bluuebunny/crossref_metadata_2025 huggingface dataset:
Default Time Copulas
Authors: Sidenius Jakob
Published: May 2010

We describe default time copulas in general and factor copulas in particular. For the latter, we describe the construction of loss distributions by recursion and its application in CDO tranche pricing. We conclude with a few remarks on the limitations of copula models.
Modeling Correlation of Structured Instruments in A Portfolio Setting
Authors: Yahalom Tomer , Levy Amnon , Kaplin Andrew S.
Published: May 2010

Traditional approaches to modeling economic capital, credit‐VaR (Value‐at‐Risk), or structured instruments whose underlying collateral is composed of structured instruments treat structured instruments as a single‐name credit instrument (i.e., a loan‐equivalent). Though tractable, the loan‐equivalent approach requires appropriate parameterization to achieve a reasonable description of the cross‐correlation between the structured instrument and the rest of the portfolio. This article provides an overview of how to calibrate loan‐equivalent correlation parameters. Results suggest that structured instruments have far higher correlation parameters than single‐name instruments.
Credit Portfolio Insurance
Authors: Islam Saiyid S.
Published: May 2010

Structured credit portfolio insurance products such as credit constant proportion portfolio insurance (CPPI) have emerged as alternative investment products to more traditional structured assets in the past few years. Related products in the form of constant proportion debt obligations (CPDOs) added further variety to the set of structured credit products. From the perspective of the investor, while credit CPPIs have the appealing feature of providing principal protection through part investment in risk‐free assets, CPDOs are designed to return high coupon payments through leveraged exposure to investment grade risk. This article compares and contrasts the similarities and differences between credit CPPIs and CPDOs, briefly describes their structures, summarizes the risks in these securities in the form of downgrade and default risk, spread risk, and interest rate risk, and presents ideas around how to model these assets in a Monte Carlo simulation framework.
Nested Simulation
Authors: Gordy Michael B. , Juneja Sandeep
Published: May 2010

We discuss nested simulation methods for computing loss distributions and Value‐at‐Risk (VaR) for a portfolio containing CDO tranches as well as single‐name CDS and bonds. When these instruments are modeled
via
correlated stochastic default‐intensity models, estimation of VaR requires a nested simulation procedure: in the outer step of the simulation one draws realizations of all default intensities up to the horizon, and in the inner step one uses simulation to reprice each instrument in the portfolio at the horizon conditional on the realized intensities. We analyze how a fixed computational budget may be allocated to the inner and the outer steps to minimize the mean square error (MSE) of the resultant estimator. We also review how importance sampling may be used to increase the efficiency of the inner step.
Equity–Credit Problem
Authors: Ayache Élie
Published: May 2010

The equity–credit problem is the problem of linking the value of a firm's equity with that of its debt. The equity–credit problem is at the core of modern corporate finance. The pricing of hybrid equity–credit derivatives such as convertible bonds involves modeling the relationship between equity and credit. We present a regime‐switching framework for equity–credit modeling.
Convertible Bonds
Authors: Ayache Élie
Published: May 2010

A convertible bond (CB) is a
corporate bond
that may be converted into a certain number of shares of the issuing company, also known as the
underlying equity
. Pricing and hedging of convertible bonds involves accounting for the volatility of the underlying stock as well as the credit risk of the firm's debt. Convertible bonds are examples of equity–credit hybrids.
Bond
Authors: Priaulet Stéphane
Published: May 2010

A
bond
, or a
debt security
, is a financial claim by which the
issuer
, or the borrower, is committed to paying back to the
bond holder
or the lender, the cash amount borrowed, called
principal
, plus periodic coupon interests calculated on this amount during a given period. It can have either a standard or a nonstandard structure. A
standard bond
is a fixed coupon bond without any embedded option, delivering its coupons on periodic dates and principal on the maturity date. Nonstandard bonds include, among others, zero‐coupon bonds, floating rate notes, inflation‐linked bonds, callable and putable bonds, and convertible bonds. Bonds are usually quoted in price, yield, or spread against an underlying benchmark bond or reference swap rate. The
quoted price
(or
market price
) of a bond is usually its
clean price
, that is, its gross price minus the accrued interest. The quoted
yield to maturity
of a bond is the discount yield that equalizes its gross price times its nominal amount to the sum of its discounted cash flows.

