I'm trying to use the Control Variates technique to reduce the variance of the estimate obtained from a Monte Carlo simulation for option pricing. As suggested in the book by Glasserman I'm using this control variate estimator

$$ \text{"option price at time 0"} \approx \hat Y = \frac 1n\sum_{i=1}^n Z_i $$

where $Z_i$ are the components of the vector $Z = Y-\theta(X-\mathbb E[X])$, with $V=e^{-rT}(S(T)-K)$ vector of discounted payoffs (outputs of the Monte Carlo simulation), $X=e^{-rT}S(T)$ and $S(T)$ is the vector of spot prices at expiry time $T$ generated in the simulation, $\theta$ is a constant chosen to be the minimzer of $Z$ that is $\theta=\dfrac{\text{cov}(Y,X)}{\text{var}(X)}$. Finally, under the risk-neutral measure $X$ is a martingale and $\mathbb E[X]=S(0)$.

The last identity comes from the previous book "the absence of arbitrage is essentially equivalent to the requirement that appropriately discounted asset prices be martingales. Any martingale with a known initial value provides a potential control variate precisely because its expectation at any future time is its initial value".

What I don't get is the basic assumption $\mathbb E[S(T)]=e^{rT}S(0)$ which implies that the spot prices will keep growing in the future ($e^{rT}$ is strictly bigger than $1$).

In the example I'm working on - option under the Schwartz model $dS = \alpha(\mu-\log S)Sdt + \sigma S dW$ - the initial spot price is $S(0)=22.93$ but almost all (98.5%) the spot prices $S(T)$ computed with the Monte Carlo simulation are smaller than $S(0)$, hence $\mathbb E[S(T)]<e^{rT}S(0)$ and $\hat Y$ is a bad estimator of the option price (exact solution is 2.08 while the control variate estimator is 5.88).

So I guess that a different $X$ has to be chosen, any idea on possible candidates?

This is the output of the Matlab code used to compute the price V of the option at time 0 using Monte Carlo simulations with the suggestion by jherek

V_MC_standard = 0.070141, std = 0.000144
V_MC_controlv = 0.070216, std = 0.000074

and this is the code

S0 = 1; % spot price at time 0
K = 1; % strike prices
T = 1/2; % expiry time
r = .1; % risk-free interest rate
alpha = .2;
sigma = 0.4;
mu = 0.3;

%% Standard Monte Carlo
N = 1e6;
X = log(S0)*exp(-alpha*T) + (mu-sigma^2/2/alpha-(mu-r)/alpha)*(1-exp(-alpha*T)) + sigma*sqrt(1-exp(-2*alpha*T))/sqrt(2*alpha)*randn(N,1);
S = exp(X);
V = exp( -r*T ) * max(0,S-K);
V0 = mean(V);
fprintf('V_MC_standard = %f, std = %f\n' , V0 , std(V)/sqrt(N) );

%% Control Variates
VC = exp(-r*T)*S; % mean(VC) == S0
C = cov(V,VC); % the covariance matrix
theta = C(1,2)/C(2,2); % the optimal theta
F = exp( exp(-alpha*T)*log(S0) + (mu-sigma^2/2/alpha-(mu-r)/alpha)*(1-exp(-alpha*T)) + sigma^2/4/alpha*(1-exp(-2*alpha*T)) );
V = V-theta*(VC-exp(-r*T)*F);
V0 = mean(V); % Controlled Monte Carlo estimate of the option value
fprintf('V_MC_controlv = %f, std = %f\n' , V0 , std(V)/sqrt(N))
  • $\begingroup$ Just a minor comment: I believe you are using this equation to compute X but I suspect the normal should be $N(0, T)$. What do you think? Then when you compute the mean, that term vanishes since its expected value is zero. $\endgroup$ – rvignolo Oct 2 '20 at 16:18
  • $\begingroup$ @rvignolo The explicit formula for the $\log$ of the spot price is $$ X_t = X_0 e^{-\alpha t} + \Big(\mu-\frac{\sigma^2}{2\alpha}-\lambda\Big)(1-e^{-\alpha t}) + \sigma e^{-\alpha t} \underbrace{\int_0^t e^{\alpha s} dW_s}_{I} $$ $I$ is the "sum" of independent Normals so it is also Normal with mean the sum of the means, ie $0$, and variance the sum of the variances, ie $\displaystyle\int_0^t e^{2\alpha s}ds=\frac{e^{2\alpha t}-1}{2\alpha}$ hence $\endgroup$ – sound wave Oct 2 '20 at 18:00
  • $\begingroup$ hence $I \sim N\Bigg(0,\dfrac{e^{2\alpha t}-1}{2\alpha}\Bigg) \sim \sqrt{\dfrac{e^{2\alpha t}-1}{2\alpha}} N(0,1)$ and $$ \sigma e^{-\alpha t} I \sim \sigma \sqrt{e^{-2\alpha t}\frac{e^{2\alpha t}-1}{2\alpha}} N(0,1) \sim \sigma\sqrt{\dfrac{1-e^{-2\alpha t}}{2\alpha}} N(0,1)$$ $\endgroup$ – sound wave Oct 2 '20 at 18:00
  • $\begingroup$ Obvious first place to look is the solution of the PDE...did you derive it yourself or get one from a Numerical Recipes sort of text? $\endgroup$ – Chris Oct 4 '20 at 5:49
  • $\begingroup$ @Chris The solution is derived by Schwartz himself at page 5 here sci-hub.st/10.1111/j.1540-6261.1997.tb02721.x $\endgroup$ – sound wave Oct 4 '20 at 6:00

Some of the assumptions here are wrong. The issue here is that $$S_0 \neq e^{-rT} E[S],$$ but $$F = E[S].$$

And thus Z should be Z=V-theta*(VC-exp(-rT)*F). If you output mean(VC) it's very clear.

It suggests that the choice of parameters for the Schwartz model are not consistent with the interest rate r, unless a non-zero convenience yield is expected.

  • $\begingroup$ Thank you for help, I tried your formula for Z but then mean(Z) is exactly equal (up to 10 decimal digits) to V_exact, and by lowering the number of simulations N to 1e1 the precision increases (equal up to 14 decimal digits). It seems quite strange, is this how it is supposed to work? $\endgroup$ – sound wave Oct 5 '20 at 11:50
  • $\begingroup$ The formula $S_0 = e^{-rT} E[S]$ is written in the book by Glasserman look here i.imgur.com/zlBFzTB.png $\endgroup$ – sound wave Oct 5 '20 at 11:51
  • $\begingroup$ The formula $F=S_0 e^{rT}$ implies that spot prices increase in time, but usng the parameters estmated from market data, they decrease if $T$ is low, increase if $T$ is higher, I think this is due to the fact that the market data that I use to estimate the parameters don't always increase, they are the first 200 monthly oil spot prices downloadable from here eia.gov/dnav/pet/pet_pri_spt_s1_m.htm (sheet Data 1) $\endgroup$ – sound wave Oct 5 '20 at 11:59
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    $\begingroup$ The equality is expected since the control variate is essentially the same as the payoff (minus a constant). If you were pricing an option instead of a forward contract, you would see a difference. $\endgroup$ – jherek Oct 5 '20 at 12:04
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    $\begingroup$ The result can not be negative, you must have made a mistake somewhere, perhaps max(F-K,0)*... $\endgroup$ – jherek Oct 10 '20 at 18:52

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