Implied Volatility, Fundamental solutions, asymptotic analysis and - PowerPoint PPT Presentation

Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop, September 11, 2008 Peter Laurence Dipartimento di Matematica e Facolt´ a di Statistica, Uni. Roma 1 Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 1/4

Main aims of this contribution Outline of talk Rapid review Implied Volatility, local volatility, mimicking behaviour. Practitioners like closed form formulas for calibration. Review Heat Kernel approach to solving stochastic volatility models. Hagan-Lesniewski, Henry-Labordère. Refined asymptotics for a class of generalized SABR like models. Joint work with Gérard Ben Arous and Tai-Ho Wang. Influence of curvature. Interaction between symmetry and heat kernel approach. Distance function: Work in progress with Matveev and Wang. Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 2/4

Quick Overview 1 The goal, from PDE point of view, is to solve parabolic equations in one or several spatial dimensions. On [0 , T ] , where T is the maturity of European option, solve: u t + a ij ( x ) u x i x j + b i u x i − ru = 0 , x ∈ Ω ⊂ R n , t ∈ [0 , T ] u ( x , T ) = ψ ( x ) final condition The matrix { a ij } is usually degenerate, so the operator above is often not uniformly parabolic. Researchers in PDE are used to seeing the equation expressed as initial value (rather than final value) problem. This can be achieved, by making the change of variables: τ = T − t . The problem then reads as: u τ − a ij ( x ) u x i x j − b i u x i + ru = 0 , x ∈ Ω ⊂ R n , t ∈ [0 , T ] u ( x , 0) = ψ ( x ) initial condition Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 3/4

Fundamental solution We are interested in finding a fundamental solution of such parabolic equations: F ( x , t, ξ, T ) , x , ξ ∈ Ω , t ∈ [0 , T ] Often Ω is R n or R n + . Here F satisfies the parabolic equation in the variables x and t and has a delta function final condition: F ( x , T, ξ, T ) = δ ξ ( x ) • We may wish to add additional boundary conditions, such as in the case of the valuation of barrier options, and in this case, we seek the Green’s function, rather than the Fundamental solution. . In the context of heat kernels we are then led to consider the Dirichlet heat kernel and the Neumann heat kernel. • Degeneracy : In mathematical finance, additional subtleties arise due to the degeneracy of the principal part at the boundary (Stochastic CEV models) or in the entire domain (Asian Options). Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 4/4

Finance in the News Quote from article by Mike Giles and Ronnie Sircar in Siam NEWS, October 2007: “ The major challenges in computational finance arise not from difficult geometries, as in many physical problems, but from the need for rapid calculation of an EXPECTATION or the solution of its associated Kolmogorov partial differential equation.” “ Efficiency is at the forefront, because models are re-estimated as new market data arrives and calibration (or “marking to market”) embeds the expectation/PDE calculation in an iterative solution to an inverse problem”’ Interpretation of last sentence: from traded market prices back out parameters in coefficients of parabolic operator and/or back out the functional form of these coefficients. Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 5/4

Closed Form or quasi-closed form solutions: A Mathematician’s Toolbox Recent years have seen surge in attempts to find closed form or quasi-closed form solutions to certain parabolic problems arising in option pricing. In today’s talk, we will concentrate on one of these, asymptotics . The other approach, which we will not discuss in detail, for lack of time,is: 1) Changes of Variables, Transformation Groups and Lie Symmetry Analysis. Literature (Partial List): Albanese and Kusnetsov: Reducing time homogeneous one dimensional diffusions to standard form and solving via special functions. Transformations of Markov Processes and Classification Scheme for Solvable Driftless Diffusions. Preprint 2005 Carr, Laurence and Wang: Reducing time inhomogeneous diffusions to standard form. Via Lie symmetry considerations. Comptes Rendus de l’Académie des Sciences, 2006. Linetsky: Time homogeneous one dimensional diffusions. Approach via eigenfunction expansions. Int. J. Theor. Appl. Finance 7 (2004). Ait-Sahalia: Annals of Statistics, 2007 “Closed-Form Likelihood Expansions for Multivariate Diffusions”. Reduction method to heat equation with lower order terms.(related to Lie). Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 6/4