A decline (rise) in interest rates will cause a rise (decline) in bond prices, with highest volatility occurring for long‐maturity bonds and bonds with low coupons. A bond price risk is most often measured by the bond interest rate sensitivity or
duration
. The duration concept is an instantaneous risk measure. It assumes a flat yield curve, solely affected by parallel shifts with small magnitude. Relaxing the assumption of small shifts in the yield curve leads to the introduction of a convexity adjustment in the price risk measure.
LIBOR
Rate
Authors: Traven Sergei
Published: May 2010

LIBOR rates are the most widely used reference rates in interest rate derivative contracts. The calculation of LIBOR rates is described and forward rate agreements (FRAs) are discussed.
Caps and Floors
Authors: Mercurio Fabio
Published: May 2010

Caps and floors are call/put options on LIBOR rates. We describe the Black formulas used for their valuation in the market and provide a formal justification for such a use. We also show the corresponding market volatility smiles and describe an extension of Black's formula that accommodates them.
Stochastic Volatility Interest Rate Models
Authors: Andreasen Jesper
Published: May 2010

This article considers stochastic volatility models for the term
structure of interest rates
.
H
eath–
J
arrow–
M
orton Approach
Authors: Björk Tomas
Published: May 2010

We give an overview of the Heath–Jarrow–Morton (HJM) approach used in interest‐rate modeling. In particular, we cover absence of arbitrage, the HJM drift condition, the Musiela parameterization, consistency, and existence of finite Markovian representations.
Forward and Swap Measures
Authors: Jamshidian Farshid
Published: May 2010

Forward and swap measures are defined and some of their main applications illustrated. The development is based on the construction of general numeraire measures from a state price density. Using the numeraire‐invariance theorem and the fact that forward asset prices are martingales under the forward measure, we indicate how a call option is replicated when the forward asset price process is a diffusion. The standard pricing formula is derived in the deterministic‐volatility case. Caps and swaptions are treated similarly, using that forward Libor and swap rates are martingales under the forward and swap measures, respectively. As another application, we calculate, in the continuous case, the risk‐neutral drift of the instantaneous forward rate and the terminal‐measure drift of the Libor forward rate.
Cox–Ingersoll–Ross (
CIR
) Model
Authors: Alfonsi Aurélien
Published: May 2010

This article intends to give a short presentation of the Cox– Ingersoll– Ross (CIR) model, giving the main features that have contributed to its success. It provides the basic mathematical knowledge on this model and the fundamental pricing results for caps and swaptions. Finally, it hints at practical questions such as calibration and Monte Carlo simulation.
QuadraticGaussian Models
Authors: Durand Philippe
Published: May 2010

Quadratic Gaussian (QG) models are factor models for the pricing of interest‐rate derivatives, where interest rates are quadratic functions of underlying Gaussian factors. We discuss analytical properties of these models and their application to the pricing of interest‐rate derivatives. Owing to its capacity to fit to implied volatility smiles and to correlation term structures, the QG model has been used for pricing and hedging exotic interest‐rate options.
Markovian Term Structure Models
Authors: Andreasen Jesper
Published: May 2010

We survey results for Heath–Jarrow–Morton (HJM) interest rate models with separable volatility, which, contrarily to the general HJM and LIBOR market models, admit a Markovian representation with respect to a finite number of state variables.
Alternating Direction Implicit (
ADI
) Method
Authors: Villeneuve Stéphane
Published: May 2010

In the Black–Scholes model option values are characterized as solutions of certain partial differential equations for European options or partial differential inequalities for American options. In general and in particular for American options, there is no closed‐form formulae and the option values have to be evaluated numerically, for example using the finite difference method. Unfortunately, when considering the pricing of options depending on several assets, the finite difference method suffers from the curse of dimensionality.

To circumvent this difficulty, it has been known for more than 50 years that the alternating direction implicit (ADI) algorithm is an efficient procedure for solving a large‐scale system of linear equations arising from the finite difference discretization of elliptic or parabolic equations. We describe here the ADI methods and discuss some of their features.
Conjugate Gradient Methods
Authors: Wan Justin W.L.
Published: May 2010

In option pricing on the basis of the Black–Scholes model, the value of an option is governed by a partial differential equation. Numerical approximations of the solution by the finite difference or finite element methods on a mesh of discrete asset prices result in a linear system to be solved in each time step. In this article, we introduce the conjugate gradient (CG) methods for fast solution of the option pricing equations. CG methods generate “optimal” approximate solutions by performing simple vector operations and matrix–vector multiply. We derive the method for symmetric problems and extend the idea to the nonsymmetric case. Convergence and preconditioning issues are discussed. As an example, a preconditioned CG method is used to efficiently price European options in a general exponential Levy model.
Multigrid Methods
Authors: Oosterlee Cornelis W.
Published: May 2010

In this article, we discuss multigrid methods for partial differential operators appearing in computational finance. We focus on multigrid for anisotropic problems, for the pricing of multiasset options, for linear complementarity problems for American options, and multigrid as a preconditioner to address the robustness of iterative solution methods.
Crank–Nicolson Scheme
Authors: Giles Michael B.
Published: May 2010

We describe the Crank–Nicolson method for the numerical solution of parabolic partial differential equations, its numerical properties and its application to the Black–Scholes equation. We also present the Rannacher start‐up procedure that is required to achieve second‐order accuracy for the option value and its first and second derivatives, and discuss extensions to nonlinear and multifactor applications.
Sparse Grids
Authors: Gerstner Thomas , Griebel Michael
Published: May 2010

