Euler Monte Carlo Discretization and Antithetic Sampling

Summary

The document frames Monte Carlo pricing of a European option under geometric Brownian motion using an Euler time-step approximation. At each step, the asset value is updated using drift and a normally distributed shock; discounted terminal payoffs are averaged to estimate the option price. It then raises a variance-reduction question: pairing each sampled normal shock with its negative appears to produce an implausible estimate under the discretized scheme, despite working with the SDE’s explicit solution.

The author suspects the antithetic partner should be formed by negating the entire sequence of shocks for a path rather than alternating signs one shock at a time. The document supplies no answer or numerical comparison, so it does not establish which implementation caused the reported error. It serves as a useful problem statement about pairing whole simulated paths and understanding how a discretization responds to correlated random inputs; it is not a complete pricing recipe or a demonstrated result.

Key ideas

• Euler discretization approximates each asset-price step using drift and a normal shock.
• Monte Carlo option pricing averages discounted payoffs at maturity.
• Antithetic sampling pairs random inputs to reduce estimator variance.
• The document questions whether to negate each shock individually or the full path’s shock sequence, but gives no resolution.

Tags

# Euler discretization of SDE, combined with antithetic sampling


# Euler discretization of SDE, combined with antithetic sampling












let's say we have a GBM $dS_t = r S_t dt + \sigma S_t dW_t$, where $W_t$ is standard Brownian motion, and we have an European option $C$ with payoff $f(S_T)$. I want to use an Euler discretization scheme to compute the price of $C$. So let's say we fix $N$ (number of time steps), $T$ (final time), and we let $\Delta t= T/N$.

So we can write $$ S_{(j+1)\Delta t}=S_{j\Delta t}(1 + r\Delta t + \sigma\sqrt{\Delta t}Z_j),$$ where $Z_j$ are independent standard normal variables. Using a simple Monte Carlo method, we can compute the price of $C$ by averaging the discounted $f(S_T)$, as usual.

What I wanted to do was to use antithetic sampling to reduce the variance of the Monte Carlo simulation. Thus, I have a generator that returns independent normal variables $Z$, and in antithetic sampling, I sample $-Z$ after each $Z$. When I do this I get a completely incorrect price using the above scheme (this works fine when using Monte Carlo simulation with the explicit solution of the SDE instead of Euler discretization).

What is wrong? I'm guessing that antithetic sampling should be done on the whole family $(Z_0,Z_1,...,Z_{N})$ instead of doing it on each individual normal, but I'm having troubles understanding the theory behind this. Any help is appreciated!

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.