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A Black-Scholes market is considered in which the underlying economy, as modeled by the parameters and volatility of the processes, switches between a finite number of states. The switching is modeled by a hidden Markov chain. European options are priced and a Black-Scholes equation obtained. The approximate valuation of American options due to Barone-Adesi and Whaley is extended to this setting.

Option pricing, Regime switching, Hidden Markov chain model, Esscher transform, MEMM, G13,

'Pairs Trading' is an investment strategy used by many Hedge Funds. Consider two similar stocks which trade at some spread. If the spread widens short the high stock and buy the low stock. As the spread narrows again to some equilibrium value, a profit results. This paper provides an analytical framework for such an investment strategy. We propose a mean-reverting Gaussian Markov chain model for the spread which is observed in Gaussian noise. Predictions from the calibrated model are then compared with subsequent observations of the spread to determine appropriate investment decisions. The methodology has potential applications to generating wealth from any quantities in financial markets which are observed to be out of equilibrium.Pairs trading, Hedge funds, Spreads,

We present a new framework for fractional Brownian motion in which processes with all indices can be considered under the same probability measure. Our results extend recent contributions by Hu, Øksendal, Duncan, Pasik-Duncan, and others. As an application we develop option pricing in a fractional Black-Scholes market with a noise process driven by a sum of fractional Brownian motions with various Hurst indices.

Abstract-In this paper we solve a finite-horizon partially observed risk-sensitive stochastic optimal control problem for discrete-time nonlinear systems and obtain small noise and small risk limits. The small noise limit is interpreted as a deterministic partially observed dynamic game, and new insights into the optimal solution of such game problems are obtained. Both the risk-sensitive stochastic control problem and the deterministic dynamic game problem are solved using information states, dynamic programming, and associated separated policies. A certainty equivalence principle is also discussed. Our results have implications for the nonlinear robust stabilization problem. The small risk limit is a standard partially observed risk-neutral stochastic optimal control problem.

Communicated by James Serrin, October 21, 19711. Introduction. Two person zero-sum differential games can be considered as control problems with two opposing controllers or players. One player seeks to maximize and one to minimize the pay-off function. The greatest pay-off that the maximizing player can force is termed the lower value of the game and similarly the least value which the minimizing player can force is called the upper value. Our objective is to determine conditions under which these values coincide.In the case of two person zero-sum matrix games, von Neumann showed that if the players are allowed "mixed strategies," i.e., probability measures over the pure strategies, then the values of the game will coincide. By analogy, the authors in collaboration with L. Markus [1] introduced relaxed controls into differential game theory; for a full discussion of relaxed controls the reader is referred to [1].In this announcement, we define notions of strategy and values and relate these to the approaches adopted by Fleming [3], [4] and Friedman [5]. Using relaxed controls we are able to show that if the "Isaacs condition" (3) is satisfied then the upper and lower values are equal, so that the game has value. In particular, if the players are allowed relaxed controls then the game always has value. Detailed proofs of these results will appear in a later publication [2].