Sabr Models and generalized SABR models Sabr Model in it’s original form (Hagan and Woodward, Hagan, Kumar, Lesniewski and Woodward, Andreasen-Andersen), with F t forward price. dF t = F β t y t dW 1 t dy t = αy t dW 2 t < dW 1 t , dW 2 t > = ρdt Calibrates to smile, but for only one maturity. ”Dynamic Sabr Model” dF t = γ ( t ) C ( F t ) y t dW 1 t dy t = ν ( t ) y t dW 2 t < dW 1 t , dW 2 t > = ρ ( t ) dt, time dependent parameters. Calibratable to implied volatility surface for several maturities. Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 7/4

Mean Reverting Sabr models λ -Sabr model (incorporating mean reversion dF t = νC ( F t ) y t dW 1 t dy t = κ ( θ − y t ) dt + γy t dW 2 t < dW 1 t , dW 2 t > = ρdt Introduced by Henry-Labordère (2005). dF t = C ( F t ) y δ t dW 1 t dy t = κ ( θ − y t ) dt + y δ t dW 2 t < dW 1 t , dW 2 t > = ρdt The homogeneous "delta-geometry" introduced by Bourgade and Croissant (2005). Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 8/4

Generalized Sabr models dF t = γ ( t ) C ( F t ) y δ t dW 1 t dy t = κ ( θ − y t ) dt + w ( y t ) dW 2 t < dW 1 t , dW 2 t > = ρdt Generalized class of SABR models introduced by G. Ben Arous, P . Laurence, TH Wang (2008): Try to understand how the asymptotics depends on function w ( y ) . Also, refine and generalize existing asymptotics. Osajima (2007): Approach to SABR models using Malliavin calculus. Easily quantifiable results only for original (lognormal) SABR models. Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 9/4

Approaches to the asymptotic expansions, I Three (loosely speaking) different approaches: Approach I Original Hagan et al. approach, eg Willmott Smile paper 2002: Key idea: Bypass fundamental solution and aim directly for the call option prices, using that at time zero,from Tanaka’s formula one can derive (Dupire 1996) � T � � F − K ) + + C 2 ( ¯ Call ( ¯ F, T ) = ( ¯ y 2 F ( F T ) | F 0 = ˆ F ) E ¯ T δ ¯ F ; α 0 = α 0 � T � F − K ) + + C 2 ( ¯ = ( ¯ y 2 p ( ˆ y, ¯ F ) ¯ F, ˆ F, ¯ y, T ) d ¯ y 0 � �� � ↓ = � T � �� � F − K ) + + C 2 ( ¯ := ( ¯ P ( ˆ y, ¯ F ) F, ˆ F, T ) 0 Function P above, in the backward variables, satisfies a backward Kolmogorov equation, final condition P ( f, y, T ) = y 2 δ ¯ F ( f ) P T + 1 2 C 2 ( f ) y 2 P ff + C ( f ) αρyP αf + α 2 y 2 P yy + drift terms = 0 Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 10/4 2 subject to the final condition delta function one variable.

Approaches to the asymptotic expansions, II & III Approach II Geometric Approach Geometric/Analytic approach This approach was introduced by Lesniewski in a lecture at the Courant Institute. Uses the hyperbolic plane. Generalized by several papers: Introduction of McKean heat kernel (see page 50), ie. fundamental solution of heat equation in hyperbolic plane, plays a central role. Henry-Labordere (2005) introduces the use of the heat kernel expansion and the λ Sabr model Geometric/Stochastic approach Ground breaking work by Varadhan (1965) and then by Molchanov (1975)and Azencott (1981), Ben Arous (1989). This is followed by the work by Bourgade and Croissant (2005) who apply Molchanov’ s results to stochastic volatility models. Approach III Malliavin stochastic calculus of variations based approach, based on work of Bismut, Kusuoka, Malliavin, due to Takahashi (1999-2008) at al.(also Fournier-Lions et al. for calculation of Greeks) and, more recently, by Osajima (2007, 2008), based on work of Kusuoka. • Also, Fouque, Papanicolaou, Sircar fast mean reverting SV models, and Alan Lewis Fourier Transform based methods. Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 11/4