Published online by Cambridge University Press: 07 October 2011
The Black–Scholes model makes a number of simplifying assumptions. Among these are that the asset price has independent Gaussian returns and constant volatility. We shall focus here on relaxation of these assumptions by allowing volatility to be randomly varying, for the following reason: a well-known discrepancy between Black–Scholes-predicted European option prices and market-traded options prices, the implied volatility skew, can be accounted for by stochastic volatility models. That is, this modification of the Black–Scholes theory has a posteriori success in one area where the classical model fails.
In fact, modeling volatility as a stochastic process is motivated a priori by empirical studies of stock price returns in which estimated volatility is observed to exhibit “random” characteristics. As we will see, stochastic volatility has the effect of thickening the tails of returns distributions compared with the normal distribution, and therefore modeling more extreme stock price movements. Stochastic volatility modeling is a powerful modification of the Black–Scholes model that describes a much more complex market.
In Chapter 1, we introduced the notation and tools for pricing and hedging derivative securities under a constant volatility lognormal model (1.2). This is the simplest continuous-time example of pricing in a complete market, while pricing in a market with stochastic volatility is an incomplete markets problem, a distinction we shall explain further below, and one which has far-reaching consequences, particularly for the hedging problem and the problem of parameter estimation.
To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Find out more about the Kindle Personal Document Service.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.