Financial market with no riskless (safe) asset

TL;DR

This paper explores financial markets lacking a riskless asset, deriving option pricing equations for various risky asset dynamics including diffusions, jump-diffusions, stochastic volatility, and fractional motions, while establishing no arbitrage and completeness conditions.

Contribution

It provides the first comprehensive derivation of option pricing models in markets without a riskless asset across multiple complex asset dynamics.

Findings

Derived Black-Scholes-Merton equations for risky assets without a riskless asset.

Established no arbitrage and market completeness conditions for each case.

Extended option pricing theory to markets with fractional and Rosenblatt motions.

Abstract

We study markets with no riskless (safe) asset. We derive the corresponding Black-Scholes-Merton option pricing equations for markets where there are only risky assets which have the following price dynamics: (i) continuous diffusions; (ii) jump-diffusions; (iii) diffusions with stochastic volatilities, and; (iv) geometric fractional Brownian and Rosenblatt motions. No arbitrage and market completeness conditions are derived in all four cases.