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2020 | OriginalPaper | Chapter

4. Option Markets, Valuation, and Hedging

Author : Jiří Witzany

Published in: Derivatives

Publisher: Springer International Publishing

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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.

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Footnotes
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.”
 
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Metadata
Title
Option Markets, Valuation, and Hedging
Author
Jiří Witzany
Copyright Year
2020
Publisher
Springer International Publishing
Book
Derivatives

Print ISBN: 978-3-030-51750-2

Electronic ISBN: 978-3-030-51751-9

Copyright Year: 2020

DOI
https://doi.org/10.1007/978-3-030-51751-9_4