### Black-Scholes Option Pricing Model

A mathematical model of a financial market containing certain derivative investment instruments. From the model, one can deduce the Black-Scholes formula, which gives the price of European-style options. It is a partial differential equation, now called the Black-Scholes equation, which governs the price of the option over time. The key idea behind the derivation was to hedge perfectly the option by buying and selling the underlying asset in just the right way, consequently "eliminating risk." This hedge is called delta hedging and is the basis of more complicated hedging strategies such as those engaged in by Wall Street investment banks. The hedge implies there is only one right price for the option and it is given by the Black-Scholes formula.

The Black-Scholes model calculates the price of a call option as:

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

Where:

C = Theoretical call premium
S = The current stock price
K = The option striking price
r = risk-free interest rate
t = time until option expiration
N = cumulative standard normal distribution
e = 2.7183
s = standard deviation of stock returns
ln = natural logarithm
d1 = [ln(S/K) + (r + S2/2)t] ÷ s√t
d2 = d1 - s√t

Assumptions in using this model:

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

Most companies pay dividends to their share holders, so this might seem a serious limitation to the model considering the observation that higher dividend yields elicit lower call premiums. A common way of adjusting the model for this situation is to subtract the discounted value of a future dividend from the stock price.

2) European exercise terms are used

European exercise terms dictate that the option can only be exercised on the expiration date. American exercise term 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. This limitation is not a major concern because very few calls are ever exercised before the last few days of their life. This is true because when you exercise a call early, you forfeit the remaining time value on the call and collect the intrinsic value. Towards the end of the life of a call, the remaining time value is very small, but the intrinsic value is the same.

3) Markets are efficient

This assumption suggests that people cannot consistently predict the direction of the market or an individual stock. The market operates continuously with share prices following a continuous Itô process. To understand what a continuous Itô process is, you must first know that a Markov process is "one where the observation in time period t depends only on the preceding observation." An Itô process is simply a Markov process in continuous time. If you were to draw a continuous process you would do so without picking the pen up from the piece of paper.

4) No commissions are charged

Usually market participants do have to pay a commission to buy or sell options. Even floor traders pay some kind of fee, but it is usually very small. The fees that Individual investor's pay is more substantial and can often distort the output of the model.

5) Interest rates remain constant and known

The Black and Scholes model uses the risk-free rate to represent this constant and known rate. In reality there is no such thing as the risk-free rate, but the discount rate on U.S. Government Treasury Bills with 30 days left until maturity is usually used to represent it. During periods of rapidly changing interest rates, these 30 day rates are often subject to change, thereby violating one of the assumptions of the model.

6) Returns are lognormally distributed

This assumption suggests, returns on the underlying stock are normally distributed, which is reasonable for most assets that offer options.

Source: Kevin Rubash, Bradley University