This paper explains how to price a flexible forward, a foreign‑exchange (FX) hedging contract that guarantees a fixed rate but lets the holder pick the delivery date within a preset window. Because that choice of timing is like an American option (the holder can exercise anytime in a window), the contract’s value depends on how volatility varies with strike and time. The authors build a model and numerical tools that respect these market features while remaining fast enough for practical use.

The researchers use a time‑inhomogeneous Heston model. In plain terms, this is a stochastic‑volatility model in which the volatility itself can change randomly and some model parameters can vary with time. They keep the model tractable by writing a recursive, matrix Riccati solution for the joint characteristic function. They also extend a known decomposition method to time‑dependent model coefficients and derive a Volterra integral equation that defines the early‑exercise surface (the rule for when it is optimal to exercise).

To compute prices they evaluate the expectation in the decomposition by two spectral methods. One is a double‑cosine (COS) expansion of the transition density. The other is a damped‑Sinc (DSINC) local basis scheme. DSINC is designed to be more accurate and to avoid Gibbs oscillations that can plague COS when the variance process is difficult (for example when the Feller ratio is low or the volatility‑of‑volatility is large). They benchmark both spectral methods against a penalty‑iteration finite‑difference solver (MCS‑ADI). Both spectral methods price a contract in about 1–2 seconds, roughly ten times faster than the finest finite‑difference grid. DSINC gives a median accuracy improvement over COS by about a factor of twelve in the tests the authors report.

The paper also discusses model choices and calibration. The time‑inhomogeneous Heston model is chosen because it can match the forward skew seen in FX markets and produce realistic term structure across maturities. To avoid overfitting, the authors use a hybrid parameterization: the long‑run variance level θ(t) is modeled smoothly (for example with splines), while correlation ρ(t) and volatility‑of‑volatility ξ(t) are taken piecewise constant over standard traded tenors. For typical flexible forwards with windows up to 1–2 years they assume deterministic interest rates, which they argue makes little difference in pricing at these tenors.