Quantitative Finance and Risk Management

The following sections are included:

• Dedication
• Table of Contents
• Acknowledgments for the 1^st and 2^nd Editions

The following sections are included:

• Who/ How/What, “Tech. Index”, Messages, Personal Note
• Summary Outline: Book Contents
• Note for the 2^nd Edition (2015)
• Climate Change Risk Management – NEW TOPIC
• Evaluation – Where do we stand? What about the future?

In this overview, we look at some general aspects of quantitative finance and risk management. There is also some advice that may be useful. A reminder: the footnotes in this book have interesting information. They function as sidebars, complementing the text.

This practical and amusing (but dead-serious) exercise will give you some glimmer in what it can be like carrying out a few activities in practical finance regarding a little data, analysis, systems, communication, and management issues. The exercise is illustrative without being technical. There are important lessons, the most important being communication. The idea is not just to read the exercise and chuckle, but actually try to do it…

In this chapter, we begin the analysis of standard or “plain-vanilla” equity options. Similar remarks will hold for other options, e.g. foreign exchange (FX) options. Just for balance, we treat some topics in the FX options chapter that are directly applicable to equity options and vice-versa. We will treat the subject qualitatively, reserving the formalism for later chapters (see especially Ch. 42).

In this chapter we discuss foreign-exchange FX derivatives. We start with FX forwards and simple FX options. We give some practical details for FX options, including hedging with Greeks. We introduce volatility skew (or smile). We give some examples of pricing exotic barrier FX options. We present the “two-country paradox”. We discuss quanto options and correlations, FX options in the presence of stochastic interest rates, and comment on numerical codes and sanity checks.

In this chapter, we consider volatility skew for equity options. We also include some formalism and simple skew models aimed at providing physical insight. Volatility skew refers to the strike dependence of the volatility. For example, some S+P 500 index option volatility data as a function of the strike E using the Black-Scholes model are shown below…

In this chapter, we discuss the construction of the forward-rate curves needed especially for pricing interest-rate derivatives. We begin with a discussion of the input rates to the forward curve construction models, and then discuss the mechanics.

The following sections are included:

• Interest Rate Swaps: Pricing and Risk
• Interest Rate Swaps: Pricing and Risk Details
• Cross-Currency Swaps
• Credit Default Swaps (CDS)
• References

Bonds are debt instruments of many different sorts issued to raise money. Issuers of bonds include corporations, governments, municipalities, and agencies. In this book, an issuer is labeled as ABC and an investor as X. Bonds are obligations of ABC to pay back to X the borrowed money (called the “notional” or “par” amount) at the maturity date of the bond and in some cases earlier. In addition, the bonds have coupons (so called because the investor used to clip off “coupons” as pieces of paper to get paid)…

The following sections are included:

• Introduction to Caps
• The Black Caplet Formula
• Non-USD Caps
• Relations between Caps, Floors, and Swaps
• Hedging Delta and Gamma for Libor Caps
• Hedging Volatility and Vega Ladders
• Matrices of Cap Prices
• Prime Caps and a Vega Trap
• CMT Rates; Volatility Dependence of CMT Products
• References

We described interest-rate caps in the last chapter. A cap is a collection or basket of options (caplets), each written on an individual forward rate. A swaption, on the other hand, is one option written on a collection or a basket of forward rates, namely all the forward rates in a given forward swap. The fact that the swap option is written on a composite object means that correlations between the individual forward rates are critical for swaptions…

The following sections are included:

• Introduction to Portfolio Risk Using Scenario Analysis
• Definitions of Portfolios
• Definitions of Scenarios
• Many Portfolios and Scenarios
• A Scenario Simulator
• Risk Analyses and Presentations

This chapter contains a case study of a complicated equity option called a CVR that was an important part of an M&A deal. Although the events happened long ago, some of the analysis is quite general and could be relevant in other contexts. The CVR will be considered in some depth in order to give an idea of the complexity that sometimes occurs. Many of the topics in this chapter are quite general. A variety of interesting theoretical points arose while pricing the CVR. These included conditions under which an option will or not be extended in time…

In this chapter we consider two more case studies of structured exotic products. We first consider DECs and synthetic convertibles, including simple and complex varieties. We then go on to consider an exotic equity call option with a variable strike and expiration.

