Exchange Options

Jamshidian, Farshid (2007): Exchange Options.

Abstract

The contract is described and market examples given. Essential theoretical developments are introduced and cited chronologically. The principles and techniques of hedging and unique pricing are illustrated for the two simplest nontrivial examples: the classical Black-Scholes/Merton/Margrabe exchange option model brought somewhat up-to-date from its form three decades ago, and a lesser exponential Poisson analogue to illustrate jumps. Beyond these, a simplified Markovian SDE/PDE line is sketched in an arbitrage-free semimartingale setting. Focus is maintained on construction of a hedge using Ito's formula and on unique pricing, now for general homogenous payoff functions. Clarity is primed as the multivariate log-Gaussian and exponential Poisson cases are worked out.

Numeraire invariance is emphasized as the primary means to reduce dimensionality by one to the projective space where the SDE dynamics are specified and the PDEs solved (or expectations explicitly calculated). Predictable representation of a homogenous payoff with deltas (hedge ratios) as partial derivatives or partial differences of the option price function is highlighted. Equivalent martingale measures are utilized to show unique pricing with bounded deltas (and in the nondegenerate case unique hedging) and to exhibit the PDE or closed-form solutions as numeraire-deflated conditional expectations in the usual way. Homogeneity, change of numeraire, and extension to dividends are discussed.

• Item Type: MPRA Paper
• Institution: University of Twente
• Language: English
• Keywords: Hedging; self-financing trading strategy; numeraire invariance; predictable representation; unique pricing; arbitrage-free; martingale; homogeneous payoff; Markovian; It\^o's formula; SDE; PDE; geometric Brownian motion; exponential Poisson process
• Subjects: G - Financial Economics > G1 - General Financial Markets > G13 - Contingent Pricing ; Futures Pricing

References

Black, F., M. Scholes, M.: The Pricing of Options and Corporate Liabilities. Journal of Political Economics, 81, 637-59, (1973).

Delbaen, F. Schachermayer, W.: The Mathematics of Arbitrage, Springer (2006).

Duffie, D.: Dynamic Asset Pricing Theory, third edition, Princeton University Press (2001).

El-Karoui, N., Geman, H., Rochet, J.C.: Change of numeraire, change of probability measure, and option pricing, Journal of Applied Probability 32, 443-458 (1995).

Harrison, M.J., Kreps , D.M.: Martingales and arbitrage in multiperiod securities markets. J. Econ. Theory 20, 381-408 (1979).

Harrison, M.J., Pliska, S.: Martingales and stochastic integrals in the theory of continuous trading. Stochastic Processes Appl, 11, 215-260 (1981).

Jamshidian, F: Options and Futures Evaluation with Deterministic Volatilities. Mathematical Finance 3 (2), 149-159 (1993).

Margrabe, W.: The Value of an Option to Exchange One Asset for Another. Journal of Finance 33, 177-86 (1978).

Merton, R: Theory of Rational Option Pricing. Bell Journal of Economics 4(1), 141-183 (1973).

Neuberger, A.: Pricing Swap Options Using the Forward Swap Market. IFA Preprint (1990).