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16 The Brace, Gatarek and Musiela model

16 The Brace, Gatarek and Musiela model

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Part Three ﬁxed-income modeling and derivatives

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

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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Part Three ﬁxed-income modeling and derivatives

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

• 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).

625

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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Part Three ﬁxed-income modeling and derivatives

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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Part Three ﬁxed-income modeling and derivatives

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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Part Three ﬁxed-income modeling and derivatives

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

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