Contact  information:

The course will take place on Tuesday and Thursday 1.30 - 2.50 PM in the Bendheim Center for Finance.
Office: ORFE E416. Office telephone number: 258 5916. Course homepage: http://www.princeton.edu/~rcont/

Course description:

While option pricing theory deals with valuation of derivative instruments given a stochastic process for the underlying asset, model calibration is about identifying the (unknown) stochastic process of the underlying asset given information about prices of options, a more difficult and often ill-posed problem. This course is an introduction to theoretical, numerical and empirical aspects of model calibration. We will review different solutions, using probabilistic, PDE and dynamic programming methods, paying attention to numerical implementation of solutions. Along the way we will cover some aspects of the theory of regularization of ill-posed inverse problems, an active branch of applied mathematics. Examples of numerical performance on empirical data of the algorithms discussed will be given during the course.

Course outline:

• Reminders on option pricing:

Arbitrage pricing in diffusion models: PDE and martingale methods.
The Black Scholes model: Black Scholes PDE. Black Scholes formula for a call option.
Black-Scholes implied volatility. Smiles and skew. Implied volatility surfaces.

• The calibration problem:

Historical vs market-implied parameters.
The historical estimation approach : advantages and  drawbacks. The motivation for implied calibration.
Formulation of the calibration problem in terms of risk neutral probabilities. Link with inverse moment problems.
Formulation of the calibration problem in terms of local volatility functions.  Link with inverse PDE problems.
 
• Model calibration as an inverse problem:

Well posed and ill posed inverse problems. Existence, uniqueness and stability.
Remedies for ill-posedness: least squares solutions, pseudo-solutions and regularization schemes.
Regularization of ill-posed linear inverse problems: truncated SVD, Tikhonov regularization, spectral filtering.
Tikhonov's approach to regularization of ill posed inverse problems.
Determination of the regularization parameter: a priori, a posteriori and error-free parameter choice.
A priori choice: error minimization.
A posteriori choice: Morozov's discrepancy method.
Error-free choice: cross validation. Bakushinski's theorem.
Model calibration as an optimization problem.

Implied distributions and state price densities:

State price densities. The Breeden-Litzenberger formula. Implied distributions.
Cumulant and Edgeworth expansions. Implied skewness and kurtosis.
Hilbert space expansion methods ( Abken & Ramamurtie, Madan ).
Non parametric regression ( Ait Sahalia & Lo ).
Regularization by  smoothness penalties ( Jackwerth & Rubinstein ).
Regularization by relative entropy minimization (Stutzer, Buchen & Kelly, Avellaneda).
Weighted Monte Carlo calibration (Avellaneda et al).
Numerical implementation of Monte Carlo calibration algorithms.
Arbitrage constraints. Convergence conditions.
 

• Implied trees and implied diffusions:


Implied trees: theory and implementation.
Backward and forward Kolmogorov equations. Dupire's formula and implied Markov diffusions.
Interpretation in terms of butterfly spreads. Informational content of European options.
Dupire's formula in implied volatility coordinates. Relation between implied and local volatility.
Numerical implementation of  Dupire's formula.
Instability of Dupire's approach. Regularization by interpolation/ smoothing

• Calibration of local volatility surfaces: variational approach.


Regularized methods: constrained optimization and Lagrangian method.
Classical versus "augmented" Lagrangian.
Tikhonov regularization: Lagnado & Osher, Berestycki & Crepey.
Numerical implementation: large scale gradient descent.
Choosing the regularization parameter.
Spline representations ( Jackson et al, Coleman et al). Link with Tikhonov regularization.
 

• Calibration of local volatility surfaces: stochastic control approach.


Controlled diffusions & Hamilton Jacobi Bellman equations: a brief overview.
Volatility calibration as a constrained stochastic control problem ( Avellaneda, Friedman, Holmes, Samperi).
Dual problem and Lagrangian formulation. Hamilton Jacobi Bellman equation. Duality gap.
Regularization methods ( Samperi ).  Numerical implementation.
 

• Other  issues:


The role of asymptotic and qualitative analysis of model properties.
Penalization vs regularization.
Calibration of stochastic volatility / jump models. Calibration of models for interest rate derivatives.
Calibration with American options as input prices. Calibration of multi-asset models.

Course material:

Here are some slides in Postscript format.

Part 1, 6 Nov 2001 : Option pricing and the calibration problem.

Part 2, 8 Nov 2001: Model calibration as an inverse problem.

Part 3, 13 & 15 Nov 2001: Regularization of   ill-posed linear problems.

Part  4, 20 & 22 Nov 2001: Implied distributions and state price densities.

Part  5: 27 & 29  Nov 2001: Implied diffusions. Dupire's formula. Implied trees.

Part 6: Dec 4 & 6, 2001: Variational  methods for volatility calibration.

Part 7: Dec 11 & 13, 2001: Stochastic programming approach to volatility calibration.

