Posted by
andrea-110 on
Feb 29, 2008; 8:58pm
URL: http://quantlib.414.s1.nabble.com/Volatility-Surface-Interpolation-tp5741p5748.html
Ferdinando Ametrano wrote:
> On Wed, Feb 6, 2008 at 6:16 PM, Luigi Ballabio <luigi.ballabio@gm...> wrote:
> > Question for the volatility experts: I know that a decreasing variance
> > makes no sense---if it's the actual variance of an underlying. Since
> > we're talking of smiled volatility here [...] should we still
> > require monotonicity along each strike?
>
> we shouldn't in my opinion. I would remove the require condition.
>
> mimicking the no-smile reasoning: if the ATM forward level was
> constant between 2 dates then you should have increasing variance and
> increasing option values in order to avoid arbitrage. But if the ATM
> changes this is not true anymore.
>
> ciao -- Nando
Hi,
I think the requirement still holds but it has to be done in a world
without "interest rate and dividends".
Without interest rates and dividends this has to be true (regardless of them smile) because the
price of a call is convex and the underlying process is a martingale. (this for each constant strike
level).
With proportional dividends (as yield or as discrete amounts), the strike has to be a function of time.
If F(t) is the forward at time t, then
For each k (= moneyness) the function
vol(t, k * F(t)) ^ 2 * t
has to be increasing.
In case you have a matrix, it is a bit tricky because you normally have strikes that do not follow
these lines, so you need to interpolate.
Andrea
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