انتخاب روش قیمت گذاری اروپا و توقف با هر دو هزینه های مبادله ثابت و متناسب

In this paper we provide a systematic treatment of the utility based option pricing and hedging approach in markets with both fixed and proportional transaction costs: we extend the framework developed by Davis et al. (SIAM J. Control Optim., 31 (1993) 470) and formulate the option pricing and hedging problem. We propose and implement a numerical procedure for computing option prices and corresponding optimal hedging strategies. We present a careful analysis of the optimal hedging strategy and elaborate on important differences between the exact hedging strategy and the asymptotic hedging strategy of Whalley and Wilmott (RISK 7 (1994) 82). We provide a simulation analysis in order to compare the performance of the utility based hedging strategy against the asymptotic strategy and some other common strategies.

The break-through in option valuation theory started with the publication of two seminal papers by Black and Scholes (1973) and Merton (1973). In both papers the authors introduced a continuous time model of a complete friction-free market where the price of a stock follows a geometric Brownian motion. They presented a self-financing, dynamic trading strategy consisting of a riskless security and a risky stock, which replicates the payoff of an option. Then they argued that the absence of arbitrage dictates that the option price be equal to the cost of setting up the replicating portfolio. In the presence of transaction costs in capital markets the absence of arbitrage argument is no longer valid, since perfect hedging is impossible. Due to the infinite variation of the geometric Brownian motion, the continuous replication policy incurs an infinite amount of transaction costs over any trading interval no matter how small it might be. A variety of approaches have been suggested to deal with the problem of option pricing and hedging with transaction costs. A great deal of them are concerned with the ‘financial engineering’ problem of either replicating or super-replicating the option payoff. These approaches are mainly preference-free models where rehedging occurs at some discrete time intervals whether or not it is optimal in any sense. However, common sense tells us that an ‘optimal’ hedging policy should achieve the best possible tradeoff between the risk and the costs of replication. Recognizing the fact that risk preferences differ among individuals, the following conclusion becomes obvious: in pricing and hedging options one must consider the investor's attitude towards risk. In modern finance it is customary to describe risk preferences by a utility function. Expected utility theory maintains that individuals behave as if they were maximizing the expectation of some utility function of the possible outcomes. Hodges and Neuberger (1989) pioneered the option pricing and hedging approach based on this theory. The key idea behind the utility based approach is the indifference argument: the writing price of an option is defined as the amount of money that makes the investor indifferent, in terms of expected utility, between trading in the market with and without writing the option. In many respects such an option price is determined in a similar manner to a certainty equivalent within the expected utility framework, which is a well grounded pricing principle in economics. The difference in the two trading strategies, with and without the option, is interpreted as ‘hedging’ the option. The utility based approach proved to be probably the most successful approach to option hedging with transaction costs. Using simulation analysis, Mohamed (1994), Clewlow and Hodges (1997), and Martellini and Priaulet (2002) demonstrated that the utility based approach achieves excellent empirical performance judging against the best possible tradeoff between the risk and the costs of a hedging strategy. Hodges and Neuberger (1989) introduced the approach with a fairly general transaction costs structure. However, they carried out computations of the optimal hedging strategies and option prices in a market with only proportional transaction costs, without really presenting the continuous time model and the numerical procedure. Davis et al. (1993) rigorously developed the model of Hodges and Neuberger (1989) for a market with proportional transaction costs only. They showed that in this case the problem amounts to a stochastic singular control problem that was formulated by Davis and Norman (1990). They proved that the problem has a unique solution. They also proved the convergence of discretization schemes employed in the numerical procedure. Further contributions to the study of the utility based option pricing approach in a market with proportional transaction costs was made by Clewlow and Hodges (1997), Whalley and Wilmott (1997), Constantinides and Zariphopoulou (1999), Andersen and Damgaard (1999), and some others. In practice, transactions often involve both fixed and proportional costs. However, very little has been done from both the theoretical and empirical sides for this transaction costs structure: Clewlow and Hodges (1997) presented the results of numerical computations of the optimal hedging strategy for a 3-period model in a market with both fixed and proportional transaction costs, without, again, really presenting the continuous time model and the numerical procedure for this case. Whalley and Wilmott (1994) provided an asymptotic analysis of the model of Hodges and Neuberger (1989) for any linear transaction costs structure, assuming that transaction costs are small. Martellini and Priaulet (2002) presented also the comparison of the performance of some simple hedging strategies in the presence of a fixed fee component. In this paper we attempt to fill this gap by providing a systematic treatment of the utility based option pricing and hedging approach in the market with both fixed and proportional transaction costs. The introduction of fixed transaction costs in addition to proportional transaction costs makes the utility maximization/optimal portfolio selection problem more complicated. In this case, due to the presence of a fixed transaction fee irrespective of the size of transaction, the optimal control strategy is discontinuous as opposed to the case with proportional transaction costs only. Unlike the case with only proportional transaction costs where the optimal strategy is described by two free boundaries, the optimal strategy with both fixed and proportional transaction costs turns out to be characterized by four free boundaries. Moreover, the formulation of the optimal portfolio selection problem where each transaction has a fixed cost component requires application of stochastic impulse control theory 1 as opposed to stochastic singular control theory. Consequently, the solution procedure and numerical algorithm to compute the expected utility and optimal trading strategy with both fixed and proportional transaction costs are different from those with only proportional transaction costs. The reader is reminded that in the problem with only proportional transaction costs one uses the gradient constraints to detect the two free boundaries (see, for example, Davis et al. (1993) and Davis and Panas (1994)). In contrast, in the problem with both fixed and proportional transaction costs one needs to employ the maximum utility operator to detect two of the free boundaries and the value matching conditions to detect the other two free boundaries. The paper is organized as follows. In Section 2 we generalize the framework developed by Davis et al. (1993) and formulate the option pricing and hedging problem in the market with both fixed and proportional transaction costs. In Section 3 we present the result of the asymptotic analysis provided by Whalley and Wilmott (1994) on the optimal hedging strategy in a market with both fixed and proportional transaction costs. In Section 4 we propose an original numerical procedure for computing option prices and corresponding optimal hedging strategies. The results of our numerical computations are presented in Section 5. Here we begin our presentation with a study of how the utility based option price depends on the level of the investor's risk aversion. Then we proceed to a detailed study of the optimal hedging strategy. We consider the case of a holder of a short European call option and find that the utility based option price is always above the corresponding Black–Scholes price and is an increasing function of the option holder's risk aversion. As risk aversion decreases, the utility based option price approaches a horizontal asymptote located above the Black–Scholes price. According to the utility based approach, the qualitative description of the optimal hedging strategy is as follows: do nothing when the hedge ratio lies within a so-called no transaction (NT) region and rehedge to the nearest target boundary inside the NT region as soon as the hedge ratio moves out of the NT region. Since hedging is widely used to reduce risk, the knowledge of the optimal hedging strategy in the presence of transaction costs is of great practical interest. With this in mind, we provide a careful analysis of the numerically computed optimal hedging strategy and find that it has two key elements: specific forms of the NT region and the region between the target boundaries, and a volatility adjustment. In particular, we focus our attention on the important differences between the optimal hedging strategies obtained using the exact numerics and the asymptotic analysis and propose the general specification of the optimal hedging strategy. To the best of the authors’ knowledge, this is the first detailed study of the exact optimal hedging strategy with quantification of the sizes of the NT region and the region between the target boundaries, and the volatility adjustment. In Section 6 we provide a simulation analysis in order to compare the performance of the numerically calculated optimal hedging strategy against the asymptotic strategy and some other common strategies. We find that the exact utility based hedging strategy outperforms all the others. The results of our simulation analysis highlight, among other things, the deficiencies of the asymptotic strategy. Section 7 summarizes the paper.

