# Asymptotic behavior of theta of vanilla call option

It is well known that the theta of call option is always negative. Also, the theta of (at the money call option) goes to infinity as the time approaches to the maturity. On the other hands, (ITM and OTM) call option has zero theta at the maturity. This can be easily checked by BS formula. Here, i am wondering that the above fact also holds for other models (eg. CEV or advanced models).

As i know, the theta of call option under CEV is given by  where X is a strike price and $Q(w; v, λ)$ is the complementary distribution function of a non-central chi-square law with v degrees of freedom and non-centrality parameter λ.

• Could you please be a bit clearer with your notation of the CEV; perhaps give the model dynamics explicitly. And as petercarr points out, it might be good to drop the risk free rate and dividend yield in your formulas. With CEV, if I remember correctly, you need a correction term for the 'strictly local martingale scenario'. – Kiwiakos Dec 31 '14 at 0:36

## 1 Answer

I think you need to ask your question differently to get better answers than mine. Your Black Scholes part has two problems. First positive infinity should be negative infinity. Second, you are assuming zero dividends in Black Scholes but you are assuming a possibly positive div yield q in the CEV part. If the div yield q is sufficiently positive in the Black Scholes model, it leads to theta switching sign. My advice is to ask your question under zero rates and dividends. If you get an answer, then follow up with positive int rate and or positive dividend yield.

• Wow, Peter: Great to have you here :-) A few years ago we had lunch together in London and were emailing a few times after that :-) My name is Holger von Jouanne-Diedrich and I was working for Siemens at the time. Now I have been a professor for about 1.5 years :-) Hope you are well! – vonjd Dec 30 '14 at 18:18
• Malliavin calculus give a better tools to investigate the greeks. – Zbigniew Dec 31 '14 at 2:23