Unique Option Pricing Measure with neither Dynamic Hedging nor Complete Markets

• DOI: http://doi.org/10.1111/eufm.12055
• Date: 01 March 2015
• Published date: 01 March 2015
• Author: Nassim Nicholas Taleb

Unique Option Pricing Measure with

neither Dynamic Hedging nor

Complete Markets

Nassim Nicholas Taleb

School of Engineering, NYU, & Former Option Trader

E-mail: nnt1@nyu.edu

Abstract

Proof that under simple assumptions, such as constraints of Put‐Call Parity, the

probability measure for the valuation of a European option has the mean derived

from the forward price which can, but does not have to be the risk‐neutral one,

under any general probability distribution, bypassing the Black‐Scholes‐Merton

dynamic hedging argument, and without the requirement of complete markets and

other strong assumptions. We confirm that the heuristics used by traders for

centuries are both more robust, more consistent, and more rigorous than held in the

economics literature. We also show that options can be priced using infinite

variance (finite mean) distributions.

Keywords: option theory, derivatives, risk management, hedging

JEL Classification: G13, D81, A14

1. Background

Option valuations methodologies have been used by traders for centuries, in an effective

way (Haug and Taleb, 2010). In addition, valuations by expectation of terminal payoff

forces the mean of the probability distribution used for option prices be be that of the

forward, thanks to Put‐Call Parity and, should the forward be risk‐neutrally priced, so will

the option be. The Black Scholes argument (Black and Scholes, 1973; Merton, 1973) is

held to allow risk‐neutral option pricing thanks to dynamic hedging, as the option

becomes redundant (since its payoff can be built as a linear combination of cash and the

underlying asset dynamically revised through time). This is a puzzle, since: 1) Dynamic

Hedging is not operationally feasible in financial markets owing to the dominance of

portfolio changes resulting from jumps, 2) The dynamic hedging argument doesn’t stand

mathematically under fat tails; it requires a very specific‘Black Scholes world’with many

impossible assumptions, one of which requires finite quadratic variations, 3) Traders use

the same Black‐Scholes ‘risk neutral argument’for the valuation of options on assets that

Peter Carr, Marco Avellaneda, Hélyette Geman, Raphael Douady, Gur Huberman, Espen

Haug, and Hossein Kazemi.

European Financial Management, Vol. 21, No. 2, 2015, 228–235

doi: 10.1111/eufm.12055

© 2015 John Wiley & Sons Ltd