Modern option pricing models are usually regarded as some of the most complex mathematical models in applied finance. However, these models have enabled financial analysts and savvy investors to calculate the value of stock options with alarming accuracy. Most of the modern option pricing models employed in determining the value of stock options are rooted in the Black and Scholes model which was developed by Myron Scholes and Fisher Black. The Black and Scholes model was first published in 1973 in the Journal of Political Economy in a paper titled “The Pricing of Options and Corporate Liabilities”. Although the Black and Scholes model has been criticized by several scholars in finance, there is no doubt that the model transformed the face of finance especially with regards to pricing of options. The Black and Scholes model has not only passed the market test, but has also resulted in a tremendous increase in derivatives trade worldwide.
Introduction
Black and Scholes pricing model is a model that was developed by Scholes and Black in 1973. It is useful in pricing both call and put options. The model assumes that the option can only be exercised when it expires and at the exercise price only. This model also assumes that the risk-free return rate and the price volatility will remain constant throughout the period. This model also assumes that there are no dividend payments. The Black and Scholes model has been criticized basing of these assumptions. Black and Scholes pricing model is given by;
C = SN (d1) –Xe-rTN (d2)
Where;
C is the call option price
S is the current stock price
X is the exercise price of the option
r is the risk-free rate of interest
T is the time to expiration
N is the area under the normal curve
d1 = {ln (S/X) + (r+δ2/2) T}/δT1/2
d2 = d1 - δT1/2
δ is the standard deviation
ln is the natural logarithm
This paper intends to calculate the call option’s price given the values of the five variables that influence the price of options.
C = SN (d1) –Xe-rTN (d2)
Where;
C is the call option price
S is the current stock price which is $60
X is the exercise price of the option which is $60
r is the risk-free rate of interest which is 12%
T is the time to expiration which is 6 months = 0.5 years
N is the area under the normal curve
d1 = {ln (S/X) + (r+δ2/2) T}/δT1/2
d2 = d1 - δT1/2
δ is the standard deviation which is √0.09 = 0.3
ln is the natural logarithm
d1 = {ln (60/60) + (0.12+0.09/2) 0.5}/0.3*0.51/2 = 0.39
d2 = 0.778 – 0.3*0.51/2 = 0.18
N (d1) = 0.6517
N (d2) = 0.5714
C = 60(0.6517) –60e-0.12*0.5(0.5714)
C = 6.81
Conclusion
Black and Scholes pricing model is an options pricing model that was developed in 1973. It is useful in pricing both call and put options. The model assumes that the price of call options is influenced by five variables; the current stock price, the time to expiration, the exercise price, the risk-free rate of interest and the underlying stock price volatility. All the variables have a direct relationship with call option’s price apart from the exercise price which has an inverse relationship. This model has several assumptions. The model assumes that the option can only be exercised when it expires and at the exercise price only. This model also assumes that the risk-free return rate and the price volatility will remain constant throughout the period. This model also assumes that there are no dividend payments.
References
Brigham, E. F., & Ehrhardt, M. C. (2010). Financial Management Theory and Practice (13 ed.). London: Cengage Learning.
Kim, S. H., & Kim, S. H. (2006). Global corporate finance: text and cases (6, illustrated ed.). New York: John Wiley & Sons.
Kolb, R., & Overdahl, J. A. (2009). Financial Derivatives: Pricing and Risk Management (illustrated ed.). New York: John Wiley & Sons.
Shim, J. K., & Siegel, J. G. (2008). Financial Management (3, illustrated, revised ed.). New York: Barron's Educational Series.
