Valuing American options and Flexible Forwards contracts in time-dependent models
For practitioners pricing FX hedging instruments with early-exercise features, the paper provides fast and accurate methods that capture volatility skew and term structure, while revealing that the early-exercise surface is nonlinear in variance.
The paper prices American options and flexible forwards under a time-inhomogeneous Heston model using a Volterra integral equation and two spectral methods (COS and DSINC). DSINC achieves about 12x better median accuracy than COS, and both methods run in 1-2 seconds, roughly 10x faster than a finite-difference solver.
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.