Abstract
Options can be compared to forward contracts where one of the counterparties pays a premium for the option to settle or not to settle. Options have become popular both on the OTC and on the organized exchange markets, but their valuation is more complex than in the case of forwards. It requires the underlying asset price volatility as a new input into the valuation models that have, at the same time, become a new market variable. We will explain how value options in the relatively elementary framework of binomial trees and in the theoretically more advanced context of stochastic asset price modeling. The last section will look at the issue of option portfolio hedging using the concept of so-called Greek letters.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Note that \( {e}^{\sigma \sqrt{\delta t}}=1+\sigma \sqrt{\delta t} \), for a small δt. In fact, the factors \( u=1+\sigma \sqrt{\delta t} \) and \( d=1-\sigma \sqrt{\delta t} \) could be alternatively used with the same asymptotic results.
- 2.
Note that the recombining property will unfortunately be violated in the step following the dividend payment even if the multipliers u and v are constant. The recombining property will hold in the subsequent steps but the number of nodes will double.
- 3.
If ϵ is an infinitesimal, then \( \sqrt{\varepsilon} \) is infinitely larger, yet still infinitesimal. Consequently, there are infinite multiples of δt that are still infinitesimal.
- 4.
According to Jensen’s inequality, if X is a random variable and φ(X) a convex function, then E[φ(X)] ≥ φ(E[X]).
- 5.
The mean E[ST] and variance \( \operatorname{var}\left[{S}_T\right]=E\left[{S}_T^2\right]-E{\left[{S}_T\right]}^2 \) are obtained simply by integrating the lognormal density function multiplied by ST and \( {S}_T^2 \). We will perform a similar integration when proving the Black-Scholes formula.
- 6.
The cumulative distribution function N(x) = Pr [X ≤ x] where X~N(0, 1) can be evaluated in Excel as NORMSDIST while the inverse function N−1(α) is evaluated as NORMSINV. The cumulative distribution function and its inverse are also often denoted as Φ and Φ−1.
- 7.
The other technical boundary conditions are f(S, t) → 0 as S → 0 and f(S, t)/S → 1 as S → ∞ for t ∈ [0, T].
- 8.
Some dealers managing large and complex portfolios monitor even the third-order derivative of the portfolio value with respect to the underlying price, which is called the speed.
- 9.
Note that there is no real Greek letter called “vega.” The name might have been introduced erroneously since the Greek letter v looks like a Latin “Vee.”
References
Albeverio, S., Fenstad, J. E., Hoegh-Krohn, R., & Lindstrom, T. (1986). Nonstandard methods in stochastic analysis and mathematical physics. New York: Dover Publications – Mineola.
Arlt, J., & Arltová, M. (2007). Ekonomické časové řady. Prague: Grada Publishing.
Black, F. (1976, March). The pricing of commodity contracts. Journal of Financial Economics, 3, 167–179.
Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637–659.
Cipra, T. (2008). Finanční ekonometrie. Praha: Ekopress.
Cox, J. C., Ross, S. A., & Rubinstein, M. (1979). Option pricing: A simplified approach. Journal of Financial Economics, 7, 229–264.
Cutland, N. J., Kopp, P. E., & Willinger, W. (1991). A nonstandard approach to option pricing. Mathematical Finance, 1(4), 1–38.
Garman, M. B., & Kohlhagen, S. W. (1983). Foreign currency option values. Journal of International Money and Finance, 2(3), 231–237.
Herzberg, F. (2013). Stochastic calculus with infinitesimals. New York: Springer.
Hull, J. (2018). Options, futures, and other derivatives (10th ed.). Upper Saddle River, NJ: Prentice Hall.
Keisler, H. J. (1976). Foundations of infinitesimal calculus (Vol. 20). Boston: Prindle, Weber & Schmidt.
Keisler, H. J. (2013). Elementary calculus: An infinitesimal approach. Courier Corporation.
Kopp, E., Malczak, J., & Zastawniak, T. (2013). Probability for finance. Cambridge: Cambridge University Press.
Merton, R. C. (1973). Theory of rational option pricing. Bell Journal of Economics and Management Science, 4, 141–183.
Nelson, E. (1977). Internal set theory: A new approach to nonstandard analysis. Bulletin of the American Mathematical Society, 83(6), 1165–1198.
Robinson, A. (1966). Nonstandard analysis. Amsterdam: North-Holland.
Shreve, S. (2004). Stochastic calculus for finance II – Continuous time models. New York: Springer.
Shreve, S. (2005). Stochastic calculus for finance I – The binomial asset pricing model. New York: Springer.
Vopěnka, P. (2010). Calculus infinitesimalis – Pars Prima. OPS-Nymburk.
Vopěnka, P. (2011). Calculus infinitesimalis – Pars Secunda. OPS-Nymburk.
Wilmott, P. (2006). Paul Wilmott on quantitative finance (2nd ed.). New York: Wiley.
Witzany, J. (2008). Construction of equivalent martingale measures with infinitesimals. Charles University, KPMS Preprint, 60, 1–16.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2020 The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Witzany, J. (2020). Option Markets, Valuation, and Hedging. In: Derivatives. Springer Texts in Business and Economics. Springer, Cham. https://doi.org/10.1007/978-3-030-51751-9_4
Download citation
DOI: https://doi.org/10.1007/978-3-030-51751-9_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-51750-2
Online ISBN: 978-3-030-51751-9
eBook Packages: Economics and FinanceEconomics and Finance (R0)