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

Efficient option pricing with transaction costs

A fast numerical algorithm is developed to price European options with proportional transaction costs using the utility-maximization framework of Davis (1997). This approach allows option prices to be computed by solving the investor’s basic portfolio selection problem without insertion of the option payoff into the terminal value function. The properties of the value function can then be used to drastically reduce the number of operations needed to locate the boundaries of the no-transaction region, which leads to very efficient option valuation. The optimization problem is solved numerically for the case of exponential utility, and comparisons with approximately replicating strategies reveal tight bounds for option prices even as transaction costs become large. The computational technique involves a discrete-time Markov chain approximation to a continuous-time singular stochastic optimal control problem. A general definition of an option hedging strategy in this framework is developed. This involves calculating the perturbation to the optimal portfolio strategy when an option trade is executed

Migration, credit markets, moral hazard, interlinkage.

A fast numerical algorithm is developed to price European options with proportional transaction costs using the utility maximization framework of Davis (1997). This approach allows option prices to be computed by solving the investor's basic portfolio selection problem, without the inser- tion of the option payo into the terminal value function. The properties of the value function can then be used to drastically reduce the number of operations needed to locate the boundaries of the no transaction region, which leads to very e cient option valuation. The optimization problem is solved numerically for the case of exponential utility, and comparisons with approximately replicating strategies reveal tight bounds for option prices even as transaction costs become large. The computational tech- nique involves a discrete time Markov chain approximation to a continuous time singular stochastic optimal control problem. A general de nition of an option hedging strategy in this framework is developed. This involves calculating the perturbation to the optimal portfolio strategy when an option trade is execute

Option pricing with transaction costs using a Markov chain approximation

An efficient algorithm is developed to price European options in the presence of proportional transaction costs, using the optimal portfolio framework of Davis (in: Dempster, M.A.H., Pliska, S.R. (Eds.), Mathematics of Derivative Securities. Cambridge University Press, Cambridge, UK). A fair option price is determined by requiring that an infinitesimal diversion of funds into the purchase or sale of options has a neutral effect on achievable utility. This results in a general option pricing formula, in which option prices are computed from the solution of the investor's basic portfolio selection problem, without the need to solve a more complex optimisation problem involving the insertion of the option payoff into the terminal value function. Option prices are computed numerically using a Markov chain approximation to the continuous time singular stochastic optimal control problem, for the case of exponential utility. Comparisons with approximately replicating strategies are made. The method results in a uniquely specified option price for every initial holding of stock, and the price lies within bounds which are tight even as transaction costs become large. A general definition of an option hedging strategy for a utility maximising investor is developed. This involves calculating the perturbation to the optimal portfolio strategy when an option trade is executed

Hedging in fractional Black-Scholes model with transaction costs

We consider conditional-mean hedging in a fractional Black-Scholes pricing model in the presence of proportional transaction costs. We develop an explicit formula for the conditional-mean hedging portfolio in terms of the recently discovered explicit conditional law of the fractional Brownian motion

Option Pricing with Transaction Costs Using a Markov Chain Approximation

An e cient algorithm is developed to price European options in the pres- ence of proportional transaction costs, using the optimal portfolio frame- work of Davis (1997). A fair option price is determined by requiring that an in nitesimal diversion of funds into the purchase or sale of options has a neutral e ect on achievable utility. This results in a general option pricing formula, in which option prices are computed from the solution of the investor's basic portfolio selection problem, without the need to solve a more complex optimisation problem involving the insertion of the op- tion payo into the terminal value function. Option prices are computed numerically using a Markov chain approximation to the continuous time singular stochastic optimal control problem, for the case of exponential utility. Comparisons with approximately replicating strategies are made. The method results in a uniquely speci ed option price for every initial holding of stock, and the price lies within bounds which are tight even as transaction costs become large. A general de nition of an option hedg- ing strategy for a utility maximising investor is developed. This involves calculating the perturbation to the optimal portfolio strategy when an option trade is executed

Conditional-Mean Hedging Under Transaction Costs in Gaussian Models

We consider so-called regular invertible Gaussian Volterra processes and derive a formula for their prediction laws. Examples of such processes include the fractional Brownian motions and the mixed fractional Brownian motions. As an application, we consider conditional-mean hedging under transaction costs in Black-Scholes type pricing models where the Brownian motion is replaced with a more general regular invertible Gaussian Volterra process.Comment: arXiv admin note: text overlap with arXiv:1706.0153