- Priv.-Doz. Dr. Matthias Ehrhardt (TU Berlin, WIAS Berlin)

Exotic options derivatives that are especially designed for customers needs.
For these non-standard options the well-known transformations of the Black-Scholes
equation to the heat equation do not work any more.
Hence, one has to discretize the Black-Scholes equation directly.
If one simply applies standard numerical methods for parabolic equations without
taking special care of the financial meaning of the variables then
one faces severe problems.

Here, in this thesis, we will show how to modify standard numerical schemes,
e.g. by nonlocal discretization approaches,
to remove unrealistic restrictions on the time step,
preserve the positivity of the solution obeying a discrete maximum principle,
adequate treatment of non-smooth initial data
(pay-off function), etc. while keeping all the stability properties
of the original scheme.
As an illustrative example we consider a discretely monitored barrier option,
a knock-and-out European Put option.

positivity-preserving finite difference schemes, M-matrices, nonlocal discretization, nonstandard finite difference methods (NSFD), discrete maximum principle, spurious oscillations, discretely monitored Barrier options, Box scheme, exponentially-fitted method, error damping, non-smooth initial conditions

- F. AitSahlia, L. Imhof and T.L. Lai,
*Fast and accurate valuation of American barrier options*, Journal of Computational Finance**7**(2003), 129-145. - L.B.G. Andersen,
*Monte Carlo simulation of barrier and lookback options with continuous or high-frequency monitoring of the underlying asset*, Working Paper, General Re Financial Products Corp., New York, 1996. - P. Boyle and S.H. Lau,
*Bumping up against the barrier with the binomial method*, Journal of Derivatives**2**(1994), 6-14. - P. Boyle and Y. Tian,
*An explicit finite difference approach to the pricing of barrier options*, Applied Mathematical Finance**5**(1998), 17-43. - M. Broadie, P. Glasserman and S. Kou,
*A continuity correction for discrete barrier options*, Mathematical Finance**7**(1997), 325-349. - M. Broadie, P. Glasserman and S. Kou,
*Connecting discrete and continuous path-dependent options*, Finance and Stochastics**3**(1999), 55-82. - M. Broadie and Y. Yamamoto,
*A Double-Exponential Fast Gauss Transform Algorithm for Pricing Discrete Path-Dependent Options*, Operations Research**53**(2005), 764-779. - P. Carr,
*Two extensions to barrier option valuation*, Applied Mathematical Finance**2**(1995), 173-209. - T.H.F. Cheuk and T.C.F. Vorst,
*Complex barrier options*, Journal of Derivatives**4**(1996), 8-22. - M. Costabile,
*A discrete-time algorithm for pricing double barrier options*, Decisions in Economics and Finance**24**(2001), 49-58. - D. Davydov and V. Linetsky,
*Structuring, pricing and hedging double-barrier step options*, Journal of Computational Finance**5**(2001), 55-88. -
D.J. Duffy,
*A Critique of the Crank-Nicolson Strengths and Weaknesses for Financial Instrument Pricing*, Technical Article, WILMOTT magazine, 2004, pp. 68-76 -
D.J. Duffy,
*Robust and Accurate Finite Difference Methods in Option Pricing - One Factor Models*, Working Report, Datasim, 2001. - G. Fusai, S. Sanfelici and A. Tagliani,
*Practical problems in the numerical solutions of PDEs in finance*, Rendiconti per gli Studi Economici Quantitativi**2001**(2002), 105-132. - G. Fusai, I.D. Abrahams and C. Sgarra,
*An exact analytical solution for discrete barrier options*, Finance and Stochastics**10**(2006), 1-26. - B. Gao, J. Huang and M.G. Subrahmanyam,
*The valuation of American barrier options using the decomposition technique*, Journal of Economic Dynamics and Control**24**(2000), 1783-1827. - E. Haug,
*Closed form valuation of American barrier options*, Inter. Journal of Theoretical and Applied Finance**4**(2001), 355-359. -
A.Q.M. Khaliq, D.A. Voss and M. Yousuf,
*Pricing exotic options with L-stable Padé schemes*, Journal of Banking & Finance and Applied Mathematics**31**(2007), 3438-3461. - A. Kolkiewicz,
*Pricing and hedging more general double-barrier options*, Journal of Computational Finance**5**(2002), 1-26. - Y.K. Kwok,
*Mathematical Models of Financial Derivatives*, second edition, Singapore, Springer, 2003. - A. Li,
*The pricing of double barrier options and their variations*, Advances in Futures and Options Research**10**(1999), 17-41. - L.S.J. Lou,
*Various types of double-barrier options*, Journal of Computational Finance**4**(2001), 125-138. - J.C. Ndogmo and D.B. Ntwiga,
*High-order accurate implicit methods for the pricing of barrier options*, arXiv, 2007. - G. Pacelli and L.V. Ballestra,
*Pricing Double-Barrier Options Using the Boundary Element Method*, (July 18, 2009). Available at SSRN: http://ssrn.com/abstract=1492653 -
A. Pelsser,
*Pricing double barrier options using Laplace transforms*, Finance and Stochastics**4**(2000), 95-104. -
D. Pooley, P.A. Forsyth, K. Vetzal and R.B. Simpson,
*Unstructured meshing for two asset barrier options*, Applied Mathematical Finance**7**(2000), 33-60. -
M. Reimer and K. Sandmann,
*A discrete time approach for European and American barrier options*, Working Paper, Rheinische Friedrich-Wilhelms-Universität, Bonn, 1995. -
D. Rich,
*The mathematical foundations of barrier option-pricing theory*, Advances in Futures and Options Research**7**(1991), 267-311. -
P. Ritchken,
*On pricing barrier options*, Journal of Derivatives**3**(1995), 19-28. -
M. Rubinstein and E. Reiner,
*Breaking down the barriers*, Risk**4**(1991), 28-35. -
S. Sanfelici,
*Galerkin infinite element approximation for pricing barrier options and options with discontinuous payoff*, Decisions in Economics and Finance**27**(2004), 125-151. -
J. Sidenius,
*Double barrier options: valuation by path counting*, Journal of Computational Finance**1**(1998), 63-79. -
A. Tagliani,
*Discrete Monitored Barrier Options by Finite Difference Schemes*, Tenth Workshop in Quantitative Finance, Milano, January 29-30, 2009. -
B.A. Wade, A.Q.M. Khaliq, M. Yousuf, J. Vigo-Aguiar and R. Deininger,
*On Smoothing of the Crank-Nicolson scheme and higher order schemes for pricing barrier options*, Journal of Computational and Applied Mathematics**204**(2007), 144-158. -
P.G. Zhang,
*Exotic Options*, World Scientific, 2nd edition, Singapure 1998. -
R. Zvan, P.A. Forsyth and K.R. Vetzal,
*Discrete Asian barrier options*, Journal of Computational Finance**3**(1999), 41-67. -
R. Zvan, K.R. Vetzal and P.A. Forsyth,
*PDE methods for pricing barrier options*, Journal of Economic Dynamics and Control**24**(2000), 1563-1590.

ehrhardt@math.uni-wuppertal.de