A consistent stable numerical scheme for a nonlinear option pricing model in illiquid markets

Markets liquidity is an issue of very high concern in financial risk management. In a perfect liquid market the option pricing model becomes the well-known linear Black-Scholes problem. Nonlinear models appear when transaction costs or illiquid market effects are taken into account. This paper deals with the numerical analysis of nonlinear Black-Scholes equations modeling illiquid markets when price impact in the underlying asset market affects the replication of a European contingent claim. Numerical analysis of a nonlinear model is necessary because disregarded computations may waste a good mathematical model. In this paper we propose a finite-difference numerical scheme that guarantees positivity of the solution as well as stability and consistency. © 2011 IMACS. Published by Elsevier B.V. All rights reserved.This paper has been supported by the Spanish Department of Science and Education grant TRA2007-68006-C02-02 and the Generalitat Valenciana grant GVPRE/20081092.Company Rossi, R.; Jódar Sánchez, LA.; Pintos Taronger, JR. (2012). A consistent stable numerical scheme for a nonlinear option pricing model in illiquid markets. Mathematics and Computers in Simulation. 82(10):1972-1985. https://doi.org/10.1016/j.matcom.2010.04.026S19721985821

Nonlinear Parabolic Equations arising in Mathematical Finance

This survey paper is focused on qualitative and numerical analyses of fully nonlinear partial differential equations of parabolic type arising in financial mathematics. The main purpose is to review various non-linear extensions of the classical Black-Scholes theory for pricing financial instruments, as well as models of stochastic dynamic portfolio optimization leading to the Hamilton-Jacobi-Bellman (HJB) equation. After suitable transformations, both problems can be represented by solutions to nonlinear parabolic equations. Qualitative analysis will be focused on issues concerning the existence and uniqueness of solutions. In the numerical part we discuss a stable finite-volume and finite difference schemes for solving fully nonlinear parabolic equations.Comment: arXiv admin note: substantial text overlap with arXiv:1603.0387

High order compact finite difference schemes for a nonlinear Black-Scholes equation

A nonlinear Black-Scholes equation which models transaction costs arising in the hedging of portfolios is discretized semi-implicitly using high order compact finite difference schemes. In particular, the compact schemes of Rigal are generalized. The numerical results are compared to standard finite difference schemes. It turns out that the compact schemes have very satisfying stability and non-oscillatory properties and are generally more e±cient than the considered classical schemes.Option pricing, transaction costs, parabolic equations, compact finite difference discretizations

Pricing American Options by the Black-Scholes Equation with a Nonlinear Volatility Function