The following sections are included:

• Contingent Caps
• Digital Options: Pricing and Hedging
• Historical Simulations and Hedging
• Yield-Curve Shape and Principal-Component Options
• Principal-Component Risk Measures (Tilt Delta etc.)
• Hybrid 2-Dimensional Barrier Options—Examples
• Reload Options
• References

The following sections are included:

• TIPS (Treasury Inflation Protected Securities)
• Municipal Derivatives, Muni Issuance, Derivative Hedging
• Difference Option on an Equity Index and a Basket of Stocks
• Resettable Options: Cliquets
• Power Options
• Path-Dependent Options and Monte Carlo Simulation
• Periodic Caps
• ARM Caps
• Index-Amortizing Swaps
• A Hypothetical Repo + Options Deal
• Convertible Issuance Risk
• References

In this chapter, we begin the discussion of barrier options using the path integral Green function formalism. Barrier options are options that have the underlying process constrained by one or several boundaries called barriers. Usually these options are European options—that is, there is only one exercise date at which the option may pay off. If the underlying crosses a barrier, a “barrier event” occurs, and the option changes its character. For example, the option may disappear, be replaced by another option, be replaced by cash, etc. The barrier is usually continuous although sometimes it is discrete and it usually exists over the whole option period until exercise. The barrier event usually is defined to occur the first time the underlying crosses the barrier. Usually the barrier is a constant value…

In this chapter, we treat double barrier options. These are options that depend on one underlying variable, but which have two barriers (upper and lower). The underlying process starts between the two barriers. The idea is in the figure…

Our purpose in this chapter is to set up the mathematics used to describe the two-dimensional hybrid option formalism. We discussed these options in Ch. 15. The barrier options are assumed European with a single continuous barrier. There are two underlying variables…

Options often contain arithmetic averaging features. In this chapter, we use path-integral techniques to obtain some general results. The reader is referred to the chapters on path integrals (Ch. 41-45) for more details…

In this chapter, we look at fat tails in distributions of underlying variable moves from a practical perspective. We are especially concerned with obtaining some sort of volatility for fat tails. We introduce the idea using some examples, and deal with practical questions at the end.

The following sections are included:

• The Importance and Difficulty of Correlation Risk
• One Correlation in Two Dimensions
• Two Correlations in Three Dimensions; the Azimuthal Angle
• Correlations in Four Dimensions - Picture
• Correlations in Five and Higher Dimensions
• Spherical Representation of the Cholesky Decomposition
• References

In this chapter, we first consider various methods for dealing with a matrix of stressed correlations. We start with scenario analysis to define target stressed correlations, motivated by data (see also Ch. 37). We introduce the concept of the average correlation stress. Naturally, these are target correlation stresses for which the correlation matrix will not be positive definite. The technique of finding an optimal positive-definite approximation to a non-positive-definite target correlation matrix is treated in the next chapter…

In this chapter, we deal with the problem of finding an optimal positive-definite (PD) approximation for a given correlation matrix. Consider a non-positive-definite (NPD) correlation matrix (ρ)^NPD. We call NPD matrices “illegal” and positive-definite matrices “legal”. Correlation matrices that are NPD can arise from various sources. As discussed before, stressed correlation matrices are desirable to probe correlation risk. Such stressed matrices are produced by moving the individual correlation matrix elements ρ_αβ from their current values Current by amounts Δρ_αβ…

In this chapter, we look at some models for the underlying dynamics of market correlations, and for correlation uncertainties. This includes the time dependence of correlations. Our point of view is that correlations have an intrinsic meaning, independent of historical time series. As an analogy, volatility is often treated as a dynamical variable in stochastic volatility models, independent of historical time series. In a similar way, correlations can be treated as dynamical variables. In the last chapter, we were concerned with a different issue, namely given some correlations, to find ways to consistently stress the given correlations. Our purpose here is to investigate some possibilities for the underlying dynamics of the correlations themselves…

