The Contingent-Claims Arms Race
Tuning the parameters for fixed-income option models is as important as choosing the
model itself.
Roland Lochoff
(Reprinted with permission from the Journal of Portfolio Management )
ROLAND LOCHOFF is director of fixed income services at BARRA in
Berkeley (CA 94704).
Although the cold war is over, “rocket” scientists continue
to extend and elaborate fixed-income contingent-claims models
for evaluating embedded options. The race centers on whose
model is bigger and better.
In general, this means that the model is more complex —
although not necessarily more effective. The models may have
different philosophical starting points, but many of the
improvements involve adding a function or an extra equation
where there was previously a fixed assumption. Increased
flexibility is good, but estimation of the parameters that feed
the model remains the most important decision the modeler
will make.
A FRAMEWORK FOR VIEWING
CONTINGENT-CLAIMS MODELS
The modeler of the future course of interest rates to value
interest rate options must make several choices and abide by
certain financial truths to make the model work. First, no-
arbitrage opportunities must be assumed. Second,
schematically, the models must choose two of three choices as
inputs, with the third determined as an output:
• The term structure of rates.
• The term structure of volatility.
• The expected change in forward rates.
These ingredients are the legs of the “assumption triangle”
shown in Exhibit 1. Any two legs will determine the third.
There are several variants on each of these legs, which can be
increasingly refined. For example, the term structure of
volatility could have one or more factors; it could be path-
dependent, or independent, and involve several additional
equations that may refer to the assumptions in the other legs of
the assumption triangle.
EXHIBIT 1
THE ASSUMPTION TRIANGLE
Once the legs of the triangle are chosen, the technique to
estimate actual values numerically must be chosen. Methods
include partial differential equations, interest rate trees, finite
difference or grid methods, Monte Carlo approaches, or path
integrals. The method chosen is not related to the choice of
model. As an example, a Cox, Ingersoll, and Ross [1985]
model may be implemented through a tree or through
differential equations.
WHERE DO WELL-KNOWN MODELS FIT IN
THIS FRAMEWORK?
There are three possible ways of combining the triangle. In
practice, only two of the three combinations have been used:
• Term structure models and equilibrium models, which
choose the change in forward rates and the term structure
of volatility to imply the allowable forms of the term
structure of rates.
• Relative valuation models, which choose the initial term
structure of rates and the term structure of volatility, and do
not need to specify expected changes in forward rates.
The first approaches to analyzing fixed-income options
were the term structure models developed by Vasicek [1977],
Dothan [1978], Richard [1978], Brennan and Schwartz [1979],
and Schaefer and Schwartz [1984]. These models all invoke a
no-arbitrage condition, following Black-Scholes [1973], but
they are not relative valuation models. They describe the
behavior of the entire term structure, and price bonds as well as
options. Because they are not relative valuation models, they
all require risk premiums and mean reversion parameters as
input.
A second approach, related to the term structure models, is
the equilibrium model developed by Cox, Ingersoll, and Ross
in 1985. Their model is effectively another term structure-type
model, but they derive it explicitly from an equilibrium model
of the economy. This equilibrium model is more firmly
grounded in economic theory, but, in practice, it shares the
strengths and weaknesses of the other term structures.
Relative valuation models represented by Ho and Lee
[1986] and Heath, Jarrow, and Morton [1992] price options
relative to a given term structure. They give no insight into the
term structure itself, and price options only relative to market
prices of the underlying instrument. The advantage of relative
valuation is precisely that they do not require estimating
expected returns — a difficult task.
None of the models contradicts each other. So if, for
example, we input the Cox, Ingersoll, and Ross (CIR) term
structure and volatility into Heath, Jarrow, and Morton (HJM),
the HJM option price should equal the CIR price. After all,
there can only be one arbitrage-free price.
All these fixed-income option modeling approaches share
some basic assumptions. They all assume that market prices
are arbitrage-free. They all make assumptions about term
structure volatility.
CALIBRATING THE MODEL
To get a sense of which inputs are important, let’s explore
how several models react to changes in inputs. We start with
extremely simple models and proceed to the most complex. In
all cases it becomes clear that volatility assumptions matter
much more than the quality of the model.
We will start with the simplest model we can imagine. In
this model, interest rates change so that a non-callable bond is
priced anywhere from $95 to $105 with uniform probability.
Suppose we also have a bond that is callable at par in this
interest rate environment. Exhibit 2 shows that the expected
value of the option in this framework is $1.36.
