The Skorokhod Embedding Problem and Model-Independent Bounds for Option Prices

This set of lecture notes is concerned with the following pair of ideas and concepts: 1. The Skorokhod Embedding problem (SEP) is, given a stochastic process X=(X t ) t≥0 and a measure μ on the state space of X, to find a stopping time τ such that the stopped process X τ has law μ. Most often we take the process X to be Brownian motion, and μ to be a centred probability measure. 2. The standard approach for the pricing of financial options is to postulate a model and then to calculate the price of a contingent claim as the suitably discounted, risk-neutral expectation of the payoff under that model. In practice we can observe traded option prices, but know little or nothing about the model. Hence the question arises, if we know vanilla option prices, what can we infer about the underlying model?

[1]  A. Skorokhod,et al.  Studies in the theory of random processes , 1966 .

[2]  L. Dubins.  On a Theorem of Skorohod , 1968 .

[3]  D. H. Root.  The Existence of Certain Stopping Times on Brownian Motion , 1969 .

[4]  H. Rost.  The stopping distributions of a Markov process , 1971 .

[5]  Itrel Monroe,et al.  On Embedding Right Continuous Martingales in Brownian Motion , 1972 .

[6]  J. Kiefer.  Skorohod embedding of multivariate RV's, and the sample DF , 1972 .

[7]  R. Chacon,et al.  One-dimensional potential embedding , 1976 .

[8]  H. Rost,et al.  Skorokhod stopping times of minimal variance , 1976 .

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

[10]  L. Rogers,et al.  Diffusions, Markov processes, and martingales , 1979 .

[11]  J. Azéma,et al.  Une solution simple au probleme de Skorokhod , 1979 .

[12]  J. Azéma,et al.  Sur l'integrabilite uniforme des martingales continues , 1980 .

[13]  R. Bass.  Skorokhod imbedding via stochastic integrals , 1983 .

[14]  Edwin A. Perkins,et al.  The Cereteli-Davis Solution to the H1-Embedding Problem and an Optimal Embedding in Brownian Motion , 1986 .

[15]  I. Gyöngy.  Mimicking the one-dimensional marginal distributions of processes having an ito differential , 1986 .

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

[17]  Leonard Rogers,et al.  A Guided Tour through Excursions , 1989 .

[18]  M. Yor,et al.  Continuous martingales and Brownian motion , 1990 .

[19]  Quelques inégalités avec le temps local en zero du mouvement Brownien , 1992 .

[20]  Jean Bertoin,et al.  Representation of Measures by Balayage from a Regular Recurrent Point , 1992 .

[21]  Bruno Dupire.  Pricing with a Smile , 1994 .

[22]  Bruce D. Grundy,et al.  General Properties of Option Prices , 1996 .

[23]  optionsHaydyn,et al.  Robust hedging of barrier , 1998 .

[24]  David Hobson,et al.  The maximum maximum of a martingale , 1998 .

[25]  David Hobson,et al.  Volatility misspecification, option pricing and superreplication via coupling , 1998 .

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

[27]  S. Shreve,et al.  Robustness of the Black and Scholes Formula , 1998 .

[28]  L. Rogers,et al.  Diffusions, Markov Processes and Martingales: References , 2000 .

[29]  J. L. Pedersen,et al.  The Azéma-Yor embedding in non-singular diffusions , 2001 .

[30]  L. Rogers,et al.  The maximum maximum of a martingale constrained by an intermediate law , 2001 .

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

[32]  J. L. Pedersen,et al.  The minimum maximum of a continuous martingale with given initial and terminal laws , 2002 .

[33]  H. Föllmer,et al.  Stochastic Finance: An Introduction in Discrete Time , 2002 .

[34]  Jan Dhaene,et al.  The Concept of Comonotonicity in Actuarial Science and Finance: Applications , 2002 .

[35]  An optimal Skorokhod embedding for diffusions , 2002, math/0212174.

[36]  M. Yor,et al.  Making Markov martingales meet marginals: with explicit constructions , 2002 .

[37]  Stephen Figlewski,et al.  Assessing the Incremental Value of Option Pricing Theory Relative to an Informationally Passive Benchmark , 2002 .

[38]  Svante Janson,et al.  Volatility time and properties of option prices , 2003 .

[39]  David Hobson,et al.  Skorokhod embeddings, minimality and non-centred target distributions , 2003 .

[40]  J. Obłój.  The Skorokhod embedding problem and its offspring , 2004, math/0401114.

[41]  Jan Ob lój.  The Skorokhod embedding problem and its offspring ∗ , 2004 .

[42]  M. Yor,et al.  An explicit Skorokhod embedding for the age of Brownian excursions and Azéma martingale , 2004 .

[43]  Arbitrage bounds for volatility derivatives as a free boundary problem , 2005 .

[44]  David Hobson,et al.  Local martingales, bubbles and option prices , 2005, Finance Stochastics.

[45]  Tai-Ho Wang,et al.  Static-arbitrage upper bounds for the prices of basket options , 2005 .

[46]  Laurent Cousot,et al.  Conditions on Option Prices for Absence of Arbitrage and Exact Calibration , 2006 .

[47]  Fima C. Klebaner,et al.  A Family of Non-Gaussian Martingales with Gaussian Marginals , 2006 .

[48]  Expensive martingales , 2006 .

[49]  Mark H. A. Davis,et al.  THE RANGE OF TRADED OPTION PRICES , 2007 .

[50]  On fake Brownian motions , 2008 .

[51]  J. Albin.  A continuous non-Brownian motion martingale with Brownian motion marginal distributions☆ , 2008 .

[52]  Roger Lee,et al.  Hedging variance options on continuous semimartingales , 2010, Finance Stochastics.

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

[54]  Jan Oblój,et al.  Robust Hedging of Double Touch Barrier Options , 2008, SIAM J. Financial Math..

[55]  David Hobson,et al.  ROBUST BOUNDS FOR FORWARD START OPTIONS , 2012 .