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

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_{Xa}. 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.

• Compute the optimal b and the variance reduction factor in using this estimator instead of the sample mean.
• 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 and variance 1.
   • Calculate the optimal mean and use this estimator instead of the sample mean.
   • Implement the calculation for a = 3 and 8.
   • 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

dX_t = ( X_t)dt + dW_t, X₀ = x.

Assume for the following the model parameters = 2, = 120 and = 25 and x = 100.

• Simulate 10 paths of the Ornstein-Uhlenbeck process, using a time horizon of 1 and 1000 time steps and plot them.
• 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 r_fix. 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

dr_t = ( r_t)dt + dW_t, r₀ = 0.02.

with parameters = 0.7, = 0.05 and = 0.006 and notional N = $10,000.

• Assume that the fixed rate r_fix = 4%, what is the expected payoff of an interest rate swap at maturity T = 3?
• If you want to issue an interest rate swap with maturity T = 2 trading at par (i.e., with value 0), which fixed rate r_fix 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 [email protected] 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