Doutoramento em Matemática Aplicada à Economia e à GestãoIn this thesis we are concerned with the study of American-style options in presence of variable transactions costs. This leads to consider some generalized Black-Scholes equations with a nonlinear volatility function depending on the product of the underlying asset price and the second derivative of the option price. Mathematically, this involves the study of a free boundary problem for a nonlinear parabolic equation. The fully nonlinear character of the corresponding differential operator induces increased difficulties. By overcoming adequately those difficulties, we obtain qualitative and quantitative results regarding both types of American-style options, that is put and call options, as described next. Firstly, we investigate the qualitative and quantitative behaviour of a solution to the problem of pricing American style perpetual put options. We assume the option price is a solution to a stationary generalized Black-Scholes equation with a nonlinear volatility function. We prove existence and uniqueness of a solution to the free boundary problem. We derive a single implicit integral equation for the free boundary position and a closed form formula for the option price. It is a generalization of the well-known explicit closed form solution derived by Merton for the case of constant volatility. We also present results of numerical computations for the free boundary position, option price and their dependence on model parameters. Secondly, we analyze a nonlinear generalization of the Black{Scholes equation for pricing American-style call options, with nonlinear volatility. This model generalizes the well-known Leland model with constant transaction costs. Due to the fully nonlinear nature of the differential operator that appears in the model, the direct computation of the nonlinear complementarity problem becomes harder and unstable. Therefore, we propose a new approach to reformulate the nonlinear complementarity problem in terms of the new transformed variable for which the differential operator has the form of a quasilinear parabolic operator. We derive the nonlinear complementarity problem for the transformed variable in order to apply the Gamma transformation for American style options. We then solve the variacional problem by means of the modi ed projected successive over relaxation (PSOR) for constructing an effective numerical scheme for discretization of the Gamma variacional inequality. Finally, we present several computational examples of the nonlinear Black- Scholes equation for pricing American-style call options in the presence of variable transaction costs.Esta dissertação incide sobre o estudo de opções americanas admitindo a existência de custos de transação variáveis. Tal estudo leva-nos a considerar equações de Black-Scholes generalizadas, com uma função de volatilidade não linear que depende do produto do preço do ativo subjacente e da segunda derivada do preço da opção, o que, do ponto de vista matemático, implica a análise de um problema de fronteira livre para uma equação parabólica não linear. O carácter não linear do operador diferencial correspondente gera dificuldades acrescidas. Contudo, um estudo adequado a condição de não linearidade permite-nos estabelecer resultados qualitativos e quantitativos sobre os dois tipos de opções americanas, mas precisamente, opções de venda e de compra, conforme descrito a seguir. Em primeiro lugar, investigamos o comportamento qualitativo e quantitativo de uma solução do problema de apreçamento de opções de venda perpétuas do tipo americano. Assumimos que o preço da opção é uma solução para uma equação de Black- Scholes generalizada estacionária com uma função de volatilidade não linear. Provamos existência e unicidade de uma solução do problema da fronteira livre. Derivamos uma equação integral implícita para o valor de fronteira livre e uma solução de forma fechada para o preço da opção. É uma generalização da conhecida solução de forma fechada explícita derivada por Merton para o caso de volatilidade constante. Também apresentamos resultados de cálculo numérico para o valor de fronteira livre, assim como para preço da opção e sua dependência dos parâmetros do modelo. Em segundo lugar, analisamos uma generalização não linear da equação de Black-Scholes para o apreçamento de opções de compra de tipo americano, com volatilidade não linear. Este modelo generaliza o conhecido modelo de Leland com custos de transação constantes. Devido à natureza totalmente não linear do operador diferencial que aparece no modelo, o cálculo direto do problema de complementaridade não linear torna-se mais difícil e instável. Portanto, propomos uma nova abordagem para reformular o problema de complementaridade não linear em termos de uma nova variável para a qual o operador diferencial tem a forma de um operador parabólico quase-linear. Derivamos o problema de complementaridade não linear para a variável transformada afim de aplicar a transformação Gama para opções de tipo americano. Em seguida, resolvemos o problema variacional por meio do relaxamento projetado sucessivo modificado (PSOR) para construir um esquema numérico eficaz para discretização da desigualdade variacional Gama. Finalmente, apresentamos vários exemplos computacionais da equação não linear de Black-Scholes para apreçamento de opções de compra no tipo americano em presença de custos de transação variáveis.info:eu-repo/semantics/publishedVersio

Analytical and numerical results for American style of perpetual put options through transformation into nonlinear stationary Black-Scholes equations

We analyze and calculate the early exercise boundary for a class of stationary generalized Black-Scholes equations in which the volatility function depends on the second derivative of the option price itself. A motivation for studying the nonlinear Black Scholes equation with a nonlinear volatility arises from option pricing models including, e.g., non-zero transaction costs, investors preferences, feedback and illiquid markets effects and risk from unprotected portfolio. We present a method how to transform the problem of American style of perpetual put options into a solution of an ordinary differential equation and implicit equation for the free boundary position. We finally present results of numerical approximation of the early exercise boundary, option price and their dependence on model parameters

Reduced Order Models for Pricing European and American Options under Stochastic Volatility and Jump-Diffusion Models

European options can be priced by solving parabolic partial(-integro) differential equations under stochastic volatility and jump-diffusion models like Heston, Merton, and Bates models. American option prices can be obtained by solving linear complementary problems (LCPs) with the same operators. A finite difference discretization leads to a so-called full order model (FOM). Reduced order models (ROMs) are derived employing proper orthogonal decomposition (POD). The early exercise constraint of American options is enforced by a penalty on subset of grid points. The presented numerical experiments demonstrate that pricing with ROMs can be orders of magnitude faster within a given model parameter variation range