Volatility Derivatives Modeling Bruno Dupire Bloomberg NY NYU, January, 2005 Outline I Products II Models III The Skorohod Problem IV Lower Bound V Conclusion Volatility Products Historical Volatility Products Historical variance: OTC products: Volatility swap Variance swap
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NYU, January, 2005
Simple product, but complex mix of underlying and volatility:
Call option has :
These sensitivities vary through time and spot, and vol :
To playpure volatility games (eg bet that S&P vol goes up, no view on the S&P itself):
Under BS: dS = σS dW,
For all S,
The log profile is decomposed as:
In practice, finite number of strikes CBOE definition:
Put if Ki<F,
We can buy today a PF which gives VIX2T1 at T1:
buy T2 options and sell T1 options.
The difference between both sides depends on the variance of PF (vol vol), which is difficult to estimate.
Much less transaction costs on F than on PF (by a factor of at least 20)
instead of F by PF!
can be estimated by combining the historical volatilities of F and Spot VIX.
Seemingly circular analysis :
F is estimated through its own volatility!
From now on, we assume 0 interest rates, no dividends and V is the quadratic variation of the price process (not of its log anymore)
For a given probability density function such that
find a stopping time t of finite expectation such that the density of W stopped at t is
Let us consider a continuous martingale X
Dambis, Dubins-Schwarz (DDS):
In a weaker form,
Given a market smile, we can compute the final density φ for the spot. Thanks to DDS:
processes that fit market prices
brownian motion stopped by a stopping time solution of the Skorokhod problem for φ
Use for instance.
and a lower variance.
plot: other solution [AY]
Same marginal distribution on the spot axis for all solutions
Root’s stopping time (var=0).
Same expected value only for AY’s, but higher variance.
Then for ,
Which is a quantity observable from current option prices
with f(x,0) = f(x)
And the replicating cost of is
finances exactly the replication of f until
dominates which has a market price
Where if and otherwise
Where is the price of for variance v and is the market implied variance for strike K