7.5Valuation Techniques in the Financial Services Industry

Investment banks, commercial banks, and a number of institutional investors have made

sizable investments in the technology of pricing derivatives. Cheap computing power,

in the form of workstations and parallel processing mainframes, and expensive brain

power are the main elements of this technology.

Numerical Methods

The techniques for valuing virtually all the new financial instruments developed by

Wall Street firms consist primarily of numerical methods; that is, no algebraic

formula is used to compute the value of the derivative as a function of the value of the

underlying security. Instead, a computer is fed a number corresponding to the price of

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the underlying security along with some important parameter values. The computer then

executes a program that derives the numerical value of the derivative and, sometimes,

additional information such as the number of shares of the underlying security in the

tracking portfolio.

Binomial-Like Numerical Methods.These numerical methods fall into several

classes. One class is the binomial approach used throughout this chapter. On Wall

Street, some investment banks employ the binomial approach to value callable cor-

porate bonds (see exercise 7.5) for their clients. This binomial approach also is com-

monly used in spreadsheets by recent M.B.A. graduates working in corporate

finance, sales, and trading, who need to find “quick and dirty” valuation results for

many derivatives.

One potential hitch in the binomial method is that it is necessary to specify the val-

ues of the underlying security along all nodes in the tree diagram. This requires esti-

mation of uand d, the amounts by which the security can move up and down from

each node. There are several ways to simplify the process. First, the binomial method

can be used to approximate many kinds of continuous distributions if the time periods

are cut into extremely small intervals. One popular continuous distribution is the log-

normal distribution. The natural logarithm of the return of a security is normally dis-

tributed when the price movements of the security are determined by the lognormal

distribution. Once the annualized standard deviation,

, of this normal distribution is

known, uand d are estimated as follows.

ueTN

1

d

u

where

Tnumber of years to expiration

Nnumber of binomial periods

eexponential constant (about 2.7183)

Thus

T

square root of the number of years per binomial period

N

Depending on the problem, the available computing power, and the time pressures

involved, the valuation expert can model the binomial steps, T/N,as anywhere from

one day to six months.²⁷

Numerical analysis of derivatives with binomial approaches is often modified

slightly to account for inefficient computing with the binomial approach. For one, the

value of the derivative is affected more by what happens to the underlying asset in the

near term and in the middle of the binomial tree, because that is where the risk-neutral

probabilities are greatest and risk-free discounting has the least impact on the current

value of the derivative. Hence, derivative valuation experts typically modify the bino-

mial method to compute near-term future values and future values in the middle of the

²⁷Chapter

8 describes how to estimate from historical data.

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250Part IIValuing Financial Assets

binomial tree more intensively and thus more precisely. Often, the tree is modified to

a rectangular grid shape.²⁸

Simulation.An additional method that Wall Street firms use to value derivatives

is simulation, a method that uses random numbers obtained from a computer to gen-

erate outcomes and then averages the outcomes of some variable to obtain values.

Simulations are generally used when the derivative value at each node along the tree

diagram depends only on the simulation path taken, rather than on a comparison

with action-contingent values of the derivative along other paths followed by the

underlying asset. Derivatives that can be valued with simulation property include

forwards and European options, but they exclude, for example, mortgages and Amer-

ican put options.

Asimulation starts with the initial price of the underlying security. Then, a random

number generated by a computer determines the value for the underlying security at

the next period. This process continues to the end of the life of the derivative. At that

point, a single path for the underlying security over a number of periods has been con-

structed and an associated path for the derivative (whose value derivesfrom the under-

lying security) has been computed. Usually, one generates derivative values along the

single path by discounting back, one period at a time, the terminal value of the deriv-

ative. The discounting is made at the short-term risk-free rate, which may itself have

its own simulated path!

Asecond path for the underlying security is then generated, the derivative value is

calculated for that path, and an initial derivative value for the second path is computed.

This process continues until anywhere from 100 to 100,000 simulated paths have been

analyzed. The initial value of the derivative over each path is then averaged to obtain

the fair market value of the derivative.

The Risk-Free Rate Used by Wall-Street Firms

All derivative valuation procedures make use of a short-term risk-free return. The most

commonly used input is LIBOR. While most academic research focuses on the short-

term U.S. Treasury bill rate as “the risk-free rate,” this research fails to recognize that

only the Treasury can borrow at this rate while arbitrage opportunities between a track-

ing portfolio and a mispriced derivative may require uncollateralized risk-free borrow-

ing by a high-quality borrower. The rate at which such borrowing takes place is much

closer to LIBOR than to the T-bill rate.²⁹

²⁸Other

approaches are also available for valuing derivatives. One method, developed by Schwartz

(1977), models the process for the underlying security as a continuous process and uses stochastic

calculus and no-arbitrage conditions to derive a differential equation for the derivative. It is possible to

solve the differential equation with numerical methods that have been devised in mathematics, including

the implicit finite difference method. Explicit methods of solving the differential equation are tied

directly to the risk-neutral valuation method.

²⁹A

further discussion of this subject is found in Grinblatt (1995). This chapter’s results on currency

forward rates require longer-term risk-free rates. Implied zero-coupon interest rates in the LIBOR market

(known as Eurocurrency rates) appear to be the correct interest rates to use for this relation. For

horizons in excess of one year, the fixed rate on one side of an interest rate swap that is exchanged for

LIBOR provides the correct risk-free rate for the equation in Result 7.3.

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