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2023 | OriginalPaper | Chapter

10. Volatility Smile Trading

Author : Ilia Bouchouev

Published in: Virtual Barrels

Publisher: Springer Nature Switzerland

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Abstract

We study the problem of the volatility smile and the strategy of vega trading. We demonstrate limitations of conventional paradigms to the oil market and use a more flexible framework of diffusion processes. We apply the technique of perturbation methods to develop a novel quadratic normal model. This model corrects the Bachelier formula for skewness and kurtosis with the three model parameters mapped to the three primary option benchmarks in the oil market.

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Footnotes
1
We only provide a simple illustration of this topic. For a more advanced exposition, we refer to Rebonato (2004), Gatheral (2006), and Derman and Miller (2016).
 
2
For brevity, we omit the rigorous derivation of this approximation. It is again based on the Taylor series expansion for both formulas, but with more cumbersome mathematical details which are less essential for our goals. We refer to Schachermayer and Teichmann (2008) and Grunspan (2011) for a more detailed comparison of the two formulas, and more advanced approximation formulas.
 
3
Recall that we are using zero interest rate. In the general case, deltas must be discounted.
 
4
The method of linearization and the related parametrix method for the diffusion equation were originally applied to option pricing in Bouchouev (1998, 2000).
 
5
Quadratic parametrizations of the local volatility function have been considered by several authors, including Ingersoll (1997), Zühlsdorff (2001), Andersen (2011), and Carr et al. (2013), but resulting option pricing formulas are considerably more complicated.
 
6
The use of three parameters that correspond to the first three moments of the price distributions is often used by market-makers to characterize the dynamics of the volatility smile in financial markets. For example, Castagna and Mercurio (2007) developed a popular vanna-volga model for options in foreign exchange markets.
 
Literature
go back to reference Bouchouev, I. (1998). Derivatives valuation for general diffusion processes, Proceedings of the Annual Conference of the International Association of Financial Engineers, New York, USA, pp. 91–104. Bouchouev, I. (1998). Derivatives valuation for general diffusion processes, Proceedings of the Annual Conference of the International Association of Financial Engineers, New York, USA, pp. 91–104.
go back to reference Bouchouev, I. (2000, August). Black-Scholes with a smile. Energy and Power Risk Management, pp. 28–29. Bouchouev, I. (2000, August). Black-Scholes with a smile. Energy and Power Risk Management, pp. 28–29.
go back to reference Carr, P., Fisher, T., & Ruf, J. (2013). Why are quadratic normal volatility models analytically tractable? SIAM Journal on Financial Mathematics, 4(1), 185–202.MathSciNetCrossRefMATH Carr, P., Fisher, T., & Ruf, J. (2013). Why are quadratic normal volatility models analytically tractable? SIAM Journal on Financial Mathematics, 4(1), 185–202.MathSciNetCrossRefMATH
go back to reference Castagna, A., & Mercurio, F. (2007). The vanna-volga method for implied volatilities. Risk, 20(1), 106–111. Castagna, A., & Mercurio, F. (2007). The vanna-volga method for implied volatilities. Risk, 20(1), 106–111.
go back to reference Gatheral, J. (2006). The volatility surface: A Practitioner’s guide. Wiley. Gatheral, J. (2006). The volatility surface: A Practitioner’s guide. Wiley.
go back to reference Grunspan, C. (2011). A note on the equivalence between the normal and the lognormal implied volatility: A model free approach, SSRN. Grunspan, C. (2011). A note on the equivalence between the normal and the lognormal implied volatility: A model free approach, SSRN.
go back to reference Ingersoll, J. E., Jr. (1997). Valuing foreign exchange rate derivatives with a bounded exchange process. Review of Derivatives Research, 1, 159–181.CrossRefMATH Ingersoll, J. E., Jr. (1997). Valuing foreign exchange rate derivatives with a bounded exchange process. Review of Derivatives Research, 1, 159–181.CrossRefMATH
go back to reference Rebonato, R. (2004). Volatility and correlation: The perfect hedger and the fox. Wiley.CrossRef Rebonato, R. (2004). Volatility and correlation: The perfect hedger and the fox. Wiley.CrossRef
go back to reference Schachermayer, W., & Teichmann, J. (2008). How close are the option pricing formulas of Bachelier and Black-Merton-Scholes? Mathematical Finance, 18(1), 155–170.MathSciNetCrossRefMATH Schachermayer, W., & Teichmann, J. (2008). How close are the option pricing formulas of Bachelier and Black-Merton-Scholes? Mathematical Finance, 18(1), 155–170.MathSciNetCrossRefMATH
go back to reference Zühlsdorff, C. (2001). The pricing of derivatives on assets with quadratic volatility. Applied Mathematics Finance, 8(4), 235–262.CrossRefMATH Zühlsdorff, C. (2001). The pricing of derivatives on assets with quadratic volatility. Applied Mathematics Finance, 8(4), 235–262.CrossRefMATH
Metadata
Title
Volatility Smile Trading
Author
Ilia Bouchouev
Copyright Year
2023
DOI
https://doi.org/10.1007/978-3-031-36151-7_10