CMT Rates and Volatility Dependence of CMT Products

Caps and swaps are also written on CMT (Constant-Maturity Treasury) rates and on CMS (Constant-Maturity Swap) rates15’ ^iii. A CMT rate has a definite maturity

l 3 Models, Rigor, and Clients: These seemingly crude approximations for Prime caps

may offend the rigorously minded quant. Be aware that on the desk, empirical models are sometimes employed regardless of “theory”. There may be no alternative. What would be your theory of Prime rate dynamics? By the way, you need an indicative Prime cap price right after lunch, because that is when the salesman is going to call up the client.

l 4 Definitional Traps for Risk: Prime vega ambiguity is not the only example of why it

is important to know the details of how the risk is defined. Once you find out, it is a good idea to document things and then periodically monitor the situation in case something changes.

l 5 CMT Rates and CMT Derivatives: The CMT rates are obtained by fitting the

treasury curve, and are contained in a weekly Federal Reserve Bank “H15” report.

Interesr-Rare Caps 133 z (e.g. z = 5 for 5-yr CMT). We need the maturity- zCMT forward rate at time t . There is an extra complexity, namely a state dependence is present for the various possible values of a forward CMT rate at a given time16.

A major consequence or complication is that the CMT rates are volatility dependent. The forward CMT rate is a composite rate that depends on the diffusion probability of getting to a given value at time t

.

For example, if a short- rate model is used for the underlying dynamicsl7, this diffusion probability depends on the short-rate volatility. The diagram gives the setup:

I

I ^{Forward CMT} Rate Kinematics

r = 0 axis I I

Derivatives written on CMT rates are used by, for example. insurance companies that have products such as SPDAs that have payouts based on CMT, for example the 5-yr CMT. If this rate goes up, the company loses money. A CMT cap can protect against this risk. CMT products are also used as hedging vehicles for mortgage-related activities, since mortgage rates are correlated with treasury rates. CMT caps are illiquid and broker quotes can remain unchanged for long periods of time.

l 6 Simplified CMT Models: Sometimes the complexity of the node dependence of CMT

rates is ignored. See Hogan et a1 (ref).

Short-rate Formalism: The interested reader might consult Ch. 43 for some details.

17

-

1

The forward C M T rate (of maturityz, at time t , at node

a )

is called FiLL ( t ; Node,). The nodes arise from a discretization of rates at each time for implementing numerical algorithms.

The C M T rate FJ2T ( t ; Node,

)

and the short rate r ( t ; Node,

)

are shown.

The discount factor P('+') ( t ; Node,

)

from maturity date t

+ ^z

back to time t at node

a

is also shown along the discount factor P ^('I

1

^{( t ; Node,}

)

corresponding to an intermediate date. r,

.

C M T rates can be built up from short rates using the same short-rate model used to price other derivatives, for consistency.

This forward C M T rate is equal to the coupon that is obtained from setting the corresponding forward treasury coupon bond to par at the forward node a , namely I?Li ( t ; Node,

)

= I00

.

The bond

I?$,!

( t ; Node,

)

depends on the state at time r because the forward discount factors used to construct this bond depend on the state. There are other treasury-market related correctionsI8.

Explicitly, the discount factorslg P ^('1

1

(?;Node,) back from times t, to time t at specific node

a

are used to construct the bond" with coupon C::),

r+r

BK!

( t ;

Node, )

= Ci:;

.P@'

⁾ ( t ;

Node, ) +

100 ^.

P'"')

( t

; Node, )

⁽ 1 0.6)

At par (i.e. 100) for the bond, :C: by definition is (?;Node,), i.e.

Given a model for the discount factors, in this way we obtain the C M T rates.

l 8 Rep0 and Auction Complexities: Because CMT rates are connected with the treasury

market, some complexities of repo (including forward repo curves) and auction effects enter CMT calculations.

l 9 Discount Factors: For CMT we use the treasury discount factors, for CMS Libor we

would use Libor discount factors.

2o Kinematics: In practice for CMT and CMS, we may need to put in extra factors (e.g. a factor !h for semi-annual coupons), the rate convention (e.g. 30/360, money market), etc.

Interest-Rare Caps I35

CMS Rates

To get a CMS rate we would set the appropriate forward Libor swap to par (zero value) at each node at time t and follow the same procedure.

