D: 13TH 2021 PLEASE SOLVE STEP BY STEP AND PLEASE PROVIDE MATLAB CODE WITH OUTPUT SCREENSHOTS: 1. Consider a stochastic volatility model for stock price [here the volatility is denoted by 0]: dS = rsdt + 0S+dW deta(0-0) dt + dB+ Here r, a,, are all positive constants, and (W, B) is a two-dimensional Brownian motion with covariance matrix 1 A In this model the volatility 0 is itself a (mean-reverting) stochastic process. We are interested in estimating the price of a vertical spread with maturity T. That is, v = eE[(ST K)* (ST K)*], 0 < K1 < K2 By It^o formula, if we define Y = logs, then dY = (r 0) dt + 0dW We would like to compare the results from the following two schemes. 1. Estimate v by straightforward Euler scheme on (Yt, Ot) 2. Estimate v by straightforward Euler scheme on (Yt, 0+) combined with control variates. Please read the following text for some general discussion on how to construct control variables. Suppose that the log-stock price process Y = logSt, satisfies the SDE dy = (r-20) dt + 0,dW where 6+ is the volatility of the stock price. This volatility 0 can be very general: a constant, a deterministic function, a function of St, or itself a stochastic process. Suppose we are interested in estimating the price of, say, a call option with strike K and maturity T [that is, the payoff (S - K)+ = (exp(Y) K)+]. The Euler scheme on Y+ will be Y' + ti+1 - tiv - ti Zi+1 and Y' to = Yo Now introduce an artificial stochastic process Y by 1 Y- = Y- ti+1 ti +r- (r 0) (t+1 t) + t+1 t Z+1 and Y, - - to = Yo This artificial process uses the same sequence {Zi}, and a constant volatility [o is usually chosen as a typical value of 0]. One can think of Y as the logarithm of the price of a virtual stock that follows a classical geometric Brownian motion with drift r and volatility o. Then we can use the discounted payoff of the call option for this virtual stock, namely, er (exp(YT) - K)+ as the control variable. The expected value of this control variable is just the classical Black-Scholes call price with parameter So, r, , K, T. In the simulation, divide the time interval [0, T] into m equal-length subintervals. That is, t = T, i = 0, 1, 2, 3,. ... .......... m Write a MATLAB function that compares schemes (1) and (2). The function should have input parameters r, a, , , So, 00, K1, K2, T, m, p, and sample size n. Report your estimates and standard errors for r = 0.1, a = 3, = 0.2, B = 0.1, So 20,00= 0.25, T = 1, K1 = 20, K2 = 22, p = 0.5, m = 50, n = 10000. = Please also indicate which value of o you have used for the artificial process Y and which control variable you have used. Explain your motivation for doing so briefly. Note: For the control variate method, just set b* = 1 for simplicity