Black-Scholes Option Pricing Model

The Black-Scholes Option Pricing Model is an approach for calculating the value of a stock option. This article presents some detail about the pricing model. 
Model didn't appear overnight, in fact, Fisher Black started out working to create a valuation model for stock warrants. This work involved calculating a derivative to measure how the discount rate of a warrant varies with time and stock price. The result of this calculation held a striking resemblance to a well-known heat transfer equation. Soon after this discovery, Myron Scholes joined Black and the result of their work is a startlingly accurate option pricing model. Black and Scholes can't take all credit for their work, in fact their model is actually an improved version of a previous model developed by A. James Boness in his Ph.D. dissertation at the University of Chicago. Black and Scholes' improvements on the Boness model come in the form of a proof that the risk-free interest rate is the correct discount factor, and with the absence of assumptions regarding investor's risk preferences.
The Black-Scholes formula for the price of a call option is:

C = S * N(d1) - K * (e ^ -rt) * N (d2)

and put option price is:

P = K * (e ^ -rt) * N (d2) - S * N(-d1)

 

     ln (S / K) + (r + (sigma) ^ 2 / 2) * t
d1 = --------------------------------------
              sigma * sqrt(t)

d2 = d1 - sigma * sqrt(t)


Where:
C = theoretical call premium
S = current stock price
N = probability that a value less than “x” will occur in a standard normal distribution
t = time until option expiration
r = risk-free interest rate
K = option strike price
e = the constant 2.7183..
sigma = standard deviation of stock returns (usually written as lower-case 's')
ln() = natural logarithm of the argument
sqrt() = square root of the argument
^ means exponentiation (i.e., 2 ^ 3 = 8)

In order to understand the model itself, we divide it into two parts. The first part, SN(d1), derives the expected benefit from acquiring a stock outright. This is found 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 model, K(e^-rt)N(d2), gives the present value of paying the exercise price on the expiration day. The fair market value of the call option is then calculated by taking the difference between these two parts.
The Black-Scholes Model makes the following assumptions.

  • The stock pays no dividends during the option's life
  • European exercise terms are used
  • Markets are efficient
  • No commissions are charged
  • Interest rates remain constant and known
  • Returns are log-normally distributed

The Black-Scholes model is used to calculate a theoretical call price (ignoring dividends paid during the life of the option). The model is based on a normal distribution of underlying asset returns.  A lognormal distribution has a longer right tail compared with a normal, or bell-shaped, distribution. The lognormal distribution allows for a stock price distribution of between zero and infinity (i.e. no negative prices) and has an upward bias (representing the fact that a stock price can only drop 100% but can rise by more than 100%).

Like the rest of the option pricing models, σ (the volatility of stock price) is the most critical parameter in Black-Scholes model. It is obvious to think that as the volatility increases for a stock, the price of a call option will also increase, along with the risk. Apart from the volatility the key determinants for pricing options are stock price S, strike price K, time to expiration T, and short-term (risk free) interest rate r. Similar to volatility, higher interest rates yield higher risk and higher call option prices. And as the strike price gets closer to stock price, call option price should be higher. And finally the longer the time to maturity, the higher the option price.

Following demonstrates the direction of movement of option prices when determinants are changed.

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Views about the future direction of a stock (i.e. whether it will go up or down in the future and by how much) are completely irrelevant to the option pricing. Significantly, the expected rate of return of the stock is not one of the variables in the Black-Scholes model (or any other model for option valuation). The important implication is that the value of an option is completely independent of the expected growth of the underlying asset (and is therefore risk neutral).

Here's a link to an excel file which dynamically shows the outcome of a call option price given different parameters.

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