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International Journal of Research in Management, Science & Technology (E-ISSN: 2321-3264) Vol. 3, No. 1, April 2015 Available at www.ijrmst.org Analysis of Price Using Black Scholes and Greek Letters in Derivative European Option Market C. Rajanikanth#1, Dr. E. Lokanadha Reddy*2 # Associate Professor, Department of MBA, Sri Venkateswara College of Engg. & Tech. (Autonomous), Chittoor, Andhra Pradesh. India * Professor of Economics, Sri Venkateswara College of Engg. & Tech. (Autonomous), Chittoor, Andhra Pradesh, India # [email protected] * [email protected] Abstract – The measurement of risk in the market is so crucial work and it cannot be used to denote for future expectations. The market is completely moving with random, wandering and it is efficient in changing of volatility. The Chicago board of trade first introduced derivatives and later it was developed all around the world and most popularly used to hedge the risk. The American option can be sold at any time before maturity, but European can sell at only maturity date, these made the researchers to concentrate to find the price of an option in the future and its sensitivity through Greek letters. This study is considered on Black-Scholes option pricing and risk measure by using Delta, Vega, Theta, Rho, Gamma. Keywords – Black-Scholes, Delta, Vega, Theta, Rho and Gamma. I. INTRODUCTION Black-Scholes model for calculating of an option was introduced in 1973 in a paper entitled, "The Pricing of Options and Corporate Liabilities" published in the Journal of Political Economy. This method developed by famous economists – Fischer Black, Myron Scholes and Robert Merton – is perhaps the most well-known options pricing model. Scholes and Merton were awarded the Nobel Prize in Economics for their work in finding a new method to determine the value of derivatives in the year 1997.B-S model is used to calculate the price of European put and call options. Greek letters in option pricing are commonly used as the sensitivities of an option price relative to changes in the value of either a state variable or a parameter (Hull, 2009). Greek letter measures a different dimension to the risk in an option position and, by analyzing Greek letters, financial institutions can successfully manage their risk. The option hedge ratio defined as the rate of change of option price to the underlying price, Black and Scholes, (1973) model offers an graceful and effective way for option pricing and option hedging 2321-3264/Copyright©2015, IJRMST, April 2015 since it can give an analytic solution for option price, as well as Greek letters, even though this model could make certain pricing bias in realistic market. The blackScholes formula thus has been regarded as a benchmark for option valuation and option hedging, and accepted by many financial professionals including practitioners who seek to manage their risk exposure. Characteristically, option trader would use the Greek letters under Black-Scholes framework as a benchmark for properly adjusting option position so that all risks are manageable. II. REVIEW OF LITERATURE Xisheng Yu and Xiaoke Xie(2013): Greek letter measures the sensitivity of an option price with respect to the change in the value of a given underlying parameter such as underlying asset’s price, value, time etc. Analysis of Greek letters for European call and put options within the Black-scholes. Hong-Yi Chen, Cheng-Few Lee and Weikang Shih (2008): Explained about the derivations of Greek letters for call and put options on both dividends-paying stock and nondividends stock and the relationship between Greek letters. Beata Stehlikova: Explained sensitivities by using binomial option pricing model and relation with different Greeks. Dr. S. Saravanan and G. Pradeep Kumar (2012): Accuracy of the Black-Scholes option pricing model with relation market prices and estimated the stock option contracts