FINM32000 - Homework 6 - Solved

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

                                                              d            j = 1,2

with risk-free interest rate r = 0.05, and constant volatilities σ[1] = 0.3, σ[2] = 0.2.

The time-0 prices are  = 110. The P-Brownian motions W[i] and W[j] have 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(ΣWTW Cov(WT )Σ>.

Consider a basket  of one-half of a share of each stock.

(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 covariance 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.

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

Consider the “geometric basket” .

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

 

and variance

 

(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).

(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 CˆMcv,βˆ from

cv,βˆ √ L6.6 or L6.7. Use the (asymptotically valid) standard error ˆσM / 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.

(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.

(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.)

(c)    Run 100000 importance sampling simulations, using the specific drift adjustment calculatedin (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.

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.