In this chapter, we discuss begin a discussion of VAR, an acronym for Value at Risk. The “Plain-Vanilla VAR” (PV-VAR) - with its incarnations as Monte Carlo PV-VAR, Historical VAR, and a quadratic form (QPV-VAR) - is a standard risk measure that we discuss first. PV-VAR is rather blunt and unrefined. In the next chapter 27, we will discuss refinements. The first refinement stage defines the “Improved Plain-Vanilla VAR” (IPV-VAR). Further refinements produce “Stressed VAR” (S-VAR) and “Enhanced/Stressed VAR” (ES-VAR)…

In this chapter, we discuss various stages of refinements of the plain-vanilla VAR discussed in Ch. 26. Increasingly realistic aspects will be included, with the final aim to obtain a risk measure that is more useful in active risk management. The first set of improvements give what is termed in this book “Improved Plain Vanilla VAR” (IPV-VAR). We then list further improvements to produce “Stressed VAR” (S-VAR) and finally “Enhanced/Stressed VAR” (ES-VAR). We close with some miscellaneous topics including subadditivity issues, and also an integrated form of VAR for which Expected Shortfall is a special case.

In this chapter, we present a formal functional derivation of the VAR and CVAR equations for the linear case. Again, CVAR = Component VAR. We pay particular attention to the CVAR volatility. The derivation is done for in the continuous multivariate framework. This shows that CVAR uncertainties are present in the limit of an infinite-length Monte-Carlo (MC) simulation run. We indicate extensions for non-linear exposures (convexity) to VAR, as discussed in the last chapter. We end with a summary of the extension to multiple time steps.

The following sections are included:

• The Component VAR (CVAR) Volatility with Two Variables
• Geometry, Math: Risk Ellipse, VAR Line, CVAR, CVAR Vol

In this chapter, we consider additional topics related to applications of VAR and CVAR for corporate-level risk management. We first discuss aggregation issues. We then discuss implied correlations between business unit P&Ls. We end with a consideration of aged inventory.

In this chapter we discuss credit risk. First we discuss issuer credit risk for bonds or other securities. We will also consider the relation of issuer credit risk and market risk, and discuss why and how sometimes these are calculated separately. We present a straightforward method of defining a unified credit + market risk measure…

This short and non-technical chapter contains some observations on models with an emphasis on risk. Model Quality Assurance will be treated in the next chapter. You can read this chapter without having to read the rest of the book.

We have discussed models for various markets and purposes, and we have spent some time talking about model risk. In this chapter, we deal with procedures and activities designed to cope with some aspects of model risk, variously denoted as “Model Quality Assurance”, “Model Review”, or “Model Validation”. We use Model Quality Assurance or Model QA for short. This is partly because the term “Quality Assurance” is used across the software industry…

This chapter contains a qualitative and non-technical overview of risk with Systems. Before starting, it should be emphasized that each organization tries to produce (or buys and customizes) the best systems it can, consistent with time pressures and resource constraints. There are very successful systems that work well, and are used every day. However, the development of systems can be problematic. This essay deals with some reasons for these problems.

This chapter is concerned with an overview of strategic directions for numerical financial computing. Parallel processing will be emphasized. The need is for rapid and cost-effective valuation of large portfolios of options, mortgage-backed securities, bonds, etc. in order to enhance competitiveness in trading, risk management, and sales. A description is given of the utility of parallel-processing computers, distributed workstation environments, and technological advances pertaining to new directions in financial numerical computing…

A complex and knotty problem faced by financial risk management at the corporate level pertains to obtaining consistent, reliable, and complete financial data. Data problems produce sources of uncertainty for risk management. Specific issues with data are discussed in other chapters in this book. Here, we deal with some overall issues in a qualitative fashion…

