Valuing American options and Flexible Forwards contracts in time-dependent models

2026-06-25

AI Summary

This study addresses the efficient pricing of American options and flexible forwards under a time-inhomogeneous Heston model, tackling challenging scenarios such as volatility skew in foreign exchange rates and low Feller ratios. The authors extend the integral equation decomposition approach—previously limited to constant coefficients—to the time-varying setting, formulating a Volterra equation that characterizes the early exercise surface. They combine recursive matrix Riccati solutions with spectral methods to evaluate conditional expectations. A damped Sinc (DSINC) local basis spectral method is proposed, achieving pricing within 1–2 seconds while improving accuracy by approximately twelvefold compared to the classical COS method and significantly outperforming refined finite difference benchmarks. The work further uncovers the highly nonlinear dependence of the early exercise surface on variance, offering a more robust and efficient numerical framework for pricing complex derivatives.

Abstract

A flexible forward (FF) is a customized FX hedging instrument that guarantees a fixed exchange rate while letting the holder choose the delivery date within a pre-agreed window. It is therefore an American-style option on timing, and its valuation must respect the volatility skew of the underlying currency pair. We price FF contracts (and, more generally, American options) under a time-inhomogeneous Heston model which captures the forward-skew term structure while preserving analytical tractability through a recursive (matrix) Riccati solution for the joint characteristic function. Extending the integral-equation (decomposition) approach to time-dependent coefficients, we derive a Volterra equation characterizing the early-exercise surface. The expectation in the decomposition formula is evaluated by two complementary spectral methods: a double cosine (COS) expansion of the transition density, and a damped-Sinc (DSINC) local-basis scheme that is more accurate and stays robust when a low Feller ratio or large vol-of-vol induces Gibbs oscillations in the COS series. Benchmarked against a penalty-iteration MCS-ADI finite-difference solver, both methods price a contract in about 1-2 seconds, roughly an order of magnitude faster than the finest finite-difference grid, while DSINC improves median accuracy over COS by about a factor of twelve. The experiments also show that the early-exercise surface is a substantially nonlinear function of the variance, contrary to the linear-in-variance approximation common in earlier work.

Problem

• American options
• Flexible Forwards
• time-dependent models
• volatility skew
• early-exercise surface

Innovation

• time-inhomogeneous Heston model
• Volterra integral equation
• early-exercise surface
• spectral methods
• DSINC