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CFRM505: Monte Carlo Methods in Finance (Winter 2024)
Final Exam
Practice Questions
1. Consider two independent uniform random variables, U1 ~ U (-1, 1), U2 ~ U (-2, 2). We want to estimate the probability
P (U1(2) + U2(2) > 2).
Apply conditional Monte Carlo to estimate this probability. State which of the two random variables you’re conditioning on, and explain briefly why. Derive the estimator, and state which random variable you’re simulating.
2. Consider a uniform random variable V ~ U (-3, 3). The objective is to estimate the expectation
E[h(V)],
for some function h, and we want to apply the control variates V2 and V for variance reduction. The resulting estimator should look like W = h(V) + c1 (V2 - a1 ) + c2 (V - a2 ).
Determine a1 and a2 explicitly. Write down the expressions for the best choices for the constants c1 and c2 and explain how you can numerically compute in your pilot (small sample) and real (large sample) runs.
3. Let Y ~ exp(1). And given Y = y, for y ≥ 0, X is normal N (y, 2). We want to estimate the probability
P (X > 4).
(i) First, apply conditional Monte Carlo to estimate this probability. Derive the estimator and state which random variable you have to simulate.
(ii) Explain whether the estimator in (i) can be further improved by using anti- thetics.
(iii) Following up from (i),i.e. after doing conditional Monte Carlo, let’salso apply importance sampling by changing the mean of Y to be 4. Give the estimator and what random variables you’re simulating.
4. A Binomial random variable X with parameters n ∈ N and p ∈ (0, 1), denoted by X ~ Bin(n, p), has the probability mass function (pmf):
n-x , x = 0, 1, 2, . . .
We apply exponential twisting and define the new pmf by
g(x) = M(t)/e txf(x),
where M(t) is the moment generating function of X, and t is a real-valued param- eter such that g is still a pmf.
(i) Compute M(t).
(ii) Show that the distribution corresponding to the new pmf g(x) is Binomial, and give the parameters. What’s the condition on t, if any?
(iii) Let p = 0.6 (the original success probability). For importance sampling, we want g to be the pmf for the distribution Bin(n, 0.9). What is the value of t should we select to achieve this?
5. Consider the Black-Scholes model where the underlying stock S is GBM(r, σ). A bull spread option has the following terminal payof
(K2 - K1
:0
if ST ≥ K2
if K1 < ST < K2 if ST ≤ K1 (1)
where 0 < K1 < K2 < ∞ . Consider the time-0 price of the option
C = EQ {e-rTh(ST )}.
(a) Describe a simulation algorithm using stratified sampling to estimate the price of this option. Use exactly 3 strata that correspond to the 3 scenarios (events) in (1) above. In particular, compute the probabilities p1, p2 , and p3 associated with the three strata, in terms of the normal cdf Φ . You can use any sample allocation rule. Make to give the random variable(s)/distribution(s) to be simulated, and write down the estimator(s)).
(b) Let’s estimate the Delta of this option. Use a mixture of the pathwise (PW) method and likelihood ratio (LR) method. Specifically, in the region(s) where the payof is flat, avoid the PW method and use the LR method there. State what random variable/distribution you simulate, and derive the estimator.