 # 1. Binomial Lattice Model (BLM): Sn+1 = SnYn+1, 72 &gt;0. Suppo...

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1. Binomial Lattice Model (BLM): Sn+1 = SnYn+1, 72 >0. Suppose that So = 50 (dollars per share), u = 1.3, d = .90, and p = 0.6. (a) For T 20, and K = 60, you are to estimate the expected payoff E(G) of an Asian call option, with payoff T Cr = n-1 Using Monte Carlo simulation: Generate 72 (large) ind copies of CT. denoted by X1 Xr and use the estimate Use n = 100, n = 1000, n = 10,000. (b) Suppose the risk-free interest rate r = 0.05. Compute the risk-neutral probability p* d)/(a d) and use it to estimate (via Monte Carlo) the price of this option; 1 Co E*(CT). (1+r) Use n = 10,000 iid copies of G. (c) Consider a barrier call option in which the payoff at time T = 12 is (S10 - 60)+ as long as the price Sn never falls below (c) 43 at specific times 72 = 2,4,6. Otherwise the payoff is 0. Thus the payoff C12 can be written as C12 = (S10 - 60)+ I{S 43, and S4 43, and S6 43}, where I(A} denotes the indicator rv for the event A. i. Use Monte-Carlo simulation to estimate the probability that the payoff is O: P(G1=0). Use 12 = 10,000 for the number of copies generated. ii. Suppose the risk-free interest rate r = 0.05. Estimate (via Monte Carlo) the price of this option; 1 Co (1++) E*(CT). Use 12 = 10,000 for the number of copies generated. 2. You will simulate the following simple (irreducible) Markov chain in what follows. The Markov chain {Xn 72 \0} has three states, S = {0,1,2}, and transition matrix 0.30 0 0.70 P 0 0.20 0.80 0.60 0.30 0.10 1 (a) With Xo = 0, simulate out to 72 = 1000 and estimate the long-run average: 1 N a lim N->oo N n-1 by using as an estimate 1000 1 [x 1000 n-1 Solve x = mP, for the limiting (stationary) distribution IT = and compare your estimate with the exact answer Einj. i-D (b) Solve x = wP, for the limiting (stationary) distribution IT = and compare your estimate in (a) with the exact answer (the mean of the limiting distribution) 2 [in, i=0 2

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clear; clc; close all;
%Solution to 1(a)
u=1.3; d=0.9; p=0.6;
S0=50; n=[100 1000 10000];
T=20; K=60;
E=expayoff(n,T,K,p,u,d,S0);
fprintf('\n\n(a) Expected payoff of Asian Call Option is as follows for different number of tials.\n');
[n' E']
%Solution to 1(b)
r=0.05;
pstar=(1+r-d)/(u-d);
C0=(1/power(1+r,T))*expayoff(n(3),T,K,pstar,u,d,S0);
fprintf('\n\n(b) Price of the option considerng risk neutral probability is %8.4f(n=10000)\n',C0);
%Solution to 1(c)(i)
P=prob0_barrier(n(3),p,u,d,S0);
fprintf('\n(c)(i) Probability for zero payoff under barrier call option is %8.4f(n=10000)',P);
%Solution to 1(c)(ii)
C0_barrier=(1/power(1+r,12))*expayoff_barrier(n(3),pstar,u,d,S0);
fprintf('\n\n(c)(ii) Price of the option considerng risk neutral probability is %8.4f(n=10000)\n\n',C0_barrier);...

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