How Changing Volatility Affects Option Prices and Implied Volatility
Summary
The discussion asks how volatility that changes over time affects option prices and implied volatility, and how those effects vary with maturity. One response reasons that greater uncertainty in the underlying’s path can raise an option’s value, while stressing that nonconstant volatility does not always produce a higher price than a constant-volatility model. Another answer points to the Black–Scholes dependence on total variance, expressed through volatility squared and time, to explain why uncertainty about volatility can matter more over a longer horizon.
The exchange offers intuition and a simplified example, not a general pricing derivation. It does not establish that implied volatility must rise or fall with maturity, and the first response’s initial mathematical argument is explicitly withdrawn by its author. The later explanation treats average volatility as more stable over time, but this is an interpretation rather than a universal result. Pricing under stochastic volatility depends on the model and assumptions.
Key ideas
- Option prices can respond to the path and uncertainty of volatility, not only to a fixed volatility input.
- The effect of nonconstant volatility on price may be positive or negative.
- In Black–Scholes, total variance depends on volatility squared multiplied by time.
- Longer maturities can make a given difference in volatility assumptions more consequential for price.
- The discussion provides intuition, not a general theorem about the direction of implied volatility changes.
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Full text
# Implied volatility and nonconstant volatility
# Implied volatility and nonconstant volatility
John Hull states in his text that "AS the maturity of the option is increases the percentage impact of nonconstant volatility on (option) prices becomes more pronounced, but its percentage impact on implied volatility usually becomes less pronounced."
I'm having difficulty understanding what he means here.
- Does nonconstant volatility produce higher option prices than a constant volatility would for options with the same underlying, strike and time to maturity ?
- How does a nonconstant volatility effect implied volatility / what's the relationship here ?
## Answer by nathanesau (score 4)
https://quant.stackexchange.com/a/18708
In general, $v = \frac{\partial C}{\partial \sigma} > 0$ and $\theta = \frac{\partial C}{\partial t} < 0$. If maturity $T$ increases than $C$ increases. Suppose volatility is non-constant. Then if $T$ increases, the option value is more volatile, since the stock price is more volatile. Since $v > 0$ the option price must increase. He claims that $\frac{\partial v}{\partial T} > 0$. Let $\phi(x)$ represent the standard normal density. Below I will derive $\frac{\partial v}{\partial T}$.
$$\begin{align*} v &= S\phi(d_1)\sqrt{T} \\ \frac{\partial v}{\partial T} &= 0.5T^{-0.5}S\phi(d_1) - d_1 v \cdot \frac{\partial v}{\partial T}d_1 \\ &= \frac{v}{t} \left(\frac{2 - d_3}{2} \right) \end{align*}$$
where $d_3 = d_1 - \frac{2\ln(S/K)}{\sigma\sqrt{T}}$. This term is > 0 if $d_3 < 2$. From my understanding the percentage impact of the implied volatility would decrease if the partial derivative is negative ($d_3 > 2$). If $\sigma(t)$ represents non constant volatility and $v(t) = \frac{\partial C}{\partial \sigma(t)}$ then $\frac{\partial v(t)}{\partial T}$ should be > 0.
I believe that a GARCH model (non-constant volatility) could result in higher or lower prices than the Black-Scholes formula (constant, implied volatility). For instance, look here
EDIT 1 (ignore above)
(A) "As the maturity of the option increases the percentage impact of non-constant volatility on (option) prices becomes more pronounced"
(B) "As the maturity of the option increases, the percentage impact of non-constant volatility on implied volatility usually becomes less pronounced."
- Does non-constant volatility produce higher option prices than a constant volatility would for options with the same underlying, strike and time to maturity?
- How does a non-constant volatility effect implied volatility / what's the relationship here ?
Given more time to expiration, the stock price can fluctuate more. This means that the option value can fluctuate more. Because volatility (in particular non-constant volatility) will likely change the stock price more given a longer amount of time, the option price will change more. This is how I interpret (A).
An average volatility becomes more stable over time. The implied volatility is an estimated constant volatility. Therefore, as time increases, the implied volatility will change less since the average non-constant volatility will remain more or less the same. This is how I interpret (B).
- Non-constant volatility may produce higher option prices, but this is not always true.
- I explained this question in my interpretation of (B).
## Answer by Alex C (score 3)
https://quant.stackexchange.com/a/18711
In black-scholes the option price depends not on sigma^2 but on sigma^2 T. So if volatility is going to be 20% or 21% over the next 10 years (assume for simplicity no other values are possible, just these two with equal prob, but we don't know which) then that will have a bigger impact on the option value than a 20 vs 21 uncertainty for a 1 year option. That is at least in part what is going on here.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.