Sparse grids are a numerical approach to solve multidimensional problems, such as multivariate partial differential equations and integrals. They are able to overcome the curse of dimension to a certain extent.
Optimization Methods
Authors: Coleman Thomas F. , Li Yuying
Published: May 2010

Numerical optimization plays a crucial decision support role in financial trading and portfolio management. In this article, we review three optimization support layers: portfolio management by quadratic programming, stochastic programming for portfolio optimization, and numerical model calibration. We highlight the importance and the complexity of the application of numerical optimization to quantitative finance.
Lattice Methods For Path‐Dependent Options
Authors: Kwok Yue‐Kuen
Published: May 2010

In this article, we discuss the lattice tree methods that are augmented with the forward shooting grid technique for pricing strongly path‐dependent options. Typical examples of options whose payoffs depend on the path dependence of the underlying asset price processes are Asian options, lookback options, and occupation time derivatives. The essence of the forward shooting grid approach is to append an auxiliary state vector at each node in the lattice tree that models the correlated evolution of the path dependence function with the underlying asset price process. We illustrate how to apply the forward shooting grid approach to price Asian options, lookback options, soft call requirement in callable convertible bonds, and call options with strike reset feature. Techniques of interpolating nodal values in the lattice tree and the associated convergence of numerical calculations are also discussed.
Wavelet Galerkin Method
Authors: Hilber Norbert , Reich Nils , Winter Christoph
Published: May 2010

In this article, we provide an overview of wavelet methods for asset pricing. Wavelet methods are a particular realization of the finite element method. They provide a very general PDE‐based numerical pricing technique. We describe the general setup and admissible market models. The fundamental concepts of the wavelet‐based finite element implementation are described. Furthermore, we illustrate the pricing of American contracts as well as the computation of sensitivities and Greeks. Numerical illustrations are given.
Tree Methods
Authors: Lelong Jérôme , Zanette Antonino
Published: May 2010

Tree methods are among the most popular numerical methods to price financial derivatives. Mathematically speaking, they are easy to understand and do not require severe implementation skills to obtain algorithms to price financial derivatives. Tree methods basically consist in approximating the diffusion process modeling the underlying asset price by a discrete random walk. In this contribution, we provide a survey of tree methods for equity options, which focus on multiplicative binomial Cox–Ross–Rubinstein model.
Partial Integro‐Differential Equations (PIDES)
Authors: Voltchkova Ekaterina
Published: May 2010

Option pricing in models with jumps leads to the solution of partial integro‐differential equations (PIDEs). We present the specific features of the PIDEs compared with the partial differential equations. In particular, we highlight the difficulties that arise for the numerical solution of these equations. We show that the use of standard methods, such as finite differences or finite elements, is not straightforward. We then present a simple explicit–implicit finite difference scheme for pricing European vanilla and barrier options in exponential Lévy models and survey the other existing numerical techniques. We also discuss American option pricing, which leads to variational inequalities with the same integro‐differential operator.
Method of Lines
Authors: Seydel Rüdiger U.
Published: May 2010

The valuation of vanilla options is considered under the one‐dimensional Black–Scholes model; a continuous dividend yield is allowed. Only the time variable is discretized, which defines a set of parallel “lines” (parallel to the
S
‐axis, where
S
is the price of the underlying). Replacing the time derivative of the Black–Scholes partial differential equation by a difference quotient establishes a system of coupled linear ordinary differential equations, one equation for each line. The ordinary differential equations can be solved analytically. This is done one line after the other in a backward fashion, starting with the payoff. The solution on each line serves as inhomogeneous term for the next line's equation. An approximation of the early exercise curve is obtained as well. Triple sums must be evaluated. The resulting set of involved analytical formulas is used along with extrapolation, which provides a fairly accurate approximation with only three lines. Using more lines, the accuracy of this method fulfills higher demands.
Monte Carlo Simulation for Stochastic Differential Equations
Authors: Bruti‐Liberati Nicola , Platen Eckhard
Published: May 2010

We introduce Monte Carlo simulation for stochastic differential equations, which is a much simpler task than scenario or pathwise simulation. A weak convergence criterion is used to classify various discrete‐time approximations and numerical schemes. Higher weak order Taylor schemes, implicit methods, and predictor–corrector methods are described.
Stochastic Differential Equations: Scenario Simulation
Authors: Platen Eckhard , Bruti‐Liberati Nicola
Published: May 2010

We consider pathwise converging simulation methods that use discrete‐time approximations for the solution of stochastic differential equations. These are different to Monte Carlo simulation methods, where probabilities and expectations but not sample paths are approximated. Scenario simulation is a more demanding task than Monte Carlo simulation. Various numerical schemes are examined and some examples are considered that can be handled by the simulation of discrete‐time approximations of trajectories. Higher order strong convergence and questions on numerical stability are discussed.
Pseudorandom Number Generators
Authors: L'ecuyer Pierre
Published: May 2010

