# Fin-Diff-Fin: Finite differences for Financial derivative models

### Summary

The famous Black-Scholes equation is an effective model for option pricing. It was named after the pioneers Black, Scholes and Merton suggested it 1973. The main goal of the present project is the development of effective numerical schemes for solving linear and nonlinear problems arising in mathematical theory of derivative pricing. To do so, there is the evident need to exchange ideas and skills in the field of numerical approximation of derivative pricing models. Moreover this project shall support preparing young promising scientists for their future career.

An option is the right (not the duty) to buy ("call option") or to sell ("put option") an asset (typically a stock or a parcel of shares of a company) for a price by the date . European options can only be exercised at the expiration date . For American options exercise is permitted at any time until the expiry date. The standard approach for the scalar Black-Scholes equation for European (American) options results after a standard transformation in a diffusion equation posed on a bounded (unbounded) domain.

The second problem arises when considering American options (most of the options on stocks are American style). Then one has to compute numerically the solution on a semi-unbounded domain with a free boundary. Usually finite differences are used to discretize the equation and artificial boundary conditions are introduced in order to confine the computational domain. If the solution on the computational domain coincides with the exact solution on the unbounded domain (restricted to the finite domain), one refers to these BCs as transparent boundary conditions (TBCs). While the numerical treatment of the free boundary has attracted a lot of attention and different strategies were developed less attention was paid to the accurate treatment of the artificial boundary In fact, many textbooks propose to use a homogeneous Dirichlet boundary condition at some finite distance. These artificial boundary condition are designed on a discrete level directly for the chosen scheme in order to conserve the stability and to prevent any numerical reflections at these boundaries.

Finally, we consider the adequate solution of nonlinear Black-Scholes models for American options using the Landau fixed domain transformation and an iterative operator splitting algorithm. We will consider different models (Leland, Barles/Soner, Frey/Stremme and Kratka/Jandacka/Ševčovič) where the volatility may depend on the time to expiry T-t, the asset price S and the second derivative of the option price VSS.

### Scientific Objectives

• development and analysis of advanced numerical schemes for linear and nonlinear derivative pricing models
• free boundary problems in derivative pricing models and their effective numerical tracking
• development of numerical algorithms for solving higher dimensional problems in derivative pricing - from basket options to index derivative pricing
• calibration of linear and nonlinear derivative pricing models to real market data, numerical solution of various inverse problems for parabolic PDEs

### Publications of the Project

 [AnEh07] J. Ankudinova and M. Ehrhardt, On the numerical solution of nonlinear Black-Scholes equations ( PDF-Format), Comput. Math. Appl. Vol. 56, (2008), 799-812. [Ank08] J. Ankudinova, The numerical solution of nonlinear Black-Scholes equations, Diploma thesis, TU Berlin, March 2008. [AnEh08] J. Ankudinova and M. Ehrhardt, Fixed domain transformations and split-step finite difference schemes for Nonlinear Black-Scholes equations for American Options , Chapter 8 in: M. Ehrhardt (ed.), Nonlinear Models in Mathematical Finance: New Research Trends in Option Pricing., Nova Science Publishers, Inc., Hauppauge, NY 11788, 2008, pp. 243-273. [Ehr08a] M. Ehrhardt, Nonlinear Models in Option Pricing - an Introduction, Preface in: M. Ehrhardt (ed.), Nonlinear Models in Mathematical Finance: New Research Trends in Option Pricing., Nova Science Publishers, Inc., Hauppauge, NY 11788, 2008, pp. 1-19. [Ehr08b] M. Ehrhardt (ed.), Nonlinear Models in Mathematical Finance: New Research Trends in Option Pricing, Nova Science Publishers, Inc., Hauppauge, NY 11788, 2008, 352 pages, ISBN: 978-1-60456-931-5. [EhMi08] M. Ehrhardt and R.E. Mickens, A fast, stable and accurate numerical method for the Black-Scholes equation of American options, Int. J. Theoret. Appl. Finance Vol. 11, Issue 5, (2008), 471-501. [Kil08] S. Kilianová, Stochastic Dynamic Optimization Models for Pension Planning , Ph.D thesis, Department of Applied Mathematics and Statistics, Comenius University, Bratislava, Slovakia. [Sev07] D. Ševčovič, An iterative algorithm for evaluating approximations to the optimal exercise boundary for a nonlinear Black-Scholes equation, Canad. Appl. Math. Quarterly, 15, No.1, (2007), 77-97. [Sev08] D. Ševčovič, Transformation methods for evaluating approximations to the optimal exercise boundary for a linear and nonlinear Black-Scholes equation Chapter 6 in: M. Ehrhardt (ed.), Nonlinear Models in Mathematical Finance: New Research Trends in Option Pricing., Nova Science Publishers, Inc., Hauppauge, NY 11788, 2008, pp. 173-218. [SSM09] D. Ševčovič, B. Stehliková and K. Mikula, Analytical and numerical methods for pricing derivative securities, Nakladatelstvo STU, Bratislava 2009, 200 pages. ISBN 978-80-227-3014-3 (in Slovak). [Ste08] B. Stehliková, Mathematical analysis of term structure models, Ph.D thesis, Department of Applied Mathematics and Statistics, Comenius University, Bratislava, Slovakia. [StSe09] B. Stehliková and D. Ševčovič, Approximate formulae for pricing zero-coupon bonds and their asymptotic analysis, Int. J. Numer. Anal. Modeling, Vol. 6, Nr. 2, (2009), 274-283. [Wur07] A. Würfel, Analytische und numerische Lösung der Black-Scholes Gleichung für europäische und amerikanische Basket-Optionen (Analytical and numerical solution of the Black-Scholes equation for european and american basket options), Diploma thesis (in german), TU Berlin, February 2007.

