Pricing Product Options and Using Them to Complete Markets for Functions of Two Underlying Asset Prices

Dilip B Madan,King Wang

doi:10.3390/jrfm14080355

Abstract

Options paying the product of put and/or call option payouts at different strikes on two underlying assets are observed to synthesize joint densities and replicate differentiable functions of two underlying asset prices. The pricing of such options is undertaken from three perspectives. The first perspective uses a geometric two-dimensional Brownian motion model. The second inverts two-dimensional characteristic functions. The third uses a bootstrapped physical measure to propose a risk charge minimizing hedge using options on the two underlying assets. The options are priced at the cost of the hedge plus the risk charge.

Highlights

Product Options and Functional ReplicationLet g( x, y) be a sufficiently smooth function of two variables
Derivative Product Strats, Morgan Stanley, 1585 Broadway, 5th Floor, New York, NY 10036, USA; This paper is the private opinion of the authors and does not necessarily reflect the policy and views of Morgan Stanley
Option markets considerably enhance the collection of functions of the stock price at the option maturity that may be purchased or sold

Summary

Product Options and Functional Replication

Let g( x, y) be a sufficiently smooth function of two variables. It may be written as the integral of its second order cross partials as follows. In the positive orthant relative to the point x0 , y0 , one may replicate such functions with positions in product calls at strikes ( a, b). In the second orthant relative to x0 , y0 products of puts on x and calls on y are employed. In the third orthant, it is products of puts, and, in the fourth, it is calls on x and puts on y. A classic application of functional replication in one dimension is the pricing of the variance swap contract or the computation of the VIX index. Applications would include the replication of straddles and strangles on the spread of one asset price over another

Multivariate Geometric Brownian Motion and Product Options

Fast Fourier Transform for Product Options

Multivariate Bilateral Gamma

Sample Computations Using the Multivariate Bilateral Gamma Model

Pricing Product Options Using the Physical Measure

Conclusions