HJM Models
Creating spot rate trees:
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Assume that spot interest rates are normally distributed and
approximate it with a binomial distribution.
Zero coupon bond prices and volatility of spot interest rates are
given for different maturities.
Guess spot rate values for the next period.
Absence of arbitrage allows the use of risk-neutral valuation of zero
coupon prices. If they don’t match with the given prices, iterate
spot rates until it matches.
Check that the volatility estimate matches the given volatility. If
they don’t match, iterate spot rates until it matches.
Repeat these steps throughout the tree to get the arbitrage free
evolution of the spot interest rate consistent with the dual term
structures of interest rates and volatility.
Spot Rate Tree
Pricing a European Put Option
on Treasury Bills
Alternative approach

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An alternative and more flexible approach
takes the initial term structure as given and
models the evolution of forward rates. This is
the Heath-Jarrow-Morton model (1992).
“Bond Pricing and the Terms Structures of
Interest Rates: A New Methodology for
Contingent Claims Valuation.” Econometrica
60, 77-105.
Forward Rates

The HJM model starts with the initial forward
rate curve f(0,T) for all T,
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where f(0,T) is the date-0 continuously
compounded forward interest rate for the future
interval [T,T + D],
and D is a small interval (D -> 0).
This implies
B(0,T + D) = B(0,T)e–f(0,T)D.
So the initial forward curve f(0,T) can be
backed up from the spot curve B(0,T).
Forward Rates Evolution

Assume that the change in the forward rate is
normally distributed and described by:
Df(t,T) = a(t,T)D + s(t,T)DW(t),
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where Df(t,T) = f(t + D,T) – f(t,T) ) is the change
in the forward rate over the interval t to t + D,
a(t,T) is the drift and s(t,T) is the volatility,
DW(t) is normally distributed with mean 0
and variance D,
and the evolution is under martingale
probabilities p.
Volatility Specification

For implementation, assume that the forward
rate’s volatility is of the form
s(t,T) = s  exp[– l(T – t)]
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where l  0 is a non-negative constant, volatility
reduction factor
Hence vol for the short-term forward rate (T-t
is small) is larger than that of long term. That
fits with the empirical observation.
Forward Rate f(t,T)

f(t,T) = f(0,T) + [f(D,T) – f(0,T)]
+ [f(2D,T) –f(D,T)] + …+ [f(t,T) - f(t – D,T)]
t -D
 f (0, T )   Df (v, T )
v 0

Using forward rate evolution specification
t -D
t -D
v 0
v 0
f (t , T )  f (0, T )   a (v, T )D   s (v, T )DW (v)

this describes the evolution of the forward rate at time t in
terms of the initial forward rate curve f(0,T), the drifts, and
the volatilities of the intermediate changes in forward rates.
The Spot Rate and Money
Market Account Evolution

Using definition of the spot rate, r(t) = f(t,t) gives
spot rate process
t -D
t -D
v 0
v 0
r (t )  f (0, t )   a (v, T )D   s (v, T )DW (v) for t  D

By definition, money market account’s value
A(0) =1 and A(t) = A(t – D)er(t – D)D for all t, or
t -D
 r (u ) D
A(t )  1.e u  0
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Using spot rate process, A(t) can be described by initial
forward rates, drifts, and volatilities.
Normalized bond prices

Zero-coupon bond price can be similarly
expressed in terms of initial forward
rates, drifts, and volatilities.
B (t , T )  1e
t -D  T -D
B(t , T )
 B(0, T )e
A(t )
-
T -D
 f ( t ,u ) D
u t
 t -D  T -D




-
a ( v ,u ) D D - 
s ( v ,u ) D DW ( v )




v0  u v
 v0  u v



Arbitrage-Free Restrictions
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A contribution of the Heath-Jarrow-Morton model is
to provide the restrictions needed on the drift and
volatility parameters of the forward rate process
such that the evolution is arbitrage-free.
We know that the zero-coupon bond price
processes are arbitrage-free if and only if the
normalized zero-coupon bond prices
B(t,T)/A(t) are martingales. Alternatively stated,
B(0,T) = Ep(B(t,T)/A(t)) for all T.
Arbitrage-Free Restrictions
(cont’d)

