Effects of Parameters on Black Scholes Model for European Put option Using Taguchi L27 Method

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1 Volume 119 No , ISSN: (on-line version) url: ijpam.eu Effects of Parameters on Black Scholes Model for European Put option Using Taguchi L27 Method Amir Ahmad Dar 1 and N. Anuradha 2 1 Department of Mathematics and Actuarial Science. 2 Department of Management Studies. 1,2 B S Addur Rahman Crescent Institute of Science and Technology Chennai. sagaramir200@gmail.com Abstract The European Black Scholes formula is a mathematical model used to estimate the fair price of options (call and put) based on the five factors without premium such as the underlying asset S 0 at time t = 0, the interest rate r, the strike price K, the time period T, and the volatility σ. In this present study, the aim is to investigate the effects of five factors on the put option. In order to explore the effects of input factors, the Taguchis orthogonal array, Analysis of Mean (ANOM), and Analysis of Variance (ANOVA) is used and the analysis is carried out using MINITAB software. The ANOM is used in order to identify which input factor effects more or less on put option and the percentage contribution of each input factor on put option is estimated by conducting ANOVA. From the result, it is observed that influencing parameters (input parameters) are in order of underlying asset, strike price, volatility, time period, and interest rate. AMS Subject Classification: 05B30, 47N40, 05B15 Key Words and Phrases: Black Scholes Model, Taguchi method, ANOM, ANOVA 1 11

2 1 Introduction The Black Scholes Merton model (also known as Black-Scholes model) is a very famous model in option pricing. It is widely used model for estimating the theoretical European option using the parameters such as S 0, K, r, σ, and T. The European style option only exercises only at expiry date. The Black Scholes model was developed by the three great economists - the Fischer Black, the Myron Scholes and the Robert Merton in The Scholes and Merton were awarded the Nobel price in 1977 (Black passed away two years before Scholes and Merton were awarded the Nobel price). The main assumptions of European option are: the option will expire only at expiry date and the dividends are not paid during the life of an option (see [1], [2]). This study investigates which input parameter effects more or less on European put option. In recently study Dar and Anuradha (2018) used the Taguchi L9 method in order to identify which parameter effects more or less on the Probability of default. The Black-Scholes formula for European call option were used in order to estimate the probability of default. It was investigated that the volatility affects more and interest rate affects less on the Probability of default (see[3], [4]). Also they used the Taguchi approach to identify which input factor effects more on European call option by using the ANOVA, ANOM, Tukey and Signal to noise ratio. It was identified that the underlying asset effects more on call option [5]. The Taguchi method is a design of the experiment, whose aim is to develop the relationship between the inputs (underlying asset, strike price, volatility, time period, and interest rate) and the response (put option) variables. The Taguchi method is a design of experiment (DOE) was developed by a Japanese engineer Genichi Taguchi in 1986 [6]. It is a famous statistical model (sometimes called robust design method) and also to improve the quality of manufactured goods. It is because as compared to other DOE models the number of experiments is fewer. For example, if we talk about the full factorial design (FFD), at three levels when the input parameters are five, it would take 243 trials because it will take all possible combinations but if we talk about the Taguchi method, at three levels when the input parameters are five, it would take only 27 trials. The experiments are not randomly generated but they are based on judgmental sam- 2 12

