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)

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