Appearances of pseudo-bosons from Black-Scholes equation

It is a well known fact that the Black-Scholes equation admits an alternative representation as a Schr\"odinger equation expressed in terms of a non self-adjoint hamiltonian. We show how {\em pseudo-bosons}, linear or not, naturally arise in this context, and how they can be used in the computation of the pricing kernel.Comment: In press in Journal of Mathematical Physic

Illiquid markets and Hamilton-Jabobi-Bellman equations

Mestrado em Matemática FinanceiraNesta tese, a hipótese da liquidez do activo com risco é relaxada. Assumimos que o mercado contém um investidor suficientemente grande para influenciar o preço do activo com risco. Contrariamente à equação de Black-Scholes clássica, as equações de Black-Scholes para modelos de mercados ilíquidos são não-lineares. Neste caso, é difícil garantir a existência e unicidade de solução clássica. Discutimos o conceito de soluções de viscosidade e a sua aplicação no problema proposto por Frey e Polte (2011). Wilmott e Schönbucher (2000) apresentaram um modelo de equilíbrio para mercados ilíquidos. Nós discutimos o conceito de estratégia auto-nanciada nessa abordagem e utilizamos o modelo de Wilmott-Schönbucher para estudar as consequências do comportamento colectivo nos mercados nanceiros. Derivamos a correspondente equação de Black-Scholes que é não-linear e tem condições de fronteira não usuais.In this thesis, the assumption of risky asset liquidity is relaxed. We assume that the market contains one trader suciently large to inuence the price of the risky asset. Unlike the classical Black-Scholes equation, the Black-Scholes equations from models of illiquid markets are non-linear. In this case, it is dicult to guarantee the existence and uniqueness of classical solutions. We discuss the concept of viscosity solutions and its application in the setting by Frey and Polte (2011). Wilmott and Schönbucher (2000) presented an equilibrium model for illiquid markets. We discuss the concept of self-nancing strategy in their framework and use the Wilmott-Schönbucher model to study the consequences of collective behaviours in nancial markets. We derive the corresponding Black-Scholes equation which is non-linear and has unusual boundary conditions

Correcting the Bias in the Practitioner Black-Scholes Method

We address a number of technical problems with the popular Practitioner Black-Scholes (PBS) method for valuing options. The method amounts to a two-stage procedure in which fitted values of implied volatilities (IV) from a linear regression are plugged into the Black-Scholes formula to obtain predicted option prices. Firstly we ensure that the prediction from stage one is positive by using log-linear regression. Secondly, we correct the bias (see Christoffersen and Jacobs, 2004, p.298) that results from the transformation applied to the fitted values (i.e. the Black-Scholes formula) being a highly non-linear function of implied volatility. We apply the smearing technique (Duan, 1983) in order to correct this bias. An alternative means of implementing the PBS approach is to use the market option price as the dependent variable and estimate the parameters of the IV equation by the method of non-linear least squares (NLLS). A problem we identify with this method is one of model incoherency: the IV equation that is estimated does not correspond to the set of option prices used to estimate it. We use the Monte Carlo method to verify that (1) standard PBS gives biased option values, both in-sample and out-of-sample; (2) using standard (log-linear) PBS with smearing almost completely eliminates the bias; (3) NLLS gives biased option values, but the bias is less severe than with standard PBS. We are led to conclude that, of the range of possible approaches to implementing PBS, log-linear PBS with smearing is preferred on the basis that it is the only approach that results in valuations with negligible bias

Optimal consumption and investment for markets with random coefficients

We consider an optimal investment and consumption problem for a Black-Scholes financial market with stochastic coefficients driven by a diffusion process. We assume that an agent makes consumption and investment decisions based on CRRA utility functions. The dynamical programming approach leads to an investigation of the Hamilton Jacobi Bellman (HJB) equation which is a highly non linear partial differential equation (PDE) of the second oder. By using the Feynman - Kac representation we prove uniqueness and smoothness of the solution. Moreover, we study the optimal convergence rate of the iterative numerical schemes for both the value function and the optimal portfolio. We show, that in this case, the optimal convergence rate is super geometrical, i.e. is more rapid than any geometrical one. We apply our results to a stochastic volatility financial market

Symmetry reductions of some non-linear 1+1 D and 2+1 D black-scholes models

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of requirements for the degree of Master of Science. May 30, 2016.In this dissertation, we consider a number of modi ed Black-Scholes equations being either non-linear or given in higher dimensions. In particular we focus on the non-linear Black-Scholes equation describing option pricing with hedging strategies in one case, and two dimensional models in the other. Classical Lie point symmetry techniques are employed in an attempt to construct exact solutions. Some large symmetry algebras are admitted. We proceeded by determining the one dimensional optimal systems of sub-algebras for the admitted Lie algebras. The elements of the optimal systems are used to reduce the number of variables by one. In some cases, exact solutions are constructed. For the cases for which exact solutions are di cult to construct, we employed the numerical solutions. Some simulations are observed and interpretedMT201

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