INSIDE INFORMATION AND STOCK FLUCTUATIONS

OF THE DISSERTATION Inside Information and Stock Fluctuations by Xin Guo Dissertation Director: Larry Shepp A model of an incomplete market with the incorporation of a new notion of “inside information” is posed. The usual assumption that the stock price is Markovian is modified by adjoining a hidden Markov process to the Black-Scholes exponential Brownian motion model for stock fluctuations. The drift and volatility parameters take different values when the hidden Markov process is in different states. For example, it is 0 when there is no subset of the market which has or which believes it has, extra information. However, the hidden process is in state 1 when information is not equally shared by all, and then the behavior of the members in the subset causes increased fluctuations in the stock price. This model is more realistic than the usual model where all participants in the market are assumed to have full information at all times. It is much harder to obtain explicit closed form solutions for the Russian and other types of hedge options in the new model. By extending the technique of smooth fit to allow jump discontinuities, we obtain the explicit solution. Other analytical tools are ODEs, the Laplace transform and martingale theory. Numerical simulationand discretizationof the new model under Cox-Ross-Rubinstein framework are also provided.

[1]  A. Janssen.  The distance between the Kac process and the Wiener process with applications to generalized telegraph equations , 1990 .

[2]  H. P. Jr. Mackean,et al.  Appendix : A free boundary problem for the heat equation arising from a problem in mathematical economics , 1965 .

[3]  J. Harrison,et al.  Martingales and stochastic integrals in the theory of continuous trading , 1981 .

[4]  M. Schweizer.  On the Minimal Martingale Measure and the Foellmer- Schweizer Decomposition , 1995 .

[5]  E. Platen,et al.  Principles for modelling financial markets , 1996, Journal of Applied Probability.

[6]  H. McKean,et al.  Diffusion processes and their sample paths , 1996 .

[7]  Daniel Ocone,et al.  Malliavin's calculus and stochastic integral representations of functional of diffusion processes † , 1984 .

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

[9]  M. Kac,et al.  An Explicit Representation of a Stationary Gaussian Process , 1947 .

[10]  F. Zapatero,et al.  Optimal Central Bank Intervention in the Foreign Exchange Market , 1999 .

[11]  Gerald Fortney.  A private conversation , 1997 .

[12]  Lester E. Dubins,et al.  Rises and upcrossings of nonnegative martingales , 1962 .

[13]  A Paul,et al.  SAMUELSON, . Proof that properly anticipated prices fluctuate randomly, Industrial Management Review, . , 1965 .

[14]  M. Fridman.  Hidden Markov model regression , 1993 .

[15]  T. Kurtz.  Approximation of Population Processes , 1987 .

[16]  T. Andersen.  Return Volatility and Trading Volume: An Information Flow Interpretation of Stochastic Volatility , 1996 .

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

[18]  L. Shepp.  Radon-Nikodym Derivatives of Gaussian Measures , 1966 .

[19]  Walter Willinger,et al.  The analysis of finite security markets using martingales , 1987, Advances in Applied Probability.

[20]  W. Barry.  On the Iterative Method of Dynamic Programming on a Finite Space Discrete Time Markov Process , 1965 .

[21]  P. Moerbeke,et al.  On optimal stopping and free boundary problems , 1973, Advances in Applied Probability.

[22]  D. Darling,et al.  THE FIRST PASSAGE PROBLEM FOR A CONTINUOUS MARKOFF PROCESS , 1953 .

[23]  L. J. Savage,et al.  How to Gamble if You Must: Inequalities for Stochastic Process. , 1967 .

[24]  J. Harrison.  Discrete Dynamic Programming with Unbounded Rewards , 1972 .

[25]  S. Ross.  Information and Volatility: The No-Arbitrage Martingale Approach to Timing and Resolution Irrelevancy , 1989 .

[26]  J. Michael Harrison,et al.  Arbitrage Pricing of Russian Options and Perpetual Lookback Options , 1993 .

[27]  Sid Browne,et al.  Optimal Investment Policies for a Firm With a Random Risk Process: Exponential Utility and Minimizing the Probability of Ruin , 1995, Math. Oper. Res..

[28]  H. Gerber,et al.  MARTINGALE APPROACH TO PRICING PERPETUAL AMERICAN OPTIONS ON TWO STOCKS , 1996 .

[29]  Robert J. Vanderbei,et al.  Optimal switching between a pair of Brownian motions , 1990 .

[30]  Mark H. A. Davis.  Mathematics of Financial Markets , 2001 .

[31]  Darrell Duffie,et al.  Stochastic equilibria with incomplete financial markets , 1987 .

[32]  S. Ross,et al.  Option pricing: A simplified approach☆ , 1979 .

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

[34]  D. Duffie.  Dynamic Asset Pricing Theory , 1992 .

[35]  John Jackson,et al.  Futures? , 2000 .

[36]  David M. Kreps,et al.  Martingales and arbitrage in multiperiod securities markets , 1979 .

[37]  Larry A Shepp,et al.  The Russian Option: Reduced Regret , 1993 .

[38]  Hung T. Nguyen,et al.  A course in stochastic processes , 1996 .

[39]  J. Hull.  Options, futures, and other derivative securities , 1989 .

[40]  Onésimo Hernández-Lerma,et al.  Controlled Markov Processes , 1965 .

[41]  B. Mandelbrot.  When Can Price Be Arbitraged Efficiently? A Limit to the Validity of the Random Walk and Martingale Models , 1971 .

[42]  P. Lakner.  Optimal trading strategy for an investor: the case of partial information , 1998 .

[43]  A. Kyle.  Continuous Auctions and Insider Trading , 1985 .

[44]  S. Jacka,et al.  Local Times, Optimal Stopping and Semimartingales , 1993 .

[45]  D. Ocone.  A guide to the stochastic calculus of variations , 1988 .

[46]  池田 信行,et al.  Stochastic differential equations and diffusion processes , 1981 .

[47]  D. Liu.  European Option Pricing with Transaction Costs , 2000 .

[48]  P. Moerbeke,et al.  An optimal stopping problem with linear reward , 1974 .

[49]  Jakša Cvitanić,et al.  Hedging Contingent Claims with Constrained Portfolios , 1993 .

[50]  D. Sworder.  Stochastic calculus and applications , 1984, IEEE Transactions on Automatic Control.

[51]  S. Taylor.  DIFFUSION PROCESSES AND THEIR SAMPLE PATHS , 1967 .

[52]  On the Brownian First-Passage Time Overa One-Sided Stochastic Boundary , 1997 .

[53]  Herman Chernoff,et al.  Sequential Tests for the Mean of a Normal Distribution , 1965 .

[54]  I. Karatzas,et al.  On the pricing of contingent claims under constraints , 1996 .

[55]  S. Ross,et al.  An Empirical Investigation of the Arbitrage Pricing Theory , 1980 .

[56]  Jakša Cvitanić,et al.  HEDGING OPTIONS FOR A LARGE INVESTOR AND FORWARD-BACKWARD SDE'S , 1996 .

[57]  Feller William,et al.  An Introduction To Probability Theory And Its Applications , 1950 .

[58]  Lester E. Dubins,et al.  On the distribution of maxima of martingales , 1978 .

[59]  R. Durrett.  Brownian motion and martingales in analysis , 1984 .

[60]  Larry A Shepp,et al.  A New Look at Pricing of the ”Russian Option“ , 1995 .

[61]  S. Jacka.  Optimal Stopping and the American Put , 1991 .