Comparing this with normalized bond price
process, using properties of normal
distribution, rearranging terms and taking
limits as D -> 0, we get
T
a (v, T )  s (v, T )  s (v, u )du
v
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Under the martingale probabilities, the
expected change in the forward rate must be
this particular function of the volatility. This is
the arbitrage-free forward rate drift
restriction.
Hedging Treasury Bills
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HJM gives the following expression for
zero-coupon bond’s price:
B (0, T )
exp[ - X (t , T ) r (t )
B (0, t )
 X (t , T ) f (0, t ) - a (t , T )],
B (t , T ) 
where, by definition
X(t,T) = {1 – exp[–l(T – t)]}/l and a(t,T) =
(s2/4l)X(t,T)2 [1 – exp (– 2lt)] for l > 0.
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Hedging Treasury Bills (cont’d)
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At maturity date T, parameters simplify to
X(T,T) = 0
and a(T,T) = 0
yielding B(T,T) = B(0,T)/B(0,T) = 1 as required.
In zero-coupon bond’s price, only spot rate r(t) is
random. Using a Taylor series expansion
B
B
1 2B
2
DB(t , T ) 
D  Dr 
(
D
r
)
 ...
2
t
r
2 r

where partial derivatives are analogous to delta and gamma
hedging of equity options.
Delta and Gamma for a
Zero-Coupon Bond
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Evaluating partial derivatives gives
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B
 - B(t , T ) X (t , T )  0
r
which is the hedge ratio or delta for the zerocoupon bond.
2B
2

B
(
t
,
T
)
X
(
t
,
T
)
0
2
r
which is the gamma for the zero-coupon bond.
Substituting these in bond price change
B
1
DB(t , T ) 
D  ( Hedge Ratio)  Dr  (Gamma)  (Dr ) 2  ...
t
2
Example: Delta Neutral
Hedging
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A financial institution wants to hedge a one-year
zero-coupon bond using 2 other zero-coupon bonds
with maturities of ½ year and 1½ years.
Let n1 = # of ½-year zero-coupon bonds
and n2 = # of 1½-year zero-coupon bonds.
Initial cost of the self-financed hedged portfolio
V(0) = 0.954200 + n1 0.977300 + n2 0.930850 = 0.
Rewrite our first equation as
n1 0.977300 + n2 0.930850 = –0.954200.
Hedging Zero-Coupon Bonds
Example: Delta Neutral
Hedging (cont’d)
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To be delta neutral the portfolio must be
insensitive to small changes in the spot rate
r(t).
B(0,0.5)
B(0,1.5) B(0,1)
n1
 n2

