Hedging contingent claims in markets with jumps.

摘要

Contrary to the Black-Scholes paradigm, an option-pricing model which incorporates the possibility of jumps more accurately reflects the evolution of stocks in the real world. However, hedging a contingent claim in such a model is a non-trivial issue: in many cases, an infinite number of hedging instruments are required to eliminate the risk of an option position. This thesis develops practical techniques for hedging contingent claims in markets with jumps.; A regime-switching model accommodates jumps in (i) the parameters of the stochastic process that drives the underlying asset; and (ii) the price path of the underlying asset itself. We develop numerical techniques for solving the system of partial differential equations that yields the option values in a regime-switching model. When the possible jump sizes of the asset price are drawn from a finite set, all sources of instantaneous risk from an option position can be eliminated by adding a finite number of hedging instruments. We explore a variety of dynamic hedging strategies for a market governed by such a regime-switching process, including techniques that eliminate just some, or all, of the instantaneous risk. The pricing and hedging methodologies are adapted to swing options, a path-dependent derivative traded in the energy markets.; A more realistic representation of jumps in the price path of the underlying asset is made by allowing the amplitudes to be drawn from a continuum. In this case, an infinite number of hedging instruments are required to eliminate the instantaneous risk of an option position, implying that perfect hedging is impossible, even with continuous rebalancing. We demonstrate in a jump-diffusion market that, by imposing delta neutrality and suitably bounding the jump risk and transaction costs at each instant of a continuously rebalanced hedge, the terminal hedging error can be made arbitrarily small. This theoretical treatment motivates a discretely rebalanced dynamic hedging strategy. Hedging examples are considered for options with both European and American-style exercise rights, in a jump-diffusion market with and without transaction costs. We also investigate semi-static hedging, a buy-and-hold strategy that attempts to replicate the value of a target option at some future time.; Levy processes constitute a broad class of stochastic processes that exhibit jumps---the jump-diffusion process is a representative member of this group. However, some Levy processes can generate an infinite number of small jumps over any time period. We demonstrate how our dynamic strategy for hedging under jump diffusion can be used to hedge under any Levy process.