Portfolios Generated by Contingent Claim Functions, with Applications to Option Pricing

TL;DR

This paper unifies portfolio generating functions with option pricing theory, extending the framework to directly measure portfolios from asset prices and deriving a generalized Black-Scholes PDE for these functions.

Contribution

It introduces a direct approach to measure portfolios generated by smooth functions of asset prices and links them to a generalized PDE similar to Black-Scholes, including contingent claim functions.

Findings

Portfolio values satisfy a PDE generalizing Black-Scholes.

Contingent claim functions are homogeneous of degree 1 and replicable under certain conditions.

Examples illustrate the application of the theory to various portfolio and option structures.

Abstract

This paper presents a synthesis of the theories of portfolio generating functions and option pricing. The theory of portfolio generation is extended to measure the value of portfolios generated by positive C^{2,1} functions of asset prices X_1,... , X_n directly, rather than with respect to a numeraire portfolio. If a portfolio generating function satisfies a specific partial differential equation, then the value of the portfolio generated by that function will replicate the value of the function. This differential equation is a general form of the Black-Scholes equation. Similar results apply to contingent claim functions, which are portfolio generating functions that are homogeneous of degree 1. With the addition of a riskless asset, an inhomogeneous portfolio generating function V : R^{+n} x [0, T] \to R^+ can be extended to an equivalent contingent claim function \hat{V} : R^+ x R^{+n} x [0, T] \to R^+ that generates the same portfolio and is replicable if and only if V is replicable. Several examples are presented.