nominal value for a treasury yield”’ at any maturity. Note that of course securities d o not have their original time to maturity since they were sold so the “10 year treasury” will be plotted at somewhat less than 10 yearsl3.
Forward Curves I1
0 to = “today”, the date of the input data and also the contract date for the hypothetical ( T , T
+
A T ) deposit,T = Start of the deposit time period T
+
AT = End of the deposit time period0
General Forward Rates
A generalization’” of the forward rate specifies a future time t with to
<
t<
T , at which time we determine what interest we will get paid for a deposit starting at T and ending at T+
A T , still using the input data from to today14. This interest rate we can call f ( T ) (1; t o ; AT).
We need this generalization if we simulate the forward rate moving in time r , always corresponding to a deposit from T to T+
A T , starting from today’s forward rates determined by today’s data15. The set of such forward rates at time t for different T forms the forward-rate curve at time t.
The kinematics are illustrated below:General Forward Rate Kinematics
I
‘‘Today” toI 1
Start T1 1
End T + A T1
l 4 Forward-Rate Example: For example, take a 3-month deposit of money from
T = 3/1/04 to T
+
AT = 6/1/04.
We may want to know what rate we can expect to contract as of t = 9/1/03 for this deposit, using input data from today ,,t = 6/1/03. We will wind up with a set of the possible values of this particular rate, for example using a Monte Carlo simulator of the diffusion process for the forward rate.HJM: The most complete formulation of this idea is due to Heath, Jarrow, and Morton
15
(HJM). See the references.
The Forward Rates and
ED
FuturesW e need today’s forward rate curve
{
f ( T ) ( t o ; A T ) } for different maturities { T }.
In practice16, we can choose the type of forward rate AT = 3M.
This will allow us to use data from today’s ED futures, which are 3-month rates, for constraints. Say T is one of the IMM fixing dates t,,, . Then the forward and ED future deposit start-times coincide, so the rates must (one would think) coincide. W e get the reasonable-looking constraint:The Forward vs. Future Correction
The above equation is not quite correct. There is a difference between the ED future rate and the forward rate even when the deposit start times T =
r,,,
coincide”. This correction can be modeled, and it typically turns out to behave roughly as the square of the time difference ( T-
to) .
The correction is added t o the future’s price or subtracted from the future rate to get the forward rate. For example, the mean-reverting Gaussian rate model produces”
’2
l 6 Meaning of 3M: 3M means 3 months. With the US money-market convention that one
year has 360 days, 3M means 90 days.
Futures and Margin Account Fluctuations: Consider a buyer X of ED futures. He must have a “margin account” that must maintain sufficient funds covering changes in the value of the futures. Every day, this account is “marked to market”, meaning that funds can be withdrawn by X from the account if the future price goes up (rates down), and funds must be added by X to the account if the future price goes down (rates up). This changes the economics of buying futures. The fluctuations in the margin account thus clearly increase with increased rate fluctuations (called rate volatility). The buyer of a forward rate contract does not have a margin account.
17
’’
Theoretical Calculation of the Forward vs. Futures Correction: The correction was calculated in my 1989 path integral I1 paper (Ref). It arises from straightforward integration. The “classical path” rateiCL)
about which rate fluctuations occur is the ED future rate in the instantaneous limit AT = E ->O. For small mean reversion o, the difference between the future and forward rate is quadratic in T- t~.
The parameter 0 is the Gaussian rate volatility (diffusion constant). See Ch. 43 on path integrals.Forward Curves 79
The correction can also be backed out numerically from implementing overall consistency of the forward rate curve in the futures region with other data’’. The results are roughly consistent with the model for appropriate choices of
o,
W .Use of Swaps in Generating the Curve
In general, a swap-pricing model is used along with the determination of the forward rates (we discuss the specifics of swap pricing below). Each fixed rate E y ) f o r the corresponding model par swap with zero value S, = 0 (obtained from a putative set of forward rates) is compared with data E y ) . The forward rates are than chosen such that each E, - E d , up to some small numerical error.
