Applications of the Likelihood Theory in Finance: Modelling and Pricing

TL;DR

This paper explores the deep connections between likelihood theory and financial modeling, showing how statistical concepts like experiments and tests can be applied to option pricing and market models, including convergence and approximation results.

Contribution

It introduces a novel framework linking likelihood processes with arbitrage-free asset models and option pricing, bridging statistical experiments with financial mathematics.

Findings

Filtered likelihood ratio processes represent positive prices.

Black-Scholes price interpreted as Bayes risk of a test.

Discrete models approximate continuous-time prices via LAN theory.

Abstract

This paper discusses the connection between mathematical finance and statistical modelling which turns out to be more than a formal mathematical correspondence. We like to figure out how common results and notions in statistics and their meaning can be translated to the world of mathematical finance and vice versa. A lot of similarities can be expressed in terms of LeCam's theory for statistical experiments which is the theory of the behaviour of likelihood processes. For positive prices the arbitrage free financial assets fit into filtered experiments. It is shown that they are given by filtered likelihood ratio processes. From the statistical point of view, martingale measures, completeness and pricing formulas are revisited. The pricing formulas for various options are connected with the power functions of tests. For instance the Black-Scholes price of a European option has an interpretation as Bayes risk of a Neyman Pearson test. Under contiguity the convergence of financial experiments and option prices are obtained. In particular, the approximation of Ito type price processes by discrete models and the convergence of associated option prices is studied. The result relies on the central limit theorem for statistical experiments, which is well known in statistics in connection with local asymptotic normal (LAN) families. As application certain continuous time option prices can be approximated by related discrete time pricing formulas.