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01-07-2015 | Regular Article

Choquet-based European option pricing with stochastic (and fixed) strikes

Authors: Tarik Driouchi, Lenos Trigeorgis, Yongling Gao

Published in: OR Spectrum | Issue 3/2015

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Abstract

This paper develops closed-form formulae for pricing European exchange options involving stochastic (and fixed) strikes under uncertainty based on the Choquet expected utility. We extend the benchmark models of Margrabe (J Financ 33:177–186, 1978) and Merton (Bell J Econ Manag Sci 4:141–183, 1973) using a modified pricing kernel and derive option “Greeks” and other option characteristics in an incomplete market with Choquet ambiguity. The Margrabe–Merton–Black–Scholes (MMBS) classical formulae are seen as special cases (under risk-neutrality) of our generalized framework under ambiguity/ignorance, suggesting that there could be multiple martingale-based option prices in the economy characterizing abnormally uncertain markets. We further show how standard option pricing properties (under risk) should be adjusted to account for investor ambiguity attitudes and heterogeneous beliefs (i.e., ambiguity aversion and seeking) and how such beliefs and attitudes can be extracted from observed option prices.

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Although there have been early contributions by Bachelier (1900), Bronzin (1908), Boness (1964), Samuelson (1965) and Thorp (1969), these three articles are widely seen as having created modern option pricing theory. The Margrabe (1978) framework is a generalization of the Black–Scholes–Merton (1973) models when the strike price is also stochastic.

Our analysis focuses on the derivation of a general model to price European options with stochastic strikes under Choquet ambiguity. The standard fixed strike formulae are shown to be special cases of this general model. The option to exchange one asset for another subsumes the call and put properties of options with fixed exercise prices.

In the context of real options, it is worth mentioning the works of Nishimura and Ozaki (2004, 2007) on job search and optimal investment under ambiguity and Miao and Wang (2011) on early option exercise in infinite-time under pessimism.

Although our formulae are general to capture uncertainty that goes beyond risk (e.g., including multiple-priors with ambiguous drifts), these models omit cases of fundamental uncertainty or complete ignorance (Cohen and Jaffray 1983) or situations where the underlying quantity does not follow some kind of Brownian motion (Theodossiou and Trigeorgis 2003).

Following Kast and Lapied (2010), consider—in discrete-time—any node \(n_t\) at date \(t\text{(0 }\le t\le T)\) and its possible up or down successors next period \(n_{t+1}^u \)and \(n_{t+1}^d \) (after an “up” or “down” movement, respectively) in a dynamic (non-recombining) binomial tree. The Choquet random walk is defined by a conditional capacity \(c\) which has the same magnitude of an “up” or “down” move from one period to the next such that: \(\frac{w( {n_{t+1}^u /n_t })}{w( {n_{t+1}^d /n_t })}=c, 0<c<1 \). The probability of up or down moves in discrete-time within this symmetric binomial tree is replaced by weighting function \(w\). Extended to continuous-time, this random walk converges to the Choquet Brownian motion used in our option pricing analysis. This ambiguous Brownian motion specification subsumes the standard probabilistic case.

For detailed proofs of the properties of this random walk and how it satisfies the Choquet expectation and dynamic consistency conditions see Chateauneuf et al. (2001), Kast and Lapied (2010) and Kast et al. (2014), in particular Propositions 3 and 6 in the latter.

A convex capacity on a finite set of states of nature \(S\) is a real-valued function \(w\) on the subsets of \(S\) such that:

\(\left\{ \begin{array}{l} {A\subseteq B\Rightarrow w( A)\le w( B);} \\ {w( \emptyset )=0,w( S)=1} \\ {w( A)+w( B)\le w( {A\cup B})+w( {A\cap B}),for\, all \,events\, A,B\in S} \\ \end{array}\right. \)

Contrary to the singleton-based GBM specification, \(\mu _{0 } \) and \(\sigma _0 \) are here influenced by capacity variable \(c\) and ambiguous parameters \(m_i \) and \(s_i \) which are specific to each investor or group of investors.

As perfect hedging is not feasible when states of the world are not known, we consider \(mg( {\xi , S})\mathrm{d}t+( {s-1})g( {\xi , S})\mathrm{d}Z=0, \)otherwise we are no longer in an ambiguous environment (if perfect hedging were feasible, financial crises could be averted). This market incompleteness feature, where uncertain sentiment cannot be fully “hedged”, characterizes periods of severe uncertainty.

As an example, the Black–Scholes call option formula under multiple-priors (i.e., ambiguous drift but objective volatility) can be formulated as: \( C{'}_0 =S_0 e^{\varepsilon ^{\prime }T}N( {\frac{\ln ( {\frac{S_0 }{X}})+( {r^{\prime }+\varepsilon ^{\prime }+0.5( \sigma )^2})T}{\sigma \sqrt{T} }})-Xe^{-r^{\prime }T}N( {\frac{\ln ( {\frac{S_0 }{X}})+( {r^{\prime }+\varepsilon ^{\prime }-0.5( \sigma )^2})T}{\sigma \sqrt{T} }})\), where \( r^{\prime }=r+m\frac{\left[ {r-( {\mu +m\sigma })} \right] }{\sigma }\) and \( \varepsilon ^{\prime }=\frac{m\left[ {( {\mu +m\sigma })-r} \right] }{\sigma }\) with \(-1 < m <\) 1.

It is important to clarify that option value is not necessarily deterministic at extreme \(c\) values (e.g., \(c\) near 0 or 1) as both the discount rate \(r^{\prime \prime }\) and ambiguity multiplier \( \varepsilon ''\) will behave unpredictably under such conditions (e.g., cases of negative interest rates).

For simplicity and ease of formulation, we consider \( \sigma ^{\prime }=0\).

Different ambiguity estimates are obtained for calls and puts because of differences in uncertainty perceptions regarding call and put payoffs. Call (put) option holders are concerned with upside (downside) prospects. \(c\) is extracted from call (put) option prices after inverting (the put equivalent of) Eq. (24).

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Title: Choquet-based European option pricing with stochastic (and fixed) strikes
Authors: Tarik Driouchi
Lenos Trigeorgis
Yongling Gao
Publication date: 01-07-2015
Publisher: Springer Berlin Heidelberg
Published in: OR Spectrum / Issue 3/2015
Print ISSN: 0171-6468
Electronic ISSN: 1436-6304
DOI: https://doi.org/10.1007/s00291-014-0378-3

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