Topics in Computational Finance a view from the trenches P . Hnaff - PowerPoint PPT Presentation

Topics in Computational Finance a view from the trenches P . Hénaff Télécom-Bretagne 20 March 2010

Stylized Facts about Financial Time Series Fat Tails Fitting a Density to Sample Returns Dependence Volatility Surface

Stylized Facts about Financial Time Series Fat Tails Fitting a Density to Sample Returns Dependence Volatility Surface Model Risk and Model Calibration Price and Value Calibration Issues with Complex Models

Stylized Facts about Financial Time Series Fat Tails Fitting a Density to Sample Returns Dependence Volatility Surface Model Risk and Model Calibration Price and Value Calibration Issues with Complex Models Numerical Challenges in Monte-Carlo Simulations Division of Labour Practical Advantages of MC Frameworks Open Topics for Research

Stylized Facts about Financial Time Series Fat Tails Fitting a Density to Sample Returns Dependence Volatility Surface Model Risk and Model Calibration Price and Value Calibration Issues with Complex Models Numerical Challenges in Monte-Carlo Simulations Division of Labour Practical Advantages of MC Frameworks Open Topics for Research Conclusion

Stylized Facts about Financial Time Series Fat Tails Closing price of December 2009 WTI contract.

Stylized Facts about Financial Time Series Fat Tails Time Series of Daily Return 0.10 0.05 TS.1 0.00 −0.05 2006−02−19 2006−09−08 2007−03−28 2007−10−15 2008−05−03 2008−11−20 Time

Stylized Facts about Financial Time Series Fat Tails Histogram of Daily Return Daily return 100 80 Frequency 60 40 20 0 −0.05 0.00 0.05 0.10 r

Stylized Facts about Financial Time Series Fitting a Density to Sample Returns Fitting a density to observed returns The Johnson family of distributions. X : observed Z : N ✭ 0 ❀ 1 ✮ Z ❂ ✌ ✰ ✍ ln ✭ g ✭ x � ✘ ✮✮ (1) ✕ where : ✽ u SL ❃ ♣ ❃ 1 ✰ u 2 u ✰ SU ❁ g ✭ u ✮ ❂ u SB 1 � u ❃ ❃ e u ✿ SN

Stylized Facts about Financial Time Series Fitting a Density to Sample Returns Johnson SU Distribution - December 2009 WTI contract. Johnson SU distribution 0.10 ● 0.05 ● ● ●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● y ● ● ● ● ● ● ● ● ● ● ● 0.00 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●● ● −0.05 ● ● ● −0.04 −0.02 0.00 0.02 0.04 q

Stylized Facts about Financial Time Series Fitting a Density to Sample Returns Johnson SU vs. Normal Johnson SU Normal 0.10 0.10 ● ● ● ● ● 0.05 0.05 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Sample Quantiles ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● y ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.00 ● ● ● ● 0.00 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −0.05 ● −0.05 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −0.04 0.00 0.02 0.04 −3 −1 0 1 2 3 q Theoretical Quantiles

Stylized Facts about Financial Time Series Fitting a Density to Sample Returns The Generalized Lambda Distribution Tukey’s Lambda distribution : Q ✭ u ✮ ❂ ✕ ✰ u ✕ � ✭ 1 � u ✮ ✕ (2) ✕ Generalized Lambda distribution : Q ✭ u ✮ ❂ ✕ 1 ✰ u ✕ 3 � ✭ 1 � u ✮ ✕ 4 (3) ✕ 2 where : Pr ✭ X ❁ Q ✭ u ✮✮ ❂ u

Stylized Facts about Financial Time Series Fitting a Density to Sample Returns Density of daily return fitted with Generalized Lambda density λ 1 : 1.10e−03 λ 2 : 1.31e+02 λ 3 : −1.76e−01 λ 4 : −5.32e−02 RPRS RMFMKL 30 STAR 25 20 Density 15 10 5 0 −0.05 0.00 0.05 0.10 Daily return