We review the basic principles underlying the design of uniform random number generators, their main quality requirements, theoretical analysis, and empirical testing. The main construction techniques of algorithmic generators are discussed, with an emphasis on the most common ones, the linear generators. Nonlinear generators, as well as hardware‐based generators, are also briefly discussed. For simulation, we explain why we favor generators that can offer multiple streams and substreams, and we recommend specific ones.
B
ermudan Options
Authors: Kolkiewicz Adam
Published: May 2010

Pricing Bermudan options that depend on multiple assets is a challenging computational problem. Algorithms based on a discretization of the state space typically suffer from the “curse of dimensionality”, meaning that the computational effort grows exponentially with the number of state variables. Regression‐based methods address this problem first by simulating trajectories of the underlying process and then by estimating continuation values through the regression of option values at the next time step on a set of regressors that depend on the current state. In principle, these methods can be applied to any situation where trajectories of the underlying process can be simulated, since no other information about the process is required. In particular, they can be applied to path‐dependent options.
Simulation of Square‐Root Processes
Authors: Andersen Leif B.G. , Jäckel Peter , Kahl Christian
Published: May 2010

We discuss methods for time discretization and simulation of square‐root stochastic differential equations (SDEs), both in isolation (CIR (Cox–Ingersoll–Ross) process) and as part of vector SDEs modeling stochastic volatility (Heston model). Both exact and biased discretization methods are covered.
Weighted Monte Carlo
Authors: Avellaneda Marco , Jäckel Peter
Published: May 2010

We give an overview of Monte Carlo methods that vary the weight given to individual iterations in a simulation and show how these can be used for model calibration and hedge calculations. We also discuss how these methods relate to the
control variates
technique for variance reduction.
Stochastic Mesh Method
Authors: Kolkiewicz Adam
Published: May 2010

Pricing options that depend on multiple assets and allow the owner to exercise early are challenging computational problems. Deterministic algorithms based on a discretization of the state space typically become ineffective, as the computational effort grows exponentially with the dimension of the problem. The stochastic mesh method has been designed to address these challenges. It approximates the dynamic programming representation of the price by randomly generating a mesh of points and then using weighted sums to calculate continuation values. The desirable feature of the method is that it provides lower and upper bounds for the true price and is asymptotically consistent.
Libor Market Models: Simulation
Authors: Andersen Leif B.G. , Piterbarg Vladimir V.
Published: May 2010

Monte Carlo simulation method for Libor market models of interest rates are presented. We cover basic Euler‐type schemes, more advanced schemes (such as predictor–corrector), and bias‐free discretization schemes.
Stochastic Taylor Expansions
Authors: Platen Eckhard
Published: May 2010

In quantitative finance, it is essential to be able to approximate quantities that can be represented as functions of solutions of stochastic differential equations. If a function is sufficiently smooth and the underlying process satisfies a stochastic differential equation, then one can use a stochastic Taylor expansion. Such expansions generalize the deterministic Taylor formula as well as the Itô formula. They allow various kinds of higher order approximations of solutions of stochastic differential equations.
Monte Carlo Simulation
Authors: Jäckel Peter , Platen Eckhard
Published: May 2010

In this introduction to the section on Monte Carlo simulation, we give a brief review of the history of the method, its wide application in today's sciences and engineering, a top‐level categorization of its uses, and some general background as to the individual articles' subjects in quantitative finance.
Mutual Funds
Authors: Gambera Michele
Published: May 2010

Mutual funds provide expert asset management, and give diversified exposure to investments. Open‐end mutual funds are investment companies that sell individual investors a variable number of shares of a pool of assets. Closed‐end mutual funds are just like holding companies: the number of shares is predetermined, and shares trade on the stock market. Exchange‐traded funds (ETFs) are a special type of closed‐end fund.

Equity funds invest in stocks, bond funds invest in debt instruments, and money market funds invest in short‐term bonds. Hybrid funds, also calledbalancedorasset allocation funds, invest in a mix of stocks, bonds, and money market instruments.

Open‐end funds are calledno‐loadif they only charge an annual management fee, which is a percentage of assets. A load is a commission that is charged when the investor either buys (front load) or sells (back load) shares. The loads generally compensate brokers and management companies.
Style Analysis
Authors: Meier Iwan
Published: May 2010

Style describes the exposures of an investment to different sources of risk. The style of a fund can be inferred from past returns or from portfolio holdings. Return‐based style analysis uses a constrained regression to estimate the risk exposures to different asset classes. The quality of the analysis crucially hinges on the selection of benchmark indices. An alternative approach is to perform a fundamental analysis of portfolio holdings data. Style benchmarks can be applied to evaluate the performance and to monitor the style consistency of a fund.
Sharpe Ratio
Authors: Alexander Gordon J. , Baptista Alexandre M.
Published: May 2010

