FINM32000 Homework 6 Solved

Let S be the column vector with components S[1], S[2], where the stock prices S[j] have risk-neutral dynamics

dS[j]=rS[j]dt+σ S[j]dW[j] j=1,2 t t [j]t t

with risk-free interest rate r = 0.05, and constant volatilities σ[1] = 0.3, σ[2] = 0.2.
The time-0 prices are S [1] = 100, S [2] = 110. The P-Brownian motions W [i] and W [j ] have

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correlation ρ = 0.8.
(a) Let X be the column vector with components X[1],X[2] where X[j] := logS[j]. Find the

covariance matrix of XT .

Hint: One possible approach is to write XT as a nonrandom vector plus ΣWT where Σ is the nonrandom diagonal matrix with diagonal elements σ[1],σ[2], and W is the random column vector with components W [1], W [2]. Then Cov(XT ) = E(ΣWTWT⊤Σ⊤) = ΣCov(WT )Σ⊤.

Consider a basket H := 12S[1] + 12S[2] of one-half of a share of each stock.

2. (b)  Using 10000 standard Monte Carlo simulations, estimate the time-0 price C of an option that pays (HT − 110)+ at time T = 1.0. Also give the standard error [the sample standard deviation, divided by the square root of the number of simulations] of your Monte Carlo estimate.

   You may either use a random number generator that produces normals with a given covari- ance matrix (which you found in (a)), or alternatively use a random number generator that produces independent normals which you then transform to introduce correlation.

   In either approach, each of the 10000 simulations should use just one R2-valued random vector Z of simulated normal zero-mean random variables.

3. (c)  Use 10000 antithetic pairs (Z,−Z) to estimate C, together with a standard error (L5.28).

Consider the “geometric basket” G := 􏰅S[1]S[2]􏰆1/2.

(d) The random variable log GT is normally distributed (because it’s a linear transformation of a multivariate normal vector). Show that log GT has expectation

􏰌 σ2+σ2􏰍 1 log(S[1]S[2]) + r − [1] [2] T

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and variance

σ2 + 2ρσ[1]σ[2] + σ2
[1] [2]T.

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5. (e)  Let CG be the time-0 price of a geometric basket option paying (GT −K)+ at time T.

   Express CG in terms of the function CBS defined in FINM 33000 L6.16. Specifically, fill in the blanks:

   CG =CBS( ,0,K,T, ,r, ).
   Your answer should be a general formula, in which you have not substituted 0.8 for ρ, etc.

   (You may also do the substitutions, but don’t neglect the general formula).

6. (f)  Using a geometric basket option as a control variate, run M = 10000 Monte Carlo simulations

to estimate C, together with a standard error. Use the control variate estimate Cˆcv,βˆ from cv,βˆ √ M

L6.6 or L6.7. Use the (asymptotically valid) standard error σˆM / M. See the ipynb file.

Problem 2

Let the bank account and non-dividend paying stock have risk-neutral dynamics dBt =rBtdt B0 =1

dSt = rStdt + σStdWt S0 > 0 where σ > 0 and W is a P-Brownian motion.

Consider a K-strike T-expiry vanilla call option, and let C denote its time-0 price.

1. (a)  Let S0 = 100, σ = 0.2, r = 0.02, K = 150, T = 1.

   Run 100000 ordinary Monte Carlo simulations to estimate C, together with a standard error.

2. (b)  Suppose that we sample from a new probability measure P∗, under which W now has constant

   drift λ instead of drift 0. Thus Wt = Wt∗ + λt where W∗ is a standard P∗-BM. Find the P∗-expectation E∗ST in terms of S0, r, σ, λ, and T.
   Calculate λ such that E∗ST = 165.

   (Why did we choose 165? The picture in L6.16 shows that the optimal distribution from which to sample will have a mean that is greater than the strike K. So let’s choose 10% higher than K. This will not be optimal, but we expect that it will be an improvement over ordinary Monte Carlo. There are more systematic ways to determine a reasonable drift adjustment, not utilized here.)

3. (c)  Run 100000 importance sampling simulations, using the specific drift adjustment calculated in (b), to estimate C, together with a standard error. Be aware that your zero-mean normal random draws, here, simulate increments of W∗ not W.

   Each simulation should require only one number to be generated by randn. See the ipynb file.

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Comment: of course, we do not need Monte Carlo to price a call under GBM. However, suppose you wanted to price a deep OTM option under an intractable stochastic volatility model, using importance sampling. You could still use a similar approach.

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