Black–Scholes Call Theta Near Expiry and Model Assumptions

Summary

The document questions how a vanilla call option’s theta behaves as expiry approaches under models beyond Black–Scholes, including CEV. The response first corrects two assumptions in the question: at-the-money call theta under the stated Black–Scholes setup should diverge negatively, not positively, and the setup implicitly assumes zero dividends while the CEV expression allows a dividend yield. These differing assumptions make a direct comparison unreliable.

The response also notes that sufficiently positive dividends can change the sign of Black–Scholes theta. It recommends first posing the comparison with both interest rates and dividends set to zero, then considering positive rates or dividend yield separately. The exchange does not derive the CEV asymptotics or establish behavior for advanced models, so it provides a framing correction rather than a general result. Theta near expiry depends on model assumptions and option moneyness; the supplied discussion does not fully resolve the original cross-model question.

Key ideas

• At-the-money Black–Scholes call theta near expiry is described as diverging negatively.
• The question's Black–Scholes and CEV setups use different dividend assumptions.
• A sufficiently positive dividend yield can change the sign of Black–Scholes call theta.
• The response recommends isolating zero-rate and zero-dividend assumptions before extending the comparison.
• No general CEV asymptotic result is established in the exchange.

Tags

# Asymptotic behavior of theta of vanilla call option


# 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.

## Answer by peter carr (score 4)

https://quant.stackexchange.com/a/16020

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.

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