prices helps in finding whether the stock options are properly priced. Sanjana Juneja(2013):Trading options and option strategies are based on risk factors and it can be predicted by calculating Black-Scholes and its Greeks. III. 34 STATEMENT OF THE PROBLEM International Journal of Research in Management, Science & Technology (E-ISSN: 2321-3264) Vol. 3, No. 1, April 2015 Available at www.ijrmst.org The volatility is considered an important function, this influence the buyer and sellers in option market. The indifference curves of investor are the dominating factor in purchasing and selling of Call and Put option. These calculations provides the solution to this problem by calculating B-S Model and its Greek letters. Delta e q N(d1 ) eq N(d1 ) 1 Theta IV. OBJECTIVES OF THE STUDY 1 .Analyzing price by using black-scholes model and its sensitivities. 2. Analyzing volatility and guide investors to hedge risk by estimating future price. IV. St s N(d1 ) rX e r N(d 2 ) 2 St s N(d1 ) rX e r N(d 2 ) 2 METHODOLOGY The research Study is executed by calculating the option prices using the Black-Scholes option pricing model for the Call option and Put options and its sensitivities. The secondary data collected from BSE and used Single Strike Price Movement formulas to calculate option prices and its sensitivities. Samples are taken from top most 5 companies and assuming that 10 percent is risk free and volatility is 10 percent respectively on nondividend paying stock. Gamma 1 N d1 St s 1 N d1 St s Vega Formulas/Models Used: Black-Scholes Model St N d1 Ct S t N d1 Xe r N d 2 Rho Pt Xe N d2 St N d1 rho X e r N(d 2 ) r s S ln t r 2 X d2 s , rho X e r N(d 2 ) 2 S ln t r s 2 X d1 2 s St N d1 d 1 s VI. RESULT ANALYSIS Collected data is analyzed using the above mentioned models and results drawn are tabulated as follows. TABLE-1 CALCULATION OF PREMIUM AND ITS SENSITIVITIES Companies Particulars Price of the underlying Risk-free interest rate (%) R : risk free rate of interest Strike price Annual volatility (%) 2321-3264/Copyright©2015, IJRMST, April 2015 Apollo Tyers Call Option Bank Baroda Put Option Bosch Ltd Call Option Sun TV Put Option Ambuja Cement Ltd Call Option 173.75 10.0 0.1 180 10.0 26222.20 10.0 0.1 26200 10.0 177.20 10.0 0.1 180 10.0 434.25 10.0 0.1 450 10.0 260.90 10.0 0.1 280 10.0 35 International Journal of Research in Management, Science & Technology (E-ISSN: 2321-3264) Vol. 3, No. 1, April 2015 Available at www.ijrmst.org o : volatility Time to expiration (days left) T-t : time to expiration Price of Call Option Price of Put Option Delta for Call Option Delta for Put Option Theta for Call Option Theta for Put Option Gamma for Call Option Gamma for Put Option Vega for Call Option Vega for Put Option Rho for Call Option Rho for Put Option 0.1 15 0.0411 0.10 5.61 0.063 -0.937 -0.017 0.032 0.035 0.035 0.044 0.044 0.004 -0.069 From the above table, some of the inferences drawn are mentioned below. 1. It was observed that Apollo Tyers call option value 0.10 and put value 5.61.Delta 0.063 for call showing market is increasing and for put value -0.937 and showing underlying is increasing. Theta value -0.017 and 0.032 because time to expiry is near. Gama for both call and put is 0.035 based on time to expiry. Vega is positive 0.044 when volatility increases. Both call and put value 0.004 and -0.069 changes very small friction based on 1 unit change in interest rate. 2. It was observed that Bank Baroda call option value 197.66 and put option 118.10.Delta 0.584 for call showing market is increasing and for put value-0.416 and showing underlying is increasing. Theta value 13.606 and -6.444 because time to expiry is near. Gama for both call and put is 0.001 based on time to expiry. Vega is positive 15.141 when volatility increases. Both call and put value 3.314 and -2.416 changes very small friction based on 1 unit change in interest rate. 3. It was observed that Bosch Ltd call option value 0.64 and put option 