Dynamic hedging with transaction costs

Black-Scholes and Merton options pricing model (BSM) makes assumptions such as continuous price dynamics, the possibility of continuous trading, known volatility of the asset price, and no transaction costs. However, in practice, these assumptions are satisfied only to a certain degree. When the financial crisis occurred, a dynamic hedging strategy based on small or fixed size price movements often breaks down. Therefore, we will use the Merton’s jump-diffusion model (JDM) in which the underlying asset price can exhibit jumps of random size. In the presence of random jumps, the performance of daily delta hedging will be examined using Monte-Carlo simulation and also any changes in its performance if the rebalancing frequency is increased at regular intervals first and at variable intervals. There are two main issues that must be addressed in delta hedging. The first is how often the hedging portfolio should be rebalanced, and the second is how hedging errors can be minimized. These issues are in conflict. Frequent hedges reduce errors but augment costs, while less frequent hedges result in large errors. In order to address these issues, we will analyse the difference in performance between frequent hedges and less frequent hedges. Moreover, we will take into account of transaction costs to the BSM model since in practice the transaction charges can be so high, especially in emerging markets where the roundtrip transaction costs of 1% and higher are common (Hull,2007). Traders may not be able to carry out as frequent hedges as they want, in order to be still protected against the downside risk. In studies by Derman (1999) and Bertsimas et al (2000) show that, under the gBm model, the hedging error for options is approximately proportional to the square root of the hedging frequency. In addition, the misspecification of the volatility parameter, when the volatility used to calculate the delta-hedge is different from the actual volatility of the log-normal price process, is analysed by Figlewski (1989), Crouhy-Galai (1995), Mahayni (2003) and Hayashi-Mykland (2005). Option hedging under jump diffusions is studied, among others, by Mocioalca (2007) and Kennedy et al (2009), who analyse the dynamic hedging under the Merton JDM with transaction costs by means of partial differential equation solutions and Monte- Carlo simulations. Option hedging with transaction costs under the BSM model is studied, among others, by Leland (1985), Toft (1996), Primbs-Yamada (2008). Importantly, Toft (1996) derives closed-form solution for the expected P&L and its volatility under the BSM model assuming different drift and volatility parameters under the historical and pricing measures. In this paper, we consider dynamic hedging in the presence of transaction costs. Our objective is to find out the effects of various percentage of proportional transaction costs will have on the hedging performance. In the Black-Scholes model, we assume that hedging took place continuously by rehedging an infinite number of times until expiry. With a bid-off spread which are the costs incurred in the buying and selling of the underlying, the cost of rehedging leads to infinite total transaction costs. Even with discrete hedging, the consequences of these costs associated with rehedging are important. Different investors have different levels of transaction costs. There are economies of scale so that the larger the size of the trade, the less significant is the costs. Since there are fixed and proportional cost structures, we will only focus on the latter in this paper. The modelling of transaction costs was initiated by Leland (1985). He adopted the hedging strategy of rehedging at every time step dt. He assumes that the cost of hedging v assets costs an amount kvS for both buying and selling: this models bid-ask spread, the cost proportional to the value traded. This model allows for the cost of trading in valuing the hedged position and the option positions can be valued by using the Leland’s adjusted volatility. Another objective of the paper is to examine the effectiveness of delta hedging strategy under both geometric Brownian motion model and Merton’s jump diffusion model with and without transaction costs. The aim is to find out the difference in delta hedging performance when the BSM assumption about the lognormal distribution of stock prices is violated and in fact, when prices follow jump diffusion process. The methodology is to simulate 1000 stock price paths using Monte Carlo simulation. Delta hedging performance is then measured as the ratio of the standard deviation of the cost of hedging the option to the Black-Scholes price of the option. In this respect, the distribution of profit and loss, higher moments of hedge errors and the costs will be analysed. Unlike the stop-loss strategy, its performance will get better if the hedge is monitored more frequently. Therefore, we will find out the hedging frequency that maximizes some utility function for both fixed and variable intervals. We will apply the time-based and move-based strategies rebalancing frequency approach for both BSM and JDM model. The remainder of this study is structured as follows. Section 2 discusses the theory behind dynamic hedging. Section 3 reviews the literature on the European hedging performance using different strategies. In Section 4, methodology is presented. Section 5 discusses the simulation analysis results of the European options hedging performance. Section 6 concludes the study