ETF Option Trading - Option Pricing Basics

Option Trading Basics - Option Pricing Model 'Greeks'

Option Pricing Model

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

This is my basic pricing model using an excel spreadsheet-hoadley addin combination. Regarding expiration date - technically this date is the Saturday after the 3rd Friday in the month which for July is 8-16-2008 - I have used the Friday or last trading day for the model input.



Interest Rate


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 [P price thv in the spreadsheet] 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.

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.

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.

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.

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.

Option Pricing Model - July Expiration

The spreadsheets below show the daily option pricing model 'greeks' for the july expiration for the at the money, in the money, and out of the money call - using a constant interest rate, volatility, and underlying price. Those of you using the hoadley application can easily create this by setting up your basic parameters as shown in the first 2 columns, and then using them for inputs for the remainder of the spreadsheet.

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:

delta goes up each day
gamma first goes up and then starts going down
theta first goes up and then starts going down


In The Money Call


Out Of The Money Call