Portfolios generated by contingent claim functions, with applications to option pricing

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 strictly positive $C^{2,1}$ C 2 , 1 -functions of asset prices and time directly, rather than with respect to a numéraire portfolio. If a portfolio-generating function satisfies a specific partial differential equation, then the value of the portfolio generated by that function replicates 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 ${V}\colon {\mathbb{R}}_{++}^{n}\times [0,T]\to {\mathbb{R}}_{++}$ V : R + + n × [ 0 , T ] → R + + can be extended to an equivalent contingent claim function ${{\widehat{V}}}\colon {\mathbb{R}}_{++}\times {\mathbb{R}}_{++}^{n} \times [0,T]\to {\mathbb{R}}_{++}$ V ˆ : R + + × R + + n × [ 0 , T ] → R + + that generates the same portfolio and is replicable if and only if ${V}$ V is replicable. Several examples are presented.