### Talks related to the Project

 [Ehr06] M. Ehrhardt, July 6, 2006, Finite differences for Financial derivative models, Frankfurt MathFinance Colloquium (HfB - Frankfurt MathFinance Institute (Goethe University) - Giessen University), HfB - Business School of Finance & Management, Frankfurt, Germany. (INVITED TALK) (ABSTRACT) [Sev07] D. Ševčovič, April 24, 2007, Analysis of the optimal exercise boundary for nonlinear Black--Scholes equation, Kolloquium der Arbeitsgruppe Modellierung, Numerik, Differentialgleichungen, Institute for Mathematics, TU Berlin, Germany. (INVITED TALK) [Ehr07a] M. Ehrhardt, June 20, 2007, Discrete Artificial Boundary Conditions for the Black-Scholes Equation of American Options, Department of Applied Mathematics and Statistics, Division of Applied Mathematics, Comenius University, Bratislava, Slovakia. (INVITED TALK) [Ank07a] J. Ankudinova, September 20, 2007, On the numerical solution of nonlinear Black-Scholes equations, International Conference on Mathematical Models in Life Sciences & Engineering, 19-21 September 2007, Institute of Multidisciplinary Mathematics, Universidad Politcnica de Valencia, Spain. (INVITED TALK) [Ehr07b] M. Ehrhardt, November 6, 2007, Finite differences for Financial derivative models, Workshop Numerics for Finance, Commerzbank, Frankfurt am Main, Germany. (TALK) [Ank07b] J. Ankudinova, November 15, 2007, On the numerical solution of nonlinear Black-Scholes equations, Institute for Mathematics, TU Berlin, Germany. (TALK) [Ank07c] J. Ankudinova, November 21, 2007, On the numerical solution of nonlinear Black-Scholes equations, Department of Applied Mathematics and Statistics, Division of Applied Mathematics, Comenius University, Bratislava, Slovakia. (INVITED TALK) [Ehr08a] M. Ehrhardt, April 3, 2008, Fixed domain transformations and highly accurate compact schemes for Nonlinear Black-Scholes equations for American Options, 4th Workshop Nonlinear PDEs and Financial Mathematics, April 3-4, 2008, Halmstad, Sweden. (INVITED TALK) [Ehr08b] M. Ehrhardt, October 6, 2008, On the numerical solution of nonlinear Black-Scholes equations, Johann-Wolfgang-Goethe University, Frankfurt am Main, Germany. (INVITED TALK) [Sev09] D. Ševčovič, January 28, 2009, Higher order estimates for the curvature and nonlinear stability of stationary solutions for curvature flow with triple junction, Berliner Oberseminar Nichtlineare partielle Differentialgleichungen (Langenbach-Seminar), Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin, Germany. (INVITED TALK)

#### Travel Informations: ehrhardt@math.tu-berlin.de