Option pricing models

An option pricing model is a mathematical formula or model into which you insert the following parameters:

  • underlying stock or index price
  • exercise price of the option
  • expiry date of the option
  • expected dividends (in cents for a stock, or as a yield for an index) to be paid over the life of the option
  • expected risk free interest rate over the life of the option
  • expected volatility of the underlying stock or index over the life of the option

When the formula is applied to these variables, the resulting figure is called the theoretical fair value of the option.

Pricing models used by the market

There are two main models used in the Australian market for pricing equity options: the binomial model and the Black Scholes model. For most traders these two models will give accurate enough results from which to work.

There are many other options pricing models available. Good books on the topic are Hull "Options, Futures, and other Derivatives" (Prentice Hall) and Natenberg "Option Volatility and Pricing" (Irwin).

The binomial option pricing model
First proposed by Cox, Ross and Rubinstein in a paper published in 1979, this solution to pricing an option is probably the most common model used for equity calls and puts today.

The model divides the time to an option’s expiry into a large number of intervals, or steps. At each interval it calculates that the stock price will move either up or down with a given probability and also by an amount calculated with reference to the stock’s volatility, the time to expiry and the risk free interest rate. A binomial distribution of prices for the underlying stock or index is thus produced.

At expiry the option values for each possible stock price are known as they are equal to their intrinsic values. The model then works backwards through each time interval, calculating the value of the option at each step. At the point where a dividend is paid (or other capital adjustment made) the model takes this into account. The final step is at the current time and stock price, where the current theoretical fair value of the option is calculated.

The number of steps in the model determines its speed, however most home PCs today can easily handle a model with 100 or so steps, which gives a sufficient level of accuracy for calculating a theoretical fair value.

The Black Scholes model

First proposed by Black and Scholes in a paper published in 1973, this analytic solution to pricing a European option on a non dividend paying asset formed the foundation for much theory in derivatives finance. The Black Scholes formula is a continuous time analogue of the binomial model.

The Black Scholes formula uses the pricing inputs to analytically produce a theoretical fair value for an option. The model has many variations which attempt, with varying levels of accuracy, to incorporate dividends and American style exercise conditions. However with computing power these days the binomial solution is more widely used.

The relationship between fair value and market price

Although the fair value may be close to where the market is trading, other pricing factors in the marketplace mean fair value is used mostly as an estimate of the option’s value.

Moreover, fair value will depend on the assumptions regarding volatility levels, dividend payments and so on that are made by the person using the pricing model. Different expectations of volatility or dividends will alter the fair value result. This means that at any one time there may be many views held simultaneously on what the fair value of a particular option is.

In practice, supply and demand will often dictate at what level an option is priced in the marketplace. Traders may calculate fair value on a option to get an indication of whether the current market price is higher or lower than fair value, as part of the process of making a judgement about the market value of the option.

Volatility

The volatility figure input into an option pricing model reflects the assumptions of the person using the pricing model. Volatility is defined technically in various ways, depending on assumptions made about the underlying asset’s price distribution. For the regular option trader it is sufficient to know that the volatility a trader assigns to a stock reflects expectations of how the stock price will fluctuate over a given period of time.

Volatility is usually expressed in two ways: historical and implied.

Historical volatility describes volatility observed in a stock over a given period of time. Price movements in the stock (or underlying asset) are recorded at fixed time intervals (for example every day, every week, or every month) over a given period. More data generally leads to more accuracy.

Be aware that a stock’s past volatility may not necessarily be reproduced in the future. Caution should be used in basing estimates of future volatility on historical volatility. In estimating future volatility, a frequently used compromise is to assume that volatility over a coming period of time will be the same as measured/historical volatility for that period of time just finished. Thus if you want to price a three month option, you may use three month historical volatility.

Implied volatility relates to the current market for an option. Volatility is implied from the option’s current price, using a standard option pricing model. Keeping all other inputs constant, you can put the current market price of an option into any theoretical option price calculator and it will calculate the volatility implied by that option price.

This is one of the key figures traders watch to assist them in assessing the value of an option. It is also commonly fed back into the option pricing model to calculate the option’s theoretical fair value.

Useful website links to find out more about option pricing models

If you type in "derivatives pricing model" or "options pricing model" into a good search engine, you will get many results. Here are just a few of the many sites covering this topic:

http://www.hoadley.net/options/bs.htm#Binomial
http://www.hoadley.net/options/bs.htm#Black-Scholes
http://www.schaeffersresearch.com/option/advanced/bscholes.htm