Basket Implied Volatility from Geodesics

This paper presents a deterministic numerical method to calculate the implied volatility of an option based on multiple assets using a stochastic volatility model. The approach uses Varadhan asymptotics for the diffusion kernel and involves the constrained minimization of a numerically computed geodesic length. We produce implied volatilities for baskets with 10 – 50 elements using the SABR stochastic volatility model and compare results with those obtained from Monte Carlo simulation. We find that the approach is significantly faster than Monte Carlo. We also present results using a domain decomposition preconditioner (additive Schwarz type) which significantly improves performance for the constrained minimization.