A Dynamic Hedging Strategy for Option Transaction Using Artificial Neural Networks

Transcription

1 A Dynamic Hedging Strategy for Option Transaction Using Artificial Neural Networks Hyun Joon Shin and Jaepil Ryu Dept. of Management Eng. Sangmyung University {hjshin, Abstract In order to avoid the risk caused by continuously changing option value, option issuers generally utilize the traditional Dynamic Delta Hedging (DDH) method. DDH tries to maintain risk-neutral position by adjusting hedge position according to the delta by Black- Scholes (BS) model. DDH, however, is not able to guarantee optimal hedging performance due to some impractical assumptions inherent in BS model. Therefore, this study presents a methodology for dynamic option hedging strategy using artificial neural network (ANN) to enhance hedging performance and shows the superiority of the proposed method through computational experiments. Keywords: Risk-neutral Position; Option; Hedging, Artificial Neural Network, Dynamic Delta Hedging 1. Introduction Traditional parametric approaches to option pricing and hedging have its roots in theory. While the pricing and hedging of some option types adopts numerical procedures such as Monte Carlo simulation or the binomial tree, for other types of option closed form pricing and hedging models have been derived. These pricing and hedging models are based on theoretical arguments using assumptions concerning the behavior of the underlying asset price, the riskless interest rate and so on. It is well known that the traditional parametric pricing and hedging models have systematic biases, owing to the use of simplifying and unrealistic assumptions [1-3]. Machine learning techniques like artificial neural networks (ANN) can be used successfully to estimate a pricing and hedging formula for options, with good out-of-sample pricing and delta-hedging performance. This nonparametric pricing and hedging method has the distinct advantage of not relying on specific assumptions about the underlying asset price dynamics and is therefore robust to specification errors which might adversely affect parametric models [4]. We introduce here a new dynamic hedging strategy via using ANN, which is more than meeting risk-neutral. This strategy tries to minimize the cost of hedging and allows negative cost of hedging, which can be interpreted as profit. In Section 2, we review the previous work in option pricing and hedging using neural networks. Section 3 addresses a methodology for dynamic option hedging with ANN. We describe mathematical formulation for obtaining the optimal hedging value. Section 4 presents experimental evaluation of the proposed strategy compared to DDH method. The paper is concluded with future directions in Section

2 2. Related Work Machine learning methodologies have shown its superiority in many industrial applications [5-8]. Recently machine learning approach has been adopted by financial engineering area. Lin et al. proposed hybrid intelligent approaches for dynamic financial time series predictive model and Hong et al. presented a location strategy for bank using inter-regional financial transaction network [9, 10]. The issues of applying neural networks to option pricing and hedging have been examined by several literatures in recent years. Hanke provided a new method which combines numerical approximation techniques and neural networks used to approximate formulas for option prices and derivatives [11]. White showed that the genetic adaptive neural network is able to approximate, to a high degree of accuracy, the complex, nonlinear option pricing function used to produce the simulated call and put option prices [12]. Hanke compared the pricing accuracy of the BS model to that of option pricing formulas approximated by neural networks. It is explored that even in a framework that is advantageous for the BS model, neural networks prove superior in terms of pricing accuracy [13]. Gencay and Qi studied the various neural network methodologies such as Bayesian regularization, early stopping, and bagging to improve generalization for pricing and hedging derivative securities [14]. Chen and Chang applied the genetic adaptive neural networks to the pricing and hedging of warrants via utilizing the pattern of specific warrants time value and delta behavior and showed that their method excelled the BS model in error degrees on pricing, risk exposure and profits on hedging [15]. Fliess and Join proposed a model-free approach for delta hedging using financial time series, which avoids most of the shortcomings encountered with the BS framework [16]. As seen above, although most literature is available on the pricing of options via neural networks, little attention has been paid to hedging. 3. Dynamic Option Hedging This section describes the procedure for obtaining the target hedging value, which is an alternative of delta by BS model and is proposed to reduce the total cost of dynamic option hedging Target Hedging Value Let Z denote the total cost of hedging and ht hedge value. The problem, THP that finds the optimal hedging values h t to minimize Z is as following: [THP] Minimize Subject to Z T t 1 ( ) { ( t, 0) t t 1} (1) t 2 Z C exp r Q max S X S h Ct Ct 1exp ( r ) QSt ( ht ht 1), t (2) C0 QS0h0 (3) 0 h 1, t (4) t 1 if ST X ht (5) 0 otherwise 112

3 Where S X r T C t Stock price Strike price Risk-free interest rate Maturity Cumulative cost including interest during t Unit balancing period( t ) Q h t Number of stock shares for a written option Hedging variable during t Constraint (1) defines the total cost of hedging that consists of discounted cumulative cost, option payoff and stock share position. Constraints of cumulative cost equation and initial value are given in (2) and (3). The range of hedging variable and the condition at maturity ( T ) are expressed in (4) and (5), respectively. We can solve this problem and get the using optimization solver such as LINGO or EXCEL etc. The value for training ANN. h t ht s will be used as a target S X Basic data Hint r T s H Output Layer Hedging Value Input Layer Hidden Layer 3.2. Dynamic Option Hedging with ANN Figure 1. Multilayer ANN Architecture The ANN of this paper is a feed-forward back-propagation neural netwok and consists of one input layer, one hidden layer and an output layer (see Figure. 1). The network has a number of inputs determined by the number of attributes being used for dynamic option hedging. Each input is scaled by a constant factor that is chosen to be the largest estimate of that attribute in the training data set. The input layer employs a tansig transter function while the hidden layer adopts a logistic activation function. The target output for the network is chosen to be the h t obtained by solving the problem THP. 113

