Evnine-Vaughan Associates, Inc. A Theory of Non-Gaussian Option Pricing: capturing the smile and the skew Lisa Borland Acknowledgements: Jeremy Evnine Roberto Osorio Jean-Philippe Bouchaud

Layout The Stock Price Model Option Pricing Theory – the Smile Results Option Pricing Theory – the Skew Results Conclusions

The Standard Stock Price Model

Gaussian Distribution Fokker-Planck Equation

The Generalized Returns Model

Student (Tsallis) Distribution Nonlinear Fokker-Planck

o Empirical --- Gaussian q=1.43 Tsallis Distribution (Osorio et al 2002)

More realistic model: eg moving average (work in progress ) Current Model: Extension to Model:

q=1.5 q=1 SP500 q=1.5

d=1 d=2 d=4 d=8 d=16 SP500 d=1 d=2 d=4 d=8 d=16 q=1.5 log P Y(t+d)-Y(t)

Deterministic Risk-Free Portfolio  Return = risk-free rate r Generalized Black-Scholes PDE Arbitrage Theorem: Option

1) Exploit PDE’s implied by arbitrage-free portfolios Solve PDE to get option price 2) Convert prices of assets into martingales Take expectations to get option price

Not a martingale w.r.t. measure F Martingale:

Not a martingale w.r.t. measure F Martingale: Is a martingale w.r.t. measure Q Effectively:

Stock Price Example European Call

Stock Price Example European Call Must integrate

Path Integral: Generalized Feynman-Kac Ansatz: Tsallis (Student) weights Result: Forward Equation:

Result: Generalized Feynman-Kac

Stock Price Example European Call

 Stock Price Payoff if Example European Call q = 1: P is Gaussian q >1 : P is fat tailed Student(Tsallis) dist.

T=0.6 T=0.05 Call Price Difference S(0) = $50, r= 6%, =0.3 $

T=0.1 T=0.4 Black-Scholes (q=1) volatilities implied from q=1.5 model

q=1.5

K=50, T=0.4, sigma=0.3, r=.06

q=1 q=1.5 q=1.45 q=1.4 q=1.3 K=50, T=0.4, sigma=0.3, r=.06

Volatility Smiles o Empirical implied vols __ q=1.43 implied vols

Implied Volatility JY Futures 16 May 2002 T=17 days

Implied Volatility JY Futures 16 May 2002 T=37 days

Implied Volatility JY Futures 16 May 2002 T=62 days

Implied Volatility JY Futures 16 May 2002 T=82 days

Implied Volatility JY Futures 16 May 2002 T=147 days

Term Structure q=1 (BS) q=1.4 (Vol Surface)

Example Currency Futures: Benefits of a more parsimonious model: 1)Better pricing - arbitrage opportunities 2) Better hedging q Mean square relative pricing error (500 options)

(with Jean-Philippe Bouchaud) The Generalized Model with Skew Volatility Leverage Correlation

(with Jean-Philippe Bouchaud) The Generalized Model with Skew

(with Jean-Philippe Bouchaud) The Generalized Model with Skew small

Example European Call Payoff if Integrate using Feynman-Kac and Pade expansion

Comments q=1: CEV model of Cox and Ross recovered Skew model can be mapped onto a higher dimensional free-particle diffusion in cylindrical coordinates Exact solutions in terms of hyper-geometric functions ?

T=0.1 T=0.5 T=1.0 S(0) = 50 alpha = 0. Volatility Skew :

S0=50 T=0.5 sigma=0.3 r=0.06 q=1.5

q =1

q=1.5 K=50,T=0.5,sigma=0.3,r=.06

Strike K T=.03 T=0.1 T=0.2 T=0.3 T=0.55 SP500 OX q=1.5, alpha = -1.

T=.082T=.159 T=.41 T=1.17 MSFT Nov ATM = 25.55

SABR Model

T Sigma q=1.4

Conclusions Simple – few parameters, easy to calculate Promising for describing real markets Better hedging, better pricing Possible to apply to other areas of mathematical finance Future work : Extending model for underlying Exotics And more …