HJM Models

Creating spot rate trees:

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 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, where f(0,T) is the date-0 continuously

compounded forward interest rate for the future interval [T,T + ],

and is a small interval ( 0). This implies

B(0,T + ) = B(0,T)e–f(0,T). 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:f(t,T) = (t,T) + (t,T)W(t),

where f(t,T) = f(t + ,T) – f(t,T) ) is the change in the forward rate over the interval t to t +

(t,T) is the drift and (t,T)is the volatility, W(t) is normally distributed with mean 0

and variance and the evolution is under martingale

probabilities .

Volatility Specification For implementation, assume that the

forward rate’s volatility is of the form(t,T) = exp[– (T – t)]

where 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(,T) – f(0,T)]

+ [f(2,T) –f(,T)] + …+ [f(t,T) - f(t – ,T)] Using forward rate evolution specification

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.

t

v

TvfTf0

),(),0(

)(),(),(),0(),(00

vWTvTvTfTtft

v

t

v

The Spot Rate and Money Market Account Evolution

Using definition of the spot rate, r(t) = f(t,t) gives spot rate process

By definition, money market account’s value A(0) =1 and A(t) = A(t – )er(t – ) for all t, or

Using spot rate process, A(t) can be described by

initial forward rates, drifts, and volatilities.

tforvWTvTvtftrt

v

t

v

)(),(),(),0()(00

t

u

ur

etA 0

)(

.1)(

Normalized bond prices Zero-coupon bond price can be

similarly expressed in terms of initial forward rates, drifts, and volatilities.

)(),(),(00),0(

)(

),( vWuvuvt

v

T

vu

t

v

T

vueTBtA

TtB

T

tu

utf

eTtB),(

1),(

Arbitrage-Free Restrictions 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 pricesB(t,T)/A(t) are martingales. Alternatively stated,

B(0,T) = E(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 0, we get

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.

T

v

duuvTvTv ),(),(),(

Hedging Treasury Bills HJM gives the following expression for

zero-coupon bond’s price:

where, by definition X(t,T) = {1 – exp[–(T – t)]}/ and a(t,T)

= (2/4)X(t,T)2 [1 – exp (– 2t)] for > 0.

)],,(),0(),(

)(),(exp[),0(),0(

),(

TtatfTtX

trTtXtBTB

TtB

Hedging Treasury Bills (cont’d)

At maturity date T, parameters simplify to X(T,T) = 0

and a(T,T) = 0yielding 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

where partial derivatives are analogous to delta and

gamma hedging of equity options.

...)(21

),( 22

2

r

rB

rrB

tB

TtB

Delta and Gamma for a Zero-Coupon Bond Evaluating partial derivatives gives

which is the hedge ratio or delta for the zero-coupon bond.

which is the gamma for the zero-coupon bond.

Substituting these in bond price change

0),(),(

TtXTtBrB

0),(),( 22

2

TtXTtBrB

...)()(21

)(),( 2 rGammarRatioHedgetB

TtB

Example: Delta Neutral Hedging

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 asn1 0.977300 + n2 0.930850 = –0.954200.

Hedging Zero-Coupon Bonds

Example: Delta Neutral Hedging (cont’d) To be delta neutral the portfolio must be

insensitive to small changes in the spot rate r(t).

i.e., n1 0.476635 + n2 1.29660 = –0.908041.

This gives the 2nd equation in two unknowns. Solving we get

n1 = –0.4760 and n2 = –0.5254.

0)1,0()5.1,0()5.0,0(

21

r

Br

Bn

rB

n

Example: Delta Neutral Hedging (Observations)

Thus, to hedge the 1-year zero-coupon bond: short 0.4760 of the ½-year zero-coupon bond 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 1-year zero-coupon bond being hedged.

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 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 (r)2 term in expression for price change cannot be neglected. E.g.,

If r = 0.01, then (r)2 = 0.0001 and can be ignored. If r = 1, then (r)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 A European call with a maturity of 35 days is

written on a 91-day T-bill with face value of $100.

By convention, the maturity of the underlying T-bill 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.

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) European call’s payoff at expiration is

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.

,)126,35(1000

)126,35(100)126,35(100)35(

KBif

KBifKBc

Options on Treasury Bills (cont’d) European call on a T-Bill is

c(t) = 100 B(t,Tm)N(d1) – KB(t,T)N(d2), where Tm = T + m, d1 = {ln[100 B(t,Tm)/KB(t,T)] + c

2/2}/c, d2 = d1 – c, c

2 = (2/2){1 – exp[–2(T – t)]}X(T,Tm)2, X(T,Tm) = {1 – exp[–(Tm – T)]}/

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, = 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 Option value depends on prices of zero-coupon

bonds B(t,T) and B(t,Tm) which in turn depends on the spot rate r(t). Using a Taylor series expansion

whereHedge ratio =

and Gamma =

are defined as before.

...)(21 2

2

2

r

rc

rrc

tc

c

rc

2

2

rc

Swaption 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).  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? 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) As swaption’s maturity date T.

The value of the floating rate payments isNP [1 – B(T,Tn)].

The value of the fixed rate payments

The net value of the swap at maturity is

n

jjSP TTBRN

1

).,()2/(

n

jjSPnPS TTBRNTTBNTV

1

).,()2/()],(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

Rewrite swap’s value

Expression in bracket is the value of a Treasury bond with coupon RS and face value NP. Let its value be Bc.

.0)(0

0)()()(

TVif

TVifTVTswaption

S

SS

.),(),()2/()(1

np

n

jjSPPS TTBNTTBRNNTV

Swaption (cont’d) Swaption’s payoff at maturity date T

is

The swaption is equivalent to a put option on a bond with strike price NP and can be valued by the HJM model.

.)(0

)()()(

TBNif

TBNifTBNTswaption

cP

cPcP