EXHIBIT 2
Simple Call Model
Price of Price of Contingent Expected
Non-Callable Callable Value Probability Value
95 95 0 0.091 0.00
96 96 0 0.091 0.00
97 97 0 0.091 0.00
98 98 0 0.091 0.00
99 99 0 0.091 0.00
100 100 0 0.091 0.00
101 100 1 0.091 0.09
102 100 2 0.091 0.18
103 100 3 0.091 0.27
104 100 4 0.091 0.36
105 100 5 0.091 0.45
Option Value (Undiscounted) 1.36
Standard Deviation 3.16
Now we double the volatility. Exhibit 3 illustrates the
mechanics. It should come as no surprise that the expected
value of the option has doubled.
EXHIBIT 3
Increased Volatility
Price of
Non- Price of Contingent Expected
Callable Callable Value Probability Value
90 90 0 0.091 0.00
92 92 0 0.091 0.00
94 94 0 0.091 0.00
96 96 0 0.091 0.00
98 98 0 0.091 0.00
100 100 0 0.091 0.00
102 100 2 0.091 0.18
104 100 4 0.091 0.36
106 100 6 0.091 0.55
108 100 8 0.091 0.73
110 100 10 0.091 0.91
Option Value (Undiscounted) 2.73
Standard Deviation 6.32
We now create another model that is quite different, but
equally simple. In this model, interest rates change in such a
way that the price of the non-callable bond increases linearly,
but with a probability that also increases linearly until par, at
which point it decreases linearly. Again we presume a call on
the bond at par. Exhibit 4 illustrates the mechanics of this
model.
EXHIBIT 4
Another Simple Model
Price of Non- Price of Contingent Expected
Callable Callable Value Probability Value
92 92 0.00 0.00 0.00
93 93 0.00 0.01 0.00
94 94 0.00 0.03 0.00
95 95 0.00 0.05 0.00
96 96 0.00 0.06 0.00
97 97 0.00 0.08 0.00
98 98 0.00 0.10 0.00
99 99 0.00 0.11 0.00
100 100 0.00 0.13 0.00
101 100 1.00 0.11 0.11
102 100 2.00 0.10 0.19
103 100 3.00 0.08 0.24
104 100 4.00 0.06 0.25
105 100 5.00 0.05 0.23
106 100 6.00 0.03 0.17
107 100 7.00 0.01 0.09
108 100 8.00 0.00 0.00
Option Value (Undiscounted) 1.28
Standard Deviation 3.16
Exhibit 5 contrasts the price distributions of the two simple
models. Notice that when the standard deviations of the two
simple models are equal (Exhibits 2 and 4), the expected value
of the option varies by only 8 cents ($1.36 versus $1.28).
EXHIBIT 5
PRICE DISTRIBUTIONS FOR
TWO SIMPLE MODELS
Exhibit 6 s hows the differences in the option values
calculated by the two models for different strike prices.
Clearly, the level of “at-the-moneyness” makes little difference
to the valuations.
EXHIBIT 6
OPTION VALUES FOR DIFFERENT CALL PRICES
To reiterate, the conclusions are that, in comparing two
simple models with different price distributions but identical
volatilities:
• The at-the-money option prices are very close.
• Even for in-the-money or out-of-the-money options, the
model prices are very close.
Now we have to address the hard question. These
conclusions hold with simple models, but do they apply to
more realistic models (or to the real market?). What we will do
is compare the simple model (we’ll use simple model 1) with
the CIR-based model in the BARRA B2 system. We will use
three Treasury bonds with different coupons with four years to
maturity that are callable at par in two years. To illustrate, we
assume a flat 9% yield curve and choose coupons of 8%, 9%,
and 10% to represent, respectively, out-of-the-money, at-the-
money, and in-the-money embedded options.
Exhibit 7 illustrates that naive model gives answers within
4 cents of the sophisticated model, which is our first indication
that the conclusions above apply to complicated models.
EXHIBIT 7
Comparing Option Values
Bond 1 Bond 2 Bond 3
BARRA’s Model 0.57 1.34 2.37
Simple Model 0.61 1.36 2.41
It is not our intention to claim that the BARRA B2 model
is only as good as an extremely naive model, but to show that
volatility inputs are critical. Exhibit 8 shows results derived
using the state-of-the-art HJM model valuing embedded
options under scenarios of different volatility. A startling
observation emerges: the value is linear with volatility. This
result appears to hold for any of the models mentioned at the
outset.
EXHIBIT 8
HJM MODEL VALUING EMBEDDED OPTIONS UNDER
SCENARIOS OF DIFFERENT VOLATILITY
Since volatility is critical, it is clear that specifying
volatility precisely becomes the key input to any contingent-
claims model. The models that allow more choices for
volatility are probably going to perform better over a full range
of interest rate environments than those with fewer.
