Question 2. Suppose that a stock price process is defined by
St = S0
Yt
i=1
Zi
, t = 1, 2, . . . , T,
where the random variables Zi take two possible values, u and d, with d < er < u and risk-free rate
r. The current stock price S0 is known. The natural filtration of (St)t=0,1,...,T is (Ft)t=0,1,...,T . Assume
that the discounted stock price process (Set)t=0,1,...,T , where Set = e
−rtSt, is a martingale with respect to
(Ft)t=0,1,...,T . It is not assumed that Z1, . . . , ZT are independent or identically distributed. Recall the
following probability results:
The conditional probability of an event A given Ft is defined as P(A | Ft) = E[I(A)| Ft], where I(A)
is the indicator random variable which takes the value 1 if the event A occurs and 0 otherwise.
The discrete random variables X1, . . . , Xn are independent if and only if
P(X1 = x1, . . . , Xn = xn) = P(X1 = x1) \times · · · \times P(Xn = xn)
holds for all possible values x1, . . . , xn.
(a) Show that Zt is Ft-measurable by writing it as a deterministic function of S0, . . . , St.