In this chapter, we deal with some important aspects of correlations and data. We discuss windowing uncertainties, including overlapping vs. non-overlapping windows. We discuss uncertainties due to the limited amount of data relative to the number of variables. We also discuss intrinsic dynamically generated correlation uncertainties. See also Ch. 23-25…

In this chapter, we consider the mathematics and theory for three topics. They are (1): the Wishart theorem, (2): the Fisher Transform, and (3): Implications for correlation uncertainties due to sampling error. John Wishart, in a brilliant exposition, generalized earlier work to obtain the distribution of standard deviations and correlations obtained from a sample of N measurements of p variables, assuming that all variables obey a multivariate Gaussian distribution…

In this chapter, we discuss Economic Capital, mostly in a qualitative fashion. We describe standard procedures and assumptions as well as problems and issues. Many relevant quantitative issues that play important roles in the actual calculations are discussed in detail elsewhere in the book.

In this chapter, we deal with exposure-change risk as an extension to risk calculations and Economic Capital. Most risk assessments use existing portfolios and exposures. We do want to gauge the historical accuracy of our risk assessments through backtesting. Nonetheless, we are really interested in assessing future risk. After all, the future is risky, not the past. Therefore, we are (or should be) interested in the risk due to potential changes in risk exposures, consistent with limit constraints. In this book, we use a forward + option approach in order to model this potential exposure-change unused-limit risk.

In previous chapters in this book, path-integral techniques were in fact used repeatedly for valuation. In this part of the book, we deal directly with the formalism of path integrals as applied to finance. Those who already know path integrals and who want to jump-start into finance might start with these chapters. The finance discussion is self-contained. For those who are unfamiliar with path integrals, the presentation will have appropriate background material…

Path integrals are widely used in physics for treating problems with stochastic variables. In particular, diffusion equations have path integrals as exact solutions. This chapter presents an introductory overview of the use of path integrals in options pricing. We begin with European stock options, the venerable Black-Scholes model. We exhibit how Bermuda and American options fit into the path-integral framework. A list of references is at the end of the chapter…

We present the general path-integral framework for one-factor (short-term interest rate), term-structure-constrained models. These include Gaussian, mean-reverting Gaussian (MRG), arbitrary rate-dependent volatilities, and memory effects. It is shown how the stochastic equations for rate dynamics are built directly into the path integral. Analytic results are derived by evaluating standard calculus integrals. No previous knowledge of either the models or path integrals is assumed in this chapter. Those familiar with models may regard this chapter as continuing the pedagogical introduction to path integrals. Those familiar with path integrals will benefit by this straightforward presentation of the models…

This chapter is based on my 1989 paper. My derivation of the Mean-Reverting Gaussian (MRG) analytic model reported in this chapter was done independently and roughly concurrently with other authors. The path integral method for me was merely a tool used to derive solutions to models.

This chapter presents some aspects of numerical methods for options based on path-integral techniques. We have already emphasized the connection between the binomial algorithm (or any lattice method), Monte-Carlo simulations, and path integrals.

A major topic is the Castresansa-Hogan method for discretizing path integrals. Some simplifying approximations are discussed. An iterative procedure based on “call filtering” for Bermuda options leads to a “quasi-European” approximation. The idea of “geometric volatility” is introduced. We also present an approximation to lognormal dynamics using a mean-reverting Gaussian designed to speed up calculations.

In this chapter, we describe multi-factor path integrals in an arbitrary number of internal dimensions. This is needed to describe models and correlated risk with many factors, multi-factor yield curve models, baskets of equities or FX rates, or any other problem containing multiple variables. The figure below gives the idea…

The following sections are included:

• Introduction to the Reggeon Field Theory (RFT)
• Summary of the RFT in Physics
• Aspects of Applications of the RFT to Finance
• The RFT and Describing Financial Crises (2^nd Edition)
• RFT Scaling Without Fitting for Various Markets In Crisis
• Rich-Cheap Analysis, the RFT, and Crises
• RFT - the Natural Extension to Brownian Motion
• Predicting Crises in Equity Markets; an Earthquake Analogy
• Finance Theory is Really Phenomenology
• References

