Black Scholes Option Pricing Made Easy!

The Black and Scholes Option Pricing Model

C=SN(d1)-Ke^(-rt)N(d2)

P=Ke^(-rt)N(-d2)-SN(-d1)

C=Theoretical Call Premium
P=Theoretical Put Premium
S=Current Stock Price
t=time until option expiry
K=option strike price
r=risk-free interest rate
N=cumulative standard normal distribution
e=expoential term (2.7183)
d1=(ln(S/K)+(r+s^2/2)t)/sSQRT(t)
d2=d1 - sSQRT(t)
s=standard deviation of stock returns

Delta = N(d2)

The model is divided into two parts. The first part, SN(d1) is the expected benefit from acquiring a stock outright. Derived by multiplying stock price [S] by the change in the call premium with respect to a change in the underlying stock price [N(d1)].

The second part of the equation, Ke(-rt)N(d2), gives the current value of paying the exercise price on the expiry day. The fair market value of the call option is then calculated by calculating the difference between these two parts.

Assumptions:

1) The stock pays no dividends during the option or warrant's life

Many companies pay dividends to their share holders, so this is a limitation, but the model may be adjusted by subtracting the discounted value of a future dividend from the stock price.

2) European exercise terms are used

i.e. option can only be exercised on the expiration date. American exercise terms allow the option to be exercised at any time during the life of the option, making american options more valuable due to their greater flexibility. Fortunately few calls are ever exercised before the last few days of their life, because exercising a call option early forfeits the remaining time value on the option.

3) Markets are efficient

Markets are random and people cannot consistently predict the direction of a market or stock.

4) No commissions are charged

5) Interest rates remain constant

The Black and Scholes model uses the risk-free rate to represent this rate e.g. U.S. Government Treasury Bills with 30 days left until maturity is often used to represent it.

6) Returns are lognormally distributed

i.e. returns on the underlying stock are normally distributed (Gaussian bell curve distribution).

So, to put it simply the value is determined by assuming the market is moving randomly. The probability of a price move is proportional to the volatility of the market and follows a Gaussian distribution (normal) Small moves are very probable and large moves far less probable.

If the strike price is much higher than the current price and the end-date not far away the option is almost worthless, but if the remaining time is long and the strike price below the current price, then the value is high. If risk-free interest rates are low then options increase in value whereas when high they become less attractive. If the volatility of the market is high there is more chance of reaching the strike-price so the value goes up.