Krämer, Romy ; Richter, Matthias : Impact of monotonicity in some model of inverse option pricing

Author(s):

Krämer, Romy
Richter, Matthias

Title:

Impact of monotonicity in some model of inverse option pricing

Electronic source:

application/pdf

Preprint series:

Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 17, 2006

Mathematics Subject Classification:

65J20			[ Improperly posed problems; regularization ]
47H30			[ Particular nonlinear operators ]
91B28			[ Finance, portfolios, investment ]

Abstract:

The paper considers a problem of inverse option pricing aimed at the identification of a not directly observable volatility function. Although volatilities are in general assumed to be functions of time and the current price of the underlying asset the study of purely time-dependent volatility functions in combination with maturity-dependent option prices is of interest. In the latter situation an important aspect is the calibration of the antiderivative of the volatility. This inverse problem leads to an operator equation with a forward operator of Nemytskii type generated by a monotone function of two variables. In recent literature an analysis of this forward operator and several numerical case studies have been conducted which revealed certain instability effects. Consequently, the applicability of several regularization approaches has been discussed. This paper supplements these results by answering some open questions. As the main advancement we show that the considered inverse problem is indeed well-posed in the Banach space of continuous functions over a certain time interval in the sense that the inverse operator is continuous. The proof is based on the special Nemytskii type of the forward operator and the monotonicity of its generating function. This result classifies the occuring instabilities as ill-conditioning phenomena. In practice the above-mentioned ill-conditioning effects result in strongly oscillating solutions. Therefore, we study the stabilizing effect of apriori information concerning the monotonicity of the data and the searched solution. Using again the special structure of the forward operator we propose a numerically efficient algorithm for the computation of a strictly monotonically increasing approximate solution. Additionally, the situation of discrete stochastic noise is discussed. The described algorithms are illustrated by numerical case studies.

Keywords:

Inverse problem of option pricing, volatility calibration, Nemytskii operator, regularization

Language:

English

Publication time:

11 / 2006