On martingale measures and pricing for continuous bond-stock market with stochastic bond

TL;DR

This paper investigates the impact of stochastic bond dynamics on option pricing in continuous markets, revealing non-uniqueness of martingale measures and implications for hedging and risk assessment.

Contribution

It introduces the effects of stochastic bond fluctuations on martingale measures and proposes rational choices including implied measures from bond prices.

Findings

Non-uniqueness of equivalent martingale measures due to stochastic bonds

Significant variation in martingale prices and hedging errors

Proposed rational measures including implied market prices of risk

Abstract

This papers addresses the stock option pricing problem in a continuous time market model where there are two stochastic tradable assets, and one of them is selected as a num\'eraire. It is shown that the presence of arbitrarily small stochastic deviations in the evolution of the num\'eraire process causes significant changes in the market properties. In particular, an equivalent martingale measure is not unique for this market, and there are non-replicable claims. The martingale prices and the hedging error can vary significantly and take extreme values, for some extreme choices of the equivalent martingale measures. Some rational choices of the equivalent martingale measures are suggested and discussed, including implied measures calculated from observed bond prices. This allows to calculate the implied market price of risk process.