Reducing Monte Carlo Pricing Error for OTM Options in Heston

Summary

The document addresses poor Monte Carlo pricing accuracy for deep out-of-the-money options under a stochastic volatility model. The question focuses on Euler simulation of the Heston process and asks whether importance sampling can reduce variance and speed convergence. The response instead recommends two adjustments: choose discrete rates so the time-stepped simulation preserves the forward, and use the forward price as a control variate so the simulated forward matches the market forward.

The stated rationale is that these adjustments should align absolute pricing errors for out-of-the-money and in-the-money vanilla options through put-call parity. This is a concise suggestion rather than a full implementation guide: it does not provide a discretization scheme, derivation, numerical comparison, or guidance on choosing an importance sampling distribution. The result also depends on the simulation setup and assumptions behind the forward matching and parity relationship.

Key ideas

• A time discretization should be adjusted so the simulated process preserves the market forward.
• The forward price can serve as a control variate to align the simulation’s expected forward with the market value.
• Put-call parity links absolute pricing errors for vanilla calls and puts with matching terms.
• The response offers variance-reduction guidance but no worked implementation or empirical comparison.

Tags

# Monte Carlo simulation for OTM options under stochastic volatility


# Monte Carlo simulation for OTM options under stochastic volatility












I'm looking to simulate the stochastic price and volatility process (Heston model) using some form of Euler method for Monte Carlo approximation of option prices. The results that I get are acceptable for deep in the money options and at the money options but not very satisfying at all for deep out of the money options. I want to reduce the variance for faster convergence and the importance sampling method seems suitable but the problem doesn't seem to be trivial at all.

Does anyone have an idea or reference on where to start?

## Answer by jherek (score 0)

https://quant.stackexchange.com/a/76350

You could make use of two ingredients:

- make sure your discretization in time respects the forward by adjusting the discrete rates properly (see Exact Forward in Monte-Carlo)

- use the forward price as control variate to make sure the forward of the simulation matches the market forward.

With those two, the absolute error in price of OTM and ITM vanilla options should match as per the put-call parity relationship.

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