The next Greek, I want to talk about is the Theta of an option. The Theta of an option is the negative of the partial derivative of the option price with respect to time to maturity. So therefore, mathematically speaking, Theta equals minus Delta C, Delta T, and that's for a call option. We can also compute it for a put option. If we actually go ahead and do the mathematics, compute the derivatives of the Black-Scholes formula, we will find that Theta is equal to this long expression here, where Phi is the standard normal PDF. So if you recall, N is the standard normal CDF, and Phi is the standard normal PDF. Why do we take the negative? Well, we take the negative because in practice, time goes forward. So in practice, the time to maturity of an option decreases. Suppose I have an option right now which is 200 days to maturity. Well then, tomorrow it will have 199 days to maturity. So therefore, the time to maturity is always decreasing in practice, and so it's conventional to take Theta to be the negative of the partial derivative of the option price with respect to time to maturity. Here are some figures. Again, we see Theta for European call option as a function of the stock price. K was equal to 100 in these examples. We assumed, R, the interest rate and indeed C, the dividend yield was equal to zero percent. We plot the Theta here for 0.05 years, 0.25 years, and 0.5 years. Notice number one that the Theta is negative in all cases. Now, in general, Theta will be negative for European call and put options. It's not always the case that it's negative. There are certain situations where Theta could be positive. But in general, most of the time Theta is negative. In other words, when you hold a European call and put option, you lose a little piece of money every day, if the underlying stock price does not change, and that's what Theta means. Remember, Theta is equal to minus the partial derivative of the call option price with respect to time to maturity. So as the time to maturity decreases, I lose a little piece of the value of the option, the Delta C decreases. Again, another observation is that, as the stock price moves away from the current strike of K equal to 100, we see that the Theta goes towards zero, and again, we can use our earlier examples to see why this is the case. We know in this time we'll say in the case, where we have a call option here, so we'll stick with the call option. In the case of a call option, we know that C_0 will be approximately equal to S_0 minus K. If S_0 is much bigger than K or very large, and it's going to be approximately equal to zero, If S_0 is much smaller than K. In both cases, the partial derivative of this term with respect to capital T, the time to maturity is zero. Likewise, the partial derivative of this term zero with respect to time to maturity capital T is also zero, and so that's why we see for large values of S, and for very small values of S, we see that the partial derivative with respect to time to maturity is zero, and that's why all of these curves approach zero as S moves away from the strike K. Why is the Theta most negative around the strike for short times to maturity? That is for time to maturity of two and a half weeks of 0.05 years. Well, one way to see that, is the following. Suppose I've just got one day to maturity. So I've got one day, so T is equal to one day to maturity. Well then, and suppose the stock price is equal to K. So I'm at the money. This means, I've got one day to maturity. If the stock price increases, I'm going to exercise the option and make some money. If the stock price decreases over the next day, I'm not going to exercise the money. So the option value will be non-zero at this point because over the next day there is a chance that the stock price will increase and I'll be exercising and making some money. However, imagine rolling time forward one day without changing the stock price. Well, in that case, this is going to go to zero days to maturity. S_0 is still equal to K and now the option expires worthless. So when there's just one day to maturity, the Theta is larger and more negative because I have more to lose over the next day than I would if there was one year to maturity. If there was one year to maturity, I would have 365 days left, moving time forward one day isn't really going to impact the value of the option very much at all. However, when I'm just one day left maturity, that one day encapsulates all of the value of the option, when I'm at the money, and if I rolled time forward one day without changing the stock price, I'm going to expire worthless, and therefore, receive nothing. So the Theta becomes more negative and peaked around the strike, as the time to maturity decreases towards zero. Under this plot, we see the Theta for European put options as a function of the time to maturity, we've plotted here three different option curves. One for not the money option, the blue curve, and the green and red curves for 10 percent out of the money, and 20 percent out of the money options, respectively. In fact, just so we're clear at 10 percent out of the money option, in this case, it's a European put option. So 10 percent out of the money option will have K being equal to 0.9 times S_0. So the strike is below the current stock price, and so currently it is out of the money. For the 20 percent case, we will have K equal to 0.8 times S_0. So in these cases, we see these out of the money options that they're Theta is decreasing. Let's take this green curve here. We see Theta is decreasing for awhile. It's negative and decreasing. But beyond a certain point, there becomes, it turns around and moves towards zero, and that's because it's becoming increasingly unlikely that the option will be exercised. Its value is moving towards zero, and so it's Theta will be zero. The partial derivative of zero with respect to T is equal to zero. So at this point the value is moving towards zero because it's becoming less and less likely to be exercised. The red curve corresponding to a 20 percent out of the money option has actually been turned earlier than the green curve because it's 20 percent out of the money it's further away, and so it's becoming less and less likely to be exercised at an earlier point in this green curve here. In case you're wondering, we can easily create these plots, just by using this expression here. This is an expression. So for those of you who are comfortable with coding, or Matlab, or Python, or indeed in Excel, you could create a table of values for T. Create the Thetas for these different T values and create a plot. You can easily create these kinds of curves here that I'm showing you.