A CMT Swap is a Volatility Product

A CMT swap is considerably more complicated than the plain-vanilla swaps we considered in the previous chapter because of the volatility dependence of the CMT rate. We need to compute the rates F&L ( t ; Node,

)

at the different reset times of the swaplets, and for the different nodes at those reset times. The swaplet contribution at a given node is proportional to the difference of FCMT

(4

( t ; Node,

)

and the other rate in the swap. This other rate can be fixed at some value E or can be a floating rate like Libor. In the latter case, the swap is called a CMT- Libor basis swap.

Because of the volatility dependence of the CMT rate, CMT swaps have nonzero vega. Even though there is no optionality written in the contract, because CMT swaps have vega they could be put in an “option” book.

CMT Caps

We now consider CMT caps”. We also need the index 1 specifying which CMT caplet we are talking about2*, so we write F,!& ( t ; Node,)

.

We numerically determine, at the 1’” CMT caplet maturity tl* and at the node a at t/*, if (t/*;Node,)

>

E . If so, the caplet gets a contribution proportional to the difference. The contributions from all nodes at r,* are added up to get the lfh _CMT caplet value. The CMT cap value is the sum of the CMT caplets.

Note that because a CMT swap is volatility dependent, CMT caps and floors have different volatility dependencies.

Lognormal CMT Rate Dynamics Comments: The CMT (or CMS) rates can be derived as composite rates from an underlying process as we illustrate here. Alternatively, these composite rates can be stochastically modeled directly. It should be noted that it is inconsistent to model a composite rate as lognormal and at the same time model the

“elementary” rates contained in the composite rate as lognormal. This is because the any function - even a sum - of lognormal processes is at best only approximately lognormal.

Naturally, this does not stop the market from quoting the composite CMT volatilities as lognormal.

l 2 CMT Rate Notation: Ugly as the notation looks, there is actually another attribute, namely the date to at which data are used to construct the CMT forward curve.

Numerical Considerations in Calculating CMT Derivatives

The numerical calculations for C M T products are highly numerically intensive.

This is because of the dependence on the discretization nodes of the quantities needed for the pricing. Most of the time using brute-force numerical code is spent calculating the zero-coupon bonds at each node. Once we have the zero coupon bonds at a node w e immediately get the C M T rate at that node, as we saw above.

W e can speed up the calculations. The basic idea is to use a fast analytic method based on a mean-reverting Gaussian process for calculating the C M T rate, including the volatility correction, at any future time and for any given future short-term rate. Explicitly we can use the mean-reverting Gaussian short- rate model for an approximation to the discount factors P") (t;Node,)

.

The explicit expression is given in the discussion of this model in Ch. 43.

The actual values of the future short-term rates come from a separate code, for example a lognormal short-rate process codez3. Hence n o negative rates actually appearz4.

References

i Definitions

Downes, J., Goodman, J. E., Dictionary of Finance and Investment Terms. Barron's Educational Series, Inc., 1987.

I' Pit Options

Hull, J. C., Options, Futures, and other Derivative Securities. Prentice-Hall, 1993.

'I' CMT Products

...

Hogan, M., Kelly, J., Paquette, L., Constant Maturiry Swaps. Citibank working paper, 1993.

Smithson, C., ABC ofCMT. Risk Magazine Vol. 8, p. 30, Sept. 1995.

O'Neal, M., CMT-Based Derivatives. Learning Curves, Derivatives Week. Institutional Investor, Inc. 1994. See p. 117.

23 History: I got the idea of mixing analytic and numerical methods to speed up the calculations for CMT products in 1994. The speedup over the brute-force lognormal numerical code was an order of magnitude.

24 Gaussian Models and Negative Rates: The analytic Gaussian model gives a reasonably good numerical approximation for discount factors, and thus for CMT rates.

There is some negative short-rate contribution to discount factors in Gaussian models.

However, the short-rate grid in the CMT rate calculation is lognormal, and no negative rates appear in the grid. The short rate ⁼ 0 axis is in the figure to emphasize this point.

11. Interest-Rate Swaptions (Tech. Index 5/10)

We described interest-rate caps in the last chapter. A cap is a collection or basket of options (caplets), each written on an individual forward rate. A swaption, on the other hand, is one option written on a collection or a basket of forward rates, namely all the forward rates in a given forward swap’. The fact that the swap option is written on a composite object means that correlations between the individual forward rates are critical for swaptions.

Swaptions are European if there is only one exercise date, Bermudan if there are several possible exercise dates, and American if exercisable at any time. The forward swap into which the swaption exercises can be either a pay-fixed or a receive-fixed swap2.

Swaptions are usually based on Libor. Swaptions on other rates also exist.