4 The ANN is trained with input data and target data. Input data are weakly KOSPI200 European options obtained from the Korea Exchange for the period November 2000 to September 2009 and consists of 6 types of information such as S, X, r, T (time to maturity), (volatility) and a hint ( H ). To improve the accuracy of training, we introduced a new data ( H ) that gives a hint to ANN regarding the relationship between current stock price and strike price. H is defined as following: t 1 if St X Ht 0 if St X 0.5 otherwize (6) The hint plays a important role in making the ANN take into account the price trend of underlying asset. The best value, which is integer, can be found by trial and error approach. Therefore, ANN is trained with varying values of from 0 to 7, and the best one is selected as final configuration. In this paper, the training of ANN has been carried out using neural network toolbox (nntool) available in MATLAB 7.4 software. 4. Experimental Results We evaluated the performance for the proposed dynamic option hedging strategy with ANN and compared it with that of dynamic delta hedging (DDH) method by BS model in Table 1. Throughout the experiments, as a yardstick for assessing the quality of solutions, we defined a relative deviation percentage (RDP) as follows: ZDDH ZProposed ZDDH ZProposed RDP 100(%) (7) Z DDH denotes the total cost of hedging obtained by using DDH method and denotes the one by the proposed strategy in this paper. Table I shows that the proposed dynamic option hedging strategy with ANN is superior to DDH for all moneynesses at the maturity ITM (in the money), ATM (at the money) and OTM (out of the money). Especially for OTM in training and ATM & OTM in test periods, our strategy shows considerable performance improvement compared to DDH. Table 1. Performance Comparison between Proposed Strategy and DDH Weakly KOSPI200 European Option Training period (Nov 2000 ~ Sep 2009) Test period (Oct 2009 ~ Jul 2010) Moneyness at the maturity ITM ATM OTM ) (3.22) 2) ): average of RDPs, 2): standard deviation of RDPs )

5 5. Conclusions A new dynamic hedging strategy via using ANN was proposed to minimize the total cost of hedging. Mathematical problem model, THP to obtain an optimal hedging value was developed and the optimal hedging value got fed into ANN as a target value. In addition, this paper adopted hint data ( ) to improve learning ability of ANN. The performance comparison was carried out between the proposed strategy in this paper and the traditional DDH based on BS model and showed that our strategy is superior to DDH. Other machine learning algorithms, such as reinforcement learning and support vector machine (SVM), also have attractive features and should be compared in future work with various data of international Exchange such as S&P 500 and NIKKEI 255. References [1] F. Black, Financial Analysts Journal, vol. 31, (1975), pp. 36. [2] M. Rubinstein, Journal of Finance, vol. 40, (1985), pp [3] D. Backus, S. Foresi, K. Li and L. Wu, Stern School of Business, Working Paper, (2004). [4] J. M. Hutchinson, A. W. Lo and T. Poggio, Journal of Finance, vol. 59, (1994), pp [5] A. A. Toptsis, International Journal of Advanced Science and Technology, vol. 5, (2009), pp. 51. [6] F. Kyoomarsi, H. Khosravi, E. Eslami and P. K. Dehkordy, International Journal of Hybrid Information Technology, vol. 2, (2009), pp [7] D. B. A. Mezghani, S. Z. Boujelbene and N. Ellouze, International Journal of Hybrid Information Technology, vol. 3, (2010), pp. 23. [8] K. Sarkar, M. Nasipuri and S. Ghose, International Journal of Database Theory and Application, vol. 4, (2011), pp. 31. [9] H. Y. Lin, H. Y. Chiu, C. C. Sheng and A. P. Chen, International Journal of Smart Home, vol. 2, (2008), pp. 13. [10] J. W. Hong, W. E. Hong and Y. S. Kwak, International Journal of u- and e- Service, Science and Technology, vol. 3, (2010), pp. 21. [11] M. Hanke, Journal of Computational Intelligence in Finance, vol. 5, (1997), pp. 20. [12] A. J. White, Journal of Computational Intelligence in Finance, vol. 6, (1998), pp. 13. [13] M. Hanke, Journal of Computational Intelligence in Finance, vol. 7, (1999), pp. 26. [14] R. Gencay and M. Qi, IEEE Transactions on Neural Networks, vol. 12, (2001), pp [15] A. P. Chen, and C. Chang, Web Journal of Chinese Management Review, vol. 5, (2002), pp. 1. [16] M. Fliess and C. Join, Delta hedging in financial engineering: towards a model-free approach, Proceedings of the 18th Mediterranean Conference on Control and Automation, (2010) May 1-2; Marrakech, Morocco. Authors Hyun Joon Shin He received the B.S., M.S., and Ph.D. degrees from the Department of Industrial Engineering, Korea University, Seoul, Korea in 1995, 1997 and 2002, respectively. He was a postdoctoral researcher in the Department of Industrial Engineering at Texas A&M University, College Station TX, USA and a senior researcher of Samsung Electronics. He is currently an associate professor in the Department of Management Engineering at Sangmyung University, Cheonan, Republic of Korea. His research interests include financial engineering, supply chain management and optimization. 115

6 Jaepil Ryu He received the B.S., and M.S., and Ph.D. degrees from the Department of Management Engineering, Sangmyung University, Choenan, Korea in 2009 and 2001, respectively. He is a Ph.D. candidate. His research interests include financial engineering and optimization. 116