How to Start a Stochastic Volatility Process

Abstract

The standard approach to using a stochastic volatility (SV) model for valuation of a derivative is to assume one deterministic level of volatility at time zero and define how volatility evolves over time as a random process. If this process is continuous, after a short time the stochastic volatility will still be quite concentrated around the initial level. Consequently, the generated implied volatility for very short expiries is still quite flat. As the example of mixture log-normal models show, at least two different levels of volatility are needed to generate convexity of the implied volatility smile (D. Brigo and Rapisarda 2002). To fit also more pronounced short tenor convexity, either a random start of the volatility process or high levels of mean reversion speed and volatility of volatility (generating a more jump-like process) have been proposed. (See Jacquier and Shi (2016) and further references therein for the first approach and Mechkov (2015a) for the second.) We illustrate these approaches in the context of calibrating time-dependent SV parameters. We then discuss the fitting to the implied volatility surface using a stochastic-local volatility (SLV) processes. Finally, we look at the impact on the mixing factor when mixing stochastic and local volatility models, a popular approach in the FX asset class.