bluuebunny/crossref_metadata_2025_split huggingface dataset:
Finite Difference Methods for Barrier Options
Authors: Randall Curt 
Published: May 2010

Abstract
This article describes the numerical solution of partial differential equations (PDEs) in the context of computational finance for the pricing of barrier options. Barrier options cease to exist (knockout barrier options) or only come into existence (knock‐in barrier options) when some observable market parameter crosses a designated level (the barrier). They have become ubiquitous in financial contracts for virtually all asset classes.
A brief discussion of continuously monitored barrier option pricing using PDEs describes the use of coordinate transformations to obtain smooth, rapid convergence, highly desirable properties that are difficult to obtain using traditional lattice methods.
In the majority of barrier option contracts, barrier monitoring frequency is at most daily, for example, at daily market close. For these options, pricing as if continuously monitored is almost always a gross approximation. Gradients of option value with respect to the underlying can become very strong near the barrier(s). The majority of the article is devoted to a more detailed discussion of PDE methods for the pricing of discretely monitored barrier options, including customization of the finite difference grid and time discretizations. Numerical examples illustrate how easy‐to‐implement grid structuring techniques can yield orders of magnitude reduction in finite difference truncation error and smooth, monotonic convergence.
Finite Difference Methods for Early Exercise Options
Authors: Toivanen Jari 
Published: May 2010

Abstract
Usually, options with early exercise possibility like American and Bermudan options have to be priced numerically. In this article, approaches based on partial differential operators and their finite difference discretization are considered. The early exercise possibility leads to an inequality constraint for the price of the option. Linear complementarity problem (LCP) and other formulations incorporating this constraint are discussed. Fairly standard finite difference discretizations can be used for the underlying partial differential operator. Several solution and approximation methods including the projected successive over relaxation (PSOR) method and penalty methods are described. As an example, an American put option is priced using different methods under the Black–Scholes model.
Fourier Transform
Authors: Černý AleŠ 
Published: May 2010

Abstract
This article is a concise introduction to applications of Fourier transform and fast Fourier transform (FFT) in option pricing. The first section defines the discrete Fourier transform (DFT) and states its most important properties. The second section explains how the binomial asset pricing model can be implemented by means of a circular convolution and how circular convolution can in turn be computed using three DFTs. This algorithm is generalized to the multinomial model. The third section discusses fast implementation of DFT and in particular it analyzes how the length of the input vector can (adversely) affect the speed of computation. We also discuss continuous Fourier transform and option pricing in continuous‐time affine models. We show how continuous‐time pricing formulae can be efficiently implemented for multiple strike values using FFT. Additional references and suggestions for further reading are provided.
Integral Equation Methods for Free Boundaries
Authors: Chadam John 
Published: May 2010

AbstractWe outline a unified approach for treating free boundary problems arising in finance using integral equation methods. Starting with the partial differential equations (PDE) formulations of the free boundary problems, we show how to derive nonlinear integral equations for the free boundaries in a variety of finance applications. Methods to treat theoretical (existence, uniqueness) questions and analytical and numerical approximations are sketched in this integral equation context.
Tikhonov Regularization
Authors: Crépey Stéphane 
Published: May 2010

Abstract

An important issue in quantitative finance is
model calibration
. The calibration problem is the
inverse
of the pricing problem. Instead of computing prices in a model with given values for its parameters, one wishes to compute the values of the model parameters that are consistent with observed prices. Now, it is well known to physicists that such inverse problems are typically
ill posed
. So, if one perturbs the data (e.g., if the observed prices change by some small amount between today and tomorrow it is quite typical that a numerically determined best fit solution of the calibration problem switches from one “basin of attraction” to the other; thus the numerically determined solution is
unstable
. To achieve a robust calibration, we need to introduce some
regularization
. The most widely known and applicable regularization method is
Tikhonov(–Phillips)
regularization method. In this paper we provide a survey of the Tikhonov regularization and illustrate it by applying the regularization to the problem of calibrating a local volatility model.
Fourier Methods in Options Pricing
Authors: Lord Roger 
Published: May 2010

Abstract
We provide a short introduction to Fourier option pricing methods, for both European and Bermudan options. The first part considers the pricing of European options using both Lévy inversion methods and the Carr–Madan approach. The second part considers the application of these techniques to the pricing of Bermudan and early exercise options. The key element here is to recognize that the continuation value of a Bermudan option is the convolution of the transition density with the option value at the next exercise date.
Quantization Methods
Authors: Pagès Gilles 
Published: May 2010

Abstract

In this paper, we present an overview of the recent developments in the applications of optimal quantization to solve nonlinear problems arising in finance like the pricing and hedging of American‐style options, portfolio management, nonlinear filtering, and stochastic partial differential equations (SPDEs). It is is a spatial discretization based on the nearest neighbor projection on a finite codebook “optimally fitted” to a random vector
X
.
Sensitivity Computations: Integration by Parts
Authors: Chen Nan 
Published: May 2010

Abstract
Derivative price sensitivities, or greeks, play an important role in the practice of risk management to quantify the potential effects of the changes of underlying market parameters on the values of derivatives. However, how to calculate them efficiently is a challenging problem for computational finance.
An obvious approach is to simulate replications of the model at perturbed parameters and then to use finite difference to form estimators. While this method has its own merits depending on the circumstances, it usually yields estimators with often unacceptably high variances, unless major computational efforts are made in terms of long calculation times. To obtain estimators with lower variance, traditional methods either differentiate the payoff functions of derivatives or differentiate the probability density of the underlying price. The former approach fails when the payoff functions are discontinuous while the latter meets difficulty if the explicit form of the density is not available.
The integration‐by‐parts method overcomes both shortcomings of the traditional methods. It shifts the differential operator from the payoffs to the underlying diffusions in order to remove the smoothness requirement on the payoff functions. This method can be traced back to the Malliavin calculus in the field of stochastic analysis.
Quasi‐MonteCarlo Methods
Authors: Niederreiter Harald 
Published: May 2010

AbstractThis article presents a survey of quasi‐Monte Carlo methods for multidimensional numerical integration and their applications to computational finance. Quasi‐Monte Carlo methods are deterministic versions of Monte Carlo methods, which outperform Monte Carlo methods for many types of integrals. First, a general background on quasi‐Monte Carlo methods is given. Then we describe principles for the construction of low‐discrepancy sequences, which are basic ingredients of any implementation of quasi‐Monte Carlo methods. Special emphasis is placed on the currently most powerful constructions based on the digital method and the theory of (T,s)‐sequences. Next, the important concept of effective dimension is discussed. One section is devoted to scrambling schemes for low‐discrepancy sequences. A review of applications to pricing problems, in particular for mortgage‐backed securities and options, concludes the article.
Exercise Boundary Optimization Methods
Authors: Jäckel Peter , Andersen Leif B.G. 
Published: May 2010

AbstractThe valuation of derivatives contracts with early exercise features in a Monte Carlo framework requires special techniques due to the inherent incompatibility of the framework's forward‐looking nature and the associated Hamilton–Jacobi–Bellman equation's backward‐looking nature. The approaches to handle this difficulty fall into two categories. The first one is founded on parametric representations of the valuation function (regression‐based techniques). The second approach is based on the optimization of a parametric representation of the exercise domain geometry (exercise boundary optimization methods). In this article, the mathematical foundation of exercise boundary optimization methods is explained, and a practical explanation of its algorithmic implementation is given.
Early Exercise Options: Upper Bounds
Authors: Andersen Leif B.G. , Broadie Mark 
Published: May 2010

Abstract
In this article, we discuss how to generate upper bounds for American or Bermudan securities by Monte Carlo methods. These techniques provide a useful supplement to strategies that provide lower bound estimates, allowing one both to generate valid confidence intervals for the true option price and to test the accuracy to any proposed approximation to the optimal exercise strategy.
Performance Measures
Authors: Bacmann Jean‐Francois , Scholz Stefan 
Published: May 2010

Abstract
The measurement of performance is the cornerstone of the evaluation of a fund. We start by examining the hypothesis underlying the most commonly used measure, namely the Sharpe ratio, in particular, the normal distribution of asset returns. As many asset classes, for example hedge funds, do not exhibit normally distributed returns, more recent measures have been considered. Some of them, the Sortino ratio and the Stutzer index, rely on a loss aversion principle. The more recent Omega function is a transformation of the cumulative density function that is also used.
Factor models offer the possibility to incorporate market conditions in the analysis. The performance is measured by the alpha, which corresponds to the remaining return once the influences of factors have been stripped out. A large part of the research focuses on the definition of the factors to be included in performance analysis. In the mutual fund context, four factors, including a market factor, a size factor, a book‐to‐market factor, and a momentum factor, are frequently chosen. Recent contributions are devoted to the understanding of the dynamic behavior of active managers, as well as the estimation of alpha, by trying to distinguish skill from luck. Finally, we show how performance can be measured with characteristic benchmarks when the weights of the underlying assets are known.
Stochastic Control
Authors: Elliott Robert J. , Siu Tak Kuen 
Published: May 2010

AbstractUncertainty is ubiquitous. It plays a primary role in economics, finance, and actuarial science, as many problems in these fields involve decisions under uncertainty. The modeling, analysis, and control of uncertainty are central issues in these fields. The theory of stochastic optimal control studies the control of (dynamical) systems involving uncertainty. It originates from the calculus of variations, which has a long history in mathematics of nearly 300 years or so. The theory of stochastic optimal control has expanded since the middle of the last century. These developments are partly due to the emergence of many powerful mathematical tools such as Ito's stochastic calculus. The expansion is also attributed partly to applications in fields such as engineering, aerospace, and manufacturing. Robert Merton pioneered the use of stochastic control in finance and applied it to obtain an elegant solution to the optimal portfolio problem in the early 1970s. His work stimulated the development of continuous‐time finance, and transformed the theory and practice of finance. In this article, we discuss applications of stochastic control to the optimal consumption–investment problem. We review three approaches namely, the martingale method, the dynamic programming approach, and the stochastic Pontryagin's maximum principle.
Transaction Costs
Authors: Bouchard Bruno , Jouini Elyes 
Published: May 2010

Abstract
Standard models for financial markets are based on the simplifying assumption that trading orders can be given and executed in continuous time with no friction. This assumption is clearly a strong idealization of the reality. In particular, securities should not be described by a single price but by a bid and ask curve. When bid and ask prices do not depend on the traded quantities, securities prices can be modeled in terms of proportional transaction costs. This article presents such transaction costs models as well as the associated duality theory that has been deveoped during the last decade and that has influenced the treatment of many other types of market frictions.
Behavioral Portfolio Selection
Authors: Shefrin Hersh 
Published: May 2010

Abstract
Portfolios satisfy both financial needs and psychological needs. Behavioral portfolio selection is a framework for understanding how psychology impacts investors' portfolios. Whereas risk and return are the central variables in neoclassical portfolio selection, the behavioral approach emphasizes the roles of emotions such as hope and fear, aspirations, the framing of outcomes as gains or losses relative to reference points, regret aversion, self‐control, and judgmental biases.
Drawdown Minimization
Authors: Browne Sid , Kosowski Robert 
Published: May 2010

Abstract
One measure of riskiness of an investment is “drawdown”, defined, most often in the asset management space, as the decline in net asset value from a historic high point. Another measure of riskiness that is oftentimes confused terminologically with drawdown is that of “shortfall”, which is simply the gap, or loss level of the current value from the initial or some other given value. In this article, we review the multiperiod continuous time literature on optimal portfolio choice and fund manager incentives in the presence on drawdown and shortfall minimization. The models differ in their assumptions regarding investment horizons (finite or infinite), constraints (fixed or stochastic benchmark), stochastic processes (diffusion with and without jumps) as well as objective function (general goal problem or expected utility maximization). In general, without transactions costs, the incorporation of drawdown constraints induces a portfolio insurance strategy, which, in the stationary stochastic model case, is that of a constant proportions portfolio insurance (CPPI) with different “floor” levels determined by the horizon and the objective.
Robust Portfolio Optimization
Authors: Iyengar Garud 
Published: May 2010

Abstract
We show how to formulate and solve robust portfolio selection problems. The objective of these robust formulations is to systematically combat the sensitivity of the optimal portfolio to statistical and modeling errors in the estimates of the relevant market parameters. We introduce “uncertainty structures” for the market parameters and show that the robust portfolio selection problems corresponding to these uncertainty structures can be reformulated as second‐order cone programs and, therefore, the computational effort required to solve them is comparable to that required for solving convex quadratic programs. We also survey recent literature on robust portfolio selection.
Diversification
Authors: Fernholz Robert 
Published: May 2010

Abstract

Diversification involves spreading investments among various assets in order to improve portfolio performance in some manner. Mathematical analysis of portfolio diversification was introduced in 1952 by Markowitz with his concept of mean/variance portfolio efficiency. A portfolio was called
efficient
if among all portfolios of the same assets, the portfolio had minimum variance for a given expected rate of return. Hence, in the setting of expected portfolio return and variance, portfolio diversification served to control the risk of a portfolio. In 1982, Fernholz and Shay showed that if the expected compound growth rate of a portfolio is considered rather than the expected rate of return, then portfolio diversification can increase the expected growth rate as well as control risk.
Risk Management: Historical Perspectives
Authors: Jorion Philippe 
Published: May 2010

Abstract
Financial risk management builds on the work of pioneers in financial theory. Spurred by the rapid expansion of derivatives markets, new risk‐management tools were developed in response to the need to manage financial risks better. This led to the development of position‐based risk measures such as Value at Risk (VAR), which are now widely used. Later, these methods were extended to other types of risk such as credit and operational risk. In spite of these technical advances, however, risk managers need to be keenly aware of the limitations of these models. Even so, risk management has now become an essential function for financial institutions.
Economic Capital Allocation
Authors: Kalkbrener Michael 
Published: May 2010

