Implied volatility and nonconstant volatility

Question

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.

1. Does nonconstant volatility produce higher option prices than a constant volatility would for options with the same underlying, strike and time to maturity ?

2. How does a nonconstant volatility effect implied volatility / what's the relationship here ?

Answer 1

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."

1. Does non-constant volatility produce higher option prices than a constant volatility would for options with the same underlying, strike and time to maturity?

2. 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).

1. Non-constant volatility may produce higher option prices, but this is not always true.

2. I explained this question in my interpretation of (B).

Answer 2

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.