Regularization Regularization Theory Application to Volatility Surfaces Volatility Calibration Volatility Calibration is an inverse problem of the form We are given call prices, C, and try to infer the risk-netural pdf, f, of the underlying, S Small perturbation in call prices can result in large changes in the pdf This is because the integral operation smoothes the latent pdf significantly The next two slides demonstrate the ill-posedness of the problem Parameter Instability Ill-posed inverse problems yield very different outputs for slight changes in the inputs Minimization of the least squares objective function is ill-posed Implied volatility dynamics Some portion of the movement reflects the changing market consensus on future volatility Some portion is due to temporary supply/demand mismatches, which is not as informative The graph below illustrates the inherent parameter instability in volatility parameters that are inferred from short-term options prices. This instability flows through to the liability valuation, Greeks and trading p&l. Local Minima A second, somewhat independent problem, is that the error surface, over which we search for a minimum, is replete with local minima (because some of the parameters compete with each other) This contributes to parameter instability, specifically parameter jumps The VolofVol/Mean Reversion Valley The vertical axis is the pricing error for a given set of options The axis oriented left/right is volofvol The axis oriented in/out is mean reversion Volatility Collapse: the three curves mark the boundary the pre-empts volatility path generation from hitting zero. The MR/Volofvol “best-fit” valley is on the wrong side of this boundary. Enter Regularization The key idea of regularization is to replace the ill-posed problem with a similar problem that is well-posed Tikhonov Regularization t is a Heston parameter vector, *t () is the optimal Heston vector which minimizes the above objective function at time t M is the calibration instrument data. N1 is the number of calibration instruments. * is the vector that holds the optimal regularization factors. Great care must be taken to choose the regularization factors “optimally”. N2 is the number of model parameters. In the case of the Heston model, N2=5. H() is the model price of the ith calibration instrument, given a parameter vector Regularization Parameters Weights between previous period’s volatility parameters and this period’s new data Remove the local minima problem. The Mean-Reversion/Volofvol Valley is now a Crater. Filter out short-term fluctuations in option quotes Data Sample period 12/30/2005 thru 2/28/2007 Monthly sampling frequency 15 observation dates Calibration Instruments Vanilla OTC quotes from Goldman, Sachs Maturity: 1, 5, 10, 15-year strings Skew Spot and forward ATM OTM calls/puts: K(T)=F(T)*exp(+/- vol(F,T)*T^.5) DO NOT include Asian: involves too much run-time Drift Parameters Zero rates bootstrapped from end-of-day swap rate curves (Bloomberg) Dividend yield’s inferred from forward prices Optimal Factor Selection: L-Curve Procedure L-Curves constructed with the entire sample period to circumvent the signal/noise problem Requires parameterization for several lambdas on each trading day Stability: standard deviation of change in parameters over 15 months Fit: mean root mean square pricing error over 15 months Find optimal factor for one factor, fix, then move to next optimal factor. The order is based on the instability of the parameters: stabilize the most unstable first. VolofVol Rho Mean Reversion Ideally, we would stabilize the next parameter that contributes the most to PnL instability This approach is computation infeasible, given the extensive Monte Carlo runs necessary A local optimization algorithm is used for the following results Local optimizations are faster than global optimizations Parameter Stickiness The chart below illustrates how parameter stability increases as a parameter’s regularization factor is gradually tuned up Parameter stability is defined as the standard deviation of monthly changes in each parameter. L-Curve: VolofVol The Leverage Anti-Correlation L-Curve Pricing Error: Average RMSE of Heston fit to data Parameter Instability: Standard deviation of monthly parameter movements Lambda=.00025 L-Curve: Rho The Leverage Anti-Correlation L-Curve Pricing Error: Average RMSE of Heston fit to data Parameter Instability: Standard deviation of monthly parameter movements Lambda=.00025 The Mean Reversion Strength L-Curve Pricing Error: Average RMSE of Heston fit to data Parameter Instability: Standard deviation of monthly parameter movements L-Curve: Mean Reversion Lambda=.00225 Initial and Target Volatility Factors We set the factors on the initial and target volatility parameters to 0 Generally, these parameters have significant influence on the pricing error and shape of the volatility surface Constraining their movement would compromise fit too much A principal components decomposition of volatilty surface movements shows First Component: Parallel Shifts account for 80-90% Second Component: Term Structure Shifts account for 5-10% Very similar to yield curve principal component decompositions Joint changes in the Initial and Target Volatility parameters translate approximately to parallel shifts Changes in the difference between Initial and Target Volatility parameters translate approximately to term structure shifts Summary Regularization stabilizes dynamics of model parameters at hardly any additional cost in fit This table summarizes the results of 1,400 distinct calibrations Future Work Time propagation The regularization factors in this presentation are based on monthly calibration frequency How does this change when we move to different frequencies, say weekly? Volatility Surface Sectors The factors are estimated from Goldman’s old quote set How will the factors change as we extend the calibration set to capture skew better? Update frequency How often do the optimal factors change over time? When should we not rely on regularization? Regularized Calibrations Mean Reversion VolofVol Rho Blue: Regularized