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Monte Carlo Pricing for a Piecewise-Constant Heston Model

Article Quant Q&A · Author: User2089
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Summary

The post studies option pricing under a time-dependent Heston model in which mean reversion changes across time intervals while other listed parameters are held fixed. It compares prices from an Elice method and two Monte Carlo approaches, Milstein and Broadie–Kaya, against values reported in a referenced paper. The displayed table covers several strikes for a specified set of initial conditions and model parameters.

The reported figures show Elice prices closely matching the paper’s values. The initial Milstein results are higher across the displayed strikes, while the author’s updated Broadie–Kaya results are closer to the reference. The accepted response says Broadie–Kaya works well relative to Milstein in this example. The post asks about simulation without the Feller condition, but offers no general study or proof addressing that question. Its evidence is a single parameter setup and table; it does not report simulation settings, uncertainty estimates, or broader validation.

Key ideas

  • The model lets the Heston mean-reversion parameter vary piecewise over time.
  • The post compares Elice pricing with Milstein and Broadie–Kaya Monte Carlo estimates.
  • For the displayed strikes, Elice prices closely track the cited reference values.
  • The updated Broadie–Kaya estimates are closer to the reference than the initial Milstein estimates in this example.
  • The numerical comparison does not establish general performance or resolve the no-Feller-condition question.

Tags

OptionsDerivatives pricingVolatilityStatistics
Full text
# Piecewise constant Heston model Monte Carlo simulation


# Piecewise constant Heston model Monte Carlo simulation












I am studying this time dependent Heston model \begin{equation} dS_t=(r-q) dt +\sqrt{V_t} dW_t^1 \\ dV_t=\kappa_t(\theta_t-V_t) dt + \sigma_t dW_t^2 \\ S_0=s_0\\ V_0=v_0\\ \rho_t=<dW_t^1,dW_t^2> \end{equation} I wrote a program using Elice method and tried to compare my result with Shamim Afshani paper https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1615153. Using Elice method I have a good matching with the paper, However my Monte Carlo routine seems not giving the results that I want. Below you find the example with python code.

For the initial values $S0 = 1$, $V0 = 0.1$ and the time points $t_0 = 0$, $t_1 = 1$, $t_2 = 3$ and $t_3 = 5$, we specify $r_t = 0$, $q_t = 0$, $\theta_t = 0.1$, $\sigma_t = 1$, $\rho_t = -0.9$ and $\kappa_t =\sum_{m=1} \kappa_mI_{[t_{m-1}<t\leq t_m]}$ where $\kappa_1 = 4$, $\kappa_2 = 2$ and $\kappa_3 = 1$

| Strike | Afshani price | Elice price | Monte carlo Milstein price | Monte carlo Broadie price |
| 0.5 | 0.548724 | 0.548733 | 0.551647 | 0.547670 |
| 0.75 | 0.370421 | 0.370423 | 0.376329 | 0.3697154 |
| 1 | 0.230355 | 0.230357 | 0.23865 | 0.229919 |
| 1.25 | 0.129324 | 0.129328 | 0.138600 | 0.12882076 |
| 1.5 | 0.063974 | 0.063981 | 0.072626 | 0.063716 |

Is there any research paper that studies Monte Carlo time dependant Heston model with no feller condition

## Answer by User2089 (score 0, accepted)

https://quant.stackexchange.com/a/75191

Indeed Broadie and Kaya method works fine in comparaison of Milstein method I have updated the table.

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