In this paper we provided a systematic treatment of the utility based option pricing and hedging approach in the market with both fixed and proportional transaction costs: we extended the framework developed by Davis et al. (1993) and formulated the option pricing and hedging problem in a market with both fixed and proportional transaction costs. We proposed and implemented the numerical procedure for computing option prices and corresponding optimal hedging strategies. We considered the case of a holder of a short European call option and studied how the utility based option price depends on the level of the option holder’s risk aversion. We found that the utility based option price is always above the corresponding Black–Scholes price and is an increasing function of the option holder’s risk aversion. As risk aversion decreases, the utility based option price approaches a horizontal asymptote located above the Black–Scholes price. Since hedging is widely used to reduce risk, the knowledge of the optimal hedging strategy in the presence of transaction costs is of great practical interest. With this in mind, we provided a careful analysis of the numerically computed optimal hedging strategy and found that it has two distinctive features: specific forms of the NT region and the region between the target boundaries, and a volatility adjustment. In particular, we focused our attention on the important differences between the optimal hedging strategies obtained using the exact numerics and the asymptotic hedging strategy of Whalley and Wilmott (1994). Finally, we provided a simulation analysis in order to compare the performance of the numerically calculated utility based hedging strategy against the asymptotic strategy and some other common strategies and found that the numerically calculated hedging strategy outperforms all the others. The results of our simulation analysis highlighted, in particular, the deficiencies of the asymptotic strategy.