Robust Valuation, Arbitrage Ambiguity and Profit & Loss Analysis

  • Yu-Hong Xu


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 \( \mathrm{P} \& \mathrm{L} \).


Arbitrage Risk-neutral valuation Profit & Loss Overestimation G-expectation 

Mathematics Subject Classification

60H05 60H10 91G20 91B24 



The author would also like to show many thanks to Prof. Shi-Ge Peng and Dr. Xin-Peng Li and other seminar participants of nonlinear expectation at Shandong University for helpful suggestions and particular thanks to Dr. Antoine Jacquier in Imperial College for discussions on the notion of P&L.


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Copyright information

© Operations Research Society of China, Periodicals Agency of Shanghai University, Science Press, and Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Mathematical Center for Interdiscipline Research and School of Mathematical SciencesSoochow UniversitySuzhouChina

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