Redaktion: K. Sigmund, G. Greschonig (Univ. Wien, Strudlhofgasse ...

Angewandte Mathematik, Industrie- und Finanzmathematik 135
High Order Compact Schemes for a Nonlinear Black-Scholes
equation
BERTRAM DÜRING
(gemeinsam mit M. Fournie, A. Jüngel)
Fach D193, Universität Konstanz, D-78457 Konstanz
Bertram.Duering@uni-konstanz.de
http://www.mathe.uni-konstanz.de/˜juengel/team/duering.html
Option pricing in markets with transaction costs lead to fully nonlinear Black-
Scholes equations with nonlinear volatilities depending on the second derivative
of the option price (the ’Gamma’), derived by Barles and Soner in 1998 [1].
The corresponding parabolic problem is solved using high order compact finite
difference schemes by extending the compact schemes proposed by Rigal. The
compact schemes are compared with classical finite difference schemes (explicit,
semi-implicit, Dufort-Frankel), and their properties (stability, non-oscillations)
are studied theoretically and numerically. It turns out that the proposed compact
scheme is stable and non-oscillatory for a wide range of parameters and gives
significantly better accuracy than the other schemes with comparable CPU times.
[1] G. Barles, H.M. Soner: Option Pricing with transaction costs and a nonlinear
Black-Scholes equation, Finance Stochast. 2, 369-397, 1998.
A Variational Inequality Approach to Financial Valuation of
Retirement Benefits
AVNER FRIEDMAN
(gemeinsam mit Weixi Chen)
The Black-Schole model for American put option allows one to compute the value
of an option by solving a parabolic variational inequality. The corresponding free
boundary tells one when the best time is to sell the option. In this talk we describe
an analogous approach to the problem of early retirement. We consider a
pension plan with the option of early retirement. The paid benefits Ψ� S� t� are the
larger of two quantities:(i) a guaranteed sum A, (ii) a multiple of the salary S� t�
at the time of retirement. The financial value of the retirement plan, V � V � S� t�
then satisfies a variational inequality, which is derived by combining ideas from
actuarial mathematics with the stochastic nature of S� t� . We determine the curve
V � S� t��� Ψ � S� t� , which is the optimal time for early retirement (from a financial
point of view) and, in particular, study its asymptotic behavior near the end-period
t � T of the pension plan.