Empirical studies indicate that the asset volatility is a random process and, in general, cannot be described by a single number. Yet, option pricing models assuminga deterministic asset volatility originated by Black-Scholes (1973) and Merton (1973)(BSMM) are very popular in practice because of their relative simplicity. In principle, vanilla option traders understand the limitations of BSMM and know how to adjust the model and manage their risks appropriately. However, more complex financial products, such as forward-starting options, or options on the realized variance of an asset, derive their value from the asset volatility rather than its price, so that for pricing and risk-managing of such products traders tend to use stochastic volatility models.
We note that there exists a large class of the so-called local volatility models (LVM), originated by Cox (1975) in a parametric form (the constant elasticity of variance model) and by Dupire (1994) in a non-parametric form (the local volatility surface model), which specify that the asset volatility depends only on the asset price and time, so that all the uncertainty in the volatility dynamics is driven by the uncertainty of the asset price. Trading experience suggests that although most of these models can explain today’s market data for simple options almost perfectly, they tend to have poor predictive and explanatory powers, and are not satisfactory for risk-managing of complex trades.
Using a stochastic volatility model in practice consists of two major steps: first, adjusting model parameters to fit vanilla options prices (model calibration), and, second, applying the calibrated model to compute the value and risk parameters of complex trades. The first step is important becausewewant to express risks of complex trades in terms of risks of liquid vanilla options, which we will subsequently use to hedge against these risks. As a result, it is important that our model is consistent with the values of these liquid vanilla options. The second step is typically achieved through numerical solution of the corresponding partial differential equations (PDEs), or Monte Carlo (MC) simulations of the corresponding stochastic differential equations (SDEs). It is common for academics to concentrate only on the first part by deriving closed-form formulas for vanilla option values and using them to estimate model parameters. However, for stochastic volatility models to be useful in practice, we have to formulate the pricing problem in either PDE or MC (or both) frameworks and to ensure that the chosen stochastic volatility model allows robust implementation of the appropriate numerical algorithms.
