Robust valuation and risk measurement under model uncertainty

TL;DR

This paper develops a robust framework for valuation and risk measurement in financial markets with mean-volatility uncertainty using Peng's G-stochastic calculus, deriving model-independent pricing bounds and arbitrage conditions.

Contribution

It introduces a G-stochastic calculus-based model for markets with mean and volatility uncertainty, deriving model-independent option pricing bounds and arbitrage definitions.

Findings

Derived upper option prices using G-Brownian motion.

Established arbitrage conditions under model uncertainty.

Interpreted the super-hedging term as maximum profit/loss.

Abstract

Model uncertainty is a type of inevitable financial risk. Mistakes on the choice of pricing model may cause great financial losses. In this paper we investigate financial markets with mean-volatility uncertainty. Models for stock markets and option markets with uncertain prior distribution are established by Peng's G-stochastic calculus. The process of stock price is described by generalized geometric G-Brownian motion in which the mean uncertainty may move together with or regardless of the volatility uncertainty. On the hedging market, the upper price of an (exotic) option is derived following the Black-Scholes-Barenblatt equation. It is interesting that the corresponding Barenblatt equation does not depend on the risk preference of investors and the mean-uncertainty of underlying stocks. Hence under some appropriate sublinear expectation, neither the risk preference of investors nor the mean-uncertainty of underlying stocks pose effects on our super and subhedging strategies. Appropriate definitions of arbitrage for super and sub-hedging strategies are presented such that the super and sub-hedging prices are reasonable. Especially the condition of arbitrage for sub-hedging strategy fills the gap of the theory of arbitrage under model uncertainty. Finally we show that the term $K$ of finite-variance arising in the super-hedging strategy is interpreted as the max Profit\&Loss of being short a delta-hedged option. The ask-bid spread is in fact the accumulation of summation of the superhedging $P\&L$ and the subhedging $P\&L $.