Positive Theta

[gver]

Hi .. The theta for deep ITM Puts is positive even when you are long .. This is primarily associated to the cost of carry .. I was hoping if someone could provide a detailed intuitive and mathematical explanation .. Thanks ..

 

[aaronhotchner]

First, you mention that theta is positive "even" if you are long, but it would "only" be positive if you are long as it must necessarily be negative if you are short.

If you want a mathematical explanation, go look up the formula for theta on Wikipedia. It's a straightforward calculation.

The intuitive explanation is the following:
Assume, for simplicity, S = 0.01, K = 100, vol = 20% (anything sufficiently small works), r = whatever, say 5%.
The value of the put is derived from its expected future payoff discounted at r. This put is so ridiculously in the money, however, that it will basically always be in the money regardless of expiry date. Since the price of the option is what it is worth today, and theta only really makes sense in valuing a European option, having to wait to get that $99.99 payoff sucks. The option is worth essentially $99.99 * e^(-rt) since the payoff is all but guaranteed and we only collect at expiration. Every day that passes reduces t, which increases the value of e^(-rt), which increases the value of the option. In situations where a put is extremely in the money, the American put is generally optimal to exercise rather than wait/sell for this very reason.

 

[gver]

Thanks for the example, makes sense now. however, for the 'even' long thing, theta (extrinsic value, decreases with time) is negative for long positions in both call and put .. Its only gamma and vega which is positive .. Theta is negative for short positions in an option .. The reason I wrote that and the confusion ..

 

[aaronhotchner]

That's sort of an example of my point: theta can be either positive or negative for the long, you don't really know until you've calculated it. But you're talking about ITM puts where it's positive if you're long... this means it must negative if you're short. Likewise, theta can be (and usually, but obviously not always, is) positive if you're short. It's best to go through the reasons why Greeks have their sign instead of rote memorization.

 

[C S]

It helps to compare the graphs of price versus spot for the American put and European put. You'll see that the American put price graph coincides with the payoff graph for very small spot, say S < S_0 (this has to do with getting the payoff earlier and it's time-value of money versus any possible gain from waiting). The European price graph looks like the American price graph for S > S_0 but for S < S_0 cuts through the payoff graph instead of coinciding. So for very small spot the European price is actually less than the payoff. As expiry approaches, the price graph has to evolve to the payoff, and so for small spot, the price is actually increasing, making theta positive.