Assignment 3, MAT4372 Winter 2020 Due by Tuesday, April 13, 9:00 am by email 1. In a binomial pricing model, suppose Mo..., M. be a martingale and let A,...,A.- be an adapted process (A, depends only on the first n coin toss). Prove that the discrete stochastic integral Sn = {4,(M.;+1 M)) is a martingale. 2. (i) In the continuous models for option pricing the risk neutral probability measure P is chosen such that the discounted price X(t) = exp(-rt)X(t) will become a martingale. Assuming the model dX(t) = X(t)dt +oX(t)dW(t) Use Ito's lemma to show that dX(t) = X(O)(c-r)dt + odW()) (ii) Show that M(t) = exp(-qW(t) -0.52t) is a martingale. This means that E(Mt + h)|M(t)) = M, (iii) Show that M(1) >0 and E(M(t)) = 1. This shows that P(A)=E(M(t)/(WE A)) is a probability measure (this is called an equivalent martingale measure). 3. Download the recent (the last 1000 days) daily closing price for S&P500 (denote by Se t =1,2,...,1000). You can use Yahoo finance, for example. (i) Plot the time series plot (in Ruse say ts.plot(x)) for the daily closing price and the log of the price. (ii) Calculate the return and draw and plot the time series of returns. Use both definition S S S -1 -1 = log(S./S-1). (iii) Assuming independence between errors estimate o? Var(ur). Here is an estimator: on] (u u)? (iv) Use Black-Schole formula with r = 0.03 and o as above to calculate the price of the option E(S1000 - 1500) Notice that in here Soin here is the value of the stock price in day 1000, i.e., S1000- (v) Draw the qq plot to check for normality of returns. Use the existing data to find 5% and 1% quantiles for the returns, 4. Consider the optimization problem we discussed in class for an N-period binomial model with the utility function U(x) = x/p, where p0 and E(M(t)) = 1. This shows that P(A)=E(M(t)/(WE A)) is a probability measure (this is called an equivalent martingale measure). 3. Download the recent (the last 1000 days) daily closing price for S&P500 (denote by Se t =1,2,...,1000). You can use Yahoo finance, for example. (i) Plot the time series plot (in Ruse say ts.plot(x)) for the daily closing price and the log of the price. (ii) Calculate the return and draw and plot the time series of returns. Use both definition S S S -1 -1 = log(S./S-1). (iii) Assuming independence between errors estimate o? Var(ur). Here is an estimator: on] (u u)? (iv) Use Black-Schole formula with r = 0.03 and o as above to calculate the price of the option E(S1000 - 1500) Notice that in here Soin here is the value of the stock price in day 1000, i.e., S1000- (v) Draw the qq plot to check for normality of returns. Use the existing data to find 5% and 1% quantiles for the returns, 4. Consider the optimization problem we discussed in class for an N-period binomial model with the utility function U(x) = x/p, where p
