Calibration of a Hybrid Local-Stochastic Volatility Stochastic Rates Model with a Control Variate Particle Method

Abstract

We propose a novel and generic calibration technique for four-factor foreign-exchange hybrid local-stochastic volatility models (LSV) with stochastic short rates. We build upon the particle method introduced by Guyon and Henry-Labordère [Nonlinear Option Pricing, Chapter 11, Chapman and Hall, 2013] and combine it with new variance reduction techniques in order to accelerate convergence. We use control variates derived from: a calibrated pure local volatility model; a two-factor Heston-type LSV model (both with deterministic rates); the stochastic (CIR) short rates. The method can be applied to a large class of hybrid LSV models and is not restricted to our particular choice of the diffusion. However, we address in the paper some specific difficulties arising from the Heston model, notably by a new PDE formulation and finite element solution to bypass the singularities of the density when zero is attainable by the variance. The calibration procedure is performed on market data for the EUR-USD currency pair and has a comparable run-time to the PDE calibration of a two-factor LSV model alone.

Strong Convergence Rates for Euler Approximations to a Class of Stochastic Path-Dependent Volatility Models

“…These values are consistent with empirical observations in equity and FX markets and are close to the calibrated values in Table 2 in [11] and Table 1 in [13].…”

“…In case of no running maximum component, the leverage function σ that is consistent with vanilla prices is given by the ratio between a calibrated Dupire local volatility and the square-root of the conditional expectation of the squared stochastic volatility [23,24,50]. In practice, the leverage function is defined on a grid of points (20 points per year in time and 30 points in space usually suffice for an acceptable calibration error [11,24]), interpolated flat-forward in time and using cubic splines in spot, and extrapolated flat outside an interval. Hence, there is no significant loss of generality from a practical point of view in making these two assumptions.…”

Not so Particular about Calibration: Smile Problem Resolved

“…Thus, Vasicek parameters: κ = 1, σ = 0.5%, r 0 = 1.5%. The conditional and exact distribution methods are given by equations (5) and (9) respectively. G-HL Particle method is described in (3).…”

Convergence of an Euler Scheme for a Hybrid Stochastic-Local Volatility Model with Stochastic Rates in Foreign Exchange Markets

“…Hence, we assume that σ is bounded, Hölder continuous in t and Lipschitz in S t . According to [11], for the leverage function to be consistent with call and put prices, it has to be given by the formula (2.9), which depends on the calibrated Dupire local volatility. In practice, the local volatility function usually arises as the interpolation of discrete values obtained from a discretized version of Dupire's formula.…”

“…There has also been an increasing interest in the calibration to vanilla options of models with stochastic and local volatility and stochastic interest rate dynamics. For instance, Guyon and Labordére [20] examined Monte Carlo-based calibration methods for a 3-factor SLV equity model with a stochastic domestic rate and discrete dividends, while Cozma, Mariapragassam and Reisinger [11] and Deelstra and Rayee [13] examined 4-factor hybrid SLV models with stochastic domestic and foreign rates and proposed different approaches to calibrate the leverage function. Hambly, Mariapragassam and Reisinger [21] took a different path and discussed the calibration of a 2-factor SLV model to barriers.…”