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Almost-Exact Simulation Scheme for Heston-type Models: Bermudan and American Option Pricing

Mara Kalicanin Dimitrov, Marko Dimitrov, Anatoliy Malyarenko, Ying Ni

TL;DR

This work extends the Almost-Exact Simulation (AES) scheme to Bermudan and American option pricing under Heston-type models, including the double Heston variant. AES leverages sampling from the noncentral chi-square distribution for the CIR variance to avoid variance negativity and discretization bias, delivering an almost exact path simulation that outperforms Euler in accuracy and efficiency, especially when the step count matches exercise dates. The paper analytically derives AES for the double Heston model and validates its performance through extensive numerical experiments, showing significant gains in accuracy and reductions in computational time compared to Euler, with robust results for both in-the-money and at-the-money options. These findings suggest AES as a practical bias-free alternative for Monte Carlo pricing in multi-factor stochastic volatility models, with potential extensions to higher dimensions and machine learning-augmented acceleration.

Abstract

Recently, an Almost-Exact Simulation (AES) scheme was introduced for the Heston stochastic volatility model and tested for European option pricing. This paper extends this scheme for pricing Bermudan and American options under both Heston and double Heston models. The AES improves Monte Carlo simulation efficiency by using the non-central chi-square distribution for the variance process. We derive the AES scheme for the double Heston model and compare the performance of the AES schemes under both models with the Euler scheme. Our numerical experiments validate the effectiveness of the AES scheme in providing accurate option prices with reduced computational time, highlighting its robustness for both models. In particular, the AES achieves higher accuracy and computational efficiency when the number of simulation steps matches the exercise dates for Bermudan options.

Almost-Exact Simulation Scheme for Heston-type Models: Bermudan and American Option Pricing

TL;DR

This work extends the Almost-Exact Simulation (AES) scheme to Bermudan and American option pricing under Heston-type models, including the double Heston variant. AES leverages sampling from the noncentral chi-square distribution for the CIR variance to avoid variance negativity and discretization bias, delivering an almost exact path simulation that outperforms Euler in accuracy and efficiency, especially when the step count matches exercise dates. The paper analytically derives AES for the double Heston model and validates its performance through extensive numerical experiments, showing significant gains in accuracy and reductions in computational time compared to Euler, with robust results for both in-the-money and at-the-money options. These findings suggest AES as a practical bias-free alternative for Monte Carlo pricing in multi-factor stochastic volatility models, with potential extensions to higher dimensions and machine learning-augmented acceleration.

Abstract

Recently, an Almost-Exact Simulation (AES) scheme was introduced for the Heston stochastic volatility model and tested for European option pricing. This paper extends this scheme for pricing Bermudan and American options under both Heston and double Heston models. The AES improves Monte Carlo simulation efficiency by using the non-central chi-square distribution for the variance process. We derive the AES scheme for the double Heston model and compare the performance of the AES schemes under both models with the Euler scheme. Our numerical experiments validate the effectiveness of the AES scheme in providing accurate option prices with reduced computational time, highlighting its robustness for both models. In particular, the AES achieves higher accuracy and computational efficiency when the number of simulation steps matches the exercise dates for Bermudan options.
Paper Structure (8 sections, 23 equations, 4 figures, 6 tables, 2 algorithms)

This paper contains 8 sections, 23 equations, 4 figures, 6 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Bermudan and American options under Heston-type Models
  3. Derivation of AES Schemes
  4. Evaluation of AES Schemes
  5. AES under the Heston model
  6. AES under the Double Heston Model
  7. Conclusion and Future Work
  8. Appendices

Figures (4)

  • Figure 1: Relative Errors of Bermudan option prices with parameters given in Eq. \ref{['eq:ParametersFellerConditionDoesNotHold']} calculated using AES under the Heston Model. The results highlight the accuracy of the AES scheme across different initial asset prices and exercise dates.
  • Figure 2: Relative Errors of biweekly Bermudan option values with parameters given in Eq. \ref{['eq:ParametersFellerConditionDoesNotHold']} calculated under the Heston Model using AES and Euler schemes for different initial asset prices $S_0$. The AES scheme consistently achieves lower errors for in-the-money and at-the-money options, especially as the time to maturity increases.
  • Figure 3: Comparative analysis of AES and Euler schemes using twice as many steps for Euler scheme simulation: (a) Accuracy difference, (b) Computation time difference, (c) Memory usage difference.
  • Figure 4: Relative Errors of American put option prices calculated using AES under different parameter conditions: (a) Parameters satisfying the Feller Condition, (b) Parameters not satisfying the Feller Condition. The AES scheme achieves high accuracy with a small number of steps.