3 pling. It reduces time, resources and cost that is why the industry is using the Taguchis method in order to reduce the time, cost and sources. The Taguchi orthogonal array design is using to organise the input factorss affecting the process and the levels of which they should be varied. The Taguchi method applied to biotechnology, engineering and marketing, Physics etc. In order to identify which parameter effects more on European put option, we need a data that is given in table 1. The data that are given in table 1 are sufficient to estimate the value of the put option by using the Black Scholes formula (equation 1) [7]. In order to develop a DOE by using Taguchi s method. The following points are necessary: 1. Define the response variable, in this study the response variable is the value of the European put option. 2. Select the input variables, there are five input variables that are: underlying asset, strike price, interest rate, time period and volatility in order to estimate the value of put option. 3. Select the number of levels, in this study we choose three levels as shown in table Select the orthogonal array, it is based on the 2nd and 3rd point. In this study we want to conduct an experiment on put option in order to understand the influence of five independent with each having three set values on a put option, then L27 orthogonal array might be the right choice. 5. Assigning the five independent variables to each column. 6. Conduct the experiment. 7. Analysis the data (the put option). Therefore, it is necessary for an option holder to know which parameter effects more and how much it effects on the put option. In order to know about it, the ANOM describes the best combinations of the parameters where the value of European put option is maximum and also it recognize which input parameter affects more significantly on the put option. The ANOVA used in order to measure the percentage contribution of each input factor on the put option. 3 13

4 2 Taguchi Method a Design of Experiment (DOE) Taguchi method is a popular statistical model developed by G. Taguchi and Konishi [8]. At starting it was used for only improving the quality of products (mainly manufactured goods). Nowadays it is used in every field in order to minimize the number of trials, time, cost and resources. This method is based on the orthogonal array experiments which give a much-reduced variance for the experiment with the optimum setting. Taguchi orthogonal array design is a type of design that is based on a design matrix and it allows you to consider a selected subset of combinations of various factors at different levels. It is the balanced and ensures that all levels of all parameters are considered equally. In this study, the five parameters are varied at three levels and on the bases of levels and parameters, the orthogonal array L27 is selected. Before conducting the experiment it is necessary to know about European put option and how to calculate the European put option. European Put option: It is a contract that gives rights to owner, but not obligation to sell the underlying asset S 0 at a specified price (strike price K) within a specified time T. It will not exercise before the maturity date. The buyer of the put option believes that the price of an underlying asset goes down in future date. The buyer of the put option will decide whether to exercise or not because he is having the rights. At the end of period the two possibilities will happen, a) At the expiry date T, if the price of the underlying asset S T is less than strike price K. Then the put option is exercised, i.e. the holder buys the underlying asset from the market price S T and sell it to the writer at price K. b) If the price of an underlying asset is greater than the strike price, then he will not exercise it (see [9], [10]). The payoff at time t=1 is: Max (K-S T,0) The Black and Scholes developed a formula in order to estimate the values of European call and put option in The Black-Scholes formula for European put option without dividend paying is: p = Kexp( rt)n( d 2 ) S 0 N( d 1 ) [1] Where N(*) is the standard cumulative distribution function 4 14

5 d 1 = (ln(s 0 /K) + (r + σ 2 /2)T )/(σ T ) d 2 = d 1 σ T The experiment runs with five parameters at three levels are determined by using the Taguchi L27 orthogonal array. The MINITAB software determines 27 trails instead of 243 as per FFD. The five parameters in this study are: underlying asset S 0, strike price K, interest rate r, time period T and volatility σ at three levels are summarized in table 1. The data that is given in table 1 is enough to calculate the value of European call option by using equation (1). Table 1: Selected process parameters and levels Levels Parameters S 0 K r σ T % 10% % 20% % 30% 3 The Taguchi L27 orthogonal array approach is appropriated for experimentation and the experimental matrix along with result (value of European put option using equation (1) is shown in table 2. Table 2: Taguchi L27 experimental design matrix with result Experiment Number S 0 K r σ T put option V alue % 10% % 10% % 10% % 20% % 20% % 20% % 30% % 30% % 30% % 30% % 30% % 30% % 10% % 10% % 10% % 20% % 20% % 20% % 20% % 20% % 20% % 30% % 30% % 30% % 10% % 10% % 10%

