Discrete Artificial Boundary Conditions for highorder methods to solve nonlinear BlackScholes equation
Master's Thesis in Financial Mathematics, Halmstad University, Sweden
Supervisor:
Description
Due to transaction costs, illiquid markets, large investors or risks from an unprotected portfolio the assumptions in the classical BlackScholes model become unrealistic and the model results in strongly or fully nonlinear, possibly degenerate, parabolic diffusionconvection equations,
where the stock price, volatility, trend and option price may depend on the time, the stock price or the option price itself.
In this thesis we will be concerned with several models from the most relevant class of nonlinear BlackScholes equations for American options with a volatility depending on different factors, such as the stock price, the time, the option price and its derivatives.
We will analytically approach the option price by following the ideas proposed by Ševčovič
[Sev07, Sev08] and transforming the free boundary problem into a fully nonlinear nonlocal parabolic equation defined on a fixed, but unbounded domain.
We will present the results of a splitstep finite difference scheme for various volatility models including the Leland model, the Barles and Soner model and the Risk adjusted pricing methodology model.
As an possible outlook we will construct (discrete) artificial boundary conditions
for the considered nonlinear parabolic partial differential equation, following the approach
of Ehrhardt et al. [Ehr97, Ehr01, EhMi08].
Let us note that very recently using Lie group analysis Bordag [LB10] found
families of exact solutions to a nonlinear riskadjusted pricing methodology model (RAPM)
of the BlackScholes equation.
This riskadjusted pricing methodology model (RAPM) incorporates both transaction
costs and the risk from a volatile portfolio.
This solution will be used as a reference for the numerical tests.
Keywords
References:
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Nonlinear Models in Mathematical Finance: New Research Trends in Option Pricing,
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