Hedging in a Financial Market with Regime-Switching
MetadataShow full item record
It is well-known that in the complete standard financial market model driven solely by Brownian motion, one can always hedge a given contingent claim starting from an appropriate initial wealth. In other words, there always exists an initial investment amount and a trading strategy from which one can produce enough wealth to pay off a given contingent claim. Furthermore, one can always produce just the right amount of wealth to settle the claim. That is, one exactly hedges, rather than super-hedges (produces excess wealth), a contingent claim in a standard financial market. In more general market models, such as those where there are constraints on the amount one can invest or when there are other processes driving the randomness in the market, it is not immediately obvious whether one can exactly hedge, or even merely super-hedge, a given contingent claim. We consider hedging problems in a generalization of the well-studied standard Brownian motion market model, namely the regime-switching market model. A standard Brownian motion market model is not very robust as it can only handle small-scale persistent changes in market behaviour. The regime-switching market model, on the other hand, is able to handle large-scale occasional changes in market behaviour, along with small-scale persistent changes, by using a continuous-time Markov chain in addition to a multi-dimensional Brownian motion to drive the randomness in the market. This generalization comes at a cost, however; adding the additional source of randomness renders the financial market incomplete. El-Karoui and Quenez and Cvitanic and Karatzas solved hedging problems in an incomplete Brownian motion market model by introducing the cumulative consumption process and the space of dual processes. These tools allowed them to show that one could in fact promise to super-hedge a given contingent claim in their incomplete market. We use these same tools to handle the incompleteness of the regime-switching market, along with more advanced stochastic analysis, namely the study of discontinuous local martingales, to handle the discontinuity of the paths of the Markov chain. We show that under a certain integrability condition, one can always promise to super-hedge a given contingent claim in a financial market with regime-switching. Furthermore, we characterize both the minimum initial wealth, called the price of the contingent claim, and the trading strategy needed to hedge the contingent claim. We further generalize the problem of hedging in a regime-switching market model by including convex portfolio constraints, introduced by Cvitanic and Karatzas, and margin requirements, of the kind introduced by Cuoco and Liu. These additions allow us to model, for example, markets where there are restrictions on investments and transactions fees on the purchase or sale of stock. Once again, we show that in such a market, under a certain integrability condition, one can always promise to super-hedge a given contingent claim. Furthermore, we characterize the minimum initial wealth and the trading strategy needed to hedge the contingent claim. We then show that under specific optimality conditions, one can exactly hedge a given contingent claim without producing an excess amount of wealth at the end of trade. In other words, we provide conditions that allow one to almost surely hedge a contingent claim without requiring them to consume wealth through a cumulative consumption process. Lastly, we address the problem of approximate hedging in a regime-switching market model, where one tries to hedge a given contingent claim with initial wealth less than the price of the claim. Since the price of the contingent claim is the minimal initial wealth one needs to almost-surely hedge the claim, if one were to begin trading with an initial wealth lower than this price, there is a non-zero probability of them failing to settle the claim. In this case, an investor should trade in an optimal way so that their expected loss from hedging is minimized. We use this approach to solve the approximate hedging problem in a regime-switching market model with portfolio constraints and margin requirements. Using convex duality and tools of non-smooth convex analysis, we show that there does exist an optimal trading strategy that minimizes a specific cost criterion when starting from a lower initial wealth than the price of the given contingent claim.
Cite this version of the work
Adam Daniel Gomes (2021). Hedging in a Financial Market with Regime-Switching. UWSpace. http://hdl.handle.net/10012/17626