Paper: arXiv 2308.13717
Authors: Ricardo T. Fernholz, Robert Fernholz
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.
Complexity vs Empirical Score
- Math Complexity: 8.5/10
- Empirical Rigor: 2.0/10
- Quadrant: Lab Rats — theoretically deep, empirically untested
Why this score: The paper presents a strong mathematical synthesis connecting stochastic portfolio theory (portfolio generating functions) with contingent claim replication and option pricing PDEs. However, it is purely theoretical with no empirical backtesting or real-market data application included in the excerpt.
Research Flowchart
flowchart TD
A[Research Goal: Synthesize portfolio generating functions & option pricing without numeraire] --> B[Inputs: Asset prices X_1...X_n, Riskless Asset, & C2,1 Functions]
B --> C[Extend Portfolio Generation to Inhomogeneous Functions & Contingent Claims]
C --> D[Formulate Generalized Black-Scholes PDE for Replication]
D --> E[Key Outcome: Portfolio value replicates function value iff PDE is satisfied]