> An implied volatility surface is used to to price exotic options
> consistently with the market expectations represented by a
> set of known european options values.
> Since the implied volatility surface is usually not flat, it is
> used to represent deviation from Gaussian behavior of the underlying,
> while preserving risk-neutral valuation.
My understanding, shared also among various academics, is that the use
of implied volatilities is not a consistent method for pricing options.
Independently of whether the process is Gaussian or not (see comment
above) there is NO REASON why it is risk-neutral. It is in fact easy to
see why. The volatility surface changes with time (you estimate it today
but then tomorrow the market can change it). In other words, by using
implied volatilities one has no idea of the risk exposed, and no idea of
how to construct a replicating portfolio. Previsibility is lost. There
might be a martingale measure, but simply we do not know it.
Please correct me if I am wrong.
I know that many practisioners use implied volatilities, but at the
moment there is no mathematical justification for it and it is a very
difficult theoretical problem. My colleagues here told me that there are
many experts are looking on the issue at the moment.
Best regards
Vassilis
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By definition you use the implied vol surface to price options. Which one you
use for risk management is obviously another question - I know people who risk
manage with one single vol - usually atm - and hence take a P&L impact whenever
they trade, and others risk manage at the vol of the strike.
The fact that the surface changes over time is not too bad - if you are doing a
vega hedge you are protected to the first order, you just did not charge time
decay if there were any convexity (compare IR models where in a one factor model
you are hedged against movements of the slope of the yield curve, but where you
do not accrue time decay for these movements either).
The method you use mainly depends on your view how the implied volatility
surface is going to change if spot moves. The assumption that vol of the strike
does not change is in principle arbitrageable - you just sell the expensive vol
and buy the cheap vol so that you are Theta neutral, and with Black-Scholes
Theta = .5 * Vol^2 * Spot^2 * Gamma + carry
you see that you can own Gamma for free -> arbitrage.
This arbitrage however is only a sure thing if you believe that vols do not
change - if they do you are taking a risk. Consequence -> the hedge you are
doing might well be the one proposed by a yet to be developed stoch vol model
and hence not too bad....
s
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