Option Pricing in Markets with Unknown Stochastic Dynamics

TL;DR

This paper develops a Bayesian framework for arbitrage-free option pricing in markets with unknown parameters, showing convergence of Bayesian prices to classical models under certain limits and highlighting differences in jump markets.

Contribution

It introduces a Bayesian approach to option pricing with unknown parameters, establishing arbitrage-free measures and analyzing convergence to classical prices in different market models.

Findings

Bayesian prices converge to Black-Scholes prices with true volatility in high frequency limit.

In Merton markets, Bayesian prices do not converge in finite time but do in the long-term limit.

The approach models incomplete markets due to prior distribution choices.

Abstract

We consider arbitrage free valuation of European options in Black-Scholes and Merton markets, where the general structure of the market is known, however the specific parameters are not known. In order to reflect this subjective uncertainty of a market participant, we follow a Bayesian approach to option pricing. Here we use historic discrete or continuous observations of the market to set up posterior distributions for the future market. Given a subjective physical measure for the market dynamics, we derive the existence of arbitrage free pricing rules by constructing subjective option pricing measures. The non-uniqueness of such measures can be proven using the freedom of choice of prior distributions. The subjective market measure thus turns out to model an incomplete market. In addition, for the Black-Scholes market we prove that in the high frequency limit (or the long time limit) of observations, Bayesian option prices converge to the standard BS-Option price with the true volatility. In contrast to this, in the Merton market with normally distributed jumps Bayesian prices do not converge to standard Merton prices with the true parameters, as only a finite number of jump events can be observed in finite time. However, we prove that this convergence holds true in the limit of long observation times.