Options Pricing Model

The purpose of this hub is to describe the inputs to an options pricing model, defining what each one of these inputs are, and giving an example of the output from the model, or what are called the option greeks.

Additionally, I am going to show you the changes in the option pricing model for the at the money call, in the money call, and out of the money call for the last 13 days of an expiration period, and what happens to the theoretical price of the option, with a constant interest rate, volatility, and underlying price.

Understanding these kinds of things are very important to the options trader, and the decisions that they will make knowing that as expiration is approached, all time value will be taken out of the option price.

The option pricing model is based on a formula used to calculate a theoretical value of an option price using current underlying prices - the variables in the formula are:

underlying price

option strike price

time remaining until expiration

risk free interest rate

volatility measured by annual standard deviation

Interest Rate

The pricing model defines interest as a risk free interest rate, for instance the rate from a 90 day treasury bill. You will note that changes in interest rates have relatively small changes in the theoretical value of an option, for instance if the theoretical value of the option was 2.42 with a risk free rate of 3%, the theoretical value would be 2.41 with a risk free rate of 2%.

Volatility

Volatility shows the range that an underlying price has fluctuated over a give period. The mathematical value of volatility is considered to be 'the annualized standard deviation of a stocks daily price changes'; there are two types of Volatility - Statistical Volatility and Implied Volatility.

Statistical Volatility is a measure of actual asset price changes over a specific period of time.

Implied Volatility is a measure of how much the 'market' expects underlying price to move based on the option price itself, or the volatility that the 'market' is implying the underlying will move.

The pricing model defines volatility as the annual standard deviation of the underlying price, or statistical volatility.

You can see that with a 50% statistical volatility the atm call has a theoretical value of 2.42, however if the actual price of the option was 2.70 - then this would imply a volatility of 55.89%

Volatility has a marked impact on the price of the option, for instance an underlying with the same price and same strike could give an option theoretical value at the given time of 2.42 with a volatility of 50%, however if volatility was 55% the theoretical value would be 2.66.

Theoretical Value

Theoretical value is the option price that is being solved for based on the model parameters. Although this price may not typically be the actual option price, it is relatively close, and it is still useful for doing whatif and comparative calculations for any parameter changes during expiration. In the case of large differences, understand that the market is expecting a 'bigger' move in the underlying, for instance from some pending news.

Delta

The Delta is a measure of the relationship between an option price and the underlying stock price. Call options have positive deltas, while put options have negative deltas. Technically, the delta is an instantaneous measure of the option's price change, so that the delta will be altered for even fractional changes by the underlying entity. The Delta is not a fixed percentage; changes in the price of the stock and time to expiration have an effect on the delta value.

For a call option, a Delta of .50 means a half-point rise in premium for every dollar that the stock goes up. For a put option contract, the premium rises as stock prices fall. As options near expiration, in the money contracts approach a Delta of 1.00, this being the amount by which an option's price will change for a one-point change in price by the underlying entity.

Consider that the delta for stock XYZ is 0.50: as the price of the stock changes by $2.00 the price of the options will change by 50 cents for every dollar. Therefore the price of the options will change by (.50 x 2) = 1.00. The call options will have their price increased by $1.00 and the put options will have their price decreased by $1.00.

Gamma

The rate of change in an option's delta for a one-unit change in the price of the underlying security; Gamma indicates an absolute change in delta. For example, a Gamma change of 0.150 indicates the delta will increase by 0.150 if the underlying price increases or decreases by 1.0.

Vega

The rate of change in an option's theoretical value for a one-unit change in the volatility assumption. Vega indicates an absolute change in option value for a one percent change in volatility. For example, a Vega of .090 indicates an absolute change in the option's theoretical value will increase by .090 if the volatility percentage is increased by 1.0 or decreased by .090 if the volatility percentage is decreased by 1.0.

Theta

The theoretical value of an option "erodes" or reduces with the passage of time. Theta is a measure of the rate of change in an option's theoretical value for a one-unit change in time to the option's expiration date. For example, a theta of -.25 indicates the option's theoretical value will change by -.25.

The spreadsheets below show the daily option pricing model 'greeks' for the at the money, in the money, and out of the money call - using a constant interest rate, volatility, and underlying price.

What you will specifically notice are the changes as you approach expiration - for instance consider the at the money call:

theoretical value goes down each day

delta goes down each day

gamma goes up each day

vega does down each day

theta goes up each day

IF you will now compare this to the in the money call you will see that the relationships are different - what you will be seeing are based on an option with intrinsic value instead of all time value AND how this reacts as you approach expiration and will have an option with value -vs- an option worth 0 value: