## Nonlinear Models in Mathematical Finance## New Research Trends in Option Pricing## Matthias Ehrhardt (ed.)## Nova Science Publishers, Inc., Hauppauge, NY 11788, 2008.## ISBN: 978-1-60456-931-5 |

**Subject:**
Nonlinear Black-Scholes equations have been increasingly attracting interest over the last two decades,
since they provide more accurate values than the classical linear model by taking into account more realistic assumptions,
such as transaction costs, risks from an unprotected portfolio, large investor's preferences or illiquid markets,
which may have an impact on the stock price, the volatility, the drift and the option price itself.

This book is concerned with several models from the most relevant class of nonlinear Black-Scholes equations for European and American options with a volatility depending on different factors, such as the stock price, the time, the option price and its derivatives. We will present both analytical techniques and numerical methods to solve adequately the arising nonlinear equations.

**Purpose:**
The purpose of this book is to give an overview on the current state-of-the-art
research on nonlinear option pricing.
Nonlinear models are becoming more and more important since they take into
account many effects that are not included in the linear model.
However, in practice (i.e. in banks) still linear models are used giving rise to large errors in computing
the fair price of options. Hence there exists a noticeable need for nonlinear modeling of financial products.
This book should help to foster the usuage of nonlinear Black-Scholes models in practice.

The book is designed to be consisting of a collection of contributed chapters. Outstanding experts working successfully in this challenging research area will be invited to contribute each a chapter of roughly 30-40 pages to this volume.

**Academic Level:**
The principal audience is graduate and Ph.D. students of mathematical finance,
lecture of mathematical (computational) finance and people working in banks
and stock markets that are interested in new tools for option pricing.

Nonlinear Models in Option Pricing - an Introduction(pages 1-19)

byMatthias Ehrhardt, Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany.

Abstract:Nonlinear Black-Scholes equations have been increasingly attracting interest over the last two decades, since they provide more accurate values by taking into account more realistic assumptions, such as transaction costs, risks from an unprotected portfolio, large investor's preferences or illiquid markets, which may have an impact on the stock price, the volatility, the drift and the option price itself.

This book consists of a collection of contributed chapters of well-known outstanding scientists working successfully in this challenging research area. It discusses concisely several models from the most relevant class of nonlinear Black-Scholes equations for European and American options with a volatility depending on different factors, such as the stock price, the time, the option price and its derivatives. We will present in this book both analytical techniques and numerical methods to solve adequately the arising nonlinear equations.

The purpose of this book is to give an overview on the current state-of-the-art research on nonlinear option pricing. The intended audience is on the one hand graduate and Ph.D. students of (mathematical) finance and on the other hand lecturer of mathematical finance and and people working in banks and stock markets that are interested in new tools for option pricing.

Option Pricing and Hedging in the Presence of Transaction Costs and Nonlinear Partial Differential Equations(pages 23-65)

byValeri Zakamouline, Faculty of Economics, University of Agder, Kristiansand, Norway.

Abstract:The celebrated Black and Scholes model for option pricing assumes perfect capital market where it is possible to replicate the payoff of an option by constructing a self-financing dynamic trading strategy. As a consequence, the absence of arbitrage dictates that the option price be equal to the cost of setting up the replicating portfolio. In the presence of transaction costs in capital markets the continuous replication policy is not feasible. Consequently, perfect hedging is impossible and the absence of arbitrage argument is no longer valid.

In this chapter we consider some approaches to option pricing and hedging in the presence of transaction costs. The distinguishing feature of all these approaches is that the solution for the option price and hedging strategy is given by a nonlinear partial differential equation. We start with a review of the classical Leland approach which yields a nonlinear parabolic partial differential equation for the option price, one of the first such in finance. Since the Leland's approach to option pricing has been criticized on different grounds, we present a justification of this approach and show how the performance of the Leland's hedging strategy can be improved. We also present the extensions of the Leland's approach for pricing and hedging of commodity futures contracts, path-dependent and basket options.

Then we proceed to the review of the most successful approach to option hedging with transaction costs, namely, the utility-based approach pioneered by Hodges and Neuberger in 1989. Judging against the best possible tradeoff between the risk and the costs of a hedging strategy, this approach seems to achieve excellent empirical performance. The asymptotic analysis of the option pricing and hedging in this approach reveals that the solution is also given by a nonlinear differential equation. However, this approach has one major drawback that prevents the broad application of this approach in practice, namely, the lack of a closed-form solution. The numerical computations are cumbersome to implement and the calculations of the optimal hedging strategy are time consuming.

Using the results of asymptotic analysis we suggest a simplified parameterized functional form of the optimal hedging strategy for either a single option or a portfolio of options and methods for finding the optimal parameters. Finally we provide an empirical testing of our optimized hedging strategies against some alternative strategies and illustrate that our strategies outperforms all the others.Utility indifference pricing with market incompletness(pages 67-100)

byMichael Monoyios, Mathematical Institute, University of Oxford, Oxford, United Kingdom.

Abstract:Utility indifference pricing and hedging theory is presented, showing how it leads to linear or to non-linear pricing rules for contingent claims. Convex duality is first used to derive probabilistic representations for exponential utility-based prices, in a general setting with locally bounded semi-martingale price processes. The indifference price for a finite number of claims gives a non-linear pricing rule, which reduces to a linear pricing rule as the number of claims tends to zero, resulting in the so-called marginal utility-based price of the claim. Applications to basis risk models with lognormal price processes, under full and partial information scenarios are then worked out in detail. In the full information case, a claim on a non-traded asset is priced and hedged using a correlated traded asset. The resulting hedge requires knowledge of the drift parameters of the asset price processes, which are very difficult to estimate with any precision. This leads naturally to a further application, a partial information problem, with the drift parameters assumed to be random variables whose values are revealed to the hedger in a Bayesian fashion via a filtering algorithm. The indifference price is given by the solution to a non-linear PDE, reducing to a linear PDE for the marginal price when the number of claims becomes infinitesimally small.

Pricing options in illiquid markets: symmetry reductions and exact solutions(pages 103-129)

byLjudmila A. Bordag, Halmstad University, Halmstad, Sweden

andRüdiger Frey, Department of Mathematics, Universität Leipzig, Leipzig, Germany.

Abstract:This chapter is concerned with nonlinear Black-Scholes equations arising in certain option pricing models with a large trader and/or transaction costs. In the first part we give an overview of existing option pricing models with frictions. While the financial setup differs between models, it turns out that in many of these models derivative prices can be characterized by fully nonlinear versions of the standard parabolic Black-Scholes PDE. In the second part of this chapter we study a typical nonlinear Black-Scholes equation using methods from Lie group analysis. The equation possesses a rich symmetry group. By introducing invariant variables, invariant solutions can therefore be characterized in terms of solutions to ordinary differential equations. Finally we discuss properties and applications of these solutions.Distributional solutions to an integro-differential parabolic problem arising on Financial Mathematics(pages 131-146)

byMaria C. MarianiandMichael Eydenberg, New Mexico State University, USA.

Abstract:In this chapter we study solutions to a parabolic integro-differential operator based on the Black-Scholes model of option pricing in financial mathematics. We obtain solutions under suitable conditions using the method of upper and lower solutions in conjunction with standard methods for solving nonlinear parabolic differential equations. We also consider solutions for a constant-coefficient divergence-form operator when the integration term appears in the form of a convolution. This result is applied to obtain distribution-valued solutions in a simple setting.

A semidiscretization method for solving nonlinear Black-Scholes equations: numerical analysis and computing(pages 149-171)

byLucas Jódar,Rafael CompanyandJosé Ramón Pintos, Instituto de Matemática Multidisciplinar, Universidad Politécnica de Valencia, Valencia. Spain.

Abstract:This chapter deals with the numerical solution of nonlinear Black-Scholes equations modeling option pricing problems in markets with transaction costs. Our approach is based on the semidiscretization technique with respect to the underlying asset variable, allowing several discretization schemes and it is applicable to distinct models.

The proposed method is purely nonlinear in the sense that it does not involve any linearization step, neither in the computation nor in the numerical analysis of the solution.

We pay special attention to two representative models. The first, is the one introduced by Hoggard, Whalley and Wilmott and developed by Parás and Avellaneda where the adjusted volatility depends on the sign of the Gamma of the option price. The second selected model is due to Barles and Soner which involves a nonlinear correction of volatility function depending on the Gamma of the option price in a more complex way apart from the time and the asset variable.

Consistency and stability of the proposed numerical solution are studied and illustrative numerical examples are included.Transformation methods for evaluating approximations to the optimal exercise boundary for a linear and nonlinear Black-Scholes equation(pages 173-218)

byDaniel Ševčovič, Department of Applied Mathematics and Statistics, Division of Applied Mathematics, Comenius University, Bratislava, Slovakia.

Abstract:The purpose of this survey chapter is to present a transformation technique that can be used in analysis and numerical computation of the early exercise boundary for an American style of vanilla options that can be modelled by class of generalized Black-Scholes equations. We analyze qualitatively and quantitatively the early exercise boundary for a linear as well as a class of nonlinear Black-Scholes equations with a volatility coefficient which can be a nonlinear function of 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 taking into account e.g. nontrivial transaction costs, investor's preferences, feedback and illiquid markets effects and risk from a volatile (unprotected) portfolio. We present a method how to transform the free boundary problem for the early exercise boundary position into a solution of a time depending nonlinear nonlocal parabolic equation defined on a fixed domain. We furthermore propose an iterative numerical scheme that can be used in order to find an approximation of the free boundary. In the case of a linear Black-Scholes equation we are able to derive a nonlinear integral equation for the position of the free boundary. We present results of numerical approximation of the early exercise boundary for various types of linear and nonlinear Black-Scholes equations and we discuss dependence of the free boundary on model parameters. Finally, we discuss an application of the transformation method for the pricing equation for American type of Asian options.Global in space numerical computation for the nonlinear Black-Scholes equation(pages 219-242)

byNaoyuki Ishimura, Graduate School of Economics, Hitotsubashi University, Kunitachi, Tokyo, Japan

andHitoshi Imai, Institute of Technology and Science, University of Tokushima, Tokushima, Japan.

