Please use this identifier to cite or link to this item:

`http://hdl.handle.net/10321/2348`

Title: | Lie symmetry analysis of the Black-Scholes-Merton Model for European options with stochastic volatility |

Authors: | Paliathanasis, Andronikos Krishnakumar, K. Tamizhmani, K. M. Leach, P. G. L. |

Keywords: | Lie point symmetries;Financial Mathematics;Stochastic volatility;Black-Scholes -Merton equation |

Issue Date: | 3-May-2016 |

Publisher: | MDPI |

Source: | Paliathanasis. A. 2016. Lie symmetry analysis of the Black-Scholes-Merton Model for European options with stochastic volatility. Mathematics. 4(28): 1-14. |

Abstract: | We perform a classification of the Lie point symmetries for the Black-Scholes-Merton Model for European options with stochastic volatility, σ, in which the last is defined by a stochastic differential equation with an Orstein-Uhlenbeck term. In this model, the value of the option is given by a linear (1 + 2) evolution partial differential equation in which the price of the option depends upon two independent variables, the value of the underlying asset, S, and a new variable, y. We find that for arbitrary functional form of the volatility, σ(y), the (1 + 2) evolution equation always admits two Lie point symmetries in addition to the automatic linear symmetry and the infinite number of solution symmetries. However, when σ(y) = σ0 and as the price of the option depends upon the second Brownian motion in which the volatility is defined, the (1 + 2) evolution is not reduced to the Black-Scholes-Merton Equation, the model admits five Lie point symmetries in addition to the linear symmetry and the infinite number of solution symmetries. We apply the zeroth-order invariants of the Lie symmetries and we reduce the (1 + 2) evolution equation to a linear second-order ordinary differential equation. Finally, we study two models of special interest, the Heston model and the Stein-Stein model. |

URI: | http://hdl.handle.net/10321/2348 |

ISSN: | 2227-7390 (print) |

Appears in Collections: | Research Publications (Applied Sciences) |

###### Files in This Item:

File | Description | Size | Format | |
---|---|---|---|---|

Paliathanasis_Mathematics_Vol4#2#28_Pgs1-14_2016.pdf | 830.13 kB | Adobe PDF | View/Open |

#### Page view(s)

167
checked on Aug 20, 2018

#### Download(s)

25
checked on Aug 20, 2018

#### Google Scholar^{TM}

Check
Items in DSpace are protected by copyright, with all rights reserved, unless otherwise indicated.