AbstractCapital allocation techniques are of central importance in portfolio management and risk‐based performance measurement. After a short review of the capital allocation problem and the existing literature, we present an axiomatic characterization of capital allocation. We analyze the relation between these axioms and properties of the associated risk measure and show that the allocation by the gradient (or Euler principle) is an immediate consequence of the proposed axioms. The axiomatization is applied to the most widely used risk measures in the finance industry to derive explicit capital allocation formulae for these measures.
Spectral Measures of Risk
Authors: Acerbi Carlo 
Published: May 2010

Abstract
Spectral measures of risk are the subclass of coherent measures of risk characterized by the additional properties of law invariance and comonotonic additivity. We provide a formal definition of spectral measures of risk and their connection with deviation measures known in actuarial mathematics. We discuss why this class is particularly important from the point of view of financial risk management. We finally provide consistent estimators.
Operational Risk
Authors: Chavez‐Demoulin Valérie , Embrechts Paul 
Published: May 2010

AbstractUnder the capital requirements of the Basel II regime, banks have to provide estimates of their operational risk on a yearly basis and, for larger international banks, use a Value‐at‐ Risk estimate at the confidence level of 99.9%. The so‐called loss distribution approach (LDA) allows for broad methodological flexibility for the estimation of this risk capital. In this article, we discuss some of the underlying statistical issues.
Regulatory Capital
Authors: Duellmann Klaus 
Published: May 2010

AbstractThe 2004 Basel II framework marks a milestone in the transition from the rules‐based 1988 Basel I Accord to a more risk‐sensitive and principles‐based approach to bank regulation. It significantly expands the regulatory scope by a new capital charge for operational risk and by its new three‐pillar structure consisting of minimum capital requirements, supervisory review process, and market discipline. Depending on the risk category, the minimum capital requirements of its most advanced approaches are either built on banks' internal models or on regulatory models requiring banks' own estimates of risk components as input. The evolutionary nature of the Basel II framework will encourage banks to migrate to more risk‐sensitive approaches in the future, depending on the complexity of their business and their risk management expertise.
Economic Capital
Authors: Rosen Dan , Saunders David 
Published: May 2010

Abstract

In financial institutions,
economic capital
(EC) acts as a buffer to absorb large losses, thereby protecting depositors and other claim holders and providing confidence to external investors and rating agencies on the financial health of the firm. EC covers all the risks that may force a financial institution into insolvency (e.g., market, credit, operational, liquidity, business, and strategic risks). Commonly, capital is designed to absorb
unexpected losses
, at a specified confidence level, while reserves are set aside to cover
expected losses
. Thus, EC is commonly defined as the difference between the portfolio's
Value at Risk
and the expected loss of the portfolio.
Stress Testing
Authors: Schachter Barry 
Published: May 2010

Abstract
Stress testing and scenario analysis encompass a variety of nonstatistical tools for evaluating the risk of loss, that is, reduction in market value, in a portfolio of financial instruments. Typically, these are used in conjunction with the other risk measurement methods, most notably Value at Risk. These tools are used in the construction of portfolio hedges, in the determination of regulatory and economic capital adequacy, and in setting portfolio risk limits, among others.
Backtesting
Authors: Christoffersen Peter 
Published: May 2010

AbstractWe survey methods for backtesting risk models using theex anterisk measure forecasts from the model and theex postrealized portfolio profit or loss. The risk measure forecast can take the form of a Value at Risk, an expected shortfall, or a distribution forecast. The backtesting can be seen as a final diagnostic check on the aggregate risk model carried out by the risk management team that constructed the risk model or they can be used by external model evaluators such as bank supervisors. The approaches suggested require only information on the dailyex anterisk model forecast and the dailyex postcorresponding profit and loss. In particular, knowledge about the assumptions behind the risk model and its construction is not required.
Electricity Markets
Authors: Meyer‐Brandis Thilo 
Published: May 2010

Abstract
Since the early 1990s, an increasing number of countries worldwide have liberalized their electricity power sectors. Contrary to the situation that prevailed earlier, when power sectors were not open to competition and prices were set by regulators according to the cost of generation, transmission, and distribution, electricity prices are now determined by an equilibrium of supply and demand. On one hand, electricity contracts are traded over the counter through bilateral agreements. On the other hand, the deregulation of electricity markets has led to the creation of organized electricity exchanges, where electricity is quoted almost as any other commodity. One effect of the liberalization of electricity markets is the introduction of substantial price risk with volatilities much higher than on stock markets and with distinct features like impressive price spikes.
Therefore, a precise statistical modeling of electricity price behavior is necessary for energy risk management, pricing of electricity‐related options, and evaluation of production assets. In this article, we give a short introduction to modern electricity markets, before we focus on an overview of reduced‐form models for electricity spot prices proposed in the literature.