 # Stochastic Volatility Models and Kelvin Waves

#### Stochastic Volatility Models and Kelvin Waves

Stochastic Volatility Models and Kelvin Waves We use stochastic volatility models to describe the evolution of the asset price, its instantaneous volatility, and its realized volatility. In particular, we concentrate on the Stein-Stein model (SSM) (1991) for the stochastic asset volatility and the Heston model (HM) (1993) for the stochastic asset variance. By construction, the volatility is not sign-definite in SSM and is non-negative in HM.

It is well-known that both models produce closed-form expressions for the prices of vanilla options via the Lewis-Lipton formula.

However, the numerical pricing of exotic options by means of the Finite Difference and Monte Carlo methods is much more complex for HM than for SSM. Until now, this complexity was considered to be an acceptable price to pay for ensuring that the asset volatility is non-negative.

#### We argue that having negative stochastic volatility is a psychological rather than financial or mathematical problem. In addition, advocate using SSM rather than HM in most applications.

We extend SSM by adding volatility jumps and obtain a closed-form expression for the density of the asset price and its realized volatility. We also show that the current method of choice for solving pricing problems with stochastic volatility (via the affine ansatz for the Fourier-transformed density function) can be traced back to the Kelvin method designed in the nineteenth century for studying wave motion problems arising in fluid dynamics.

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 assuming a 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 riskmanaging 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 the uncertainty of the asset prices drives all the uncertainty in the volatility dynamics.

#### Trading experience suggests that although most of these models can explain today’s market data for simple options almost perfectly. Moreover, they tend to have poor predictive and explanatory powers. Lastly, are not satisfactory for risk-managing of complex trades.

Using a stochastic volatility model in practice consists of two major steps. Firstly, adjusting model parameters to fit vanilla options prices (model calibration). Secondly, applying the calibrated model to compute the value and risk parameters of complex trades.

The first step is important. Because we want 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 thevalues 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 closedform 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.