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What is Theta?

Theta is a number that measures the change in the option value with respect to a change in the time to maturity.   That is, if theta is -10 and 0.01 of a year passes then the predicted decay in the option’s price is approximately 10 cents. 

We do not normally think of time as being an exposure that we want to hedge.  However, understanding the impact of time (or more importantly how an option price decays over time) is of the utmost importance to an option trader.  The reason is because in practice the two major drivers of option price (the underlying asset price and the volatility of the underlying asset price process) both change over time.  As a result, a complex multidimensional problem faces an option trader:  price, volatility and time.

Two components make up an option’s theta. The first arises from the assumption that the underlying asset price’s volatility increases with time (formally this increase is assumed to be linear in the square root of time).  Thus, this component becomes significant as the time to maturity gets small.  The second term, which has less impact, arises from the changing present value of the strike price as the option approaches maturity.  That is, recall that the terminal  value of an option is Max(0, S-K) or Max(0, K-S) for a call and put respectively.  (S = the underlying asset price, K = strike price).  But the strike price need only be paid at the end of the option's life which makes the present value of the strike price relevant to any decision regarding should I acquire the stock directly versus buy the option today.

A Graphical Analysis of Option Decay Dynamics

In order to gain insight into the decay dynamics we can apply the Sensitivities function of the online Calculator.

Online Exercise 1:  Theta against Time to Maturity

Click on the Option Sensitivities button and plot theta against the versus maturity for either the default case in the calculator or for our usual IBM at-the-money call option.  You can see that the largest decay effects occur for at-the-money options as the time to maturity shrinks as expected. 

Now if you change the calculator settings to work with an out-of-money (i.e., Strike price greater than the underlying stock price for a call option) serves to weaken the decay effect. 

Online Exercise 2:  Theta against Underlying Asset Price

To gain further insight into the option’s theta, plot theta against asset price.   You can see that the European put option’s theta has positive and negative regions. That is, for a European put option premium behavior over time to maturity does not always imply decay.   You are encouraged to repeat the above exercise for a European call option to observe the effects of time decay.

The positive theta region for a European put is a special case that arises when the European put is sufficiently in-the-money but cannot be exercised early. In this case the time premium for the European put is negative. This is why its price is predicted to increase with the passage of time in this region. The time premium is the difference between the option’s price and the present value of its exercised price. 

American Options and the Time Premium

A negative time premium cannot exist for an American put option.  This is because these options can be exercised at any point in time and therefore can never fall below their early exercise value (S-K or K-S for a call and put respectively).

OS Financial Trading System, PO Box 11356, Pittsburgh, PA 15238 USA, Phone 1-800-967-9897, Fax 1-412-967-5958, Email fts@ftsweb.com,

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