
S(t)=S(0)emt+σW(t), (8.5)
where W(t) is the standard Wiener process (Brownian motion), see Sec-
tion 3.3.2. This means, in particular, thatS(t) has the log normal distribution.
Consider a European option on the stock expiring at timeTwith payoff
f(S(T)). As in the discrete-time case, see Theorem 8.4, the time 0 priceD(0)
of the option ought to be equal to the expectation of the discounted payoff
e−rTf(S(T)),
D(0) =E∗
(
e−rTf(S(T)))
, (8.6)
under a risk-neutral probabilityP∗that turns the discounted stock price process
e−rtS(t) into a martingale. Here we shall accept this formula without proof,
by analogy with the discrete time result. (The proof is based on an arbitrage
argument combined with a bit of Stochastic Calculus, the latter beyond the
scope of this book.)
What is the risk-neutral probabilityP∗, then? A necessary condition is that
the expectation of the discounted stock prices e−rtS(t) should be constant
(independent oft), just like in the discrete time case, see (3.5).
Let us compute this expectation using the real market probabilityP.Since
W(t) is normally distributed with mean 0 and variance t, it has density
√^1
2 πte
−x 22 tunder probabilityP. As a result,E
(
e−rtS(t))
=S(0)E
(
eσW(t)+(m−r)t)
=S(0)
∫∞
−∞eσx+(m−r)t√^1
2 πte−x 22 t
dx=S(0)e(m−r+(^12) σ^2 )t
∫∞
−∞√^1
2 πte−(x− 2 σtt)^2
dx=S(0)e(m−r+(^12) σ^2 )t
∫∞
−∞√^1
2 πte−y 22 t
dy=S(0)e(m−r+(^12) σ (^2) )t
.
Ifm+^12 σ^2 =r, then the expectationE(e−rtS(t)) =S(0)e(m−r+
(^12) σ (^2) )t
clearly
depends ont,soS(t) cannot be a martingale underP.
However, the calculations above suggest a modificationP∗ofPthat would
make the corresponding expectationE∗(e−rtS(t)) independent oftby elimi-
nating the exponential factor e(m−r+^12 σ^2 )t. Namely, ifP can be replaced by