Bermudan Options

Almost-Exact Simulation Scheme for Heston-type Models: Bermudan and American Option Pricing ArXiv ID: 2601.00815 “View on arXiv” Authors: Mara Kalicanin Dimitrov, Marko Dimitrov, Anatoliy Malyarenko, Ying Ni 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. ...

A deep BSDE approach for the simultaneous pricing and delta-gamma hedging of large portfolios consisting of high-dimensional multi-asset Bermudan options ArXiv ID: 2502.11706 “View on arXiv” Authors: Unknown Abstract A deep BSDE approach is presented for the pricing and delta-gamma hedging of high-dimensional Bermudan options, with applications in portfolio risk management. Large portfolios of a mixture of multi-asset European and Bermudan derivatives are cast into the framework of discretely reflected BSDEs. This system is discretized by the One Step Malliavin scheme (Negyesi et al. [“2024, 2025”]) of discretely reflected Markovian BSDEs, which involves a $Γ$ process, corresponding to second-order sensitivities of the associated option prices. The discretized system is solved by a neural network regression Monte Carlo method, efficiently for a large number of underlyings. The resulting option Deltas and Gammas are used to discretely rebalance the corresponding replicating strategies. Numerical experiments are presented on both high-dimensional basket options and large portfolios consisting of multiple options with varying early exercise rights, moneyness and volatility. These examples demonstrate the robustness and accuracy of the method up to $100$ risk factors. The resulting hedging strategies significantly outperform benchmark methods both in the case of standard delta- and delta-gamma hedging. ...

A pure dual approach for hedging Bermudan options ArXiv ID: 2404.18761 “View on arXiv” Authors: Unknown Abstract This paper develops a new dual approach to compute the hedging portfolio of a Bermudan option and its initial value. It gives a “purely dual” algorithm following the spirit of Rogers (2010) in the sense that it only relies on the dual pricing formula. The key is to rewrite the dual formula as an excess reward representation and to combine it with a strict convexification technique. The hedging strategy is then obtained by using a Monte Carlo method, solving backward a sequence of least square problems. We show convergence results for our algorithm and test it on many different Bermudan options. Beyond giving directly the hedging portfolio, the strength of the algorithm is to assess both the relevance of including financial instruments in the hedging portfolio and the effect of the rebalancing frequency. ...

Optimizing Neural Networks for Bermudan Option Pricing: Convergence Acceleration, Future Exposure Evaluation and Interpolation in Counterparty Credit Risk ArXiv ID: 2402.15936 “View on arXiv” Authors: Unknown Abstract This paper presents a Monte-Carlo-based artificial neural network framework for pricing Bermudan options, offering several notable advantages. These advantages encompass the efficient static hedging of the target Bermudan option and the effective generation of exposure profiles for risk management. We also introduce a novel optimisation algorithm designed to expedite the convergence of the neural network framework proposed by Lokeshwar et al. (2022) supported by a comprehensive error convergence analysis. We conduct an extensive comparative analysis of the Present Value (PV) distribution under Markovian and no-arbitrage assumptions. We compare the proposed neural network model in conjunction with the approach initially introduced by Longstaff and Schwartz (2001) and benchmark it against the COS model, the pricing model pioneered by Fang and Oosterlee (2009), across all Bermudan exercise time points. Additionally, we evaluate exposure profiles, including Expected Exposure and Potential Future Exposure, generated by our proposed model and the Longstaff-Schwartz model, comparing them against the COS model. We also derive exposure profiles at finer non-standard grid points or risk horizons using the proposed approach, juxtaposed with the Longstaff Schwartz method with linear interpolation and benchmark against the COS method. In addition, we explore the effectiveness of various interpolation schemes within the context of the Longstaff-Schwartz method for generating exposures at finer grid horizons. ...

Efficient option pricing in the rough Heston model using weak simulation schemes ArXiv ID: 2310.04146 “View on arXiv” Authors: Unknown Abstract We provide an efficient and accurate simulation scheme for the rough Heston model in the standard ($H>0$) as well as the hyper-rough regime ($H > -1/2$). The scheme is based on low-dimensional Markovian approximations of the rough Heston process derived in [“Bayer and Breneis, arXiv:2309.07023”], and provides weak approximation to the rough Heston process. Numerical experiments show that the new scheme exhibits second order weak convergence, while the computational cost increases linear with respect to the number of time steps. In comparison, existing schemes based on discretization of the underlying stochastic Volterra integrals such as Gatheral’s HQE scheme show a quadratic dependence of the computational cost. Extensive numerical tests for standard and path-dependent European options and Bermudan options show the method’s accuracy and efficiency. ...

Finite-Difference Solution Ansatz approach in Least-Squares Monte Carlo ArXiv ID: 2305.09166 “View on arXiv” Authors: Unknown Abstract This article presents a simple but effective and efficient approach to improve the accuracy and stability of Least-Squares Monte Carlo. The key idea is to construct the ansatz of conditional expected continuation payoff using the finite-difference solution from one dimension, to be used in linear regression. This approach bridges between solving backward partial differential equations and Monte Carlo simulation, aiming at achieving the best of both worlds. In a general setting encompassing both local and stochastic volatility models, the ansatz is proven to act as a control variate, reducing the mean squared error, thereby leading to a reduction of the final pricing error. We illustrate the technique with realistic examples including Bermudan options, worst of issuer callable notes and expected positive exposure on European options under valuation adjustments. ...