The
ex post
Sharpe ratio (SR) is a measure of a portfolio's performance over an evaluation period that is expressed as the portfolio's average excess return per unit of risk. Risk is measured by the portfolio's standard deviation of returns over the evaluation period. The ratio is widely used in practice to compare a portfolio's performance relative to a market index and relative to other portfolios. Similarly, a portfolio's
ex ante
SR indicates its expected excess return per unit of risk as measured by its standard deviation of future returns. This ratio can be used to make portfolio construction decisions.
Risk–Return Analysis
Authors: Markowitz Harry M.
Published: May 2010

A risk–return analysis analyzes the trade‐off of some measure of risk and some measure of return. Almost universally, the measure of return used is the mean or expected value. Various measures of risk have been proposed including variance (or, equivalently, standard deviation), semivariance (which is the same as variance, but only considers the downside), absolute deviation, and probability of loss. This article considers the merits of each, how trade‐off curves (also known asefficient frontiers) of these are computed, and how one gets (or can get) data for these.
Expected Utility Maximization
Authors: Korn Ralf
Published: May 2010

We present the martingale method for solving the continuous‐time portfolio problem in a complete market setting. As a special application, a mean–variance problem under nonnegative wealth constraints is examined in detail.
Black–Litterman Approach
Authors: Meucci Attilio
Published: May 2010

We take the reader through the Black–Litterman approach, providing all the proofs. We show how minor modifications of the original model greatly improve its range of applications. We discuss full generalizations of this and related models
Fixed Mix Strategy
Authors: Mulvey John M. , Kim Woo Chang
Published: May 2010

An advantage of adopting multiperiod portfolio models is improved investment performance
via
the fixed mix rule. Owing to its similarity to the “buy low/sell high” strategy, it is often misunderstood that it requires mean‐reverting processes for the assets. We show that mean reversion is not necessary to benefit from the fixed mix rule,
via
a simple multidimensional geometric Brownian motion. We also list successful examples from the domain of equities, commodity futures, alternative investments, and momentum strategies.
Kelly Problem
Authors: MacLean Leonard C. , Ziemba William T.
Published: May 2010

The Kelly investment strategy, namely, the period‐by‐period maximization of the expected value of a logarithmic utility function of final wealth has many good properties and some poor ones. It maximizes final wealth asymptotically and minimizes the time to achieve asymptotically large goals. But in the long run, it is very risky because its Arrow–Pratt risk aversion is essentially zero. It is the riskiest utility function one should ever use. Negative power utility, for lognormally distributed assets, is equivalent to fractional Kelly strtegies that blend cash with Kelly optimal weighting. These strategies provide a smoother wealth path but usually end up with less final wealth. Hence there is a growth–security trade‐off.
Risk‐Sensitive Asset Management
Authors: Nagai Hideo
Published: May 2010

The problems equivalent to expected power utility maximization are discussed. The results obtained through general approach using Hamilton–Jacobi– Bellman (H–J–B) equations are overviewed while illustrating explicit calculation for linear Gaussian models. Ongoing related researches are also mentioned.
Merton Problem
Authors: Bergman Yaacov Z.
Published: May 2010

The Merton problem is casting the classical consumer theory in a continuous time framework. Given an initial wealth and a set of capital assets whose prices follow correlated Ito processes, the consumer selects a lifetime consumption program that he most prefers. He selects the program that is optimal for him out of all feasible consumption programs that he can achieve by investing his available wealth continuously in a dynamically managed portfolio of the available capital assets. When the consumer's preference over consumption programs is represented by an additive (or even recursive) utility functional, then his optimization problem can be solved by dynamic programming. This problem and its solution are described in this entry.
Mean–Variance Hedging
Authors: Schweizer Martin
Published: May 2010

Suppose discounted asset prices in a financial market are given by aP‐semimartingaleS. Mean–variance hedging is the problem of approximating, with minimal mean‐squared error, a given payoff by the final value of a self‐financing trading strategy. Mean–variance portfolio selection consists of finding a self‐financing strategy whose final value has maximal mean and minimal variance. In both cases, this leads to projecting a random variable inL2(P) onto a space of stochastic integrals ofS, and, apart from proving closedness of that space, the main difficulty is to find more explicit descriptions of the optimal integrand. Both problems have a wide range of applications, and many examples and solution techniques can be found in the literature. Nevertheless, challenging open questions still remain.
Convex Risk Measures
Authors: Föllmer Hans , Schied Alexander
Published: May 2010

We discuss the quantification of financial risk in terms of monetary risk measures. Special emphasis is on dual representations of convex risk measures, relations to expected utility and other valuation concepts, conditioning, and consistency in discrete time.
Value‐at‐Risk
Authors: Dowd Kevin
Published: May 2010

