The Black and Scholes Model

 

The Black and Scholes Option Pricing 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.

 

 

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, Ke(-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.

 

Assumptions of the Black and Scholes 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.

 

Delta:

 

Delta is a measure of the sensitivity the calculated option value has to small changes in the share price.

 

Gamma:

Gamma is a measure of the calculated delta's sensitivity to small changes in share price.

 

Theta:

Theta measures the calculated option value's sensitivity to small changes in volatility.

 

Vega:

Vega measures the calculated option value's sensitivity to small changes in volatility.

 

Rho:

 

Black and Scholes Generalized Model

 

The Black Scholes Generalized model is suitable for evaluating European style options on instruments which assume to pay a continuous dividend yield during the life of the option. Since an option holder does not receive any cash flows paid from the underlying instrument, this should be reflected in a lower option price in the case of a call or a higher price in the case of a put. The Black Scholes Generalized model provides a solution by subtracting the present value of the continuous cash flow from the price of the underlying instrument. Assumptions under which the formula was derived include:

• the option can only be exercised on the expiry date (European style);
• the underlying instrument does not pay dividends;
• there are no taxes, margins or transaction costs;
• the risk free interest rate is constant;
• the price volatility of the underlying instrument is constant; and
• the price movements of the underlying instrument follow a lognormal distribution.

 

This Financial CAD function (which is based on the Black Scholes Generalized Model) can be used to work with the following types of instruments:

 

§         Options on instruments with a continuous dividend yield

§         Options on forwards or futures

§         Options on instruments with no yield

§         Options on spot foreign exchange

 

Advantages of Black and Scholes

 

The Black and Scholes option-pricing model presents a number of advantages. The most prevalent advantage is its ease of use. It tells the user what is important not what is important. In other words, it includes the very factors that market analysts look for. Secondly, it does not promise to produce the exact prices that show up in the market, but it does a remarkable job of pricing options that meet all of the assumptions