Fixed-Income Securities

Lecture 5: Tools from Option Pricing

Philip H. Dybvig
Washington University in Saint Louis

• Review of binomial option pricing

• Interest rates and option pricing
• Effective duration
• Simulation

Copyright c Philip H. Dybvig 2000

 Option-pricing theory
The option-pricing model of Black and Scholes revolutionized a literature previ-
ously characterized by clever but unreliable rules of thumb. The Black-Scholes
model uses continuous-time stochastic process methods that interfere with un-
derstanding the simple intuition underlying these models. We will use instead the
binomial option pricing model of Cox, Ross, and Rubinstein, which captures all
of the economics of the continuous-time model but is simpler to understand and
use. The original binomial model assumed a constant interest rate, but it does
not change much when interest rates can vary.
Cox, John C., Stephen A. Ross, and Mark Rubinstein, 1979, Option Pricing: A
Simplified Approach, Journal of Financial Economics 7, 229–263
Black, Fischer, and Myron Scholes (1973), The Pricing of Options and Corporate
Liabilities, Journal of Political Economy 81, 637–654
 A simple option pricing problem in one period
Short-maturity bond (interest rate is 5%):

1 -
1.05
Long bond:

*1.15
1 
H
HHj
1
Derivative security (intermediate bond):


*
 109
? HH
j
H
103
 The replicating portfolio
To replicate the intermediate bond using αS short-maturity bonds and αB long
bonds:

109 = 1.05αS + 1.15αL

103 = 1.05αS + 1.00αL

Therefore αS = 60, αL = 40, and the replicating portfolio is worth 60+40 = 100.
By absence of arbitrage, this must also be the price of the intermediate bond.
 General single-period valuation
Compute the replicating portfolio and the price of the general derivative security
below. Assume U > R > D > 0.
Riskless bond:

1 -
R
Long bond:


* US
S 
H
HHj
DS

Derivative security:


*
 VU
??? HH
j
H
VD
 State prices and risk-neutral (artificial) probabilities
Valuation can be viewed in terms of state prices pU and pD or risk-neutral proba-
bilities πU and πD , which give the same answer (which is the only one consistent
with the absence of arbitrage):

V alue = pU VU + pD VD = R−1(πU∗ VU + πD
∗
VD )
where
R−D U −R
pU = R−1 pD = R−1
U −D U −D
are called state prices and

R−D U −R
πU = πD =
U −D U −D
are called risk-neutral probabilities. Note that the risk-neutral probabilities equate
the expected return on the two assets. Risk-neutral probabilities need not equal
the true probabilities, but most term structure models used in practice assume
they are the same.
 Multi-period valuation
One special appeal of the binomial model is that multi-period valuation is not
much more difficult than single-period valuation. This is because multiperiod
valuation simply applies the single- period valuation again and again, stepping
from maturity backwards through the interest-rate tree. Another approach that
is not limited to the binomial model is to use a simulation for valuation. This is
because we can express the value of a claim as an expectation using the artificial
probabilities. For example, we have that

∗ 1 1 1 1
E [ ... CT ]
R1 R2 R3 RT
is the value at time 0 of a claim to the cash flow CT to be received at time T ,
where Rs is one plus the spot rate of interest quoted at time s − 1 for investment
to time s. To use simulation, we simulate many realizations of the interest rates
and the final cash flow, and take the sample mean corresponding to population
statistic in the formula.
 Example: valuation of a riskless bond
Consider a two-period binomial model in which the short riskless interest rate
starts at 20% and moves up or down by 10% each period (i.e., up to 30% or
down to 10% at the first change). The artificial probability of each of the two
states at any node is 1/2. What is the price at each node of a discount bond
with face value of $100 maturing two periods from the start?
The interest tree is
*

 40%

*
 30% 
H
H
20% H
HH
j
H
*

 20%
j
10% 
H
H
j
H
0%

and the value of the bond is given by


*
 100

* 76.92 HH
69.93 
HHH
j
H
*

 100
j
90.91 
H
H
j
H
100
 Special considerations for pricing interest derivatives

