Milstein Convergence for the Heston Stochastic Volatility Model

Summary

The document asks how quickly simulated asset paths converge under a Milstein discretization of the Heston stochastic volatility model. It gives the coupled stochastic differential equations for the asset price and variance, including mean reversion in variance and correlation between the asset and variance Brownian motions. The specific question is whether the asset process has convergence order one when results from a given time mesh are compared with results from a finer mesh.

No answer, derivation, numerical experiment, or qualification is included, so the document does not establish a convergence rate. The question also leaves the convergence notion unspecified: pathwise strong error and weak error for expectations are distinct measures, and the square-root variance process creates additional considerations for discretization. The equations make the topic useful to researchers studying stochastic simulation, but readers need further analysis before applying any claimed order to option prices or other quantities derived from the paths.

Key ideas

• The Heston model couples asset returns to a mean-reverting stochastic variance process.
• The asset and variance shocks are correlated, which affects joint simulation.
• The document asks about the Milstein method's convergence rate for the asset path.
• It does not specify a convergence metric or provide an answer or supporting experiment.

Tags

# Milstein scheme for Heston model - rate of convergence


# Milstein scheme for Heston model - rate of convergence












Heston model is described by following SDE \begin{equation} \begin{aligned} dS_t &= \mu S_t dt + \sqrt{\nu_t} S_t dW^S_t \\ d\nu_t &= \kappa(\theta - \nu_t) dt + \xi \sqrt{\nu_t} dW^{\nu}_t \\ \textrm{Corr}[W^S_t, W^{\nu}_t] &= \rho \end{aligned} \end{equation}

If I use Milstein scheme for $\nu _t$ and $S_t$, what will be the rate of convergence of process $S$ (when I compare results with denser mesh)? Is it $1.0$ or no?

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