Armed with the Hardy Decomposition for option prices, it now becomes much easier to understand why the smile exists.
To be clear, options trader might use the smile to manage supply & demand, but here we discuss the mathematical basis for smile – which is important if you want to understand how to generate smile in a monte-carlo model.
The Hardy Decomposition
In a previous post (click here) I demonstrated a new and useful way of writing option prices:
Option Value = Intrinsic Value + ATM Price * HardyFactor,
where HardyFactor is well approximated by:
HardyFactor ≅ 0.3*d1*d1 – 1.07*|d1| + 1,
where
ATM Price = 0.4 * sigma * sqrt(Maturity)
d1 = (F-K)/[sigma*sqrt(Maturity)],
with F being the forward, and K being the strike, sigma being the normal vol.
Vega of an ATM option
If we take the derivative of the ATM price with respect to sigma we get:
Diff_sigma ATM Price = 0.4 * sqrt(Maturity).
This just means that that vega of an ATM option does not change as the level of volatility changes.
Vega of an OTM option
In the Hardy Decomposition the HardyFactor is dependent on sigma: as sigma increases we find that d1 decreases, and therefore that the HardyFactor increases.
Intuitively this is because d1 is a measure of how far F is from the strike K in terms of the volatility of the underlying asset: eg. if the forward F is 1 standard deviation away from the strike K and then the volatility of the asset increases, F will now be less than 1 standard deviation away from K.
All of this means that for an OTM option, say a high-strike call:
- the vega will increase when sigma increases, and
- the vega will decrease as sigma decreases.
These two facts are summarized by saying that an OTM option has positive volga, short for “positive vol-gamma”, ie the second derivative with respect to sigma is positive.
Hedging OTM options with ATM options
Suppose we buy a high-strike call, and hedge the vega by selling some amount of an ATM call such that the total vega exposure is null.
As we have just seen, when sigma increases this portfolio will become vega positive because the vega of the OTM call will increase. Likewise, when sigma decreases this portfolio will become vega negative because the vega of the OTM call will decrease.
In short, this portfolio goes long sigma when sigma goes up, and goes short sigma when sigma goes down. It’s the ideal portfolio to hold: it makes money for you whether sigma goes up or down.
Traders will say that
“you are long convexity”
and make you pay for it by making the OTM call that you would like to buy a bit more expensive than the ATM call. And they do that by increasing the implied vol.
Hence the implied vols of the OTM options, aka the wings, get lifted above those of the ATM options.
The amount the wings get lifted is a function of the amount of volatility of sigma that traders expect to happen: so we say that
“smile is a function of vol-of-vol”.
Fermat's Last Spreadsheet 