• The interest rate is not constant and varies through the tree.
• The interest rate is not an asset price (and cannot, for example, be plugged
in as the price in the Black-Scholes formula).
• The short rate is known at the start of the period and is forward-looking
(while the return on a risky asset is not known until the end of the period).
• It is usually handiest to assume risk-neutral probabilities and the volatility and
work from there.
• We usually match the initial observed yield curve one way or the other.
• One implicit or explicit input is a view on mean reversion and volatility of
interest rates.
 Valuation of interest derivatives
We value interest derivatives besides the riskless bonds using the same approach.
First we compute the value at maturity based on the contractual terms, and then
we compute the value at previous points in the tree using repeatedly the formula
for one-period valuation. If we have an early exercise option or a cash flow before
maturity, that needs to be taken into account in the one-period valuations. We
need to be careful to understand what is shown in the tree, for example, perhaps
it is the value of a live bond (that is not converted) after the coupon interest in
the period is received.
Sometimes we need to value one security first before valuing another. For exam-
ple, if we are computing the value of an option on a coupon bond, we need to
value the bond at each node in the tree before valuing an option on the bond.
To include the possibility of exercise, the value of the option will be the larger
of the value if not exercised (which comes from looking at next period’s value
in the option pricing formula) and the value if exercised now (which comes from
looking at the bond price at this node). Care must be taken to decide whether
we need to look at the bond price before or after any coupon is paid.
 In-class exercise: bond and bond option valuation
Consider a two-year binomial model. The short riskless interest rate starts at
50% and moves up or down by 25% each year (i.e., up to 75% or down to 25%
at the first change). The artificial probability of each of the two states at any
node is 1/2. What is the price at each node, of a discount bond with face value
of $100 maturing two periods from the start?
What is the value at each node of an American call option on the discount bond
(with face of $100 maturing two periods from now) with a strike price of $60
and maturity one year from now?
 Fitting an initial yield curve: using fudge factors
Suppose we write down our favorite term structure model and we find that it does
not even give the correct prices for Treasury strips. This is more than a minor
embarrassment, since it means that the pricing of derivative securities from the
model will produce an arbitrage in reality. Fortunately, we can use fudge factors
to correct the pricing of the discount bonds. Fudge factors are adjustments to
the interest rate process. The same adjustment is made to all nodes at the same
point in time.
This approach was introduced in my paper,1 which is why they are sometimes
called “Dybvig factors” (for example in the BARRA documentation).

1
P. Dybvig, Bond and Bond Option Pricing Based on the Current Term Structure, 1997, Mathematics of Derivative Securities, Michael A. H.
Dempster and Stanley Pliska, eds., Cambridge University Press.
 Fudge factors in the binomial model
In the binomial model, we can write the discount bond prices as

1 1 1 1 
 

Dom(0, t) = E ∗  ...

om om om
R1 R2 R3 om
Rt


where om indicates the original model and Rsom is one plus the interest rate at
time s in that original model. If we see instead discount rates D(0, t) in the
economy, we want to change the interest rate process to fit what is observed.
Intuitively, we want to add to each interest rate the different between the implied
forward rate in the economy and the interest rate in the original model (and this
adjustment is exact enough for many purposes). More precisely, we set

D(0, s − 1)/D(0, s)
Rs = Rsom
Dom(0, s − 1)/D om(0, s)
Note that the numerator in the adjustment is one plus the forward rate observed
in the economy, and the denominator is one plus the forward rate in our original
model.
 In-class exercise: fudge factors
Consider a two-year binomial model. Start with an original model in which the
short riskless interest rate starts at 5% and moves up or down by 5% each period
(i.e., up to 10% or down to 0% at the first change). The artificial probability of
each of the two states at any node is 1/2.
What is the price of a one-year discount bond in this original model? the two-year
discount bond?
Suppose the one-year discount rate in the economy is 6% and the two-year dis-
count rate is 7%. Compute the fudge factors and draw the tree for the adjusted
interest rate process.
 Fudge factors: good and bad points
The fudge factors do fit today’s STRIP curve of riskless bonds, but may fail to do
so tomorrow. When using fudge factors, it is necessary to re-estimate the fudge
factors every period. To the extent that the fudge factors are changing a lot over
time, there is significant volatility in interest rates that is not part of the option
pricing activity. Understating the volatility tends to underprice options and long
bonds with a lot of convexity.
Fortunately, most of the movements in the yield curve are explained well by
a single risky factor, so viewing the rest as a total surprise (by including it as
updates in the fudge factors) will not usually create big pricing errors. This is one
result that surprised me in my analysis: I originally planned to criticize models
that implicitly used fudge factors, but I actually found some support for what
they were doing. It pays to keep an open mind! I should note that it is possible
to design securities (for example an option on a spread between yields) that
eliminate the main risky factor and will be badly mispriced using fudge factors.
This idea may come in handy at some point when you are trading with people
using fudge factors in a mechanical way!
The main alternative to using fudge factors is the use of multi-factor models that
include different sources of interest risk. We might have one factor for more-or-
less parallel shifts, another for changes in the slope, and a third for curvature.
These models can be handled well by simulation (and in some cases we have
exact formulas), but tend to get messy in a binomial framework.
 Mean reversion in interest rates
The simple binomial model with equal up and down probabilities at each node
has several unrealistic features. One is that futures prices in the model have the
same sensitivity to rate shocks as the short rate, while actual short rates move
much more than forward rates. Or, to put it another way, short rates are much
more sensitive to interest shocks than are yields on long bonds.
We remedy this problem by introducing mean reversion. Interest rates move on
average towards a long-term mean. If interest rates move up above the mean,
then the mean change becomes negative and rates tend to move back (revert)
towards the mean. The usual form for mean reversion is

