Robust Valuation, Arbitrage Ambiguity and Profit & Loss Analysis

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 market and option market with uncertain prior distributions are established by Peng’s G-stochastic calculus. 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 mean uncertainty of the underlying stocks. Appropriate definitions of arbitrage for super- and sub-hedging strategies are presented such that the super- and sub-hedging prices are reasonable. In particular, 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 superhedging strategy is interpreted as the max Profit & Loss ( $$ \mathrm{P} \& \mathrm{L} $$ ) of shorting a delta-hedged option. The ask-bid spread is in fact an accumulation of the superhedging $$ \mathrm{P} \& \mathrm{L}$$ and the sub-hedging $$ \mathrm{P} \& \mathrm{L} $$ .