Briefly state the Black-Scholes result on calculation of option premium. How did Merton extend this?

Briefly state the Black-Scholes result on calculation of option premium. How did Merton extend this? 

Ans. The Black-Scholes model for calculating the premium 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. The formula, developed by three economists 

–Fischer Black, Myron Scholes and Robert Merton–is perhaps the world’s most well-known options pricing model. Black passed away two years before Scholes and Merton were awarded the 1997 Nobel Prize in Economics for their work in finding a new method to determine the value of derivatives (the Nobel Prize is not given posthumously; however, the Nobel committee acknowledged Black’s role in the Black-Scholes model). The Black-Scholes model is used to calculate the theoretical price of European put and call options, ignoring any dividends paid during the option's lifetime. While the original Black-Scholes model did not take into consideration the effects of dividends paid during the life of the option, the model can be adapted to account for dividends by determining the ex-dividend date value of the underlying stock. 

– The model makes certain assumptions, including: 

– The options are European and can only be exercised at expiration 

– No dividends are paid out during the life of the option 

– Efficient markets (i.e., market movements cannot be predicted)

– No commissions 

– The risk-free rate and volatility of the underlying are known and constant 

– Follows a lognormal distribution; that is, returns on the underlying are normally distributed. 

– The formula, shown in Figure 16.4, takes the following variables into consideration: 

– Current underlying price 

– Options strike price 

– Time until expiration, expressed as a per cent of a year Implied volatility Risk-free interest rates

C = Call premium 

S = Current stock price 

t = Time until option exercise 

K = Option striking price 

r = Risk-free interest rate 

N = Cumulative standard normal distribution 

e = Exponential term 

s = St. Deviation 

In = Natural Log

Merton explained that there is a striking similarity between options and loans. The payoff to a loan holder is equal to the pay off to an option holder. It means that when a bank makes a loan, its payoff is isomorphic to writing a put option on the asses of the borrower. Therefore we can use Black Scholes Merton valuation for risky loans.