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Stochastic Local Volatility

Carol Alexander () and Leonardo Nogueira
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Leonardo Nogueira: ICMA Centre, University of Reading and Banco Central do Brasil

ICMA Centre Discussion Papers in Finance from Henley Business School, University of Reading

Abstract: There are two unique volatility surfaces associated with any arbitrage-free set of standard European option prices, the implied volatility surface and the local volatility surface. Several papers have discussed the stochastic differential equations for implied volatilities that are consistent with these option prices but the static and dynamic no-arbitrage conditions are complex, mainly due to the large (or even infinite) dimensions of the state probability space. These no-arbitrage conditions are also instrument-specific and have been specified for some simple classes of options. However, the problem is easier to resolve when we specify stochastic differential equations for local volatilities instead. And the option prices and hedge ratios that are obtained by making local volatility stochastic are identical to those obtained by making instantaneous volatility or implied volatility stochastic. After proving that there is a one-to-one correspondence between the stochastic implied volatility and stochastic local volatility approaches, we derive a simple dynamic no-arbitrage condition for the stochastic local volatility model that is model-specific. The condition is very easy to check in local volatility models having only a few stochastic parameters.

Keywords: Local volatility; stochastic volatility; unified theory of volatility; local volatility dynamics (search for similar items in EconPapers)
JEL-codes: C16 G13 (search for similar items in EconPapers)
Pages: 23 pages
Date: 2004-09, Revised 2008-03
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