MA 573 - Computational Methods of Mathematical Finance - Assignment 5 Solved

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Worcester Polytechnic Institute                                                                                      Spring 2017

Department of Mathematical Sciences

Professor: Stephan Sturm

Teaching Assistant: Qingyun Ren



due on Wednesday, March, in Bloomberg class, to Qingyun Ren

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

                                                                                                                      (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).

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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

 

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