(model) - (data)
Cash Rates and the Front Part of the Curve
The initial part of the forward rate curve is determined by cash. The forward rate for a deposit starting today and ending in 3M is just defined as the 3M cash rate;
this starts the forward rate curve. There are some consistency issues”.
The Break or Changeover Point
In practice, a famous problem occurs at a break point or changeover point. Below this point, the forward rates are obtained with futures (with the correction) and above this point, swaps are used 2’. The forward rates tend to be quite
Numerical Value of the Futures vs. Forward Correction: Numerically the corrections are roughly (“ball-park” as they say) c 1 bp for the front contracts, but become substantial - on the order of I0 bp at for contracts settling in 5 years. Since the bid-offer spreads of plain vanilla swaps are only a few basis points, these corrections are significant.
’O Use of Cash Rates in Generating the Curve: The cash rates other than the 3M rate can also be used as inputs for generating the forward rates, especially for short-term effects (e.g. 1 week). Generally, the cash rates do not influence the curve further out than a few months much since there is a good overall consistency in the market between cash and futures. One chronic nuisance is that the combination of a 3M cash rate with a 3M forward rate starting in 3M should in principle just be the 6M cash rate. Ask your friendly curve constructor quant if he really gets that one right.
Specification of the Changeover or Break Point Between Futures and Swaps: The choice of where this changeover of the use of the futures and the swaps depends on the algorithm. It can also depend on the traders who may change it around. Usually it is between 2 and 4 years for USD. For other currencies that have fewer futures, the changeover point is earlier.
discontinuous for T around this point. Other discontinuities can also occur.
These discontinuities can lead to substantial effects in pricing some derivatives.
Curve-Smoothing Algorithms
Smoothing algorithms can be used to smooth out discontinuities in the curve generated by the above procedure”. Philosophically it is unclear whether the curve really should be made as smooth as possible. The curve acts as if it has a stored “potential energy”. In practice, if you make the curve drop in one place it tends t o bulge up in another place, somewhat like sitting on a large balloon. The smoothing algorithms can also result in smooth but noticeably large oscillations.
These oscillations affect pricing of those securities that are not in the constraint set (i.e., pricing still occurs with all the given constraints realized).
Example of a Forward Curve Here is an illustrative picture.
Smoothed Fwd Rates
/-Forward Rates O h Smoothed Fwd Rates
I
6.60
,
I6.40 6.20 6.00 5.80 5.60 5.40 5.20
ONmLobm~~c3~~30
m L o ( D m 0c u K l c u c u c u c u m Time yrs
’? How to Be Smooth: Cubic splines are a popular choice for smoothing a curve with discontinuities generated by an algorithm. Another is to calculate independently the forward-rate curves generated by futures and by swaps. These are then added together in an overlap region with weights that are chosen as some function of T. Other smoothing choices exist, some sophisticated and proprietary, which I cannot discuss here.
Forward Curves 81
The graph is of a forward 3M Libor curve without smoothing and with a simple smoothing algorithmz3. The unsmoothed curve has regions of somewhat zigzag behavior. The smoothed curve has some
oscillation^^^.
Model Risk and Curve Construction
It should be recognized that the construction of the forward-rate curve is not well posed in a mathematical sense. This is because there are an infinite number of forward rates {f“’ ( t , , ; A T ) ) for all { T } , or at least a large number at different
{ q . ) ,
but only a relatively small set of data points for constraints. Therefore different algorithms can and d o result in different curves.Later in this book, we will thoroughly discuss model risk. The construction of the forward curve is a good example where different but perfectly sane arguments can lead to different results. The differences in turn affect pricing and hedging of derivatives differently; this is part of model risk.
Rate Units and Conventions
One of the annoying features of interest-rate products is the plethora of units or rate conventions. We need to know how to translate between these conventions.
Perhaps a system uses one set of units in calculating, or a salesman wants to quote a result in a certain convention.
