Option's theta

By roma, 17 August, 2020

Theta measures the sensitivity of an option's price to changes in time, all other parameters being equal. Theta usually has a negative value if the trader owns options, and a positive value if the trader has a short option position. Figure 1 illustrates how the call option price changes as the time passes. All other parameters of the Black Scholes model are left unchanged:
- the price of the underlying asset ($50)
- strike price ($50)
- implied volatility (25%)
- interest rate (0%)

Fig. 1. ATM option price
ATM option price


You can see in the Figure that the option price loses value as the expiry approaches. The less time before expiry, the less likely it is that the option will be in the money. Theta measures the rate at which an option loses its time value. Graphically, theta is the tilt coefficient of a tangent line at a certain point in time. The option in Figure 1 is at the money, and ATM options, especially those with a close expiry date, have a high theta, i.e. every day the value of the option falls by a higher theta value.  As the expiry date approaches, the tangent line tilt coefficient (or first derivative of the option price with respect to time) increases. Consequently, the theta will increase every day and the option price will fall at a higher rate.

Example. 90 days before the expiration, the option on the stock of an energy company from the U.S. Apache Corp. is trading at $1.78, theta of option is $0.01. This means that tomorrow's option will cost $0.01 less than today, all other things being equal. Thus, 89 days prior to exercise the price of the option will be $1.77, if other parameters remain unchanged. 10 days before expiration, the option will be worth $0.59 and the value of theta will increase to $0.03.


Theta and asset price movements

A theta has a negative value for a long option position (both put and call options) - i.e. the portfolio loses value every day with all other parameters being equal. Conversely, theta is positive for a short option position. As time passes (other parameters remain unchanged), options lose value. Professional traders say that the option holder “pays theta” and the seller (i.e., the trader with the short option position) “receives theta”. Due to the fact that theta is the price for gamma (i.e., the option holder “pays theta” for the right to hedge the delta at a profit), the theta chart is similar to that of the gamma with respect to the price of the underlying asset.

Fig. 2. Gamma and spot price
Gamma and spot price

Fig. 3. Theta and spot price
 Theta and spot price


Relationship between Theta and Gamma

From a practical point of view, theta is the price for gamma. Option holders have a positive gamma and can always hedge the delta at a profit, but the free cheese is only in the mousetrap. The question arises: what does the option holder give in return for the positive gamma? The value of the positive gamma is the theta of the option, which reduces the value of the option as time passes.

Fig. 4. Gamma and expiration
 Gamma and expiration

Fig. 5. Theta and expiration
Theta and expiration

Delta hedging or gamma trading is also called realized volatility trading. Gamma trading is usually performed using ATM options with close expiry dates because they have the largest gamma and therefore the largest theta (in absolute terms). Trading the realized volatility for the buyer of options means making sure that the gamma gain, i.e. the delta of hedging, exceeds the theta. For example, if a trader paid $35.73 in theta during the day on a long option position, and earned $51 from the delta hedging, because the stock price was very volatile during the day, then we can say that the realized volatility of the stock was higher than the implied volatility. Thus, profit from the realized volatility is achieved with the help of gamma, and theta depends on the implied volatility.