A Simple Stochastic Rate Model for Rate Equity Hybrid Products

34 Pages Posted: 18 Mar 2013

See all articles by Ernst Eberlein

Ernst Eberlein

University of Freiburg

Dilip B. Madan

University of Maryland - Robert H. Smith School of Business

Martijn Pistorius

Imperial College London

Marc Yor

Université Paris VI Pierre et Marie Curie

Date Written: March 10, 2013

Abstract

A positive spot rate model driven by a gamma process and correlated to equity is introduced and calibrated via closed forms for the joint characteristic function for the rate r, its integral y and the logarithm of the stock price s under the T-forward measure. The law of the triple (r,y,s) is expressed as a nonlinear transform of three independent processes, a gamma process, a variance gamma process and a Wiener integral with respect to the Dirichlet process. The generalized Stieltjes transform of the Wiener integral with respect to the Dirichlet process is derived in closed form. Inversion of this transform using Schwarz (2005, The generalized Stieltjes transform and its inverse, Journal of Mathematical Physics, doi: 10.1063/1.1825077) makes large step simulations possible. Valuing functions are built and hedged using quantization and high dimensional interpolation methods. The hedging objective is taken to be capital minimization as described in Carr, Madan and Vicente Alvarez (2011, Markets, profits, capital, leverage and returns, Journal of Risk, 14, pp. 95-122).

Suggested Citation

Eberlein, Ernst and Madan, Dilip B. and Pistorius, Martijn and Yor, Marc, A Simple Stochastic Rate Model for Rate Equity Hybrid Products (March 10, 2013). Available at SSRN: https://ssrn.com/abstract=2234760 or http://dx.doi.org/10.2139/ssrn.2234760

Ernst Eberlein

University of Freiburg ( email )

Department of Mathematical Stochastics
Eckerstrasse 1
D-79104, Freiburg
Germany
++49 761 203 5660 (Phone)
++49 761 203 5661 (Fax)

Dilip B. Madan (Contact Author)

University of Maryland - Robert H. Smith School of Business ( email )

College Park, MD 20742-1815
United States
301-405-2127 (Phone)
301-314-9157 (Fax)

Martijn Pistorius

Imperial College London ( email )

South Kensington Campus
Exhibition Road
London, Greater London SW7 2AZ
United Kingdom

HOME PAGE: http://www3.imperial.ac.uk/people/m.pistorius

Marc Yor

Université Paris VI Pierre et Marie Curie ( email )

175 Rue du Chevaleret
Paris, 75013
France

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