Egger, Herbert ; Hein, Torsten ; Hofmann, Bernd : On decoupling of volatility smile and term structure in inverse option pricing

Author(s):

Egger, Herbert
Hein, Torsten
Hofmann, Bernd

Title:

On decoupling of volatility smile and term structure in inverse option pricing

Electronic source:

application/pdf

Preprint series:

Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 19, 2005

Mathematics Subject Classification:

35R30			[ Inverse problems for PDE ]
47J06			[ Nonlinear ill-posed problems ]
65J20			[ Improperly posed problems; regularization ]
91B28			[ Finance, portfolios, investment ]

Abstract:

Correct pricing of options and other financial derivatives is of great importance to financial markets and one of the key subjects of mathematical finance. Usually, parameters specifying the underlying stochastic model are not directly observable, but have to be determined indirectly from observable quantities. The identification of local volatility surfaces from market data of European Vanilla options is one very important example of this type. As many other parameter identification problems, the reconstruction of local volatility surfaces is ill-posed, and reasonable results can only be achieved via regularization methods. Moreover, due to sparsity of data, the local volatility is not uniquely determined, but depends strongly on the kind of regularization norm used and a good a-priori guess for the parameter. By assuming a multiplicative structure for the local volatility, which is motivated by the specific data situation, the inverse problem can be decomposed into two separate subproblems. This removes part of the non-uniqueness and allows to establish convergence and convergence rates under weak assumptions. Additionally, a numerical solution of the two subproblems is much cheaper than that of the overall identification problem. The theoretical results are illustrated by numerical tests.

Keywords:

Inverse problem, option pricing, mathematical finance, volatility identification, Black-Scholes equation, Tikhonov regularization, convergence rates

Language:

English

Publication time:

1 / 2006