Robust pricing–hedging dualities in continuous time

We pursue a robust approach to pricing and hedging in mathematical finance. We consider a continuous-time setting in which some underlying assets and options, with continuous price paths, are available for dynamic trading and a further set of European options, possibly with varying maturities, is available for static trading. Motivated by the notion of prediction set in Mykland (Ann. Stat. 31:1413–1438, 2003), we include in our setup modelling beliefs by allowing to specify a set of paths to be considered, e.g. superreplication of a contingent claim is required only for paths falling in the given set. Our framework thus interpolates between model-independent and model-specific settings and allows us to quantify the impact of making assumptions or gaining information. We obtain a general pricing–hedging duality result: the infimum over superhedging prices of an exotic option with payoff G$G$ is equal to the supremum of expectations of G$G$ under calibrated martingale measures. Our results include in particular the martingale optimal transport duality of Dolinsky and Soner (Probab. Theory Relat. Fields 160:391–427, 2014) and extend it to multiple dimensions, multiple maturities and beliefs which are invariant under time-changes. In a general setting with arbitrary beliefs and for a uniformly continuous G$G$, the asserted duality holds between limiting values of perturbed problems.

[1]  M. Yor.  DIFFUSIONS, MARKOV PROCESSES AND MARTINGALES: Volume 2: Itô Calculus , 1989 .

[2]  Yu-Jui Huang,et al.  Model-Independent Superhedging Under Portfolio Constraints , 2014 .

[3]  P. Meyer,et al.  Probabilities and potential C , 1978 .

[4]  Yan Dolinsky,et al.  Martingale Optimal Transport in the Skorokhod Space , 2014, 1404.1516.

[5]  Terry Lyons,et al.  Uncertain volatility and the risk-free synthesis of derivatives , 1995 .

[6]  Alexander M. G. Cox,et al.  Root's barrier: Construction, optimality and applications to variance options. , 2011, 1104.3583.

[7]  R. C. Merton,et al.  Theory of Rational Option Pricing , 2015, World Scientific Reference on Contingent Claims Analysis in Corporate Finance.

[8]  Erhan Bayraktar,et al.  On Arbitrage and Duality Under Model Uncertainty and Portfolio Constraints , 2014 .

[9]  H. Föllmer,et al.  Optional decompositions under constraints , 1997 .

[10]  Frode Terkelsen,et al.  Some Minimax Theorems. , 1972 .

[11]  L. Denis,et al.  A THEORETICAL FRAMEWORK FOR THE PRICING OF CONTINGENT CLAIMS IN THE PRESENCE OF MODEL UNCERTAINTY , 2006, math/0607111.

[12]  Zhaoxu Hou,et al.  Pointwise Arbitrage Pricing Theory in Discrete Time , 2016, Math. Oper. Res..

[13]  Douglas T. Breeden,et al.  Prices of State-Contingent Claims Implicit in Option Prices , 1978 .

[14]  Marcel Nutz,et al.  Superreplication under Volatility Uncertainty for Measurable Claims , 2012, 1208.6486.

[15]  F. Black,et al.  The Pricing of Options and Corporate Liabilities , 1973, Journal of Political Economy.

[16]  Jan Obłój,et al.  Arbitrage Bounds for Prices of Weighted Variance Swaps , 2010 .

[17]  Rados Radoicic,et al.  An Explicit Implied Volatility Formula , 2017 .

[18]  Jan Oblój,et al.  Robust pricing and hedging of double no-touch options , 2009, Finance Stochastics.

[19]  P. Wilmott,et al.  An Asymptotic Analysis of an Optimal Hedging Model for Option Pricing with Transaction Costs , 1997 .

[20]  V. Strassen.  The Existence of Probability Measures with Given Marginals , 1965 .

[21]  F. Delbaen,et al.  The fundamental theorem of asset pricing for unbounded stochastic processes , 1998 .

[22]  Nizar Touzi,et al.  A Stochastic Control Approach to No-Arbitrage Bounds Given Marginals, with an Application to Lookback Options , 2013, 1401.3921.

