Finance, Economics, and Mathematics

Chapter 14Risk-Neutral Economy and Zero Price of Risk

Mathematics and Financial Economics, 8 (2014), 229–239.

Abstract

The paper investigates the equilibrium in an economy in which all participants are indifferent to risk. The mechanism of asset and derivative pricing in such economy is identified. It is shown that no economy in equilibrium with stochastic interest rates can be simultaneously risk-neutral and have zero market price of risk. On the other hand, there exist equilibrium economies with risk-averse participants and zero prices of risk.

Introduction

The concept of the risk-neutral economy, that is, an economy in which all participants are indifferent to risk and only care about expected return, is often used in finance as a standard of reference. That is due to the supposition that in such an economy there is no compensation for risk and all assets have the same expected return, which is therefore equal to the risk-free rate.

In general, a complete economy (an economy that allows all derivative contracts) in which there are no opportunities for riskless arbitrage will contain a process, called the market price of risk, for each source of uncertainty it involves. If x is a Wiener process of the risk sources and λ is the vector of the corresponding market prices of risk, and if r is the short riskless rate, then the expected instantaneous rate of return μ on an asset whose exposure to the sources of risk is β satisfies the relationship

1

It is often assumed that ...