Mathematics for financeMath 194 Homework 31. 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 becomputed 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-freecontinuous 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 correspondsto t = T /nday. We showed inppeµ t dtt satisfy (ignore aclass that the real probability p = u d , u = eand d = ehigher order term t3/2 )pS0 u + (1 p)S0 d = S0 eµ tandpu2 + (1p)d2[pu + (1p)d]2 =2t.In other words, the return and volatility of the binomial model is matched with the realdata.r tShow 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 cointoss H and 1 otherwise. Si t is the stock price at the i-th periodis at the last period).p(STPGiven the binomial tree in (a), it is obvious that ln(ST /S0 ) =t nk=1 Bi t . Questiong(b) proved that the volatility under p and p˜ are the same, meaning Var[ln(ST /S0 )] =2Var[ln(ST /S0 )] = T . Therefore, the central limit theorem (let n ! 1) implies thatln(ST /S0 ) has the normal distribution N (a, 2 T ) under p˜ for some unknown constant a.2˜Show that a = E[ln(ST /S0 )] = (r2 )T . (Hint: use the fact that ln(ST /S0 ) has Gaussian˜ T ], e.g., the discount stock price is martingaledistribution and the formula S0 = e rT E[Sunder p˜.)22(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˜.