Dupire

Robust Calibration of Local Volatility Models

For the so-called local volatility model, Bruno Dupire derived a closed form solution, which, when naively applied, delivers cliffy volatility surfaces. A robust and fast parameter calibration scheme has to be found.

Problem overview

Call and/or put options on liquid assets or equity indices are traded for different expiries and for a range of strike prices. It turns out that the traded option prices do not fit into the constant volatility world of Black-Scholes but exhibit so-called “volatility smiles” or “volatility skewnesses”.

A model for which such a behavior can be obtained without the need of stochastic volatility is the local volatility model: Here the stochastic movement of the price of the underlying follows

dS=μSdt+σ(S,T)SdWdS=μSdt+σ(S,T)SdW

with the drift rate μμ , the volatility function σσ and the increment dWdW of the Wiener process. The local volatility function cannot be measured directly, but has to be identified from the quoted option prices as mentioned above. Bruno Dupire showed in 1994: If these call prices were available as a function, then σσ must satisfy

Function

When we apply this inversion formula directly, we obtain Local volatility surface by applying Dupire’s inversion formula on a 50×50 grid: Strikes from 50 to 150 percent of spot. Expiries up to 5 years. Synthetic implied (annual) volatilities between 25 and 35%. Noise level of up to absolute 0.1%.

There are mainly two reasons for this cliffy behavior already at a low noise level: (1) Differentiation per se is an instable process that leads to noise amplifying if the noise frequency is high. (2) The second derivative  term in the denominator is close to zero if the traced options under consideration are deep in the money or deep out of the money.

Results and achievements

Mathematically, two conflicting targets should be achieved. On the one hand, the fit for the traded option data should be as good as possible (“model prices close to market prices”), on the other hand, the local volatility surface should be as smooth as possible.

Nonlinear Tikhonov regularization with an appropriate regularization parameter choice fulfills the requirement of a fast and robust identification procedure. Computing time for a market data set of 2500 options was 8 seconds on a Windows 7 laptop computer.

Regularised local volatility surface which can then be used for the valuation and risk analysis of more complex (non vanilla) financial instruments.

Further Reading

Aichinger, Binder: A Workout in Computational Finance, Wiley, 2013.

Egger, Engl: Tikhonov regularization applied to the inverse problem of option pricing: convergence analysis and rates, Inverse Problems 21, 1027–1045, 2005.

UnRisk website: www.unrisk.com