Hedging in Lévy models and the time step equivalent of jumps
Ales Cerny (Cass Business School, UK)
Joint work with Stephan Denkl and Jan Kallsen

Wednesday June 4, 11:30-12:00 | session 4.7 | Hedging | room I

We consider option hedging in a model where the underlying follows an exponential Lévy process. We derive approximations to the variance-optimal and to some suboptimal strategies as well as to their mean squared hedging errors. The results are obtained by considering the Lévy model as a perturbation of the Black-Scholes model. The approximations depend on the first four moments of logarithmic stock returns in the Lévy model and option price sensitivities (greeks) in the limiting Black-Scholes model. We illustrate numerically that our formulas work well for a variety of Lévy models suggested in the literature.

Optimal Static Hedging of Currency Risk Using FX Options and FX Forwards contracts
Anil Bhatia (Tata Consultancy Services, India)
Joint work with Sanjay Bhat and Vijaysekhar Chellaboina

Wednesday June 4, 12:00-12:30 | session 4.7 | Hedging | room I

Exposure to foreign exchange (FX) risk arises when companies conduct business in multiple currencies. A typical scenario such a company encounters may involve a future receivable in foreign currency (FC) for some service/goods the company exported. The FC thus received needs to be converted to home currency (HC) at some predetermined time. However, due to uncertain fluctuations in exchange rates, the company may incur huge losses (or reduced pro?ts) at the time of conversion. Since the direction and magnitude of these fluctuations are uncertain the exchange rate is justified to be classified as a risk. The subject of analysis and mitigation of such financial risk factors comes under risk management and the specific case of exchange rate risk comes under currency risk management. The first step in a risk management process is to derive a predictive model for underlying factors such as FX interest rates. The next step is to identify an appropriate risk measure/metric, a number that can be used to quantify or summarize the effect of the risk from the company's exposure to FC. There are many risk measures available to quantify such risk exposure. Note, that these risk metrics try to capture the effect of the randomness of the risk factors on profit and loss and hence computation of risk metrics will involve probability-based methods. The final step in this process is to determine an optimal hedging strategy. An optimal hedging problem is a constrained optimization problem that minimizes a risk metric over a feasible set of positions in hedging instruments, which are typically used to update the hedging portfolio.
In this work, we consider optimal static currency hedging strategies which allows us to construct a hedging portfolio using optimal positions in FX options and Forward FX contracts respectively. First, we consider a static trading strategy and develop analytical expressions for different risk measures such as Value-at-Risk (VaR), Conditional Value-at-Risk (CVaR), Probability of Loss, and Conditional Expectation of Loss. The results presented here make no implicit assumptions about the underlying probability distribution. Next, using the expressions for risk measures we derive optimal static hedging strategies to minimize these risk measures. Finally, we illustrate the results by specializing the underlying model to some known distributions, for example geometric Brownian motion.