Abstracts

We consider the problem of optimally hedging a European contingent claim (ECC) with its underlying in a discrete-time setting. The ECC could be written on multiple underlyings and may be path-dependent. Specifically, we consider two quadratic optimal hedging strategies : minimum-variance hedging in a risk-neutral measure and optimal local-variance hedging in a market probability measure. The objective function for the former is the variance of the hedging error calculated in a risk-neutral measure and the latter optimizes the variance of the mark-to-market value of the portfolio over a trading interval in a market measure. The motivation for introducing and deriving expressions for optimum local-variance hedging are two fold. First, it is useful to consider strategies that minimize the variance of the mark-to-market value of the portfolio locally in time. Second, their analysis is simpler even for the general semi-martingale case. The main aim of the work is to derive explicit closed form solutions to hedge different types of ECCs using the above mentioned quadratic hedging schemes. To arrive at closed-form solutions, we assume geometric Brownian motion (GBM) as the model for the underlying asset prices. The different types of ECCs considered include a general path-independent ECC, exchange option to represent a multi-asset ECC and a discretely-monitored path-dependent ECC. All the hedging solutions are expressed in terms of the pricing function of the hedged ECC and the prices of the underlying assets. These explicit solutions when used instead of complex Monte-Carlo based solutions makes the proposed hedging solution well suited for computer implementation.
Yet another motive of the work is to compare the effectiveness of the two quadratic trading strategies with the standard delta hedging. Since the delta hedging strategy is s based on the theory of continuous-time trading, our intention is to show that, when trading is performed in discrete-time, quadratic optimal strategies perform better than the standard delta-hedging. The comparison is done on multiple performance measures such as the probability of loss, expected loss, different moments of the hedging error and shortfall measures such as value at risk (VaR) and conditional value at risk (CVaR). The outputs of various measures can then be used to evaluate the appropriateness of using a particular trading strategy in a given scenario. We argue that the evaluation process can be better performed if a trader relies on the results of multiple performance metrics instead of just one. Our performance evaluation results on path-independent and Exchange like options conclude that in the discrete-time setting the quadratic optimal hedging strategies outperform the delta hedging strategy. Indeed, as the re-balancing is done more sparsely, the quadratic-optimal trading schemes fair better than the standard delta-hedging.