2.70.Delta 0.288 for call showing market is increasing and for put value-0.712 and showing underlying is increasing. Theta value -0.055 and -0.005 because time to expiry is near. Gama for both call and put is 0.095 based on time to expiry. Vega is positive 0.122 when volatility increases. Both call and put value 0.021 and -0.053 changes very small friction based on 1 unit change in interest rate. 4. It was observed that Sun TV call option value 1.07 and put option 13.26.Delta 0.167 for call showing market is increasing and for put value-0.833 and showing underlying is increasing. Theta value -0.072 and 0.050 because time to expiry is near. Gama for both call and put is 0.020 based on time to expiry. Vega is 2321-3264/Copyright©2015, IJRMST, April 2015 0.1 0.1 0.1 0.1 8 15 29 1 0.0219 0.0411 0.0795 0.0027 197.66 0.64 1.07 0.00 118.10 2.70 13.26 19.02 0.584 0.288 0.167 0.000 -0.416 -0.712 -0.833 -1.000 -13.606 -0.055 -0.072 -0.000 -6.444 -0.005 0.050 -0.077 0.001 0.095 0.020 0.000 0.001 0.095 0.020 0.000 15.141 0.122 0.306 0.000 15.141 0.122 0.306 0.000 3.314 0.021 0.057 0.000 -2.416 -0.053 -0.298 -0.008 positive 0.306 when volatility increases. Both call and put value 0.057 and -0.298 changes very small friction based on 1 unit change in interest rate. 5. It was observed that Sun TV call option value 0.00 and put option 19.02.Delta 0.000 for call showing market is increasing and for put value-1.000 and showing underlying is increasing. Theta value -0.000 and -0.077 because time to expiry is near. Gama for both call and put is 0.000 because one day maturity. Vega is 0.000 when volatility low .Both call and put value 0.000 and -0.008 changes very small friction based on 1 unit change in interest rate. VII. SUGGESTIONS The investor should wait for a time to increase in underlying value to make profits in Apollo Tyres. Stike price should increase in Bank Baroda. The investor is advised to purchase a call option in Bosch Ltd and Amudha Cement Ltd. The investor can generate in-the-money in Sun TV option . By calculating sensitivities one can evaluate the price of option exactly. VIII. CONCLUSIONS Each company values are different it operates with demand factors for a particular industry. The options, either call or put in European type moves with nonlinear payoff for both parties. This makes the investors to understand how to price an option strategically and make in-the-money in the option market. The BlackScholes model gives you a price of option for forward, based on some parameters (spot price, strike price, interest, volatility etc).The Greek letters are used to understand to identify the market price fluctuation or simply it is used to calculate risk sensitivities towards 36 International Journal of Research in Management, Science & Technology (E-ISSN: 2321-3264) Vol. 3, No. 1, April 2015 Available at www.ijrmst.org price change. This paper educated the investors how to behave in the option market. REFERENCES [1] [2] [3] [4] John C. Hull “Options, Futures and other derivatives” Prentice Hall, Seventh edition. Black and Scholes, “The pricing of options and corporate liabilities” Journal of political economy, May 1973. Xisheng Yu and Xiaoke Xie” On Derivations of Black-Scholes Greek Letters” ISSN 2222-1697 (Paper) ISSN 2222-2847 (Online) Vol.4, No.6, 2013. Hong-Yi Chen, Rutgers University, Cheng-Few Lee, Rutgers University, and Weikang Shih, Rutgers University, USA 2321-3264/Copyright©2015, IJRMST, April 2015 [5] [6] [7] [8] [9] 37 “Derivations and Applications of Greek Letters – Review and Integration” 2008. Beata Stehlikova ”Black-Scholes model: Greeks - sensitivity analysis” 2015. Dr. S. Saravanan and G. Pradeep Kumar “Estimation of Stock Option Prices Using Black-Scholes Model” Volume No. 2 (2012), Issue No. 11 (Nov) ISSN 2231-5756. Sanjana Juneja “Understanding the greeks and their use to measure risk” Volume No. 3 (2013), Issue No. 10 (Oct) ISSN 2231-5756. www.nscindia.com. www.bscindia.com