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

Julien Hok () and Shih-Hau Tan ()
Additional contact information
Julien Hok: Crédit Agricole CIB
Shih-Hau Tan: Cuemacro

Decisions in Economics and Finance, 2019, vol. 42, issue 2, No 11, 609-637

Abstract: 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 $$P(t, S, r) \mathcal {Z}(t, S, r)$$P(t,S,r)Z(t,S,r), where P(t, S, r) represents the risk-neutral probability density of (S(t), r(t)) and $$\mathcal {Z}(t, S, r)$$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 (search for similar items in EconPapers)
JEL-codes: C02 (search for similar items in EconPapers)
Date: 2019
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (3)

Downloads: (external link)
http://link.springer.com/10.1007/s10203-019-00232-3 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:decfin:v:42:y:2019:i:2:d:10.1007_s10203-019-00232-3

Ordering information: This journal article can be ordered from
http://www.springer. ... ry/journal/10203/PS2

DOI: 10.1007/s10203-019-00232-3

Access Statistics for this article

Decisions in Economics and Finance is currently edited by Paolo Ghirardato

More articles in Decisions in Economics and Finance from Springer, Associazione per la Matematica
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().