# Theta

- The theta of an option is the rate of change in its value with the passage of time, assuming that other things remain the same
- For a portfolio, the theta is the rate of change in the value of the portfolio as time passes, given that other things are constant. A positive theta implies that the portfolio value will increase as the time passes, while a negative theta implies that the value will decrease with the passage of time, if there is little move in the stock price and the implied volatility
- Normally expect the theta of an option to be negative as with the passage of time, an option
- loses value
- Exception can be an in-the money European put option on a non dividend paying stock

- For
**at-the-money option,**theta increases as the expiration date nears- Theta decreases as an option which is either out of money or in the money approaches expiration

- We have theta of call given by:

- Where:

- For a put option, theta is given by:
- Where:
- S
_{0}= Stock price at time 0, i.e. present price of the stock - d
_{1}and d_{2}are as defined in the Black-Scholes Pricing formula earlier - Ïƒ = Stock price volatility
- K = Strike price
- T = Time of maturity of the option measured in years, so that 6 months will be 0.5 years
- r = Risk neutral rate of interest

- S