# The value of theta of an ATM option is proportional to the volatility, but for OTM/ITM options theta is not proportional to vol, why?

I have seen a graph of theta v/s volatility where the theta of ATM changes linearly with the volatility whereas for ITM/OTM options the theta didn't show a direct proportional relation with volatility, why is that so?

The formula for $$\Theta$$ for a European option under BSM (assuming 0 rates & dividends) is: $$\frac{S * n(d_1) * \sigma}{2\sqrt{T}}$$

For ATM options where $$d_1 \approx 0$$, the formula is locally linear in volatility. This is because $$d_1$$, given by $$\frac{ln(\frac{S}{K}) + \frac{\sigma^2 T}{2}}{\sigma\sqrt{T}}$$ is ~0 on the numerator, so changes to volatility have little effect. When S != K, the whole fraction is sensitive to volatility changes.

Example:

S = 1, K = 1.2, T = 1.

When $$\sigma = 0.2$$, $$d_1 = -0.811$$, so $$n(d_1) = 0.287$$. Giving $$\Theta = 0.0287$$

If $$\sigma = 0.3$$, $$n(d_1) = 0.359$$ and $$\Theta = 0.0539$$

All this is saying is that the partial derivative of theta w.r.t volatility != 0 for OTM options. We can formalise this:

$$\frac{d\Theta}{d\sigma} = \frac{S*n(d_1)}{2\sqrt{T}}*(1 + d_1*d_2)$$

Which is locally pretty stable for ATM options, but is much more sensitive for OTM options. You'll see the trend for this sensitivity to quickly peak as volatility rises, but after that it linearly becomes less sensitive to volatility increases as vol increases.

Use this desmos link to play around with to see how this sensitivity changes: https://www.desmos.com/calculator/exeeuuzmzk

You're right, the relationship between theta and volatility for ATM options is more linear compared to ITM/OTM options. Here's why:

Theta and Time Value:

Theta (θ) represents the rate of decay of an option's price due to time passing (time decay). Options have two components: intrinsic value (for ITM options) and time value (for both ITM and OTM options). Time value reflects the potential for the underlying asset price to move and the option to become profitable by expiration. ATM Options and Theta:

ATM (At-The-Money) options have minimal intrinsic value and rely heavily on time value for their price. As volatility increases, the uncertainty of the underlying asset's price movement goes up. This makes the time value of the option more valuable, leading to a more direct proportional increase in theta (faster decay) with increasing volatility. ITM/OTM Options and Theta:

ITM (In-The-Money) options already have some intrinsic value. Their time value is lower compared to ATM options. OTM (Out-The-Money) options have only time value. However, this time value is generally smaller than ATM options as the chance of them becoming ITM by expiration is lower. When volatility increases, the time value of ITM/OTM options can also increase. However, the impact is less pronounced compared to ATM options due to: Lower starting time value: The smaller initial time value in ITM/OTM options means there's less room for a proportional increase compared to ATM options with a higher starting time value. Limited impact on ITM options: For deep ITM options, their price is already dominated by intrinsic value. Even with a volatility increase, the change in time value has a smaller effect on overall theta. In Summary:

ATM options with a high reliance on time value exhibit a more linear relationship between theta and volatility. ITM/OTM options have a dampened effect on theta's increase with volatility due to their lower starting time value and (for ITM) influence of intrinsic value. Additional Notes:

This is a simplified explanation, and other factors like interest rates and time to expiration can also influence the relationship. Options pricing models can provide a more detailed picture, but the core concept remains – ATM options are more sensitive to volatility changes impacting theta.