Abstract:In this chapter the numerical treatment of the nonlinear Black-Scholes equation, which is global in the space variable, is discussed. Exploiting suitable transformation of variables, we do not make any truncation nor approximation of unbounded domains of the equation. Numerical implementation shows that our method and scheme are well formulated. As model problems we deal with the Black-Scholes equation with the effect of transaction costs, which is known to become nonlinear. Moreover, we apply our machinery to other singular nonlinear partial differential equation arising in the optimal investment problem.Fixed domain transformations and Split-Step Finite Difference Schemes for Nonlinear Black-Scholes equations for American Options(pages 243-273)

byJulia AnkudinovaandMatthias Ehrhardt, Institute for Mathematics, TU Berlin, Germany.

Abstract:Due to transaction costs, illiquid markets, large investors or risks from an unprotected portfolio the assumptions in the classical Black-Scholes model become unrealistic and the model results in strongly or fully nonlinear, possibly degenerate, parabolic diffusion-convection 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 chapter we will be concerned with several models from the most relevant class of nonlinear Black-Scholes 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 and transforming the free boundary problem into a fully nonlinear nonlocal parabolic equation defined on a fixed, but unbounded domain.

Finally, we will present the results of a split-step finite difference scheme for various volatility models including the Leland model, the Barles and Soner model and the Risk adjusted pricing methodology model.Pricing Hydroelectric Power Plants with/without Operational Restrictions: a Stochastic Control Approach(pages 275-303)

byZhuliang ChenandPeter Forsyth, Cheriton School of Computer Science, University of Waterloo, Waterloo, Canada.

Abstract:In this chapter, we price the value of a hydroelectric power plant under a stochastic control framework, taking into consideration the implication of operational constraints such as the ramping and minimum flow rate constraints for the purpose of environmental protection. The power plant valuation problem under the operational constraints is characterized as a bounded stochastic control problem, resulting in a Hamilton-Jacobi-Bellman (HJB) partial integrodifferential equation (PIDE). The valuation problem without the operational restrictions is characterized as an unbounded stochastic control problem; we propose an impulse control formulation, resulting in an HJB variational inequality, for the valuation problem in this scenario. We develop a consistent numerical scheme for solving both the HJB PIDE for the bounded control problem and the HJB variational inequality for the unbounded control problem. We prove the convergence of the numerical scheme to the viscosity solution of each pricing equation, provided a strong comparison result holds. Numerical results indicate that failing to consider operational constraints may considerably underestimate the value of hydroelectric power plants.Numerical solutions of certain nonlinear models in European options on a distributed computing environment(pages 305-320)

byChoi-Hong Lai, School of Computing and Mathematical Sciences, University of Greenwich, United Kingdom.

Abstract:Financial modelling in the area of option pricing involves the understanding of the correlations between asset and movements of buy/sell in order to reduce risk in investment. Such activities depend on financial analysis tools being available to the trader with which he can make rapid and systematic evaluation of buy/sell contracts. In turn, analysis tools rely on fast numerical algorithms for the solution of financial mathematical models. There are many different financial activities apart from shares buy/sell activities. However it is not the intention of this chapter to discuss various financial activities. The main aim of this chapter is to discuss a distributed algorithm for the numerical solution of a European option. Both linear and non-linear cases are considered.

The algorithm is based on the concept of the Laplace transform and its numerical inverse. The scalability of the algorithm is examined. Numerical tests are used to demonstrate the effectiveness of the algorithm for financial analysis. Time dependent functions for volatility and interest rates are also discussed. Applications of the algorithm to non-linear Black-Scholes equation where the volatility and the interest rate are functions of the option value are included. Some qualitative results of the convergence behaviour of the algorithm is examined. This work also examines the various computational issues of the Laplace transformation method in terms of distributed computing. The idea of using a two-level temporal mesh in order to achieve distributed computation along the temporal axis is introduced. Finally the chapter ends with some conclusions.}

Calibration problems in option pricing(pages 323-352)

byBertram Düring, Institute for Analysis and Scientific Computing, Vienna University of Technology, Vienna, Austria.

Abstract:We review different approaches to calibrate volatility functions in Black-Scholes type models to market data. We focus on an optimal control approach, where a regularized cost functional is minimized over a suitable set of admissible volatilities. The cost functional measures the deviations of option prices obtained from a pricing model to the given market data. We discuss the Black-Scholes case as well as the extension to pricing models in markets with frictions, e.g. models for option pricing in the presence of transaction costs.

ehrhardt@wias-berlin.de