16 The Brace, Gatarek and Musiela model
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Note that here we are using the discrete compounding deﬁnition of an interest rate, consistent
with that introduced in Chapter 13, rather than our usual continuous deﬁnition, which is more
often used in the equity world.
Let’s suppose that we can write the dynamics of each forward rate, Fi , as
dFi = µi Fi dt + σi Fi dXi .
This looks like a lognormal model but, of course, the µs and σ s could be hiding more F s. So
really it’s a lot more general than that. Similarly suppose that the zero-coupon bond dynamics
are given by
i−1
dZi = rZi dt + Zi
aij dXj
(37.7)
j =1
where
Zi (t) = Z(t; Ti ).
There are several points to note about this expression. First of all, we are clearly in a risk-neutral
world with the drift of the traded asset Z being the risk-free rate. If the bond wasn’t traded
then we couldn’t say that its drift was r.3 That’s why we couldn’t say that the drift of Fi was
r, since the forward rate is not a traded instrument.
Actually, we’ll see shortly that we don’t need a model for r, it will drop out of the analysis.
Also, it looks like a lognormal model but again the as could be hiding more Zs.
Finally, the zero-coupon bond volatility is only given in terms of the volatilities of forward
rates of shorter maturities. This is because volatility after its maturity will not affect the value
of a bond, and that’s why there is no aii term in Equation (37.7).
We can write
Zi = (1 + τ Fi )Zi+1 .
Applying Itˆo’s lemma to this we get
dZi = (1 + τ Fi )dZi+1 + τ Zi+1 dFi + τ σ i Fi Zi+1
i
ai+1,j ρ ij dt,
(37.8)
j =1
where ρ ij is the correlation between dXi and dXj .
Think of Equation (37.8) as containing three types of terms: dXi ; dXj for j = 1, . . . , i − 1;
dt.
Equating coefﬁcients of dXi in Equation (37.8) we get
0 = (1 + τ Fi )ai+1,i Zi+1 + τ Zi+1 σ i Fi ,
(aii = 0 remember), that is
ai+1,i = −
σ i Fi τ
.
1 + τ Fi
3 The drift isn’t r, of course, in the real world. But we are in the risk-neutral world for pricing, and in that world all
traded instruments have growth r.
the Heath, Jarrow & Morton and Brace, Gatarek & Musiela models Chapter 37
Equating the other random terms, dXj for j = 1, . . . , i − 1, gives
aij Zi = (1 + τ Fi )Zi+1 ai+1,j ,
that is
ai+1,j = aij for j < i.
It follows that
ai+1,j = −
σj Fj τ
1 + τ Fj
for j < i.
Finally, equating the dt terms we get
i
rZi = (1 + τ Fi )rZi+1 + τ Zi+1 µi Fi + τ σ i Fi Zi+1
ai+1,j ρ ij .
j =1
From the deﬁnition of Fi the terms including r cancel, leaving
i
i
µi = −σ i
ai+1,j ρ ij = σ i
j =1
j =1
σj Fj τ ρ ij
.
1 + τ Fj
And we are done.
The stochastic differential equation for Fi can be written as
i
σj Fj τ ρ ij
σi Fi dt + σi Fi dXi .
dFi =
1 + τ Fj
(37.9)
j =1
Equation (37.9) is the discrete BGM version of Equation (37.6) for HJM.4
Assuming that we can measure or model the volatilities of the forward rates (σi ) and the
correlations between them (ρ ij ) then we have found the correct risk-neutral drift. We are on
familiar territory; Monte Carlo simulations using these volatilities, correlations and drifts can
be used to price interest rate derivatives. In practice, one estimates the volatility functions from
the market prices of caplets. Invariably, this is how one calibrates the time-dependent functions,
rather than estimating them from historical data, say.
37.17 SIMULATIONS
We can write (37.9) as
d(log(Fi )) = σi
i
j =1
σj Fj τ ρ ij
− 12 σ i2 dt + σi dXi .
1 + τ Fj
4 Although the notation is slightly different. In the HJM analysis we wrote the random component in terms of uncorrelated
dXi , here we have a different dXi for each forward rate, but they are all potentially correlated.
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In simulating this random walk we would typically divide time up into equal intervals; we
would assume, in order to integrate this expression from one interval to the next, that the F s
(and the σ s and ρs) were all piecewise constant during each time interval. Simulation then
becomes relatively straightforward.
What is not so obvious, however, is how to present value the cashﬂows. We know from
the risk-neutral concepts that there are two aspects to calculating the expectation that is the
contracts value, and they are
• simulating the risk-neutral random walk, and
• present valuing the cashﬂows.
Well, I’ve explained the ﬁrst of these, what about the second?
