How Moneyness Shapes Option Vega in Black–Scholes
Summary
The document explains how implied volatility affects option value across moneyness in the Black–Scholes framework. Put–call parity links a deep in-the-money call to an out-of-the-money put, so their volatility sensitivities are qualitatively symmetric. This helps address the intuition that a deeply profitable option should gain substantially from added upside despite a higher chance of finishing out of the money.
The response points to vega, the price sensitivity to implied volatility, plotted against standardized log-moneyness using the forward price, strike, volatility, and time to expiry. It states that vega is practically zero for options far from at the money, including deeply in-the-money and deeply out-of-the-money contracts. Thus, volatility has little material price effect in those regions. The explanation is qualitative and refers to a chart without supplying it or deriving the formula, so it gives an intuition and diagnostic rather than a full mathematical proof.
Key ideas
- Put–call parity makes an in-the-money call’s volatility sensitivity comparable to that of an out-of-the-money put.
- Vega measures how option price changes as implied volatility changes.
- Standardized log-moneyness incorporates forward price, strike, volatility, and time to expiry.
- Vega is practically zero for options far from at the money, whether deeply in or out of the money.
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Full text
# Why do the prices of deep in-the-money options increase with volatility in the Black Scholes framework?
# Why do the prices of deep in-the-money options increase with volatility in the Black Scholes framework?
I can understand that volatility increases the value of an option when a stock is out/at the money. Then more volatility means the stock's distribution gets more upside without suffering a greater probability of ending out of the money.
But imagine that a stock was far in the money. If I have a distribution that has a low volatility and thus greater chance of ending in the money, if I was risk averse, I might prefer that to an option that has more upside but more downside as well. Using the same thought process, If I was risk neutral, I might be indifferent to each and thus they may have the same value to me.
Does this mean then that increasing volatility disproportionately increase the upside of the option relative to the increase in the probability of ending out of the money? Is there an intuitive explanation or a way to show this effect mathematically?
## Answer by Chris Taylor (score 1, accepted)
https://quant.stackexchange.com/a/61165
First note that the sensitivity of price to implied vol must be qualitatively symmetric for out-of-the-money and in-the-money options, since by put-call parity an ITM call behaves the same as an OTM put.
Second, for options that are far from at-the-money (whether they are deep ITM or deep OTM) the price does not particularly increase with volatility! You can see this by plotting the vega (derivative of price wrt implied volatility) of an option against moneyness, where moneyness is defined in terms of forward price $F$, strike $K$, implied volatility $\sigma$ and time to expiry $\tau$
$$ m= \frac{\log(K/F)}{\sigma \sqrt{\tau}} $$
Plotting vega against moneyness gives a chart like this
You can see that for deeply ITM or OTM options (e.g. $|m| > 3$) the vega is practically zero, so the option does not materially increase in price as implied volatility increases.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.