Question about option theta

Question

I have a question about a option theta.

When I evaluate the option theta of near expiry put option using Black-Scholes formula given the data as follow:

1. Index Level = 20,500
2. Strike Price = 20,000
3. Interest Rate = 0.14%
4. Dividend Yield = 3.41%
5. Volatility = 23.64%
6. Time to maturity = 0.01 (2 days)

Using data as above, the put option price is \$12.97, but the theta per one day is -\$13.39. That means after one day, the put option price becomes negative given the above parameters (except time to maturity) do not change. I wonder the reason of put option price is smaller than the absolute value of theta.

I guess the reason is that the theta tell us the linear change of put option price against the time, but the relationship of option price and time to maturity is not linear, then the above case may happen. Is my conjecture correct? And is there any financial interpretataion on the above phonomena?

Answer 1

Disclaimer: I did not check your example, i.e., that "theta * 1 day" will predict a negative option price.

Theta is the derivative with respect to time-to-maturity. It is the change of the option price with respect to an infinitessimal change in time and not with respect to a change of one day - even if the derivative is "scaled" towards a time scale having 1 day as its unit.

In other words $\int_{t}^{t+d} \frac{\partial V}{\partial \tau} d \tau = V(t+d) - V(t)$ and in general $\int_{t}^{t+d} \frac{\partial V}{\partial \tau} d \tau \neq \frac{\partial V}{\partial \tau}(t) \cdot d$, however, the latter is an approximation. Here $V(t)$ is the option value, evaluated at time $t$.