The normal mixed model

Instead of tricky integrals, price contingent claims by simulating asset price paths.

If we generate random variates $X_1, ..., X_n$ indepedent with the distribution $X / S_0(T)$ then $\bar{X} = \frac{1}{n} \sum X_i$ estimates $\pi_X (0)$ with usual CLT confidence intervals.

To generate the $X_i$ , need to simulate asset price paths $((S_1(t)), . With a Black-Scholes model $S_i (t) = exp((r- \frac{1}{2} , so we need to simulate Brownian motion.

What to do next depends on the type of claim. For example, with Euro options only need final value so only need $\tilde{W}_T ~ N(0, T)$ . But in general will need the whole price path. Will discretise into $m$ steps of size $\Delta t = T / m$ , then fill in other values via linear interpolation. This works well for complicated options. It doesn’t work well for American options because it doesn’t provide optimal strategy.

Variance reduction techniques

High accuracy required for financial application, usually want error to be less than 1 part per thousand. Standard errors $\in o(\sqrt{n})$ , so therefore useful to use variance reduction techniques.

Antithetic variates

The simplest of these is antithetic variates – if $(W_t)$ is a Brownian motion, then so is $(-W_t)$ . Then let $X = \frac{1}{2} (f(S_i^{(i')} (T)) . Variance will be reduced because prices from positive and negative paths are likely to be negatively correlated.

Control variates

Idea: Use known “model payoffs” $Y$ , similar to $X$ , with $E[Y] = \mu_Y$ . $\frac{1}{n} \sum (X_i + c(Yi - \mu_Y))$ , estimates $E[X]$ as $E[X + c(Y - .

This will be better than simple sampling if $Var(X + c(Y - \mu_Y)) \lt , ie. $Var(X) + 2cCov(X,Y) + Var(Y) \lt Var(X)$ . A pilot sample may help to estimate the best value $c^{*}$ of $c$ . Alternatively could just take $c$ to be $-1$ , ie use $\frac{1}{n} \sum (X_i - Y_i + \mu_Y)$ . Relies on $Var(X-Y)$ being smaller than $Var(X)$ .

Another approach uses several model payoffs $Y^{(1)}, Y^{(2)}, ..., with $E[Y^{(j)}] = \mu_j$ . Then estimate $E[X]$ using $\frac{1}{n} \sum (X_i + \sum c_j(Y^{(j)}_i - \mu_j))$ . This works as above. You can find the $c_j$ ’s by regressing $X_i$ on $Y^{(1)}_i, ...$ .

Importance sampling

Idea: Use $E_K [X \frac{dQ}{dR} ]$ , ie. simulate from distribution of $X under R rather than $X$ under $Q$ . Hopefully $Var_R (X will be smaller than $Var_Q (X)$ .

\[ Var_R(X \frac{dQ}{dR}) = E_R [ X^2 {(\frac{dQ}{dR})}^2] - \mu^2

We want $\frac{dQ}{dR}$ to be small. (Note: $\frac{dQ}{dR} \gt 0$ and $E_Q ). Can do this with GBM(geometric brownian motion). If $S_1(t) = S_1(0) exp( (r-\frac{1}{2}\sigma^2)t \sigma we can find an equivalent measure $R$ s.t. $\frac{dR}{dQ} = , under which $\bar{W}_T = \tilde{W}_T - ct$ is a Brownian motion. So simulating the assert price under $R$ is just a matter of simulating a GBM.

Also:

\[ \frac{dQ}{dR} = exp (-\tilde{W}_T + \frac{1}{2} c^2 T)