1. HJM Models

2. 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.

3. Spot Rate Tree

4. Pricing a European Put Option on Treasury Bills

5. 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,

6. 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 + 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).

7. 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),
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.

8. Volatility Specification
For implementation, assume that the forward rate’s volatility is of the form
s(t,T) = s  exp[– l(T – t)]
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.

9. 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)]
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.

10. 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 – D)er(t – D)D for all t, or
Using spot rate process, A(t) can be described by initial forward rates, drifts, and volatilities.

11. Normalized bond prices
Zero-coupon bond price can be similarly expressed in terms of initial forward rates, drifts, and volatilities.

12. 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 prices B(t,T)/A(t) are martingales. Alternatively stated,
B(0,T) = Ep(B(t,T)/A(t))  for all T.

13. 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
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.

14. Hedging Treasury Bills
HJM gives the following expression for zero-coupon bond’s price:
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.

15. Hedging Treasury Bills (cont’d)
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
where partial derivatives are analogous to delta and gamma hedging of equity options.

16. 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

17. 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) = n n = 0.
Rewrite our first equation as
n n = –

18. Hedging Zero-Coupon Bonds

19. 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.,
n n = –
This gives the 2nd equation in two unknowns.
Solving we get
n1 = – and n2 = –

20. Example: Delta Neutral Hedging (Observations)
Thus, to hedge the 1-year zero-coupon bond:
short of the ½-year zero-coupon bond
and short 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 So our hedge based on hedge ratio will NOT be a duration hedge .

21. 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 (Dr)2 term in expression for price change cannot be neglected. E.g.,
If Dr = 0.01, then (Dr)2 = 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.

22. 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 =
The strike discount rate and the dollar strike price are inversely related- if one goes up, the other goes down.

23. 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.

24. 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)] + 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.
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.

25. Example: Pricing Treasury Bill Call Options

26. 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
where
Hedge ratio =
and
Gamma =
are defined as before.

27. 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.

28. 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 ....

29. Swaption (cont’d) As swaption’s maturity date T.
The value of the floating rate payments is
NP  [1 – B(T,Tn)].
The value of the fixed rate payments
The net value of the swap at maturity is

30. 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.

31. 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.