# OPTIONS PRICING MODELS

There are various Option pricing models which traders utilize to arrive at the correct value of the Options they may have under study. Amongst the more popular models would be the Binomial Pricing Model and the Black & Scholes Model. Both of which are explained in some detail below:

Binomial Pricing Model: The Binomial Pricing Model is an Options pricing model which was developed by William Sharpe back in 1978. Today, we find a large variety of pricing models which differ according to their hypothesis or the underlying financial instrument upon which they are based; for instance, stock options, currency options, options on interest rates, amongst others.

The Binomial model breaks down the time to expiration into potentially a very large number of time intervals, or steps. A tree of stock prices is initially produced working forward from the present to expiration. At each step it is assumed that the stock price would move up or down by an amount calculated using volatility and time to expiration. This produces a binomial distribution, or recombining tree of underlying stock prices. This tree represents all the possible paths that the stock price could take during the life of the Option.

At the end of the tree; that is, at expiration of the Option, all the terminal Option prices for each of the possible final stock prices are known as they would simply equal their intrinsic value.

Next the Option prices at each step of the tree are calculated working back from expiration to the present. The Option prices at each step are used to derive the Option prices at the next step of the tree using risk neutral valuation based on the probabilities of the stock price moving up or down, the risk free rate and the time interval of each step. Any adjustments to stock prices (at an ex-dividend date) or Option prices (as a result of early exercise of American options) are worked into the calculations at the required point in time. At the top of the tree you are left with only one Option price.

The advantage the Binomial model has over the Black-Scholes model is that it can be used to accurately price American options. This would be because, it would be possible to check at every point in an Option's life (that is, at every step of the Binomial tree) for the possibility of early exercise. For instance, where due to a dividend or a put being deeply in-the-money the Option price at that point is less than its intrinsic value.

Where an early exercise point is found, it is assumed that the Option holder would elect to exercise and the Option price can be adjusted to equal the intrinsic value at that point. This then flows into the calculations higher up the tree and so on.

Now, the main disadvantage of the Binomial model would be its relatively slow speed. It would be great for half a dozen calculations at a time; but, even with today's fastest computers it would not be a practical solution for the calculation of thousands of prices in a few seconds; which is what would be required for the production of the animated charts in any strategy evaluation model.

The Black & Scholes Model: This model was published by Fisher Black and Myron Scholes in 1973, and was named after their respective surnames. It is one of the most popular Options pricing model; and is noted for its relative simplicity and its fast mode of calculation; quite unlike the Binomial model, it does not rely on calculation by iteration.

The Black & Scholes model is used to calculate a theoretical Call price (while ignoring dividends paid during the lifespan of the Option) using the five key determinants of an Option's price; which are, the stock price, the strike price, volatility, the time to expiration and the short term risk-free interest rate.

The original formula for calculating the theoretical Option price (or OP) is as follows:

OP = SN (d_{1}) − Xe^{−rt} N (d_{2})

Where,

d_{1} = { ln [S÷X] + [r + v²÷2] t } ÷ v √t

d_{2} = d_{1} − v √t

The variable are;

S = the stock price; X = the strike price; t = time remaining until expiration, expressed as a percent of a year; r = current continuously compounded risk free interest rate; v = annual volatility of stock price (the standard deviation of the short-term returns over one year); ln = natural logarithm; N(x) = standard normal cumulative distribution function; and e = the exponential function.

This model is based on the log normal distribution of stock prices, as opposed to a normal or bell-shaped distribution. The log normal distribution allows for a stock price distribution of between zero and infinity (that is 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%).

This would lead to the risk neutral valuation; where the expected rate of return of the stock (that is, the expected rate of growth of the underlying asset which equals the risk free rate plus a risk premium) is not one of the variables in the Black-Scholes model (or any other model for Option valuation). The important implication is that the price of an Option is completely independent of the expected growth of the underlying asset. Thus, while any two investors may disagree on the rate of return they expect on a stock, they would always agree on the fair value of the Option on that underlying asset (given an agreement to the assumptions of volatility and the risk free rate).

The key concept underlying the valuation of all derivatives is the fact that the price of an Option is independent of the risk preferences of investors and would be a risk neutral valuation. This would mean that all derivatives can be valued by assuming that the return from their underlying assets is the risk free rate.

While dividends are ignored in the basic Black-Scholes formula, there are a number of widely used adaptations to the original formula which enable it to handle discrete and continuous dividends accurately. Despite these adaptations the Black-Scholes model cannot be used to accurately price Options with an American style exercise, as it only calculates the Option price at one point of time; that is at expiration. It does not consider the steps along the way where there may be the probability of early exercise of an American option. This would be a significant limitation.

Despite the above, the main advantage of this model would be speed; as it allows the calculation of a large number of Option prices in a relatively short period of time. Since, high accuracy is not critical for American option pricing (for instance, when animating a chart to show the effects of time decay) using the Black-Scholes model would be a good alternative. But, then again the Binomial model would be advisable for the relatively few pricing and profitability numbers where accuracy may be important and speed irrelevant.

We would strongly recommend that the investor first study these investment instruments along with the above listed valuation and Option pricing models; conduct dry runs with pen and paper; understand the nuances of the dynamics of the underlying security and the connectivity between the various risks that he or she would be taking on during the pendency of a futures or options position held by him.