0
r
r
r
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i.e.,
n1 0.476635 + n2 1.29660 = –0.908041.
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This gives the 2nd equation in two unknowns.
Solving we get
n1 = –0.4760 and n2 = –0.5254.
Example: Delta Neutral
Hedging (Observations)
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Thus, to hedge the 1-year zero-coupon bond:
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short 0.4760 of the ½-year zero-coupon bond
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and short 0.5254 of the 1½-year zero-coupon bond.
To construct a synthetic 1-year zero-coupon bond, go
long in these 2 zero-coupon bonds.
By traditional theory, duration of our hedging
portfolio must equal 1 which is the duration of the 1year zero-coupon bond being hedged.
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This would give a different 2nd equation in n1 and n2.
But our hedging portfolio has duration 1.0261. So our hedge
based on hedge ratio will NOT be a duration hedge .
Gamma Hedging
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A delta neutral portfolio is constructed to be
insensitive to small changes in the spot rate r(t).
However, it is not insensitive to large changes in r(t),
where (Dr)2 term in expression for price change
cannot be neglected. E.g.,
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If Dr = 0.01, then (Dr)2 = 0.0001 and can be ignored.
If Dr = 1, then (Dr)2 = 1 is of the same order of magnitude
and cannot be ignored.
Gamma neutral portfolio is neutral to larger shocks.
To make a portfolio both delta and gamma neutral,
introduce an extra bond and proceed as before.
Options on Treasury Bills
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A European call with a maturity of 35 days is
written on a 91-day T-bill with face value of
$100.
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By convention, the maturity of the underlying Tbill is fixed at 91 days over option’s life.
Strike price is quoted as a discount rate of
4.50%. The dollar value of the strike is
K = [1 – (4.50/100)  (91/360)]  100
= 98.8625.
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The strike discount rate and the dollar strike price
are inversely related- if one goes up, the other
goes down.
Options on Treasury Bills
(cont’d)
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European call’s payoff at expiration is
100 B(35,126) - K
c(35)  
0
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if
100 B(35,126)  K
if
100 B(35,126)  K ,
where B(35,126) is the value at date 35 of a T-bill
that has 91 days left; hence it matures at date
126.
Assume that forward rates follow the
stochastic process given in Chapter 16.
=> Forward and spot interest rates are normally
distributed under the martingale probabilities.
=> Zero-coupon bond price is lognormal.
Options on Treasury Bills
(cont’d)
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European call on a T-Bill is
c(t) = 100 B(t,Tm)N(d1) – KB(t,T)N(d2),
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where Tm = T + m,
d1 = {ln[100 B(t,Tm)/KB(t,T)] + sc2/2}/sc,
d2 = d1 – sc,
sc2 = (s2/2l){1 – exp[–2l(T – t)]}X(T,Tm)2,
X(T,Tm) = {1 – exp[–l(Tm – T)]}/l.
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As T-bill price is lognormally distributed, this formula
looks similar to the Black-Scholes model.
But there are important differences. Suppose at t = 0,
the volatility reduction factor for forward rates, l = 0.
Still, the volatility is different due to the extra maturity at
option’s expiration coming from T-bill’s remaining life.
Example: Pricing Treasury Bill
Call Options
Hedging of Options on T-Bills
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Option value depends on prices of zerocoupon bonds B(t,T) and B(t,Tm) which in
turn depends on the spot rate r(t). Using a
Taylor series expansion
c
c
1  2c
2
Dc  D  Dr 
(
D
r
)
 ...
2
t
r
2 r

where
c
Hedge ratio =
r
and
2

c
Gamma =
r 2
Swaption
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A swaption is an option on an interest rate
swap. Its holder has the right to enter into a
prespecified swap at a fixed future date.
Swaption (Receive Floating, Pay Fixed).
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The swap rate is RS on a per annum basis.
Swaption expires at date T.
The swap has n fixed and floating rate payments
at dates T1, T2, . . . , Tn and ends on the last date
Tn.
Why Swaption?
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A company that is going to issue a
floating rate bond in 3 M time.
It plans to swap its liabilities into fixed
Long a swaption to pay fixed of 7% and
receive floating for 3 years
So if the prevailing swap rate is above
7%, then ....
Swaption (cont’d)
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As swaption’s maturity date T.
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The value of the floating rate payments is
NP  [1 – B(T,Tn)].
The value of the fixed rate payments
n
N P  ( RS / 2) B(T , T j ).
j 1

The net value of the swap at maturity is
n
VS (T )  N P  [1 - B(T , Tn )] - N P  ( RS / 2) B(T , T j ).
j 1
Swaption (cont’d)

As a newly issued swap at date T has zero
value, the holder will exercise swaption if
VS(T)  0. Thus swaption payoff at maturity
date T is
VS (T )
swaption (T )  
0

Rewrite swap’s value
if
VS (T )  0
if
VS (T )  0.
 n

VS (T )  N P -  N P  ( RS / 2)  B(T , T j )  N p  B(T , Tn ).
 j 1

 Expression in bracket is the value of a Treasury
bond with coupon RS and face value NP. Let its
value be Bc.
Swaption (cont’d)

Swaption’s payoff at maturity date T is
 N P - Bc (T )
swaption (T )  
0

if
N P  Bc (T )
if
N P  Bc (T ) .
The swaption is equivalent to a put
option on a bond with strike price NP
and can be valued by the HJM model.