6 3 Result and Analysis 3.1 Analysis of Mean (ANOM) The response mean (ANOM) is the average response for each combination of control parameters (factors) levels in a statistic Taguchi method. The aim of this method is to recognize which input parameter effects more significantly on put option and also it determines the top mixture/combination where the European put option gets maximum value. The delta identifies the size of effect by the taking the difference between the highest and the lowest value of average for a parameter and the rank in the response table 3 helps us to identify which parameter effects more. The parameter with the highest delta value is given rank 1, the parameter with the second highest delta is given rank 2, and so on Table 3: Response Table for Means Level S 0 K r σ T Delta Rank In table 3, the rank is based on the delta (Range = maximumminimum), higher delta means the higher efficiency. The ranking for all the input factors given in table 3. The selected numbers (bold) are the maximum in every column, it gives us the best combination. We conclude that the best combination is S 0 1*K3*r1*σ3*T1. This combination gives the maximum value of put option. If you take any possible combination from table 1, you will not get a better combination than S 0 1*K3*r1*σ3*T1. This is the only best combination where you will get the maximum value. 3.2 Analysis of Variance (ANOVA) Analysis of Variance (ANOVA) The percentage contribution comes from adding up the total adjusted sum of squares (Adj SS) and then taking each terms Adj SS and dividing by the total to get a percentage i.e. P ercentage contribution = Adj SS / total Adj SS 6 16

7 Table 4: Percentage contribution of each parameter on put option Source Adj SS Parentage contribution Rank S % 1 K % 2 r % 5 σ % 3 T % 4 Total % The percentage contribution of each parameter on put option is given in table 4. The percentage contribution of S 0, r, K, σ and T are percent, 2.27 percent, percent, 29.38percent, and 2.47 percent respectively. 4 Conclusion This study investigated that which input parameter effects more/less on European put option. In general 3 5 = 243 trials were supported to be conducted. However, only 27 trials were done bt the help of Tauchi method. The ANOM table shows us that the underlying asset S 0 is having higher delta, so it is given as rank 1. It means that underlying asset affects more on European put option. Also it described the best combination of five input factors in order to get the maximum value of the put option. The best combination of parameters is S 0 1*K3*r1*σ3*T 1. It also indicates that, the buyer will get maximum the value of put option when he/she have lower a underlying asset, higher strike price, lower interest rate, highest volatility and lower time period. The ANOVA measure the percentage contribution of each input factor and the percentage contribution of S 0, K, r, σ and T are percent, percent, 2.27 percent, percent, and 2.47 percent respectively. Both the models ANOM and ANOVA displayed the same result, that underlying asset effects more and interest rate effects less on put option. Assumption: The percentage contribution/rank will vary with the change in data set. References [1] Black, F., and Scholes, M., The pricing of options and corporate liabilities, Journal of political economy,81(3), (1973),

8 [2] Merton, R C., On the pricing of corporatedebt: The risk structure of interest rates, The Journal of Finance, 29(2), (1974), [3] Dar, A and Anuradha., Use of orthogonal arrays and design of experimanar via Taguchi L9 method in probability of default, Accounting, 4(3), (2018), [4] Dar, A. A., Anuradha, N., Afzal, S., Design of experiment on probability of default (PD). International journal of pure and Applied Mathematics, 118(10),(2018), [5] Dar, A., and Anuradha, N., An application of Taguchi l9 method in black scholes model for European call option, International Journal of Entrepreneurship, 21(3) (2017) in press. [6] Taguchi, G. Introduction to quality engineering: designing quality into products and processes. (1986). [7] Dar, A. A., Anuradha, N., Probability Default in Black Scholes Formula: A Qualitative Study. Journal of Business and Economic Development, 2(2), (2017), [8] Taguchi, G., Konishi, S., Taguchi Methods: Orthogonal Arrays and Linear Graphs-Tools for Quality Engineering, American Supplier Institute, Center for Taguchi Methods (1987). [9] Hull, J. C., Basu, S., Options, futures, and other derivatives, Pearson Education India, (2016). [10] Dar, A. A., Anuradha, N., One Period Binomial Model: The risk-neutral probability measure assumption and the state price deflator approach, International journal of mathematics Trends and Technology, 43(4), (2017),

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