## Transcribed Text

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

These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction
of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice.
Unethical use is strictly forbidden.

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);...