This is not the case if the volatility is modeled to have a random component of

# This is not the case if the volatility is modeled to

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model. This is not the case if the volatility is modeled to have a random component of its own, as in the stochastic volatility models of the next section. One disadvantage of the local volatility suface approach is that volatility and stock price changes are now perfectly correlated (positively or negatively). Empirical studies suggest that when volatility goes up, prices tend to go down and vice versa , but not that there is a perfect ( 1) inverse correlation. Short-time tight fit vs. long-time rough fit There are many competing ways to estimate the volatility surface σ ( t, S ) from traded option prices. Where the problem lies is in the stability of the fits over time: with new data a week later, the fits are often completely different, even though, given the large degree of freedom, they are very good fits to the current data. There is a tradeoff between a tight fit over a short time vs. a rough fit over a longer time, and this stability of parameters problem is an important criterion with which to assess any models. 14.4 Stochastic Volatility Models In “pure” stochastic volatility models (as opposed to local volatility surfaces), the asset price { S t , t 0 } satisfies the SDE dS t = μS t dt + σ t S t dW t , where { σ t , t 0 } is called the volatility process. It must satisfy some regularity conditions for the model to be well-defined (at the very least, it must be positive), but it does not have to be an Itˆo process: it can be a jump process, a Markov chain, etc. Unlike the local volatility surface models, the volatility process is not perfectly correlated with the Brownian motion W . Therefore, volatility is modeled to have an independent random component of its own. These models are continuous-time versions of ARCH and GARCH models developed to fit a wide range of time-series, from inflation to stock prices, by Engle and Granger, for which they won the 2003 Nobel Prize in Economics. The CH stands for conditional heteroskedasticity, which roughly means the same as random volatility. Typically, volatility is taken to be an Itˆo process, meaning it is described by a SDE driven by a second Brownian motion. Within this framework, we have PDE/tree methods available for computation. After that, we want a model in which volatility is positive. One feature that most models seem to like is mean-reversion. We need to explain carefully what we mean by mean-reversion because anything that is random and has a mean must be coming back towards its mean on occasion. The term mean-reverting really refers to the characteristic (typical) time it takes for a process to get back to its long run mean-level (assuming it has one). The AR in ARCH and GARCH stand for auto-regressive, which is another way to describe mean reversion. (The G stands for generalized). From a stochastic volatility modeling perspective, mean-reverting refers to a linear pull- back term in the drift of the volatility process itself, or in the drift of some (underlying) process of which volatility is a function. Let us denote σ t = f ( Y t ) where f is some positive function. Then mean-reverting stochastic volatility means that the SDE for Y looks like dY t = α ( m Y t ) dt +

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• Fall '11
• COULON
• Stochastic volatility, Smile Curve

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