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2023 | OriginalPaper | Chapter

12. Volatility Arbitrage and Model Calibration

Author : Ilia Bouchouev

Published in: Virtual Barrels

Publisher: 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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Footnotes
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).
 
Literature
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Metadata
Title
Volatility Arbitrage and Model Calibration
Author
Ilia Bouchouev
Copyright Year
2023
DOI
https://doi.org/10.1007/978-3-031-36151-7_12