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

This preview shows page 6 - 7 out of 9 pages.

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 +
Image of page 6

Subscribe to view the full document.

Image of page 7
  • Fall '11
  • Stochastic volatility, Smile Curve

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern

Ask Expert Tutors You can ask 0 bonus questions You can ask 0 questions (0 expire soon) You can ask 0 questions (will expire )
Answers in as fast as 15 minutes