A generalized approach to optimal hedging with option contracts

In this paper, we develop a theoretical model in which a firm hedges a spot position using options in the presence of both quantity (production) and basis risks. Our optimal hedge ratio is fairly general, in that the dependence structure is modeled through a copula function representing the quantiles of the hedged position, and hence any quantile risk measure can be employed. We study the sensitivity of the exercise price which minimizes the risk of the hedged portfolio to the relevant parameters, and we find that the subjective risk aversion of the firm does not play any role. The only trade-off is between the effectiveness and cost of the hedging strategy.     

A risk-sensitive stochastic control approach to an optimal investment problem with partial information

We consider an infinite time horizon optimal investment problem where an investor tries to maximize the probability of beating a given index. From a mathematical viewpoint, this is a large deviation probability control problem. As shown by Pham (in Syst. Control Lett. 49: 295–309, 2003; Financ. Stoch. 7: 169–195, 2003), its dual problem can be regarded as an ergodic risk-sensitive stochastic control problem. We discuss the partial information counterpart of Pham (in Syst. Control Lett. 49: 295–309, 2003; Financ. Stoch. 7: 169–195, 2003). The optimal strategy and the value function for the dual problem are constructed by using the solution of an algebraic Riccati equation. This equation is the limit equation of a time inhomogeneous Riccati equation derived from a finite time horizon problem with partial information. As a result, we obtain explicit representations of the value function and the optimal strategy for the problem. Furthermore we compare the optimal strategies and the value functions in both full and partial information cases.

---

Electronic Supplementary Material Supplementary material is available for this article at      

A PDE approach for risk measures for derivatives with regime switching

This paper considers a partial differential equation (PDE) approach to evaluate coherent risk measures for derivative instruments when the dynamics of the risky underlying asset are governed by a Markov-modulated geometric Brownian motion (GBM); that is, the appreciation rate and the volatility of the underlying risky asset switch over time according to the state of a continuous-time hidden Markov chain model which describes the state of an economy. The PDE approach provides market practitioners with a flexible and effective way to evaluate risk measures in the Markov-modulated Black–Scholes model. We shall derive the PDEs satisfied by the risk measures for European-style options, barrier options and American-style options.