This article provides a brief overview of Value at Risk (VaR). It defines the VaR and explains the parameters on which it is predicated. It explains the uses to which the VaR has been put, including its uses to determine risk targets and position limits, allocate capital, guide investment, trading and hedging decisions, remunerate traders managers, and disclose risks. It then discusses the attractions of the VaR: it is defined in terms of an easily understood metric (namely, money lost), provides a common consistent risk measure across different positions and risk factors, allows for sensible risk aggregation, and is holistic and probabilistic. It then reviews its limitations: it is “blind” to tail risks, does not satisfy the important property of subadditivity, can be unreliable when used for risk‐expected return analysis, and is vulnerable to “gaming” when used to control or remunerate traders. After discussing modern alternatives to the VaR, it addresses the (restrictive) circumstances in which a rational user might choose to use the VaR. These include the important case of probability‐of‐ruin problems, which include some capital allocation problems. It ends by briefly reviewing some of the common problems experienced by VaR users.
Expected Shortfall
Authors: Acerbi Carlo
Published: May 2010

Conditional Value at Risk (CV@R) and expected shortfall (ES) are synonymous terms defining a statistics of a random variable widely adopted as a portfolio measure of risk. It is the best‐known example of a coherent measure of risk, and is therefore a candidate alternative standard to Value at Risk for capital regulatory purposes. We discuss its definition, properties, and main applications.
Market Risk
Authors: Jorion Philippe
Published: May 2010

Modern risk management systems were developed in the early 1990s to provide centralized risk measures at the top level of financial institutions. In particular, Value at Risk (VAR) has become widely used as a statistical measure of risk based on current positions. Owing to the nature of this large‐scale aggregation problem, measuring VAR involves a number of choices and tradeoffs. VAR also has inherent limitations due to the fact that it provides a risk measure under normal but not extreme market conditions. Once put in place, however, the same risk architecture can be utilized to perform stress tests, which are essential risk tools. Generally, risk managers need to be aware of limitations in their quantitative risk models.
Models
Authors: Derman Emanuel
Published: May 2010

Although the syntax of financial modeling is inspired by and closely resembles the syntax of theoretical physics, the similarities are deceptive. The semantics of the two fields are very different. This article compares modeling approaches in physics and finance, and discusses the appropriate way to use valuation models in financial markets.
Model Validation
Authors: Crouhy Michel , Galai Dan , Mark Robert
Published: May 2010

Critical to reducing model risk is to establish a process for the independent vetting of how models are both selected and constructed. This article discusses the six phases of the model validation process.
Copulas: Estimation
Authors: Schmidt Thorsten
Published: May 2010

Copulas are a general tool for assessing the dependence structure of random variables. Important properties as well as several examples are discussed, including Archimedean copulas and the Marshall–Olkin copula. As measures of dependence, we consider linear correlation, rank correlation, the coefficients of tail dependence, and association.
Correlation Risk
Authors: Schmidt Thorsten
Published: May 2010

Correlation is a well‐established concept to capture the linear relationship between two or more variables. This article covers basic properties, fallacies of correlation, and correlation risk in financial applications.
Commodity Forward Curve Modeling
Authors: Borovkova Svetlana
Published: May 2010

In this article, the important subject of commodity forward curve modeling is addressed. Forward curves are of paramount importance in commodity markets: they provide information about the views of market participants, anticipated price trends, and expectations about future supply and demand. Futures prices provide price discovery and are essential for daily marking to market existing portfolio positions. Moreover, futures prices provide the way of calibrating derivatives pricing models under the risk‐neutral probability measure. Here, an overview of forward curve models for nonseasonal commodities such as crude oil and seasonal commodities such as natural gas or agricultural commodities is given. Also, a brief outline on how these models can be estimated from the historical data and used for applications such as risk management and derivatives pricing is given.
Commodity Price Models
Authors: Roncoroni Andrea
Published: May 2010

We propose a classification of existing arbitrage‐free stochastic models of commodity price dynamics. Focus is put on model primitives, structural elements and driving sources of noise.
Commodities and Numéraire
Authors: Geman Helyette , Shih Yih‐Fong
Published: May 2010

The goal of the article is to show that, in contrast to what has been sometimes stated, the behaviour of commodity prices over the last 19 years, namely a mean‐ reverting pattern during the nineties, then a remarkable rise during the period 2000–2008 and a collapse as of August 2008, is far from being solely explained by the moves of the dollar, the currency in which commodities are still denominated today. To make our point, we analyse three strategic commodities, crude oil, gold and copper, and show that the key features of the price trajectories remain essentially unchanged when using the dollar, the euro and the CRB index as the numeraire. On the other hand, the second part of the paper argues that the choice of the numeraire is crucial in the technical setting of options, plain‐ vanilla or complex, valuation in commodity markets and forward curve modelling.
Swing Options
Authors: Carmona René , Ludkovski Michael
Published: May 2010

Swing options are the main type of volumetric contracts in commodity markets. A swing contract gives the holder the right (but not the obligation) to adjust volume of received commodity at his or her discretion.
Liquidity
Authors: Spatt Chester S.
Published: May 2010

