The optimal multi-period hedging model of currency futures and options with exponential utility (Q2332718)

The collapse of the Bretton Woods system allows to constituting a new multi-currency system. Since 1973, the international foreign exchange market has been developing towards foreign-exchange liberalization. Exchange rate risks become prevailing, makes the fluctuation of the foreign exchange market more frequent. In the present paper the optimal cross-hedging and risk control problem of a competitive firm facing exchange risk exposure in a multi-period setting is considered. The optimal positions of currency futures and options to maximize the exponential utility of the terminal wealth by dynamic programming approach is presented. To evaluate the potential effectiveness of cross-hedging with currency futures and options, the terminal wealth, utility based on terminal wealth and the variance of the wealth accumulation path, are compared. Here, three cases are considered: hedging with futures and options, net futures hedging and no hedging. An empirical study of two cross-exchange rate of USD/EUR and CNY/USD is performed. In the Introduction of the paper the process of applying of the derivative financial instruments to hedging against the exchange rate risk is reminded. The main results of this paper are given in Section 2. The return model \[ \left\{ \begin{array}{l} r_t = \mu_t + a_t, \\ a_t = \sigma_t e_t, \\ \sigma_t^2 = \alpha_0 + \alpha_1 \sigma_{t-1}^2 + \alpha_2 a^2_{t-1}, \\ e_t \sim N(0,1) \ \ \text{ or} \ \ e_t \sim t_d, \end{array}\right. \] where \(r_t,\) \(t=1,2, \ldots, T\) denotes the return of the exchange exchange rate, \(\mu_t\) is the conditional mean return with regard, is considered. When \(e_t \sim N(0,1)\) (the standard normal distribution) the so-called GARCH-n model is obtained and when \(e_t \sim t_d\) (the Student-t distribution with degree of freedom \(d\)) is obtained. Some preliminary Lammas are proved. The methodology to solve the dynamic hedging problems with futures and options is presented in a special algorithm. In Section 3 an empirical study of a cross-hedging problem with exchange rates of CNY/USD and USD/EUR is provided. The obtained results are graphically illustrated. Models of GARCH-n and GARCH-t of the martingale distributions are considered. The performance of the derived hedging strategies on futures and options hedging, net futures hedging and no hedging hedging terms of terminal wealth, the utility based on the terminal wealth, and the variance of wealth accumulation path are compared. In the Conclusion of the paper the hedging problems are discussed.