# Importance Sampling - where to center the sampling distribution?

Consider a Monte Carlo (MC) approximation to a European call with BS parameters $r = 0.05, \sigma = 0.4, T = 10, S_0 = 50$ and $K = 95$. Consider the following results, each using 1M points:

1. plain MC: $\$21.6901 \pm \$0.1735$
2. importance sampling with the sampling mean of $S_T = K$: $\$21.7161 \pm \$0.1511$
3. importance sampling with the sampling median of $S_T = K$: $\$21.8104 \pm \$0.0650$
4. importance sampling with the sampling mode of $S_T = K$: $\$21.7801 \pm \$0.0210$

where the $\pm$ is a 95% confidence interval. The BS price is $\$21.7766$. It seems that setting the sampling mode of$S_T$to$K$offers the greatest variance reduction, but is this a general rule? I am actually a bit suspicious of importance sampling because when I use more "reasonable" parameters, importance sampling sometimes increases the variance. Indeed, again with 1M points but using$r = 0.05, \sigma = 0.2, T = 1, S_0 = 50$and$K = 50$I get 1. plain MC:$\$5.2192 \pm \$0.0144$2. importance sampling with the sampling mean of$S_T = K$:$\$5.2162 \pm \$0.0197$3. importance sampling with the sampling median of$S_T = K$:$\$5.2133 \pm \$0.0173$4. importance sampling with the sampling mode of$S_T = K$:$\$5.2207 \pm \$0.0136$with BS price$\$5.2253$.

For plain vanillas, is there a good rule on how to pick the sampling distribution? (plain vanillas because they are more tractable for me :))

• usually people carry through with the distributional assumption that underlies BS. Some seem to apply exponential change of measure via a cumulant generating function (but I can't comment on it as I have never used it). Have you considered control variate methods and otherwise QMC? – Matthias Wolf Jun 16 '15 at 6:03
• I would recommend this paper arxiv.org/abs/math/0702473 . It makes a link between importance sampling and large deviations, and give some applications to finance. – lehalle Jun 17 '15 at 6:14

Given two representations: $$C = E_f[\varphi(X)] = \int \varphi(x) f(x)dx = \int \varphi(x) \frac{f(x)}{g(x)}g(x)dx = E_g[\varphi(X)\frac{f(X)}{g(X)}]$$ The difference of the variances of the MC estimators associated with the two expression is $$Var[\widehat{C}^f_N] - Var[\widehat{C}^g_N] = \frac{1}{N}\int \varphi(x)^2 \left(1 - \frac{f(x)}{g(x)}\right) f(x)dx$$ We want this to be positive for importance sampling to improve estimation.
In your case, you can compute explictly this variance gain by setting $\varphi(x) = S_0(e^x - e^k)_+$ (where $k =\ln(K/S_0e^{rT})$ is the log-moneyness forward) and $f$, $g$ gaussian densities of parameters $(-\sigma^2/2,\sigma^2)$ and $(\mu-\tau^2/2,\tau^2)$.