We discuss the nature of liquidity as suggested by both market turmoil in 2007 and 2008 and classic academic treatments. A liquid asset is one that trades without significant movements in its price. We interpret liquidity in the context of the Kyle model, the no‐trade theorem, and adverse selection more broadly. We tie liquidity to volume and link this to whether there are close substitutes for the asset and whether liquidity is a priced‐factor. We also discuss liquidity provision, the self‐fulfilling nature of liquidity, coordination failures leading to illiquidity, and the transmission of illiquidity.
Execution Costs
Authors: Almgren Robert
Published: May 2010

Execution costs can be a significant factor in overall investment performance, but are not straightforward to model and to measure. We discuss the three aspects of quantitative modeling of trade costs. For posttrade measurement, we discuss several different benchmarks in common use. For trade optimization, we discuss the construction of optimal trajectories to minimize cost and uncertainty. Finally, we discuss some of the issues in developing and calibrating pretrade cost estimation models.
Intraday Price Efficiency
Authors: Kelley Eric K.
Published: May 2010

Empirical methodologies used to assess intraday price efficiency are described. The metrics presented here summarize the extent to which actual prices deviate from an underlying efficient price and, therefore, attest to overall market quality. Two different approaches are discussed, and their relative merits are evaluated. The first approach uses a vector autoregression model over transaction‐based returns and trade information. The second one uses simple autocorrelations of intraday midquote returns. Because it explicitly decomposes price changes into permanent (efficient)
versus
temporary (pricing error) components and it is set in transaction time as opposed to clock time, the first approach is preferred.
G
losten–
M
ilgrom Models
Authors: Glosten Lawrence R.
Published: May 2010

Glosten‐Milgrom models analyze transaction prices arising from quotes of competing risk‐neutral dealers making a market in a single security and facing both privately informed and uninformed traders. The dealers must quote bid and offer prices at which they are willing to, respectively, buy and sell. A trader arrives and decides to buy, sell, or leave. Upon seeing the outcome of the trader's decision, the dealers modify their quotes in preparation for the next arrival. In the zero dealer profit equilibrium, private information induces a positive spread between bid and offer as the bid is the revised value of the security contingent on a sale. Since the quotes are revised expectations, transaction prices form a martingale. Specific examples illustrate the general results.
Kyle Model
Authors: Back Kerry , Baruch Shmuel
Published: May 2010

The Kyle model provides a tractable framework in which liquidity and the informativeness of prices can be measured. It has played a key role in research on heterogeneous information in financial markets. The model formalizes the relation of liquidity to the average rate of flow of new information. In this article, we review discrete‐time and continuous‐time versions of the Kyle model.
Roll Model
Authors: Nimalendran Mahendrarajah
Published: May 2010

Roll's 1984 article “A Simple Implicit Measure of the Effective Bid–Ask Spread in an Efficient Market” developed a simple model to estimate the bid–ask spread using the serial covariance of observable trade prices. This model has been very useful in situations where the bid–ask spread was not directly available. More importantly, the model has been a foundation upon which several extensions and other models have emerged to estimate the spread and its determinants.
Automated Trading
Authors: Hendershott Terrence
Published: May 2010

Technology changes the way securities are traded. Both liquidity suppliers and liquidity demanders use computer algorithms to improve and automate their trading. Liquidity demanders attempt to minimize the price impact of a large order by dividing their order into small parts and executing it over time across multiple markets. Liquidity demanders do this to disguise the magnitude of their trading interest to prevent other market participants from capitalizing on their future trading demands. Liquid suppliers condition their quotes—buying and selling price schedules—on all available information. Technology allows more information to be processed more quickly. This should improve the efficiency of prices and allow for liquidity to be supplied at lower costs. While humans create and parameterize the algorithms used to trade, the actual trading process involves fewer and fewer humans.
Market Transparency
Authors: Foucault Thierry , Pagano Marco , Röell Ailsa
Published: May 2010

The article surveys theoretical and empirical work on the transparency of the securities trading process, distinguishing broadly between pretrade transparency (the display of quotes for prospective trades) and posttrade transparency (the publication of information regarding completed transactions). The impact of transparency on market liquidity, informational efficiency, and trading volume is reviewed, as well as the resulting gains and losses to different market participants.
Econometrics of Option Pricing
Authors: Renault Eric
Published: May 2010

The information content of a cross‐section of European option prices written on a given stock with a given time to maturity is summarized by the volatility smile. This article discusses how to graph the volatility smile, to interpret its asymmetry, convexity, term structure and time variation. Implied volatility, implied risk aversion, risk neutral valuation, and pricing kernels are discussed in the context of dynamic mixtures of geometric Brownian motions, possibly featuring stochastic volatility, long range dependence in volatility, and Poisson jumps. More general stochastic processes with possibly infinite activity jump processes as well as more data‐driven nonparametric approaches are also sketched.
Long Range Dependence
Authors: Comte Fabienne
Published: May 2010

