Easy Mistake With the Log-Euler Discretization On Black-Scholes

Mon, Mar 11, 2024

In the Black-Scholes model with a term-structure of volatilities, the Log-Euler Monte-Carlo scheme is not necessarily exact.

This happens if you have two assets $S_1$ and $S_2$, with two different time varying volatilities $\sigma_1(t), \sigma_2(t)$. The covariance from the Ito isometry from $t=t_0$ to $t=t_1$ reads

$$\int_{t_0}^{t_1} \sigma_1(s)\sigma_2(s) \rho ds,$$ while a naive log-Euler discretization may use $$\rho \bar\sigma_1(t_0) \bar\sigma_2(t_0) (t_1-t_0).$$ In practice, the $\bar\sigma_i(t_0)$ are calibrated such that the vanilla option prices are exact, meaning $$\bar{\sigma}_i^2(t_0)(t_1-t_0) = \int_{t_0}^{t_1} \sigma_i^2(s) ds.$$

As such the covariance of the log-Euler scheme does not match the covariance from the Ito isometry unless the volatilities are constant in time. This means that the prices of European vanilla basket options is going to be slightly off, even though they are not path dependent.

The fix is of course to use the true covariance matrix between $t_0$ and $t_1$.

The issue may also happen if instead of using the square root of the covariance matrix in the Monte-Carlo discretization, the square root of the correlation matrix is used.