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

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 usually time-consuming because of the multi-dimensional nature of the problem. In this paper, we develop a calibration technique based on a partial differential equation (PDE) approach which allows an efficient implementation. The essential idea is based on solving the derived forward equation satisfied by 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 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. Different numerical experiments are examined and compared to demonstrate the results of our theoretical analysis.

Pricing and Rick Analysis in Hyperbolic Local Volatility Model with Quasi‐Monte Carlo

“…Here we extend previous analysis to a more realistic local volatility-type diffusion, namely the hyperbolic local volatility introduced by Jäckel (2008) and widely used in the quantitative finance industry (see e.g. Bompis & Hok, 2014;Hok & Tan, 2019;Hok et al, 2018).…”

Pricing and Risk Analysis in Hyperbolic Local Volatility Model with Quasi Monte Carlo

“…[17,18] for discussions). Here we extend previous analysis to a more realistic local volatility type diffusion, namely the hyperbolic local volatility introduced in [15] and widely used in quantitative finance industry (see e.g [19,20,21]).…”

“…This model was introduced in [26]. It behaves similarly to the Constant Elasticity of Variance (CEV) model, and has been used for numerical experiments in [19,20,21]. The advantage of this model is that zero is not an attainable boundary, and that allows to avoid some numerical instabilities present in the CEV model when the underlying asset price is close to zero (see e.g.…”

Speeding up the Euler scheme for killed diffusions

“…Here ν > 0 is the level of volatility, and β ∈ (0, 1] is the skew parameter. First introduced in Jäckel [31], this behaves similarly to the constant elasticity of variance (CEV), this model and has been widely used in quantitative finance for numerical experiments in Hok et al [25,26,24]. A practical advantage of this model is that zero is not an attainable boundary, which in turn avoids some numerical instabilities present in the CEV model when the underlying asset price is close to zero (see e.g.…”