Description
1. Let X be a standard normal random variable. We want to use Monte Carlo methods to
estimate P(X ≥ a) for fixed a ∈ R by sampling X and computing the sample mean for
Y = 1l{X≥a}. For large a this is a nontrivial task as there will not just be enough sample
points that give a positive contribution. Control variates are the way out here. First we
consider the control variate estimator
1l{X≥a} − b
X − E[X]
(1)
for some constant b.
(a) Compute the optimal b
∗ and the variance reduction factor in using this estimator
instead of the sample mean.
(b) Implement the calculation for a = 3 and a = 8 and N = 100, 000 (or larger, if
needed).
2. As in problem 1, we want to use Monte Carlo methods to estimate P(X ≥ a) for fixed
a ∈ R by control variates. However we want now to use importance sampling, with X
having under the measure P˜ a normal distribution with mean µ and variance 1.
(a) Calculate the optimal mean µ
∗ and use this estimator instead of the sample mean.
(b) Implement the calculation for a = 3 and 8.
(c) Compare the convergence of different variance reduction techniques (antithetic
sampling, control variates, importance sampling) among eachother andwith the
base case (without variance reduction)
2
3. A popular stochastic process to model interest rates, exchange rates, volatility or
commodity prices is the Ornstein-Uhlenbeck (OU) process. It follow the dynamics
dXt = λ(κ − Xt) dt + σ dWt
, X0 = x.
Assume for the following the model parameters λ = 2, κ = 120 and σ = 25 and x = 100.
(a) Simulate 10 paths of the Ornstein-Uhlenbeck process, using a time horizon of 1 and
1000 time steps and plot them.
(b) Play around with the parameter and try to find out there intuitive meaning.
Which properties of the process are they describing?
4. An interest rate swap is a financial product that exchanges the interest rate gains from
the floating market rate r against those of a previously fixed rate rfix. Thus it pays
N
e
R T
0
rs ds − e
rfixT
where N is the amount notional of the contract. Assume that the interest rate follows
under the risk-neutral measure the Ornstein-Uhlenbeck dynamics
drt = λ(κ − rt) dt + σ dWt
, r0 = 0.02.
with parameters λ = 0.7, κ = 0.05 and σ = 0.006 and notional N = $10, 000.
(a) Assume that the fixed rate rfix = 4%, what is the expected payoff of an interest
rate swap at maturity T = 3?
(b) If you want to issue an interest rate swap with maturity T = 2 trading at par (i.e.,
with value 0), which fixed rate rfix do you have to choose?
Note: All programming problems should be either in Python 2.7 or Python 3.5. Matlab
and R are accepted, but no support for these languages is provided. Please comment the
programs extensively and send them in a .zip file with title Lastname HW5.zip and
suject line ”MA 573 HW5 Lastname” to Qingyun Ren qren@wpi.edu before the due
date of the homework (replacing the bold words by your actual last name). Please
provide printouts of programs amd plots that one can comment on them.
6 points per problems