Hedging strategies with a put option and their failure rates

The most popular derivatives are forwards, options and swaps. They are basic blocks for all sorts of other more complicated derivatives, and should be used prudently. Several parameters need to be determined in the processes of risk management, and it is necessary to investigate the influence of these parameters on the aims of the hedging policies and the possibility of achieving these goals (Annaert et al., 2007). However, the literature on risk management is much silent on how to optimally decide on these parameters (Annaert et al., 2007).

The problem of determining the optimal strike price and optimal hedging ratio is considered by Ahn et al. (1999), Annaert et al. (2007), Deelstra et al. (2010) and the references therein, where a put option is used to hedge market risk under a constraint of budget. The chosen option is supposed to finish in-the-money at maturity in the aforementioned papers, such that the predicted loss of the hedged portfolio is different from the realized loss. And the constraint of hedging budget is supposed to be binding in those papers, which means that the company will always spend the maximum available to buy options, such that the cost of hedging is determined by the amount of hedging budget. Whether the performance of hedging strategy is affected by the two assumptions is considered in this paper.

Following Ahn et al. (1999), Annaert et al. (2007) and Deelstra et al. (2010), the aim of hedging is to minimize the potential loss of investment under a specified level of confidence. In other words, the optimal hedging strategy is to minimize the Value-at-Risk (VaR) under a specified level of risk. However, the present paper is different from the aforementioned papers in several aspects. First, the chosen put option is not supposed to finish in-the-money at maturity in this paper. As the possibility of inexecution is taken into account, the predicted loss of hedging is closer to the realized loss. Second, the constraint of hedging budget is not supposed to be binding, such that the expenditure of hedging is not always equal to the maximum available. Third, the available put options are specified by their strike prices in a discrete manner, such that the optimal strike price can only be chosen from a predetermined finite set of strike prices, which is similar to the situation faced in real world financial market. Finally, the performances of the resulted optimal hedging strategies are investigated through hypothesis tests, where the failure of hedging means that the realized loss exceeds the level of VaR predicted by the hedging strategy. The simulated investigations indicate that the proposed method is more accurate than the method deduced from the spirit of Ahn et al. (1999), Annaert et al. (2007) and Deelstra et al. (2010).

The paper is organized as follows. Section 2 describes the stock hedging problem. Section 3 presents the main theoretical analysis of loss and its risk, where the probability that the potential loss of hedging strategy exceeds a predetermined threshold is calculated under a geometric Brownian motion. Section 4 describes two methods to determine the optimal hedging strategy, one is deduced from the spirit of the aforementioned papers, and the other is proposed in the present paper. The failure rates of hedging strategies are compared through simulations in Section 5. Section 6 gives the conclusions and discussions.

Analogously to Ahn et al. (1999) and Deelstra et al. (2010), a stock is supposed to be bought at time zero with price S 0 , and to be sold at time T with uncertain price S T . In order to hedge the market risk of the stock, the company decides to choose one of the available put options written on the same stock with maturity at time τ , where τ is prior and close to T , and the n available put options are specified by their strike prices

As the prices of different put options are also different, the company needs to determine an optimal hedge ratio h (0 ≤ h ≤ 1) with respect to the chosen strike price. The cost of hedging should be less than or equal to the predetermined hedging budget C. In other words, the company needs to determine the optimal strike price and hedging ratio under the constraint of hedging budget.

The chosen put option is supposed to finish in-the-money at maturity, and the constraint of hedging expenditure is supposed to be binding by Ahn et al. (1999), Annaert et al. (2007), Deelstra et al. (2010) and the references therein. These two assumptions are re-laxed in this pa

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