Determining volatility surfaces and option values from an implied volatility smile (Q2725579)
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scientific article; zbMATH DE number 1619438
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Determining volatility surfaces and option values from an implied volatility smile |
scientific article; zbMATH DE number 1619438 |
Statements
19 June 2003
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option pricing
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implied volatility
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path-dependence
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Determining volatility surfaces and option values from an implied volatility smile (English)
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From the assumptions of path-independence and no arbitrage, a linear Partial Differential Equation (PDE) governing the Stock Pricing Function (SPF) is derived and solved. By interpreting local volatility as an exotic derivative on the stock, a nonlinear PDE for the volatility function is developed, which is solved analytically using the known SPF. Moreover it is shown that the nonlinear PDE is both a necessary and sufficient condition for prices to be path-independent.NEWLINENEWLINENEWLINEClosed form formulae for option prices and risk-neutral densities are derived, which are consistent with a wide class of local volatility functions.NEWLINENEWLINEFor the entire collection see [Zbl 0958.00019].
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