Question 1: You were given the value of u and d in the binomial tree. This exercise will walk you through two methods for determining u and d.

As you know, the Black-Scholes model assumes that the price of a risky asset (the stock S
BS
, wit cur nt
price S0) follows a lognormal distribution. Specifically, in the Black-Scholes model, for any t > 0, ln
S
BS
t
is
normally distributed under the risk-neutral measure Q, with mean E
Q
ln
S
BS
t
= ln (S0) +
r −
1
2
σ
2
t and
variance VarQ
ln
S
BS
t
= σ

(2)
where µ˜ :=
r −
1
2
σ
2
t, σ˜
2
:= σ
2
t, r > 0 is the risk-free rate, and σ > 0 is assumed to be known (stock
volat

− 1).
The idea behind the determination of the tree parameters u and d is to choose them so that the mean and variance
(under Q) of the stock price on the tree approximately match the mean and variance of the lognormal distribution
above. There is some flexibility in this procedure. The two classical methods that we will explore below are easy
to implement: the first one assumes ud = 1 (Method 1) and the second one assumes qu =
1
2
(Meth

d 2). (a) General setup:

Consider a multi-period binomial model, with each period of length h (in fractions of a year), and assume
that the model is arbitrage-free. The known time-0 price of the risky asset (stock) is S0. At time h (end
of the first period), the stock price (Sh) is a random variable that can take the values uS0 or dS0, for some
u and d such that u > d > 0. The risk-free rate of interest is r > 0 per year, compounded continuously.
More generally, for each n ≥ 1, the stock price at the end of the n
th period is g

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