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2023 | OriginalPaper | Buchkapitel

12. Volatility Arbitrage and Model Calibration

verfasst von : Ilia Bouchouev

Erschienen in: Virtual Barrels

Verlag: Springer Nature Switzerland

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Abstract

This chapter focuses on the important problem of model calibration. We present the bootstrapping method for calibrating volatility time-dependency and back out market-implied probability distribution from option prices. We then outline a more difficult problem of reconstructing the underlying diffusion process. Some readers may find it interesting that this problem, known as the inverse problem of option pricing, in its general case presents a rare example of an unsolved mathematical problem.

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Fußnoten
1
Another example of a discrete volatility matrix is constructed in Pilipović (2007), where instead of imposing the exponential structure on local volatilities, an additional constraint is set by equating long-term local volatilities to historical realized volatilities. Our preference is to avoid explicitly tying market-implied volatility matrix to realized volatilities due to hedging imbalances and the presence of the volatility risk premium documented in Chap. 9.
 
2
See Bachelier (1900). Bachelier’s brief statement of this formula is often overlooked in the literature, where the formula is typically attributed to a more comprehensive work on this topic by Breeden and Litzenberger (1978).
 
3
The equation was presented at several conferences in 1993 and subsequently published in Dupire (1994).
 
4
This inverse problem has been extensively studied in academic literature. See, among many others, Avellaneda et al. (1997), Bouchouev and Isakov (1997, 1999), Dempster and Richards (2000), Carr and Madan (2001), Bouchouev et al. (2002), Chiarella et al. (2003), Alexander (2008), Lipton and Sepp (2011), and references therein.
 
5
For a more detailed discussion of this topic, we refer to Gatheral (2006), and Derman and Miller (2016).
 
6
For further reading on jump-diffusions and stochastic volatility models, we refer to Rebonato (2004), Javaheri (2005), Gatheral (2006), and Derman and Miller (2016).
 
Literatur
Zurück zum Zitat Alexander, C. (2008). Market risk analysis, Vol. III: Pricing, hedging and trading financial instruments. Wiley. Alexander, C. (2008). Market risk analysis, Vol. III: Pricing, hedging and trading financial instruments. Wiley.
Zurück zum Zitat Avellaneda, M., Friedman, C., Holmes, R., & Samperi, D. (1997). Calibrating volatility surfaces via relative entropy minimization. Applied Mathematical Finance, 4(1), 37–64.CrossRefMATH Avellaneda, M., Friedman, C., Holmes, R., & Samperi, D. (1997). Calibrating volatility surfaces via relative entropy minimization. Applied Mathematical Finance, 4(1), 37–64.CrossRefMATH
Zurück zum Zitat Bachelier, L. (1900). Théorie de la Spéculation, Annales scientifiques de l’Êcole Normale Supêrieure, Serie 3 17, 21–86. Bachelier, L. (1900). Théorie de la Spéculation, Annales scientifiques de l’Êcole Normale Supêrieure, Serie 3 17, 21–86.
Zurück zum Zitat Bouchouev, I., & Isakov, V. (1999). Uniqueness, stability and numerical methods for the inverse problem that arises in financial markets. Inverse Problems, 15(3), R95–R116.MathSciNetCrossRefMATH Bouchouev, I., & Isakov, V. (1999). Uniqueness, stability and numerical methods for the inverse problem that arises in financial markets. Inverse Problems, 15(3), R95–R116.MathSciNetCrossRefMATH
Zurück zum Zitat Bouchouev, I., Isakov, V., & Valdivia, N. (2002). Recovery of volatility coefficient by linearization. Quantitative Finance, 2(4), 257–263.MathSciNetCrossRefMATH Bouchouev, I., Isakov, V., & Valdivia, N. (2002). Recovery of volatility coefficient by linearization. Quantitative Finance, 2(4), 257–263.MathSciNetCrossRefMATH
Zurück zum Zitat Breeden, D. T., & Litzenberger, R. H. (1978). Prices of state contingent claims implicit in option prices. Journal of Business, 51(4), 621–651.CrossRef Breeden, D. T., & Litzenberger, R. H. (1978). Prices of state contingent claims implicit in option prices. Journal of Business, 51(4), 621–651.CrossRef
Zurück zum Zitat Carr, P., & Madan, D. (2001). Determining volatility surfaces and option values from an implied volatility smile. In M. Avellaneda (Ed.), Quantitative analysis in financial markets (Vol. II, pp. 163–191). World Scientific.CrossRef Carr, P., & Madan, D. (2001). Determining volatility surfaces and option values from an implied volatility smile. In M. Avellaneda (Ed.), Quantitative analysis in financial markets (Vol. II, pp. 163–191). World Scientific.CrossRef
Zurück zum Zitat Chiarella, C., Craddock, M., & El-Hassan, N. (2003). An implementation of Bouchouev’s method for short time calibration of option pricing models. Computational Economics, 22, 113–138.CrossRefMATH Chiarella, C., Craddock, M., & El-Hassan, N. (2003). An implementation of Bouchouev’s method for short time calibration of option pricing models. Computational Economics, 22, 113–138.CrossRefMATH
Zurück zum Zitat Dempster, M. A. H., & Richards, D. G. (2000). Pricing American options fitting the smile. Mathematical Finance, 10(2), 157–177.MathSciNetCrossRefMATH Dempster, M. A. H., & Richards, D. G. (2000). Pricing American options fitting the smile. Mathematical Finance, 10(2), 157–177.MathSciNetCrossRefMATH
Zurück zum Zitat Dupire, B. (1994). Pricing with a smile. Risk, 7(1), 18–20. Dupire, B. (1994). Pricing with a smile. Risk, 7(1), 18–20.
Zurück zum Zitat Gatheral, J. (2006). The volatility surface: A Practitioner’s guide. Wiley. Gatheral, J. (2006). The volatility surface: A Practitioner’s guide. Wiley.
Zurück zum Zitat Javaheri, A. (2005). Inside volatility arbitrage: The secrets of skewness. Wiley. Javaheri, A. (2005). Inside volatility arbitrage: The secrets of skewness. Wiley.
Zurück zum Zitat Lipton, A., & Sepp, A. (2011, October). Filling the gaps. Risk, 24(10), 78–83. Lipton, A., & Sepp, A. (2011, October). Filling the gaps. Risk, 24(10), 78–83.
Zurück zum Zitat Pilipović, D. (2007). Energy risk: Valuing and managing energy derivatives. McGraw-Hill. Pilipović, D. (2007). Energy risk: Valuing and managing energy derivatives. McGraw-Hill.
Zurück zum Zitat 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
Metadaten
Titel
Volatility Arbitrage and Model Calibration
verfasst von
Ilia Bouchouev
Copyright-Jahr
2023
DOI
https://doi.org/10.1007/978-3-031-36151-7_12