37.18 PVING THE CASHFLOWS
Before present valuing the cashﬂows (in anticipation of later averaging, and hence pricing) we
must be able to write them in terms of the quantities we have simulated, that is the forward
rates. That may be simple or hard. For the simpler instruments they are already deﬁned in terms
of these quantities. The more complicated contracts might have cashﬂows that are, in the sense
of our exotic option classiﬁcation, higher order. In the HJM framework we present valued these
cashﬂows using the average spot rate r up until each cashﬂow. In the BGM model we don’t
have such an r, of course.
In the BGM model we must present value using the discount factors applicable (for each
realization) from one accrual period to the next. That is, we present value each cashﬂow back
to the present through all of the dates Ti using the one-period discount factor at each period:5
1
.
1 + τ Fi (Ti )
This is the discrete version of the present valuing we do with the average spot rate in the HJM
model. Indeed, if you take the limit as τ → 0 in all of the above equations you will get back
to the HJM model; all the sums become integrals for example.
37.19 SUMMARY
The HJM and BGM approaches to modeling the whole forward rate curve in one go are very
powerful. For certain types of contract it is easy to program the Monte Carlo simulations. For
example, bond options can be priced in a straightforward manner. On the other hand, the market
has its own way of pricing most basic contracts, such as the bond option, as we discussed in
Chapter 32. It is the more complex derivatives for which a model is needed. Some of these are
suitable for HJM/BGM.
5 Notice that I didn’t use the word ‘step’ instead of ‘period’ here. ‘Step’ would refer to the small time step in the Monte
Carlo simulation; period refers to the time between Ti−1 and Ti .
the Heath, Jarrow & Morton and Brace, Gatarek & Musiela models Chapter 37
FURTHER READING
• See the original paper by Heath, Jarrow & Morton (1992) for all the technical details for
making their model rigorous.
• For further details of the ﬁnite maturity interest rate process model see Sandmann &
Sondermann (1994), Brace, Gatarek & Musiela (1997) and Jamshidian (1997).
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CHAPTER 38
ﬁxed-income
term sheets
In this Chapter. . .
•
•
the Chooser Range Note
the Index Amortizing Rate Swap
38.1
INTRODUCTION
We now take a close, detailed, look at a couple of particularly interesting ﬁxed income contracts.
You will fully appreciate the Visual Basic code after you have read about numerical methods
at the end of the book.
38.2
CHOOSER RANGE NOTE
Figure 38.1 shows the term sheet for a chooser range note. The vanilla range note has cashﬂows
linked to the number of days that the reference rate (typically a LIBOR rate) lies within a
speciﬁed band. In the Chooser Range Note (CRN), the band is not pre-speciﬁed in the contract
but is chosen by the contract holder at the start of each period. In the example of the term
sheet shown here there are four decisions to be made, one at the start of each period. And
that decision is not of the simple binary type (‘Do I exercise or not,’ ‘Do I pay the instalment
or not’) but is far more complex. At the start of each period the holder must choose a range,
represented by, say, its mid point. Thus there is a continuous and inﬁnite amount of possibilities.
38.2.1
Optimal Choice of Ranges?
Deciding on the optimal ranges is not as complicated as it seems, if approached correctly. The
contract is priced from the hedger’s perspective and the ranges are chosen so as to give the
contract the highest possible value. The hedging writer of the contract is exposed to risk-neutral
interest rates, and the forward curve; the contract holder will choose ranges depending on his
view on the direction of real rates. Since forward rates contain a component of ‘market price
of risk’ and since actual rates rarely show the same dramatic slope in rates and curvature as
shown in the forward curve, then it is unlikely that the holder of the contract will choose the
range that coincides with that giving the contract its highest value.
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Figure 38.1 Term sheet for a chooser range note.
38.2.2
Pricing
Introduce M as the mid point of the chosen range. To price this contract we just ask how does
its value vary with M.
Since the payoff depends primarily on the level of a short-term interest rate we can probably
just use a one-factor model. This contract is, unusually, one that has embedded decisions yet can
be priced simply in either a partial differential equation/ﬁnite-difference manner or by Monte
Carlo simulations. The former technique is often preferred when a contract contains decisions,
as discussed in Chapter 24. Because of this we can work in either a classical risk-neutral onefactor, Vasicek etc., world, or HJM or BGM. Whichever model we choose let’s just write the
value of a non-chooser range note starting at time t when the spot rate is r and having mid
point of the range M as V f (r, M, t).1
We will start by valuing the ﬁrst leg of the contract, the part expiring, in the example, after
just six months.
Choose M and value a non-chooser rate note with the same characteristics. Now vary M.
Typically you will end up with results that resemble those shown in Figure 38.2. This function
is V f (r0 , M, t0 ) with r0 being today’s spot rate and t0 being today’s date.
