Inhaltsverzeichnis - Mathematisches Institut der Universität zu Köln

DMV Tagung 2011 - Köln, 19. - 22. September
Pascal Heider
Universität Köln
Arbitrage-free Approximation of Call Price Surfaces
In this talk I present a construction of arbitrage-free call price surfaces from observed market data by
locally constrained least squares approximations. Derivatives of the surface are computed accurately so
that implied volatility, local volatility and transition probability density are obtained at no additional costs.
Observed input data are afflicted by a price uncertainty and cause an input data risk on the computed
call surface. I present a simple model for the input risk and perform an analysis to study and measure the
effect of the input risk on the surfaces. With this analysis one can determine the trust-worthiness of the
computed results and their implications on option pricing a posteriori.
Christian Jonen
Universität zu Köln
Valuing High-Dimensional American-Style Derivatives: A Robust Regression Monte Carlo
Method
Pricing high-dimensional American-style derivatives is still a challenging task, as the complexity of numerical
methods for solving the underlying mathematical problem rapidly grows with the number of uncertain
factors. In this paper we extend the important class of regression-based Monte Carlo methods for valuing
these complex financial products. The key idea of our proposed approach is to fit the continuation value
at every exercise date by robust regression rather than by ordinary least squares. By using robust regression,
we are able to get a more accurate approximation of the continuation value due to taking outliers
in the cross-sectional data into account. In order to guarantee an efficient implementation of our Robust
Regression Monte Carlo (RRM) method, we suggest a new Newton-Raphson-based solver for robust regression
with very good numerical properties. We use techniques of the statistical learning theory to prove
the convergence of our RRM estimator. In order to test the numerical efficiency of our proposed method,
we price Bermudan options on up to thirty assets. It turns out that our RRM approach shows a remarkable
convergence behavior; we get speed-up factors of up to over four compared with the state-of-the-art Least
Squares Monte Carlo method proposed by Longstaff and Schwartz (2001).
Literatur
C. Jonen (2011). Valuing high-dimensional American-style derivatives: a robust regression Monte Carlo
method. To be submitted.
F. Longstaff and E. Schwartz (2001). Valuing American options by simulation: a simple least-squares
approach. Review of Financial Studies, 14, 113 - 147.
179