E[rt+1 − rt] = k(r − rt)

When the interest rate rt is larger than the long-term mean r, then the interest
rate usually declines. When rt is smaller than r, then the interest rate usually
rises.
 Mean reversion in the binomial model
In the binomial model, suppose that the interest rate increases from r t to rt + δ
with probability π and decreases from r to rt − δ with probability 1 − π. Then
E[rt+1 − rt] = πδ + (1 − π)(−δ)
= (2π − 1)δ
For mean reversion (using the formula on the previous slide), we want to set this
mean return equal to k(r − rt). Solving
(2π − 1)δ = k(r − rt)
for π, we obtain
1 k(r − rt)
π= +
2 2δ
Without mean reversion, k = 0 and π = 1/2. With mean reversion, k > 0 and
the probability of going up is less than 1/2 when r is large but smaller than 1/2
when r is small.
 Mean reversion: a technical adjustment
If the probability on the last slide (which we use for the risk-neutral probability)
is computed to be larger than 1, we round it down to 1. And, if it is computed to
be smaller than 0, we round it up to 0. Having artificial probabilities larger than
1 or less than 0 does not make economic sense (it implicitly implies arbitrage),
and may lead to numerical instability. As a computational boon, we do not have
to compute security prices for nodes beyond where the probability goes to 1 or
0, since nodes further out do not carry any weight in valuation.
 Volatility
As in equity options, changing volatility is an important concern for pricing of
interest derivatives. The volatility of interest rates tends to be higher when
interest rates are higher, but there is not a tight connection between volatility of
rates and their levels. As a first adjustment for changing volatility, we can make
the interest rate grid more widely spaced at large rates and more narrowly spaced
at small rates. This has the advantage of keeping the interest rate positive and
also admitting that volatility tends to be higher when rates are higher.
It is possible to use a binomial model with changes in volatility as well as rates, but
it is usually simpler to use a simulation model in this case. Adding interest rate
volatility has a minor impact on the programming difficulty in simulation models
but a major impact in binomial models. The only drawback is that simulation
models tend to be much slower. I like to use between 100,000 and 1,000,000
sample paths for simulation estimates of derivative prices; practitioners often save
computer time by using fewer paths and obtain inaccurate results. Computer time
is getting cheaper and cheaper, and there is less and less excuse for making this
mistake.
 Volatility models
There are two main models of uncertain volatility used in econometrics. Both have
names you can love to hate. Stochastic volatility (SV) models have a volatility
at a point in time that is unobserved but can only be guessed from the history.
The name is inappropriate because it sounds like it could apply to any model of
volatility. ARCH models look at the best estimate of volatility conditional on the
history. ARCH models are easy to work with and in spite of appearances are no
less general than SV models (since volatility in the ARCH model at a point in time
can be interpreted as the best estimate of the unknown volatility in an SV model).
ARCH stands for AutoRegressive Conditional Heteroskedasticity; do I have to
explain why I do not like this term? There is a whole alphabet soup of various
ARCH models, for example GARCH (Generalized ARCH, which is actually less
general than ARCH), LARCH (Linear ARCH), EGARCH (Exponential GARCH),
etc.
 Sample SV and ARCH models
Here is an example of an SV model:

rt+1 = rt + kr (r − rt) + σter,t

σt+1 = (σt + kσ (σ − σt))(1 + eσ,t)
Note that there is a separate error term in the volatility equation, and we do not
know the true volatility.
Here is an example of an ARCH model:

rt+1 = rt + kr (r − rt) + σter,t

σt+1 = (σt + kσ (σ − σt)) |er,t|
In this case the volatility equation does not have its own error term and the ran-
dom part of the change in volatility depends on the error term (the shock) in the
interest rate term. If we know the parameters and the starting volatility, we can
figure out the volatility for all times from observing the interest rates. Knowing
the volatility in the model simplifies econometric estimation of the parameters
and also makes application easier.