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Calibration of local volatility model with stochastic interest rates by efficient numerical PDE methods

  • Julien HokEmail author
  • Shih-Hau Tan
Article
  • 15 Downloads

Abstract

Long-maturity options or a wide class of hybrid products are evaluated using a local volatility-type modelling for the asset price S(t) with a stochastic interest rate r(t). The calibration of the local volatility function is challenging and time-consuming because of the multi-dimensional nature of the problem. A key requirement of any equity hybrid derivatives pricing model is the ability to rapidly and accurately calibrate to vanilla option prices. In this paper, we develop a calibration technique based on a partial differential equation (PDE) approach which allows an accurate calibration and provides an efficient implementation algorithm. The essential idea is based on solving the derived forward equation satisfied by \(P(t, S, r) \mathcal {Z}(t, S, r)\), where P(tSr) represents the risk-neutral probability density of (S(t), r(t)) and \(\mathcal {Z}(t, S, r)\) the projection of the stochastic discounting factor in the state variables (S(t), r(t)). The solution provides effective and sufficient information for the calibration and pricing. The PDE solver is constructed by using ADI (alternative direction implicit) method based on an extension of the Peaceman–Rachford scheme. Furthermore, an efficient algorithm to compute all the corrective terms in the local volatility function due to the stochastic interest rates is proposed by using the PDE solutions and grid points. It reduces by one order the computations costs and then allows to speed up significantly the calibration procedure. Different numerical experiments are examined and compared to demonstrate the results of our theoretical analysis.

Keywords

Local volatility model Stochastic interest rates Hybrid Calibration Forward Fokker–Planck-type equation Alternating direction implicit (ADI)  method 

JEL Classification

C02 

Notes

Acknowledgements

The authors thank Prof. Kevin Parrott from the University of Greenwich and the participants at QuantMinds International conference 2018 for their fruitful comments. The authors also thank the referees and the associate editor for their constructive feedbacks and remarks to improve the quality of this paper.

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Copyright information

© Associazione per la Matematica Applicata alle Scienze Economiche e Sociali (AMASES) 2019

Authors and Affiliations

  1. 1.Crédit Agricole CIBLondonUK
  2. 2.CuemacroLondonUK

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