Volume 5, Issue 6
Conservative and Finite Volume Methods for the Convection-Dominated Pricing Problem

Germán I. Ramírez-Espinoza & Matthias Miltenberger

Adv. Appl. Math. Mech., 5 (2013), pp. 759-790.

Published online: 2013-05

[An open-access article; the PDF is free to any online user.]

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  • Abstract

This work presents a comparison study of different numerical methods to solve Black-Scholes-type partial differential equations (PDE) in the convection-dominated case, i.e., for European options, if the ratio of the risk-free interest rate and the squared volatility-known in fluid dynamics as Peclet number-is high. For Asian options, additional similar problems arise when the "spatial" variable, the stock price, is close to zero. Here we focus on three methods: the exponentially fitted scheme, a modification of Wang's finite volume method specially designed for the Black-Scholes equation, and the Kurganov-Tadmor scheme for a general convection-diffusion equation, that is applied for the first time to option pricing problems. Special emphasis is put in the Kurganov-Tadmor because its flexibility allows the simulation of a great variety of types of options and it exhibits quadratic convergence. For the reduction technique proposed by Wilmott, a  put-call parity is presented based on the similarity reduction and the put-call parity expression for Asian options. Finally, we present experiments and comparisons with different (non)linear Black-Scholes PDEs.

  • Keywords

Black-Scholes equation convection-dominated case exponential fitting methods fitted finite volume method Kurganov-Tadmor scheme minmod limiter

  • AMS Subject Headings

65M10 91B25

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

miltenberger@zib.de (Matthias Miltenberger)

  • BibTex
  • RIS
  • TXT
@Article{AAMM-5-759, author = {Ramírez-Espinoza , Germán I. and Miltenberger , Matthias }, title = {Conservative and Finite Volume Methods for the Convection-Dominated Pricing Problem}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2013}, volume = {5}, number = {6}, pages = {759--790}, abstract = {

This work presents a comparison study of different numerical methods to solve Black-Scholes-type partial differential equations (PDE) in the convection-dominated case, i.e., for European options, if the ratio of the risk-free interest rate and the squared volatility-known in fluid dynamics as Peclet number-is high. For Asian options, additional similar problems arise when the "spatial" variable, the stock price, is close to zero. Here we focus on three methods: the exponentially fitted scheme, a modification of Wang's finite volume method specially designed for the Black-Scholes equation, and the Kurganov-Tadmor scheme for a general convection-diffusion equation, that is applied for the first time to option pricing problems. Special emphasis is put in the Kurganov-Tadmor because its flexibility allows the simulation of a great variety of types of options and it exhibits quadratic convergence. For the reduction technique proposed by Wilmott, a  put-call parity is presented based on the similarity reduction and the put-call parity expression for Asian options. Finally, we present experiments and comparisons with different (non)linear Black-Scholes PDEs.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.12-m1216}, url = {http://global-sci.org/intro/article_detail/aamm/95.html} }
TY - JOUR T1 - Conservative and Finite Volume Methods for the Convection-Dominated Pricing Problem AU - Ramírez-Espinoza , Germán I. AU - Miltenberger , Matthias JO - Advances in Applied Mathematics and Mechanics VL - 6 SP - 759 EP - 790 PY - 2013 DA - 2013/05 SN - 5 DO - http://dor.org/10.4208/aamm.12-m1216 UR - https://global-sci.org/intro/aamm/95.html KW - Black-Scholes equation KW - convection-dominated case KW - exponential fitting methods KW - fitted finite volume method KW - Kurganov-Tadmor scheme KW - minmod limiter AB -

This work presents a comparison study of different numerical methods to solve Black-Scholes-type partial differential equations (PDE) in the convection-dominated case, i.e., for European options, if the ratio of the risk-free interest rate and the squared volatility-known in fluid dynamics as Peclet number-is high. For Asian options, additional similar problems arise when the "spatial" variable, the stock price, is close to zero. Here we focus on three methods: the exponentially fitted scheme, a modification of Wang's finite volume method specially designed for the Black-Scholes equation, and the Kurganov-Tadmor scheme for a general convection-diffusion equation, that is applied for the first time to option pricing problems. Special emphasis is put in the Kurganov-Tadmor because its flexibility allows the simulation of a great variety of types of options and it exhibits quadratic convergence. For the reduction technique proposed by Wilmott, a  put-call parity is presented based on the similarity reduction and the put-call parity expression for Asian options. Finally, we present experiments and comparisons with different (non)linear Black-Scholes PDEs.

Germán I. Ramírez-Espinoza & Matthias Miltenberger. (2020). Conservative and Finite Volume Methods for the Convection-Dominated Pricing Problem. Advances in Applied Mathematics and Mechanics. 5 (6). 759-790. doi:10.4208/aamm.12-m1216
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