When we say volatility, we typically think of a single
number, perhaps the volatility of short rates, or the implied
volatility of T-bond futures. Yet as one obvious contradiction,
we know that short rates are almost always more volatile than
long. In fact it makes more sense to think in terms of a term
structure of volatility.
Various models allow specification of the term structure of
volatility in different ways. The BARRA B2 model (a variant
of the Cox, Ingersoll, and Ross model) uses a mean reversion
parameter linked to the volatility assumption. The HJM model
allows explicit specification of this function. HJM models may
have two or more ways of specifying this volatility.
EXHIBIT 9
Two-Factor Model for Callable Government Bonds — Option Values
Coupon
6.0 7.0 8.0 9.0 10.0
25-Year Maturity
Callable at Par in 20 Years 0.43 0.82 1.25 1.80 2.41
15-Year Maturity
Callable at Par in 10 Years 0.82 1.56 2.47 3.50 4.75
10-Year Maturity
Callable at Par in 5 Years 0.57 1.41 2.55 4.16 6.32
EXHIBIT 10
“Tuned” One-Factor Model versus Two-Factor Model Differences in
Option Values
Coupon
6.0 7.0 8.0 9.0 10.0
25-Year Maturity -0.11 -0.09 0.01 0.03 0.07
Callable at Par in 20 Years
15-Year Maturity -0.02 0.11 0.12 -0.02 -0.09
Callable at Par in 10 Years
10-Year Maturity 0.00 0.05 0.00 -0.09 -0.01
Callable at Par in 5 Years
Clearly, this provides a more realistic range of outcomes in
that there are more degrees of freedom in the process.
Nevertheless, it also confronts the modeler with choices that
may not be obvious and presents the risk of straying into the
realm of misplaced rigor.
Exhibits 9 and 10 compare the performance of a “well-
tuned” two-factor model and a well-tuned one-factor model.
For this range of contingent claims, the extra complexity of the
two-factor model (with carefully chosen parameters) gains us
little.
As you can see in this example, the contingent claims are
long-dated options in relatively long bonds. The one-factor
model is tuned so that it gives good performance for bonds like
these. For a long bond with a short call, this single-factor
model may not have the flexibility to stretch to all ranges of
outcomes with the same precision as a two-factor model. And,
for example, for an option on the spread between short and
long rates, the parameters chosen for the long option example
may be completely inappropriate.
We have tuned the B2 model so that it performs best in
areas where options can have the most material effect on the
value and risk of the assets it covers. Portfolios that contain a
high proportion of derivative instruments, where the value of
the contingent claim can be a high proportion of the value of
the total underlying, will probably need a two- (or more) factor
model to get reasonable values.
We hope we have made the case that while option
modeling technology has become very advanced and useful, it
is also surrounded by a body of cant that can confound the
unsuspecting practitioner.
To conclude:
• Volatility is the most important model parameter.
• Even bad models can be tuned to give good results for most
simple options.
• Good models are good because they work over a wider
range of options.
REFERENCES
Black, F., and M. Scholes. “The Pricing of Options and Corporate
Liabilities.” Journal of Political Economy, Vol. 81 (May-June 1973), pp.
637-654.
Brennan, M.J., and E.S. Schwartz. “A Continuous Time Approach to the
Pricing of Bonds.” Journal of Banking and Finance, Vol. 3 (July 1979),
pp. 133-155.
Cox, John C., Jonathan E. Ingersoll, Jr., and S.A. Ross. “A Theory of the
Term Structure of Interest Rates.” Econometrica, Vol. 53, No. 2 (March
1985).
Dothan, U. “On the Term Structure of Interest Rates.” Journal of
Financial Economics, Vol. 6 (March 1978), pp. 59-70.
Heath, D., R. Jarrow, and A. Morton. “Bond Pricing and the Term
Structure of Interest Rates: A New Methodology for Contingent Claim
Valuation.” Econometrica. Vol. 60 (1992), pp. 77-105.
Ho, T.S.Y., and S.B. Lee. “Term Structure Movements and Pricing
Interest Rate Contingent Claims.” Journal of Finance, Vol. XLI, No. 5
(December 1986).
Richard, S. “An Arbitrage Model of the Term Structure of Interest Rates.”
Journal of Financial Economics, Vol. 6 (1978), pp. 33-57.
Schaefer, S., and E.S. Schwartz. “A Two-Factor Model of the Term
Structure: An Approximate Analytical Solution.” Journal of Financial and
Quantitative Analysis, Vol. 19, No. 4 (December 1984).
Vasicek, O. “An Equilibrium Characterization of the Term Structure.”
Journal of Financial Economics, Vol. 5 (November 1977), pp. 177-188.