The following sections are included:

• Explicit Time Scales Separating Dynamical Regions
• The Macro-Micro Yield-Curve Model: Ch. 48-50
• Further Developments for the Macro-Micro Model: Ch. 51
• A Function Toolkit: Ch. 52
• References

The following sections are included:

• Summary of this Chapter
• The Problem of Kinks in Yield Curves for Models
• Introduction to this Chapter
• Statistical Probes, Data, Quasi-Equilibrium Drift
• Yield-Curve Kinks: Bête Noire of Yield Curve Models
• EOF / Principal Component Analysis
• Simpler Lognormal Model with Three Variates
• Wrap-Up and Preview of the Next Chapters
• Appendix A: Definitions and Stochastic Equations
• Appendix B: EOF or Principal-Component Formalism
• “Sub-Period Centered” Principal Components
• Figures: Multivariate Lognormal Yield-Curve Model
• References

The following sections are included:

• Summary of this Chapter
• Introduction to this Chapter
• Short and Long Time Scales
• Cluster Decomposition Analysis and the SMRG Model
• Third-Order Correlation Functions
• Historical Quasi-Equilibrium Yield Curve Path for Data
• Large Value of the Mean Reversion in this Model
• Physical Picture - How Interest Rates Really Seem to Behave
• Other Statistical Tests and the SMRG Model
• Principal Components (EOFs) and the SMRG Model
• Wrap-Up for this Chapter
• Appendix A: Definitions and Stochastic Equations
• Appendix B: The Cluster-Decomposition Analysis (CDA)
• Figures: Strong Mean-Reverting Multifactor Yield-Curve Model
• References

The following sections are included:

• Summary of this Chapter
• Introduction to this Chapter
• Spectral Decomposition and Time Scales
• The Macro Component and Macroeconomics
• The Micro Component and Short-Term Trading Activity
• Prototype: Prime (Macro) and Libor (Macro + Micro)
• Details of the Macro-Micro Yield-Curve Model
• Model for the Random Macro Time Step Dynamics
• Wrap-Up of this Chapter
• Appendix A. No Arbitrage and Yield-Curve Dynamics
• Figures: Macro-Micro Model
• References

The following sections are included:

• Summary of This Chapter
• Using SSA to determine the Macro Component
• Intuition: Short to Long Times - Volatility, No-Arbitrage
• The Macro-Micro Model applied to FX and Equity Markets
• Formal Developments in the Macro-Micro Model
• No Arbitrage and the Macro-Micro Model: Formal Aspects
• Hedging, Forward Prices, No Arbitrage, Options (Equities)
• Satisfying the Interest-Rate Term-Structure Constraints
• Chaos and the Macro-Micro Model
• Technical Analysis and the MM Model
• The Macro-Micro Model and Data
• Finance Models Related to the Macro-Micro Model
• Macroeconomics, Economics Literature, Macro-Micro Model
• References

In this chapter, a “toolkit” of functions potentially useful for analyzing business cycles is presented. These functions could then form part of the macro component of financial markets operating over long time scales. These functions may also be useful on shorter time scales for trading, as described at the end of this chapter…

The following sections are included:

• Summary: Climate Change Risk Management
• Climate Change Risk Management – Formal Structure
• Finance, Economics, Risks, and Opportunities
• Positive Opportunities in Mitigating Climate Change
• Long-term discounting for climate change impacts in the future
• Climate-Change-Induced Economic and Financial Crises?
• Business, Investors, Regulators, and Climate Change Risks
• Economic Models Related to Climate Change Impacts
• Wrap Up of Climate Risk Management
• Epilogue
• Appendix I. The Physical Science Basis of Climate Change
• Appendix II. Impacts of Climate Change and Vulnerabilities
• Appendix III. Mitigation and Adaption for Climate Change
• Appendix IV. Risk from Climate Contrarian Obstruction
• References

The following sections are included:

• INDEX: CH. 1 – CH. 52
• INDEX: CLIMATE CHANGE RISK MANAGEMENT, CH. 53