STAD70H3S Name: ____________________________ Test 2: Mar 04, 2016 Student #: _________________________
1) Assume asset S follows simple geometric Brownian motion (GBM) with annual volatility σ=20%. Moreover, let the annual interest rate be r=5% and S0=$100.
a) Consider a derivative that pays off $1 in one year if the asset price drops below
$90 AND then rebounds above $90 before the end of the year . Use Monte Carlo simulation with path discretization to estimate the price of this option; simulate n=10,000 paths with m=50 points per path. Provide the estimated value and a 95% confidence interval around it.
b) Consider another derivative that knocks in if the asset price drops below $90, in which case it pays off the absolute difference between the asset price and $90 in
one year, i.e. the payoff is |S1 90|, min0t1{St}90. Use Monte Carlo
0, otherwise
simulation with n=10,000 to estimate the price of this option without path discretization (hint: simulate pairs of the minimum & final price of each path). Provide the estimated value and a 95% confidence interval around it.
2) Consider the stochastic process {Rt } which follows a Cox-Ingersoll-Ross (CIR) model, i.e. it follows the stochastic differential equation: dRt ( Rt )dt Rt dWt , with
R0 2 and 3, 1, 1
a) Use Euler discretization to simulate n=10,000 paths of this process from time
t 0 1 , where each path has m=50 steps. Plot a histogram of the simulated final
values at time 1.
b) It can be shown that, given R0 and for any t 0 , the CIR random variable Rt
multiplied by freedom df
4 follows a non-central χ2 distribution with degrees of 2 (1 e t )
4 2
4R et
0 , i.e.
and non-centrality parameter ncp
R ~ non- central 2 (df, ncp) . To verify your discretization from part a) is
2 (1 e t )
correct, create a QQ-plot of the sample quantiles of the R1 ’s you generated with
the theoretical quantiles from the appropriate non-central χ2 (Hint: use the R function qchisq(p,df,ncp), where df and ncp are as given above, and p is the vector of empirical cumulative probabilities of your simulated values).
4 2(1et) t