[23]  F. Delbaen,et al.  A general version of the fundamental theorem of asset pricing , 1994 .

[24]  Gianluca Cassese,et al.  ASSET PRICING WITH NO EXOGENOUS PROBABILITY MEASURE , 2007 .

[25]  F. Knight.  The economic nature of the firm: From Risk, Uncertainty, and Profit , 2009 .

[26]  N. Touzi,et al.  On the Robust superhedging of measurable claims , 2013, 1302.1850.

[27]  P. Mykland.  Financial options and statistical prediction intervals , 2003 .

[28]  Xiaolu Tan,et al.  Tightness and duality of martingale transport on the Skorokhod space , 2015, 1507.01125.

[29]  Sergey Nadtochiy,et al.  Robust Trading of Implied Skew , 2016, 1611.05518.

[30]  M. Avellaneda,et al.  Pricing and hedging derivative securities in markets with uncertain volatilities , 1995 .

[31]  B. Bouchard,et al.  ROBUST FUNDAMENTAL THEOREM FOR CONTINUOUS PROCESSES , 2014, 1410.4962.

[32]  Matteo Burzoni,et al.  Universal arbitrage aggregator in discrete-time markets under uncertainty , 2014, Finance Stochastics.

[33]  HEAT KERNEL INTEREST RATE MODELS WITH TIME-INHOMOGENEOUS MARKOV PROCESSES , 2010, 1012.1878.

[34]  J. Obłój,et al.  PERFORMANCE OF ROBUST HEDGES FOR DIGITAL DOUBLE BARRIER OPTIONS , 2012 .

[35]  Svante Janson,et al.  The inverse first-passage problem and optimal stopping , 2015, 1508.07827.

[36]  Nicolas Perkowski,et al.  Pathwise superreplication via Vovk’s outer measure , 2015, Finance Stochastics.

[37]  Paul A. Samuelson,et al.  Rational Theory of Warrant Pricing , 2015 .

[38]  David Hobson,et al.  Robust hedging of the lookback option , 1998, Finance Stochastics.

[39]  H. Soner,et al.  Quasi-sure Stochastic Analysis through Aggregation , 2010, 1003.4431.

[40]  Halil Mete Soner,et al.  Robust hedging with proportional transaction costs , 2013, Finance Stochastics.

[41]  H. Soner,et al.  Martingale optimal transport and robust hedging in continuous time , 2012, 1208.4922.

[42]  Bruno Bouchard,et al.  Arbitrage and duality in nondominated discrete-time models , 2013, 1305.6008.

[43]  Frank Riedel,et al.  Financial economics without probabilistic prior assumptions , 2015 .

[44]  N. Touzi,et al.  The maximum maximum of a martingale with given $\mathbf{n}$ marginals , 2012, 1203.6877.

[45]  K. Bichteler,et al.  Stochastic Integration and $L^p$-Theory of Semimartingales , 1981 .

[46]  Beatrice Acciaio,et al.  A MODEL‐FREE VERSION OF THE FUNDAMENTAL THEOREM OF ASSET PRICING AND THE SUPER‐REPLICATION THEOREM , 2013, 1301.5568.

[47]  Leonard Rogers,et al.  Robust Hedging of Barrier Options , 2001 .

[48]  Zhaoxu Hou,et al.  Robust pricing and hedging under trading restrictions and the emergence of local martingale models , 2014, Finance Stochastics.

[49]  Yan Dolinsky,et al.  Corrigendum to “Martingale optimal transport in the Skorokhod space” [Stochastic Process. Appl. 125(10) (2015) 3893–3931] , 2016 .

[50]  A. Shiryaev,et al.  Limit Theorems for Stochastic Processes , 1987 .

[51]  Vladimir Vovk,et al.  Continuous-time trading and the emergence of probability , 2009, Finance and Stochastics.

[52]  M. Beiglböck,et al.  Model-independent bounds for option prices—a mass transport approach , 2011, Finance and Stochastics.