Mathematics for finance

Mathematics for finance Math 194 Homework 3 1. Textbook Vol I, Chapter 2, Exercise 2.3. 2. Textbook Vol I, Chapter 2, Exercise 2.4. 3. Textbook Vol I, Chapter 2, Exercise 2.6. 4. Let Mn be a martingale such that M0 = 0. Show that (a) E[Mn ] = 0; (b) Cov(Mn+1 , Mn ) = E[Mn2 ]. 5. (a) Background : The stock (without dividend paying) return µ and its volatility can be computed as follows. Suppose a sequence of historical prices Si is observed on daily basis. ui is defined as the continuous compound return ln(Si+1 /Si ) on day i (e.g., Si+1 = Si eui ). Under the assumption that ui are i.i.d. random variables for all i, the daily return µ = E[ui ] and the daily volatility = Std[ui ] (e.g., standard deviation). The daily risk-free continuous compound return is r (e.g., $1 becomes $er one day later). Show that Var[ln(ST /S0 )] = 2 T (Stock price ST for day T ). (b) Binomial Tree Construction : Consider the stock price from the time 0 to T (in days). In the n-periods binomial tree, each period corresponds to t = T /n day. We showed in p p eµ t d t t satisfy (ignore a class that the real probability p = u d , u = e and d = e higher order term t3/2 ) pS0 u + (1 p)S0 d = S0 eµ t and pu2 + (1 p)d2 [pu + (1 p)d]2 = 2 t. In other words, the return and volatility of the binomial model is matched with the real data. r t Show that the risk neutral probability is p˜ = e u d d and under this risk neutral measure, the volatility of the binomial model does not change, ignoring a higher order term t3/2 (Hint: use the Taylor expansion). (c) ST Distribution under p˜: Denote Bi t be the random variable taking 1 when i-th coin toss H and 1 otherwise. Si t is the stock price at the i-th period is at the last period). p(STP Given the binomial tree in (a), it is obvious that ln(ST /S0 ) = t nk=1 Bi t . Question g (b) proved that the volatility under p and p˜ are the same, meaning Var[ln(S T /S0 )] = 2 Var[ln(ST /S0 )] = T . Therefore, the central limit theorem (let n ! 1) implies that ln(ST /S0 ) has the normal distribution N (a, 2 T ) under p˜ for some unknown constant a. 2 ˜ Show that a = E[ln(S T /S0 )] = (r 2 )T . (Hint: use the fact that ln(ST /S0 ) has Gaussian ˜ T ], e.g., the discount stock price is martingale distribution and the formula S0 = e rT E[S under p˜.) 2 2 (d) Show that ST = S0 e(r 2 )T + T z with the standard normal random variable z ? N (0, 1) under p˜. ST is said to satisfy the lognormal distribution under p˜.