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2023 | OriginalPaper | Buchkapitel

8. Options and Volatilities

verfasst von : Ilia Bouchouev

Erschienen in: Virtual Barrels

Verlag: Springer Nature Switzerland

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Abstract

This chapter summarizes the main building blocks that make up the business of volatility trading. It starts by covering remarkable contributions of Louis Bachelier whose a century-old pricing formula is still being used by oil traders. The classical Black-Scholes-Merton framework of option replication is then presented in a more general setting of diffusion processes. We highlight the importance of distinguishing between three commonly used types of volatility: local volatility, realized volatility, and implied volatility.

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Fußnoten
1
Bachelier (1900).
 
2
Black and Scholes (1973), and Merton (1973).
 
3
Some energy commodities, such as natural gas or power prices, are more susceptible to large short-term spikes where diffusions are often combined with jump processes, but this modeling paradigm is more complex as options can no longer be dynamically replicated with futures. We discuss some limitations of diffusions in Chap. 12.
 
4
Black (1976).
 
5
For the type of options studied by Bachelier, the discounting was not necessary as the option premium was netted against settlement at expiry. As a result, in his original derivation the interest rate was ignored. We will explain shortly why a similar assumption can be made for pricing many oil options. The discounting factor brings in an additional time dependency resulting from the present value of money, rather than from the evolution of the variance.
 
6
Greeks are discussed in many standard derivatives textbooks, such as Alexander (2008) and Hull (2018). For their practical interpretation, see also Leoni (2014).
 
7
Vega is not a letter of a Greek alphabet but it was adopted by option traders for its phonetic similarity.
 
8
See, for example, Wilmott et al. (1993).
 
9
Barone-Adesi and Whaley (1987).
 
10
For example, if daily prices are used then realized volatility is the standard deviation of returns multiplied by \( \sqrt{250} \) given approximately 250 trading days in a year, and for weekly prices, it is multiplied by \( \sqrt{52} \).
 
11
See Samuelson (1965).
 
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 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 Barone-Adesi, G., & Whaley, R. E. (1987). Efficient analytic approximation of American option values. Journal of Finance, 42(2), 301–320.CrossRef Barone-Adesi, G., & Whaley, R. E. (1987). Efficient analytic approximation of American option values. Journal of Finance, 42(2), 301–320.CrossRef
Zurück zum Zitat Black, F. (1976). The pricing of commodity contracts. Journal of Financial Economics, 3(1/2), 167–179.CrossRef Black, F. (1976). The pricing of commodity contracts. Journal of Financial Economics, 3(1/2), 167–179.CrossRef
Zurück zum Zitat Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. The Journal of Political Economy, 81(3), 637–654.MathSciNetCrossRefMATH Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. The Journal of Political Economy, 81(3), 637–654.MathSciNetCrossRefMATH
Zurück zum Zitat Hull, J. C. (2018). Options, futures, and other derivatives (10th ed.). Pearson.MATH Hull, J. C. (2018). Options, futures, and other derivatives (10th ed.). Pearson.MATH
Zurück zum Zitat Leoni, P. (2014). The Greeks and hedging explained. Palgrave Macmillan.CrossRef Leoni, P. (2014). The Greeks and hedging explained. Palgrave Macmillan.CrossRef
Zurück zum Zitat Merton, R. C. (1973). Theory of rational option pricing. The Bell Journal of Economics and Management Science, 4(1), 141–183.MathSciNetCrossRefMATH Merton, R. C. (1973). Theory of rational option pricing. The Bell Journal of Economics and Management Science, 4(1), 141–183.MathSciNetCrossRefMATH
Zurück zum Zitat Samuelson, P. A. (1965). Proof that properly anticipated prices fluctuate randomly. Industrial Management Review, 6(2), 41–49. Samuelson, P. A. (1965). Proof that properly anticipated prices fluctuate randomly. Industrial Management Review, 6(2), 41–49.
Zurück zum Zitat Wilmott, P., Dewynne, J., & Howison, S. (1993). Option pricing: Mathematical models and computation. Oxford Financial Press.MATH Wilmott, P., Dewynne, J., & Howison, S. (1993). Option pricing: Mathematical models and computation. Oxford Financial Press.MATH
Metadaten
Titel
Options and Volatilities
verfasst von
Ilia Bouchouev
Copyright-Jahr
2023
DOI
https://doi.org/10.1007/978-3-031-36151-7_8