# Black Scholes Model

In their 1973 paper, The Pricing of Options and Corporate Liabilities, Fischer Black and Myron Scholes published an option valuation formula that today is known as the Black-Scholes model. It has become the standard method of pricing options. The Black-Scholes model is a tool for equity options pricing. Options traders compare the prevailing option price in the exchange against the theoretical value derived by the Black-Scholes Model in order to determine if a particular option contract is over or under valued, hence assisting them in their options trading decision. This model is based on following Assumptions:

1. The rates of return on a share are log normally distributed. 2. The value of the underlying share and the risk free rate are constant during the life of the option. 3. The market is efficient and there are no transaction costs and taxes. 4. There is no dividend to be paid on the share during the life of the option.

The Black-Scholes formula calculates the price of a call option to be:

C = S N(d1) - X e-rT N(d2)

where

| C = price of the call option|

| S = price of the underlying stock|

| X = option exercise price|

| r = risk-free interest rate|

| T = current time until expiration|

| N() = area under the normal curve|

| d1 = [ ln(S/X) + (r + σ2/2) T ] / σ T1/2|

| d2 = d1 - σ T1/2 |

Put-call parity requires that:

P = C - S + Xe-rT

Then the price of a put option is:

P = Xe-rT N(-d2) - S N(-d1)

Let us take a simple example to understand this better:

I am interested in writing a six months call option on a particular share, which is currently selling for Rs 120. The volatility of the share returns is estimated as 67 per cent. I would like the exercise price to be Rs 120. The risk free rate is assumed to be 10 percent. How much premium should I charge for writing the call option? Sol : Let us first calculated d1 and d2 :...

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