Stochastic Control Methods: Hedging in a Market Described by Pure Jump Processes

Abstract

This paper considers the asset price movements in a financial market with a risky asset and a bond. The dynamics of the risky asset, modeled by a marked point process, depend on a stochastic factor, modeled also by a marked point process. The possibility of common jump times with the price is allowed. The problem studied is to determine a strategy maximizing the expected value of a utility function of the hedging error. Two different approaches are considered: an Hamilton Jacobi Bellmann equation is studied for a simplified model and a contraction technique is introduced for a more general model.

Partially informed investors: hedging in an incomplete market with default

“…the mispricing that they might induce; see Runggaldier [18]. As already observed in Geradi and Tardelli [10], on very small time scales actual prices do not really change continuously over time, but rather at discrete random points in time in reaction to trades and/or to significant new information. Moreover, the advent of intraday information on financial asset price quotes and the increasing number of studies on market microstructure leads us to describe the prices as processes that are piecewise constant and jump at irregularly spaced random times in reaction to trades or to significant new information.…”

“…Proof. Taking into account that S t = S 0 + t 0 R S u− (e φ − 1)m(du, dφ) in order to obtain (9) and (10), recall the results of the previous lemma and note that…”

Utility Maximization in a Pure Jump Model With Partial Observation

“…In Gerardi and Tardelli [17], taking into account a similar model, the stochastic factor X is supposed to be a nontradable asset, and the payoff is written on this stochastic factor. The optimal strategy is found with an explicit representation for the value function.…”

Partially informed investors: hedging in an incomplete market with default

“…In this paper the prices are studied by introducing a model similar to that presented by Tardelli [20], [21], and Gerardi and Tardelli [10]. In particular, the dynamics of the underlying asset price S is described by a pure jump process driven by two point processes and by the default indicating process.…”