We define the notion of long‐range dependence and give examples of discrete‐ and continuous‐time stochastic models with long‐range dependence.
Generalized Method of Moments (
GMM
)
Authors: Hall Alastair R.
Published: May 2010

This article describes the basic framework for statistical estimation and inference using the generalized method of moments and also illustrates the types of empirical models in finance to which these techniques have been applied.
Duration Models
Authors: Werker Bas J.M.
Published: May 2010

Duration models appear when studying the moment in time that certain events occur. Next to general applications in economics and medical sciences, their financial applications are in the more specific field of market microstructure (transaction times of assets or derivatives in a given market), and also in corporate governance (tenure of management).
Entropy‐Based Estimation
Authors: Kitamura Yuichi , Stutzer Michael
Published: May 2010

Efficient solutions to various estimation problems in financial modeling can be found by constrained optimization of entropy subject to linear constraints. Two such problems are the estimation of risk‐neutral probabilities needed for derivative security valuation and the estimation of parameters in econometric models amenable to general method of moments (GMM) estimation. The statistical theory of large deviations provides a frequentist interpretation of the entropy methods.
Risk Measures: Statistical Estimation
Authors: Gourieroux Christian
Published: May 2010

This article presents the main risk measures suggested in the current regulations for finance (Basel 2) and insurance (Solvency 2) to fix the reserves, which are the Value‐at‐Risk (VaR), Tail‐VaR, and the distortion risk measures (DRM). These measures can be used in a conditional or an unconditional framework. We explain how they can be estimated and give the distributional properties of the estimator. Finally, we interpret the selection of reserve levels as a control problem and explain how this interpretation can be used to check the methodologies proposed by banks or insurance companies to the regulators.
Market Microstructure Effects
Authors: Russell Jeffrey R.
Published: May 2010

Empirical studies on market microstructure examine high‐frequency, often trade‐by‐trade dynamics of prices and quotes. This article discusses available data types for different markets, data sources, and properties of high‐frequency data that are important from an econometric modeling perspective. The most common modeling approaches are introduced, discussed, and compared.
Econometrics of Diffusion Models
Authors: Aït‐Sahalia Yacine
Published: May 2010

We review some of the econometric and statistical methods available to estimate and test continuous‐time diffusion models from discrete observations, with a particular emphasis on likelihood methods.
Garch Models
Authors: Teräsvirta Timo
Published: May 2010

This article contains a brief account of univariate and multivariate autoregressive conditional heteroskedasticity (ARCH) and generalized autoregressive conditional heteroskedasticity (GARCH) models.
Realized Volatility and Multipower Variation
Authors: Andersen Torben G. , Todorov Viktor
Published: May 2010

This article reviews basic notions of return variation in the context of a continuous‐time arbitrage‐free asset pricing model and discusses some of their applications. We first define return variation in the infeasible continuous‐sampling case. Then we introduce realized measures obtained from high‐frequency observations which provide consistent and asymptotically normal estimates of the underlying return variation. The article discusses applications of these measures for reduced‐form volatility modeling and forecasting as well as testing for the presence of jumps.
Autoregressive Moving Average (ARMA) Processes
Authors: Brockwell Peter J.
Published: May 2010

Autoregressive moving average (ARMA) processes {Yt,t = 0,  ± 1, …} have played a key role as stationary models for time series observed at regularly spaced times. They constitute a very convenient parametric family, capable of exhibiting a broad range of dependence structures and marginal distributions. Well‐established prediction, model‐selection, and estimation techniques are also available and the asymptotic properties of the customary estimators are well understood. Continuous‐time ARMA (denoted CARMA) processes play an analogous role in continuous time. Although continuous‐time data is very rarely available, it is often the case that observations of a time series at discrete times can be fruitfully regarded as sampled values of an underlying continuous‐time process. The continuous‐time framework is very convenient for the modeling of high‐frequency data and for dealing with irregularly spaced discrete‐time data. Many of the problems of mathematical finance are formulated most conveniently in a continuous time setting, and in constructing more complex continuous‐time models for financial data, CARMA processes also play a role. Although ARMA and CARMA processes with infinite second moments can be defined we restrict our attention here to second‐order processes, for which these moments are finite. This article outlines the structure and essential properties of such processes.
Measurements Errors
Authors: Jacod Jean
Published: May 2010

The main types of errors encountered in financial data are described, with emphasis on the “microstructure noise” and some hints at how these errors are dealt with.
Stylized Properties of Asset Returns
Authors: Cont Rama
Published: May 2010

Nonparametric or semiparametric statistical analysis of time series of financial asset—stock, index, exchange rate—returns has revealed a set of model‐free statistical properties, often expressed in qualitative terms, which are shared across a wide range of financial markets and instruments. We give an overview of these properties, commonly known as “stylized empirical facts”: unconditional and conditional distributions, tail properties, volatility clustering, and linear and nonlinear dependence of returns in time and across stocks.