General references :

There exists no book on model calibration covering the material in this course. Most mathematical finance textbooks completely avoid the question.
Some of the practical problems encountered in model calibration and solutions used by practitioners are covered in :

Riccardo Rebonato : Volatility and Correlation In the Pricing of Equity, FX and Interest Rate Options, Wiley, 2000.

In the context of stochastic volatility models some topics are discussed in:

Fouque, Papanicolaou & Sircar (2001): Derivatives in financial markets with stochastic volatility, Cambridge University Press.

However there are many good books on general aspects of ill posed inverse problems and their regularization if you are interested in this topic:

AN Tikhonov and VY Arsenin, Solutions of Ill Posed Problems,W.H. Winston, Washington DC, 1977.

A N Tikhonov et al: Numerical methods for the solution of ill-posed problems, Dordrecht ; Boston : Kluwer Academic Publishers, c1995.

H.W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems, Kluwer Academic Publishers, Dordrecht, 1996.

Articles:

• Abken, Peter A., Dilip B. Madan, and Sailesh Ramamurite, 1996a, Estimation of risk-neutral and statistical densities by hermite polynomial approximation: with an application to eurodollar futures options, Working paper, Federal Reserve Bank of Atlanta.
• Ait Sahalia, Y & Lo, A ( 1998)

Nonparametric Estimation of State-Price Densities Implicit in Financial Asset Prices,
Journal of Finance, 1998, 53, 499-547
• Andersen & Andreasen:

Jump diffusion models: volatility smile fitting and numerical methods for pricing.
Journal of Computational Finance.
• Andersen & Brotherton Ratcliffe (1997) The equity option volatility smile: an implicit finite difference approach,

Journal of Computational Finance, Vol 1, No 2.

• Avellaneda M., Friedman C, Holmes & Samperi: Calibrating volatility surfaces via relative entropy minimization.

Applied Mathematical Finance, March 1997.
• Marco Avellaneda, Robert Buff, Craig Friedman, Nicolas Grandchamp, Lukasz Kruk, and Joshua Newman :

Weighted Monte Carlo: A New Technique for Calibrating Asset­Pricing Models,
International Journal of Theoretical and Applied Finance , 2001.
• Bodhurta & Jermakyan (1999):  Non parametric estimation of an implied volatility surface.

Journal of Computational Finance, Vol. 2, 1999, Number 4, 29-60.
• Buchen PW, Kelly M (1996) "The maximum entropy distribution of an asset inferred from option prices",

J. Fin. Quant. Anal. 31(1), 143-159.
• Berestycki H, Busca J, & Florent I (2000):  An inverse parabolic problem arising in finance

Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 331 (12 part 1)  pp. 965-969.
• Berestycki H, Busca J, & Florent I (2001):  Asymptotics and calibration of local volatility models, Quantitative Finance, forthcoming.
• Bouchouev & Isakov :     Uniqueness and stability for inverse problems that arise in financial markets.

Inverse Problems (1999), R95-R116.
• Busca, J. & R Cont (2001): Implied volatility surfaces in exponential Lévy models, Working Paper.
• Coleman, Li & Verma (1999):  Reconstructing the unknown volatility function.

Journal of Computational Finance Vol. 2, Number 3, 1999, 77-102.
• Cont, R. (1998) Beyond implied volatility. Rapport interne CMAP.
• Cont, R. & Da Fonseca, J. (2001): Deformation of implied volatility surfaces: an empirical approach, in: Takayasu (Ed.), Empirical Finance, Tokyo: Springer.
• Cont, R. & Da Fonseca, J. (2001): Dynamics of implied volatility surfaces, Rapport interne CMAP.
• Corrado Ch. and T. Su, (1997)  Implied Volatility Skews and Stock Index Skewness and Kurtosis Implied by SP500 Options prices

Journal of Financial Research, XIX, No 2, 175-192.
• Derman, Kani & Kamal ( 1997 ): Trading and hedging of local volatility, Journal of Financial Engineering, Vol 6, No 3, Sept 1997, p 233 - 270.
• Emanuel Derman, Iraj Kani and Neil Chriss: Implied Trinomial Trees of the Volatility Smile, The Journal of Derivatives, Summer 1996.
• Jackson, Suli & Howison (1999):  Computation of deterministic volatility surfaces.

Journal of Computational Finance Vol. 2, Number 2, 1998-99, 5-32
• Jackwerth & Rubinstein 91996):  Recovering probability distributions from option prices, Journal of Finance,  Vol 51, 1611-1631.
• Benjamin Jourdain and Laurent Nguyen (2001): Minimisation de l'entropie relative par méthode de Monte-Carlo

Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 332 (4) (2001) pp. 345-350
• Lagnado & Osher (1997):  A technique for calibrating derivative security pricing models: numerical solution of an inverse problem.

The Journal of Computational Finance, 1(1), Fall 1997.
• M. Rubinstein (1994) Implied binomial trees. Journal of Finance, 49(3):771-819, 1994
• Samperi, D. (2001): Model calibration using entropy and geometry, preprint.
• M Stutzer (1997)  A simple nonparametric approach to derivative security valuation,

Journal of Finance, Vol 101, No 5, 1633-1652.