Let 5 be the convention for a rate. This means there are assumed to be Ndays-, days per year (e.g. 360, 365), a frequency
4
of compounding ( I Annual, 2 Semiannual, 4 Quarterly, 12 Monthly), and the number n, of compounding periods in a year (numerically n, =4 ) .
For example, 5 = O . O S / y r with convention SA365 means4
= 2 , Nda,,_, = 365.
If time differences are measured in calendar days, the convention is called “actual”. The relation between the rate23 Illustration Only: The curve is meant to be illustrative and was produced by some software I wrote. The data are not current. The smoothing algorithm is a simple technique invented for the book. Your algorithm will no doubt be much better. See the next footnote.
24 Which Curve is Better: Smoothed or Unsmoothed? Say for argument that you have this great whiz-bang smoothing technique but the rest of the world uses unsmoothed rates (or “inferior” smoothing techniques). Then you will be off the market - which is after all determined by the rest of the world - for some OTC products that are sensitive to the regions of differences of the smoothed and unsmoothed rates. Conversely, if the world smoothes the rates and you don’t, you will also be off the market. Again, the smoothing algorithm, when used, is not unique. Have fun.
conventions is given by the requirement that, independent of convention, the same physical interest is produced in one year with Dt, yr calendar days, viz
The logic is as follows. First, assume Dt, yl’ = Ndays-l and take
(
= 1. For an initial amount $N, at the end of the year of Ndays-l days, by definition the 5 convention produces interest q$N. If the frequency4
= 2 , at the end of Ndays-l/2 days the 5 convention produces interest 5$N/2. This interest is reinvested for the remaining half of the Ndays-l days, resulting in compounding.Similarly, at the end of Ndaysl days, with
f2
= 1 ther2
convention produces interest r2$N. Hence at the end of Ndays-l the r2 convention produces interestr2 .$N . Generalizing the logic gives the result. Equivalently, we have Ndays-l
days-2
(7.4)
For example, if the r2 convention is Q360 with
(
= 4 , Ndays_* = 360, we get yz = 0.0490/ y r.
This is ten basis points less than for the 5 convention. Such an amount is definitely significant, being larger than the bid-ask spread for many swaps in the market.See the footnote for another conventionz5 called 30/360.
Compounding Rates
Compounding rates is related to rate conventions, but the emphasis is different.
Instead of coming up with two representations of the same rate, we wish to generate a composite rate from two other rates. For example, suppose that we
25 30/360 Day Count: This assumes that there are 30 days per month and 360 days per year. So the number of days between two calendar dates apart by (Nyears Nmonhs Ndays) is calculated as 360*Nyf,,
+
30*Nmonh,+
Nda.
Corrections are made as follows: If the first date is on the 3 I s , reset it to the 30”. IfYthe first date is Feb. 28 and it’s not a leap year or if the first date is Feb. 29, reset it to the “30th of February”. If the second date is on the 3 lst, and first date is on the 30” or 3 Is‘, reset the second date to the 30th.Forward Curves 83 have two neighboring 3M forward rates. We can find an equivalent 6M forward rate that generates the same interest as applying the first 3M forward rate and reinvesting the interest in compounding with the second 3M forward rate.
Suppose that r, = f ( T = t u ) (to ; 3M) is the 3M forward rate starting at time t, and ending at t, = t,
+
3M , while rm = f (T=rm) ( t o ; 3M) is the 3M forward rate starting at t, and ending at tb = t,+
3M.
The equivalent 6M forward rate --
f (T=tu) ( t o ; 6 M ) starting at t, and ending at tb = t,+
6M (usingN,, = 360 for rates, with time differences "actual") is given by
Rate Interpolation
Often we need to interpolate rates. A curve-generating algorithm may, for example, generate 3M forward rates at IMM dates. However, we need 3M rates at arbitrary times. Usually, simple linear interpolation suffices. So, if is the IMM forward rate starting at tl-l, and f , is the IMM forward rate starting at t, , then the 3M forward rate f a starting at an intermediate time tl-l