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LOCALIZED RADIAL BASIS FUNCTIONS FOR NO-ARBITRAGE PRICING OF OPTIONS UNDER STOCHASTIC ALPHA–BETA–RHO DYNAMICS

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Mathematical finance

Published online by Cambridge University Press:  19 August 2021

N. THAKOOR Open the ORCID record for N. THAKOOR [Opens in a new window]
N. THAKOOR*
Affiliation:
Department of Mathematics, University of Mauritius, Reduit80837, Mauritius
*
e-mail: [email protected]
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Abstract

Closed-form explicit formulas for implied Black–Scholes volatilities provide a rapid evaluation method for European options under the popular stochastic alpha–beta–rho (SABR) model. However, it is well known that computed prices using the implied volatilities are only accurate for short-term maturities, but, for longer maturities, a more accurate method is required. This work addresses this accuracy problem for long-term maturities by numerically solving the no-arbitrage partial differential equation with an absorbing boundary condition at zero. Localized radial basis functions in a finite-difference mode are employed for the development of a computational method for solving the resulting two-dimensional pricing equation. The proposed method can use either multiquadrics or inverse multiquadrics, which are shown to have comparable performances. Numerical results illustrate the accuracy of the proposed method and, more importantly, that the computed risk-neutral probability densities are nonnegative. These two key properties indicate that the method of solution using localized meshless methods is a viable and efficient means for price computations under SABR dynamics.

Keywords

stochastic alpha–beta–rho modellocal radial basis functionsfinancial optionsno-arbitrage prices

MSC classification

Primary: 65M06: Finite difference methods
Secondary: 91G20: Derivative securities
Type
Research Article
Information
The ANZIAM Journal , Volume 63 , Issue 2 , April 2021 , pp. 203 - 227
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Copyright
© Australian Mathematical Society 2021

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