Vega of options in same expiry

Quantitative Finance Level 1

Which of the following options (on the same expiry) has the largest vega when the stock is trading at 100?

Put on the 100 strike Put on the 120 strike Call on the 80 strike Call on the 120 strike

1 solution

Chew-Seong Cheong
Mar 19, 2015

For a same underlying stock, strike price and expiration date, the vega of its call is same as the vega of its put. Vega is largest when the option is near at the money. The following graph from www.OptionTradingTips.com illustrates this point:

What is the reason that the vega is largest when the option is near at the money?

Calvin Lin Staff - 6 years, 2 months ago

Vega is given by (see Black-Scholes Model ):

$\nu = \sqrt{\frac{t}{2\pi}}Se^{-\frac{d_1^2}{2}}\quad \text{where }d_1 = \frac {1}{\sigma\sqrt{t}} \left[ \ln{\left(\frac{S}{K}\right)+\left(\frac{r+\sigma^2}{2}\right)t}\right]$

This shows that vega is maximum when $S=K$ , that is at the money.

Chew-Seong Cheong - 6 years, 2 months ago

Yes, that's going by the formulaic derivation.

Is there an intuitive reason that this statement should be true?

Calvin Lin Staff - 6 years, 2 months ago

@Calvin Lin – Thanks for the encouragement.

$\text{Option value = Intrinsic value + Time value}$ , for small time value, $V \approx S - K$ . Since $S$ and $K$ are close to normally distributed, $V$ is also close to normal and has largest value at its mean, that is $S-K=0$ or at the money. Therefore, $\partial V$ is largest at the money and so is vega $\nu$ .

Chew-Seong Cheong - 6 years, 2 months ago

@Chew-Seong Cheong – Hm, not quite. The intrinsic value is fixed, for a fixed underlying price and a fixed strike. They are not normally distributed.

Vega arises from the time value, and so if you assume that the time value is 0, then the vega would be 0.

Calvin Lin Staff - 6 years, 2 months ago

Because volatility can swing the price either ways, resulting in increase in the probability of option getting ITM/OTM. And thus an increase in premium due to volatility which can be ascribed to vega.

Ashish Jha - 1 month, 2 weeks ago

To clarify, I'm not asking why options have (positive) vega.

I'm asking why the ATM option has the highest vega, and if there's any intuitive way to visualize it.

There actually is a way to do so, by considering the probability distribution of stock prices as volatility increases.

Note that you cannot increase the probability of option getting both ITM and OTM (since these sum to 1). Also, that probability is related more to delta than to vega (and yes, volatility affects delta).

Calvin Lin Staff - 1 month, 2 weeks ago

Vega tells us an option's sensitivity to implied volatility. So, intuitively speaking the deep OTM and ITM options' sensitivity to change in the implied volatility is lower than ATM options' sensitivity. A change in implied volatility would not make a deep OTM option suddenly become at the money or in the money, similarly a change in IV would not make a deep ITM option suddenly worthless. But for at the money options, that are on the verge of being worthless or valued, the change in IV would impact it greatly than the other two classes of options. And thus ATM options have a higher sensitivity to implied volatility, which in turn results in higher vega.

Ashish Jha - 1 month, 2 weeks ago

@Ashish Jha – Be careful with (the phrasing of) your logic. While I can guess what you're trying to convey, that's based on my experience with options (and some mind-reading skills).

Some concerns are:

Remember that vega is the increase in price per % volatility. That's not really expressed in what you've said.
Saying "a change in implied volatility" doesn't tell me if you're talking about an increase or decrease in volatility, and hence even if price changes, I don't know if you're arguing for a positive or negative vega.
Remember that an ITM option (say call) has a corresponding OTM put option, and these have the same vega. You seem to be making an argument that OTM has positive vega and ITM has negative vega.
"But for at the money options, that are on the verge of being worthless or valued" -> ATM options are very rarely "on the verge of being worthless". EG See straddle approximation formula .
You have not demonstrated which have a higher sensitivity -> why is the absolute change in price more significant ATM than anywhere else?

Calvin Lin Staff - 1 month, 2 weeks ago

@Calvin Lin – Hey, would you be able to suggest some books that might help in a better understanding of options? As of now i've read Hull(not completed though), and Trading option greeks by dan passarelli. Do you suggest some other readings or lecture videos that might be available on youtube?

Ashish Jha - 1 month, 2 weeks ago

Vega of options in same expiry

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