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Refining Heston Monte Carlo Time Steps to Meet an Error Target

Article Quant Q&A · Author: Hans
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Summary

The document asks whether option maturity should determine the time-step size when pricing with Monte Carlo under the Heston stochastic-volatility model, including whether longer-dated options or steps that grow toward maturity can use coarser grids. It cautions against choosing a step size from maturity or the long-run variance distribution alone.

Instead, the proposed starting point is to refine the simulation grid, increasing paths and time steps until an error measure falls below the required threshold. A simple illustrative measure is the change between consecutive runs. The answer notes that convergence can be accelerated, but does not detail those methods or prescribe an error metric. It recommends establishing a robust uniform-step method first, then investigating efficiency improvements such as non-uniform steps; the appropriate grid remains problem- and accuracy-dependent.

Key ideas

  • Monte Carlo time-step size should be judged against a defined pricing accuracy target.
  • Refine the number of paths and time steps until the chosen error measure is acceptable.
  • Differences between consecutive runs are offered as a simple possible error measure.
  • Establish a reliable uniform time grid before investigating non-uniform steps.
  • The document does not specify a universal grid or explain the referenced efficiency methods.

Tags

OptionsDerivatives pricingVolatilityStatisticsBacktesting
Full text
# Time Step Size for Heston Model for Different Option Maturity


# Time Step Size for Heston Model for Different Option Maturity












Suppose we are to price with Monte Carlo method two options differing only in the maturity time, with the same, say, call option payoff, or Asian option payoff with a fixed averaging window, with the underlying stock price following the Heston model. We know the distribution of variance in the Heston model approaches a stationary one as time approaches infinity. To achieve the same accuracy,

1) can we use longer time step size for the option with longer maturity than the one with shorter maturity?

2) What about using variable time step size with step size growing towards the time of maturity?

## Answer by Will Gu (score 1)

https://quant.stackexchange.com/a/31345

The time step size is more like a byproduct of the convergence of Monte Carlo simulation rather than something to be decided a priori. It depends on the accuracy threshold. In general, one should keep refining the grid (number of paths * number of time steps) until her error metric goes under the threshold. To the simplest, this metric could be the difference between two consecutive runs.

As a side note, there are many ways to speed up the convergence, I'll leave it out for now.

UPDATE:

I described the general approach in determining the optimal uniform time step. It may differ problem by problem, depends on the accuracy demand. On the other hand, here's a paper that looks relevant to what OP was looking for. I think a good approach would be to first establish robust method with uniform time step, and then explore ways to improve efficiency (non-uniform time step being one of them).

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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