ÖOur Quantitative Solutions Last revised: 6/28/00

The William Margrabe Group, Inc. ("the Group") can be your complete source for advanced quantitative solutions to your more technical problems related to

models for

pricing derivatives
market risk management
credit risk management

every sort of structure in the Derivatives Dictionary and more –

plain vanilla options
contingent and optional payoffs
multivariate and rainbow derivatives (more than one underlying)
hybrid (underlying risks of different types, e.g., energy and currency)
exotic derivatives, the wilder, the better

– based on a wide range of underlying risk factors:

interest rates
bond prices
commodity prices
currencies exchange rates
equity prices and/or
indexes or combinations of any of the above.

We can help you

develop
benchmark
test and
document

a variety of models based on solutions to the Black-Scholes partial differential equation

"Analytical" (there's always a numerical algorithm) and analytical approximations (Whaley)
Binomial (CRR and JR) and Trinomial
Explicit, Implicit, and Explicit-Implicit (including Crank-Nicolson)
Monte Carlo (one to n dimensions, two, three, or four variants of each)
Numerical quadrature (Romberg, Gaussian, ...)

as well as Interest Rate Derivative models, including

Black caplet and swaption
Black-Derman-Toy
Black-Karasinski
Brace-Gatarek-Musiela
Ho-Lee
Heath-Jarrow-Morton (normal, CIR, and lognormal; binomial and Monte Carlo)
Hull-White (extended Vasicek)
Vasicek and
others, as needed.

We pride ourselves on producing the best model for your specific need. For example, suppose you need a model for pricing equity options. Depending on the specific situation, you may want steps in price, log of price, or some other transformation of price.You may want binomial or finite difference. If binomial, then CRR, JR, or any of an infinite number of others. If finite difference, then explicit, implicit, or mixed. You may want to specify a single volatility, a term structure of volatilities, or a grid of volatilities. You may want to specify a single rate of interest or a term structure. You may want to assume that dividends are a continuous yield or in discrete lumps. If a continuous yield, then maybe a single yield or maybe a term structure.

Based on Derivatives and Risk Management Experience since 1973, we analyze your problems and deliver our expertise in the form of

Written Products
Presentations
Software Solutions

ÖWritten Products

We can help you evaluate and document all aspects of your models and systems:

Contract and Confirm
Underlying risk factor(s)
Mathematical derivation
Computational algorithm
Source code
Catalog entry in software library
User interface
Inputs from market sources
Comparison of outputs with our benchmarks for the same model
Sales literature
Publications in the trade press

For a sample of our technical writing over nearly two decades, refer to the following publications:

"Selecting and Customising Equity Derivatives for Investment Funds." In Equity Derivatives; Applications in Risk Management and Investment. London: Risk, 1997.
"The Best of Both Worlds: Turnkey and Customization," Derivatives Strategy, June 1996, pp. 47, 50, 51.
"Triangular Equilibrium and Arbitrage in the Market for Options to Exchange Two Assets." The Journal of Derivatives 1 (Fall 1993).
Executive Stock Options: Benefits, Costs, and Implications for Investors. (with Eric Sorensen and Tad Flynn) Salomon Brothers Inc (October 1993).
"Binomial Pricing of Exploding Options," in Topics in Money and Securities Markets Bankers Trust Co. (no. 49, May 1989).
"The Value of an Option to Exchange One Asset for Another." Journal of Finance 33 (March 1978), pp. 177-186.

ÖPresentations

We offer presentations on a variety of technical subjects, including

Elements of Derivatives

Linear Derivatives. Defines and discusses pricing of forward, futures, FRA, and swap contracts, their pricing, and risk management. (One day)
Introduction to Options. Call and put option contracts, put-call parity and binomial pricing, Greeks and risk management. (One day, based on my presentation in the Salomon Brothers training program)
Derivatives Pricing Boot Camp. Four days of concentrated indoctrination and mental training. If you survive the camp, you'll hold your head a little higher and know that you can survive anything the enemy throws at you. Must pass a test in math, probability, and programming to get in. Must pass a test on a wide range of option pricing models to get out.
Derivatives for Poets. A nonmathematical introduction to the language, history, sociology, and psychology of derivatives. For those with dominant left brains. (One day)

