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 SBS, wit current price S0) follows a lognormal distribution. Specifically, in the Black-Scholes model, for any t > 0, ln (SBSt) is normally distributed under the risk-neutral measure Q, with mean EQ[ln (SBSt)] = ln (S0) + (r 1/2 2) t and variance VarQ(ln (SBSt)) = 2 t:

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 (Method 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 uS0or 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 given by

Snh= ZS(n1)h,

where Z is a random variable that can take the values u or d (0, u).

(i) Compute the risk-neutral mean and variance (i.e., under Q) of the random variable Y1 := Sh/S0 in terms of u, d and rh.

(ii)Let T be a positive integer and SThbe the stock price at the end of the Tthperiod in the binomial model. Show that EQ[STh] = EQ[SBSTh] always holds. In other words, the means of the binomial model and the Black Scholes model always match.

(iii) Suppose that we keep t = Th fixed, and let h 0 (at the same time T ). Intuitively explain why SThis approximately lognormally distributed.

Hint: You may use the Central Limit Theorem, which says that the sum of many iid random variables is approximately normally distributed.

(b) Method 1

In the first method, we assume ud = 1. The choice of u and d are given by

(i) Derive an expression for qu:= Q [Sh= uS0] = Q [Y1= u], as a function of the time steph.

(ii) Show that qu 1/2 as h 0.

(iii)Show that

(iv) Based on (iii), show that ln(STh) has asymptotically the same variance as ln(SBSTh), ash 0.

Remark: As a consequence,SThhas asymptotically the same distribution asSBSTh, since they are bothlognormally distributed with the same mean and the same -parameter.

(v) Given the assumption of no-arbitrage, what condition(s) must the model parameter > 0satisfy, for a given h? What if h 0?

(c) Method 2:

Consider the same one-period binomial model as in (a). Instead of assuming that u and d are chosen such that ud = 1 as in Method 1, assume now that u and d are chosen such that qu = Q [Sh= uS0] = 1/ 2 , that is, the probability of an up-jump is 1/ 2 under the risk-neutral measure.

(i) Given the assumption that qu = 1/ 2 , show that there exists > 0 such that u = e rh + and d = e rh .

(ii) Determine , u and d so that the risk-neutral variance (i.e., under Q) of the random variable Y1matches the risk-neutral variance of SBSh/S0.

(ii) Show thatShow that EQ[(STh)2] = EQ[(SBSTh)2] for any positive integer T.

Remark: Similarly to Method 1, because of the Central Limit Theorem and the matching moments, ST h has asymptotically the same distribution as S BS T h .

(iii)In this model, does the assumption of no arbitrage imply any restriction on the choice of the model parameter > 0, for a given h?