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Local Volatility Surface and No-arbitrage

In: Fitting Local Volatility Analytic and Numerical Approaches in Black-Scholes and Local Variance Gamma Models

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  • Andrey Itkin

Abstract

Suppose that the local volatility function σ(K, T) is somehow known. Also, suppose that given σ(K, T), there exist theoretical option prices that solve the Dupire equation derived in the previous chapter. Finally, suppose that for a given set of option strikes K and maturities T these theoretical prices exactly coincide with the corresponding market prices. Then one can build a local volatility surface by using the values of σ(K, T) at the given set of [K, T], and say that this surface represents the given set of the market quotes. Note, that in practice, we usually solve the inverse problem. This is, given the market quotes, find the local volatility function that being used in the Dupire equation produces the theoretical option prices such that the difference between this set of option prices and the corresponding market prices reaches minimum under some suitable norm…

Suggested Citation

  • Andrey Itkin, 2020. "Local Volatility Surface and No-arbitrage," World Scientific Book Chapters, in: Fitting Local Volatility Analytic and Numerical Approaches in Black-Scholes and Local Variance Gamma Models, chapter 2, pages 13-23, World Scientific Publishing Co. Pte. Ltd..
    • Handle: RePEc:wsi:wschap:9789811212772_0002
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    More about this item

    Keywords

    Local Volatility; Stochastic Clock; Geometric Process; Gamma Distribution; Piecewise Linear Volatility; Variance Gamma Process; Closed Form Solution; Fast Calibration; No-Arbitrage;
    All these keywords.

    JEL classification:

    • C6 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling
    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics
    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques

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