Intermediate Derivatives Analysis

Credit Risk and Credit Derivatives. Two days of intuitive explanation (morning) and analytical exploration (afternoons) of credit risk (day one) and credit derivatives (day two). Take them together, or separately.
The Only Pricing Model You'll Ever Need. Shows how to use the Black-Scholes-Merton model (and Margrabe's extension to two dimensions of risk) to price options on bonds, commodities, currencies, equities, interest rates, and indexes. Explains how to use this simple model to obtain precise option values. (One day)
Bootstrap Methods of Yield Curve Estimation Includes linear interpolation of spot and par coupon rates and exponential interpolation of zero prices. (Half day)
Introduction to Option Pricing Algorithms. Introduces analytic, binomial, trinomial, finite difference, and Monte Carlo solutions to option pricing problems. Does not include models that are peculiar to the fixed income markets (e.g., HW and HJM). Not an introduction to options. (Two days)
Introduction to Fixed Income Option Pricing Algorithms. Introduces analytic, binomial, trinomial, finite difference, and Monte Carlo solutions to fixed income option pricing problems. Includes only models that are peculiar to the fixed income markets (e.g., HW and HJM). Not an introduction to options. (Two days)
An Introduction to the Binomial HJM Model. This includes a working spreadsheet model for students to take away and study. (One day)
Monte Carlo Implementation of the HJM Model. Includes a working spreadsheet model for students to take away, study, and expand. (One day)
An Introduction to the Hull-White Model. This includes a working spreadsheet model for students to take away and study. (One day)

Advanced Derivatives Analysis
Advanced "Black-Scholes" Derivatives Pricing. Covers methods of derivatives pricing, including analytic, binomial, trinomial, finite difference, and Monte Carlo. Includes fixed income and other risks, applications to cases. (Five days)
The Wand and the Wizard. A demonstration of where eight methods of pricing exotic options in the Black-Scholes setting ought to give the the same answer, but sometimes break down badly. Learn why it happens and how to avoid it. (Half day)
Beware of Greeks. An explanation of the practical problems associated with half-a-dozen ways of computing Greeks. (Half day)
When Bad Models Happen to Good People. Tales of derivatives disasters that had their roots in that fertile ground which is a rich mix of bad models, greedy traders, and overmatched controllers and internal auditors. (Half day)
Multifactor Pricing Models for Interest Rate Derivatives. Derivative of theory and practice for multifactor HJM Monte Carlo models.

We've been presenting complicated information in academic and business forums for more than two decades.

Prospective clients:

For a schedule of public presentations of these talks, please check our "Derivatives Calendar".
If you want to sponsor one of these presentations at your firm, for your employees, only, please email our Presentations Department or call us in Area Code 914 at telephone number 738-3309 to receive more information.

ÖSoftware Solutions

Prospective Software Buyers

We prefer to guide our customers toward existing, off-the-shelf, low-cost software solutions, whenever possible, which is most of the time. Do you want to acquire some Excel add-in software to price a product or include in a risk management system? As you search the market for software, you might want to check our links at www.FreeOptionPricing.com to sites offering free and commercial software, including Java applets.

However, if you can't find what you want there or elsewhere, ask us. We conducted (circa 6/1/98) extensive testing of eight commercial packages of Excel add-ins for pricing options. We have built or reviewed software for several major derivatives dealers. We may be able to

help you find a vendor who offers the software solution you need
help you evaluate the quality of software you are considering, or
deliver customized software solutions on those rare occasions when you can't find anything that fits or feels right.

Software Vendors

Do you have a high quality, free and commercial software – including Java applets – for derivatives pricing or risk management? Tell us. We'd like to consider it when solving our customers' problems, and perhaps include a hot link to your web site.

The Group can provide you also with

models from which your people can develop algorithms
algorithms from which your people can produce code
documentation of models that you are coding
verification of models you are delivering or receiving.

Click here to email Dr. Risk or the William Margrabe Group

ABOUT CONSULTING AT THE WILLIAM MARGRABE GROUP, INC.:
Investment,
Risk Management,
Derivatives, and
Financial Engineering

Our other web sites:

www.FreeOption
Pricing.com
Free option pricing calculators from here and around the world.

www.RiskManagement
Digest.com
Summaries
of the best articles
from the best publications
in the risk management trade press.

www.Derivatives
Digest.com
Summaries
of the best articles from the best publications
in the derivatives trade press.

www.AskDrRisk
.com
Answers to your questions about Investment,
Risk Management,
Derivatives, and
Financial Engineering