Clearly, there is an value of M, M ∗ in the ﬁgure, which gives the contract its highest value,
V ∗ . This is how much we must sell the ﬁrst leg of the contract for. If we sell it for less than
this we run the risk of the holder of the contract choosing this very M ∗ , so that we would
lose money. However, by selling for this amount we can only beneﬁt if the holder chooses a
1
The r represents either a short rate or the random factor in HJM or BGM.
ﬁxed-income term sheets Chapter 38
0.06
V*
0.05
Vf(r, M, t)
0.04
0.03
0.02
M*
0.01
0
0
0.02
0.04
0.06
0.08
0.1
M
0.12
0.14
0.16
0.18
0.2
Figure 38.2 How the ﬁrst leg of the contract varies with M.
value different from M ∗ . Remember we are valuing in the risk-neutral world because we will
be hedging, whereas the holder is more concerned with the real behavior of rates.
The second, third, and fourth legs are slightly more complicated. We were able to value the
ﬁrst leg because we know r today. To value the second etc. legs we have to imagine ourselves
at different levels of r at the start of the range period. For example, to ﬁnd the value of the
second leg now, six months before it starts, we must value a ﬁxed range note with different
starting values for r, as explained above (by introducing the mid point M). This function of r
then becomes the ﬁnal condition for a differential equation or simulation over the six months
from now to the start of the second leg.
If we denote the start of the later legs by t1 , t2 and t3 , we must ﬁnd the functions V f (r, M, ti ).
This is the value of a non-chooser range note. We then take its maximum
V (r, ti ) = max V f (r, M, ti ) .
M
We use this as the ﬁnal condition for valuing back from time
ti to the present, t0 .
38.2.3
Differences Between Optimal
for the Writer and the Buyer
This contract requires the holder to make four decisions during its life. Each of these four decisions involves choosing the
mid point of an interest rate range, a continuous spectrum of
possibilities.
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0.08
0.07
0.06
0.05
0.04
0.03
Forward Rates
Six-month LIBOR
0.02
0.01
0
0
1
2
3
4
5
6
7
8
9
10
Figure 38.3 Typical forward rate curve at the start of the contract’s life, and typical evolution of
actual short-term interest rates over its life.
0.08
0.07
0.06
0.05
0.04
0.03
Forward Rates
Six-month LIBOR
0.02
0.01
0
0
1
2
3
4
5
6
7
8
9
10
Figure 38.4 The price-maximizing ranges will depend on the risk-neutral, forward rate curve.
(Schematic only, the choice will also depend on the volatility of the curve.)
ﬁxed-income term sheets Chapter 38
0.08
0.07
0.06
0.05
0.04
0.03
Forward Rates
Six-month LIBOR
0.02
0.01
0
0
1
2
3
4
5
6
7
8
9
10
Figure 38.5 The ranges chosen by the holder are more likely to represent the best guess at the
evolution of actual rates.
In the ﬁrst ﬁgure, Figure 38.3, we see the forward rate curve as it might be at the start of
the contract’s life. The shape of this curve is more often than not upward sloping, representing
adjustment for the price of risk. One expects a higher return for holding something for a
longer term.
The ﬁgure also shows a possible evolution of short-term interest rates. It is this path which
determines, in part, the ﬁnal payoff. Notice how the path of rates does not follow the forward
curve. Obviously it is stochastic, but it does not usually exhibit the rapid growth at the short end.
The second ﬁgure. Figure 38.4, shows a plausible choice of price-maximizing ranges. These
will naturally be dependent upon the forward curve. (The ﬁgure is schematic only. The actual
ranges ‘chosen’ by the writer when maximizing the price will depend on the volatility of interest
rates as well.)
The ﬁnal ﬁgure, Figure 38.5, shows the ranges as chosen by the holder of the contract. He
makes a decision about each range at the start of each new period. Of course, his choice will be
closely related to where the short-term rate is at that time, with some allowance for his view.
Clearly there is great scope for a signiﬁcant difference between the price-maximizing choice
and the ﬁnal choices made by the holder. Our concept applies equally well to this case as
to the exercise of American options, discussed in Chapter 63. The writer of the option can
expect a windfall proﬁt which depends on the difference between the holder’s strategy and the
price-maximizing strategy.
38.3
INDEX AMORTIZING RATE SWAP
In Figure 38.6 we see a term sheet for an index amortizing rate swap as described in Chapter 32.
Here we see the mathematics and coding for such a contract.
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Figure 38.6 Term sheet for a USD index amortizing swap.
Valuing such an index amortizing rate swap is simple in the framework that we have set
up, once we realize that we need to introduce a new state variable. This new state variable
is the current level of the principal and we denote it by P . Thus the value of the swap is
V (r, P , t).
The variable P is not stochastic: It is deterministic and jumps to its new level at each resetting
(every quarter in the above example). Since P is piecewise constant, the governing differential