In this note we give pricing formulas for different instruments linked to rate futures (euro-dollar futures). We provide the future price including the convexity adjustment and the exact dates. Based on that result we price options on futures, including the mid-curve options.Interest rate futures, options on futures, HJM, one factor model.

Henrard Marc

English

Research Papers in Economics

EURODOLLAR FUTURES AND OPTIONS: CONVEXITY ADJUSTMENT IN
HJM ONE-FACTOR MODEL
MARC HENRARD
Abstract. In this note we give pricing formulas for diﬀerent instruments linked to rate futures
(euro-dollar futures). We provide the future price including the convexity adjustment and the
exact dates. Based on that result we price options on futures, including the mid-curve options.
1. Introduction
This note is dedicated to euro-futures in the HJM framework with deterministic volatility. The
futures we intend to price are the Libor-futures as traded on CME for USD and on LIFFE for
EUR. The dates related to those futures are based on the third Wednesday of the month
1, which is
the start date of the Libor rate underlying the future. We denote that rate by L. This rate is ﬁxed
at a spot lag prior to that date. In EUR and USD this lag is two business days and the ﬁxing take
place on the Monday. The maturity date of the Libor rate is three month2 after the start date.
On the ﬁxing date at the moment of the publication of the Libor rates the future price is 1−L.
Before that moment, the price evolves with demand and oﬀer. Every day the closing price is used
for margining. The margining process consists in receiving the diﬀerence in price between the
closing price of the day and the closing price of the previous day (or the transaction price on the
trade date). The price are multiply by the nominal and divided by four. The one fourth represent
the three month period as part of the year, as the Libor rates are quoted on annual basis.
The ﬁrst task of this note is to compute the fair futures price (in the arbitrage free sense) from
this description. This question is certainly not new and answers are abundant also. We refer to [4,
Section 12.4] and [6, Section 11.5] for the theoretical framework, [5] for a formula in the extended
Vasicek model and [7] for a more sophisticated approach including the volatility smile.
The speciﬁcity of this note is to provide explicit formula for the general HJM framework with
deterministic volatility (than can be found in [7] and [6] under the name of Gaussian HJM) and to
include the exact futures dates. All the formulas in the mentioned references do not diﬀerentiate
between the ﬁxing date and the start date.
The second task will be to price the options on futures as traded on CME. Those options are not
subject to margining them-self. The premium is paid up-front. If the option is exercised, one enter
into a future trade at a price equal to the strike (and no cash is exchanged). The value resulting
from the exercise comes through the margining of the futures.
The exercise date is set before or on the ﬁxing date of the future. Only the standard quarterly
futures (March, June, September and December) are used as underlying of the options. The most
popular options, called quarterly options, have their exercise date set on the ﬁxing date. Another
type of options, called serial options, have their expiry one or two months before the ﬁxing date.
The last type of options, called mid-curve options, have their expiry one or two year before the
ﬁxing date.
Those characteristics create a technically quite complex product. Even if we don’t take into
account the underlying future convexity adjustment, there is a (second) convexity adjustment. the
Date: First version: March 24, 2004; this version: March 6, 2005.
Key words and phrases. Interest rate futures, options on futures, HJM, one factor model.
JEL classiﬁcation: G13, E43.
AMS mathematics subject classiﬁcation: 91B28, 91B24, 91B70.
1All dates mentioned in this note are adjusted in some way in the case they fall on a non-good business day.
2Futures exists also on the one month Libor, but the most popular are on three month.
12 M. HENRARD
option is on a forward rate and not on a deposit. The pay-oﬀ is not paid at the deposit maturity,
not even at its start or ﬁxing but at the option expiry that can be several years before.
For those options, we also provide explicit formulas in the HJM framework. We don’t know
any reference with similar formulas. All the exact dates of the options are also used in this case
(expiry date, ﬁxing date, start date and end date). In [6] a general theoretical formula is provided
when the option exercise date is equal to the future ﬁxing date and is equal to the start of the
underlying deposit. We specialize all the results for the extended Vasicek or Hull-White model.
2. Model and hypothesis
We model interest rate products. The base assets are P(t,u), the price in t of the zero-coupon
bond paying 1 in u. We describe them for all 0 ≤ t,u ≤ T, where T is some ﬁxed constant.
We work in a Heath-Jarrow-Morton [2] one factor model framework (see for example the chapter
Dynamical term structure model in [4]). By this we mean we have a model with the following
properties. The function P is positive and regular enough so that it can be written as
P(t,u) = exp

−
Z u
t
f(t,s)ds

.
Let A = {(s,u) ∈ R2 : u ∈ [0,T] and s ∈ [0,u]}. We work in a ﬁltered probability space
(Ω,{Ft},F,Preal). The ﬁltration Ft is the (augmented) ﬁltration of a one-dimensional standard
Brownian motion (Wreal)0≤t≤T.
H: There exists σ : [0,T]2 → R+ measurable and bounded3 with σ = 0 on [0,T]2\A such that
for some process (rs)0≤t≤T, Nt = exp(
R t
0 rsds) forms with some measure N a numeraire
pair4 (with Brownian motion Wt),
df(t,u) = σ(t,u)
Z u
t
σ(t,s)ds dt − σ(t,u)dWt
dPN(t,u) = PN(t,u)
Z u
t
σ(t,s)ds dWt
and rt = f(t,t).
The notation PN(t,s) designates the numeraire rebased value of P, i.e. PN(t,s) = N
−1
t P(t,s). To
simplify the writing in the rest of the paper, we will use the notation
ν(t,u) =
Z u
t
σ(t,s)ds.
Note that ν is increasing in u, measurable and bounded. Moreover for t > u, ν(t,u) = 0.
In the case of the extended Vasicek model, the volatility function is given by ν(s,t) = σ
a(1 −
exp(−a(t − s))). We will analyse this model for σ constant and time-dependent. In the time-
dependent version, the volatility parameter is piecewise constant with σ(s) = σi for s ∈ [si−1,si)
where 0 = s0 < s1 < ··· < si < ··· < T.
3. Preliminary results
We now state two technical lemmas, the proof of which can be found in [3]. Similar formulas
can be found in [1, (3.3),(3.4)] in the framework of coherent interest-rate models.
Lemma 1. Let 0 ≤ t ≤ u ≤ v. In a HJM one factor model, the price of the zero coupon bond can
be written has,
P(u,v) =
P(t,v)
P(t,u)
exp
Z u
t
(ν(s,v) − ν(s,u))dWs −
1
2
Z u
t
 
ν2(s,v) − ν2(s,u)

ds

.
3Bounded is too strong for the proof we use, some L1 and L2 conditions are enough, but as all the examples we
present are bounded, we use this condition for simplicity.
4See [4] for the deﬁnition of a numeraire pair. Note that here we require that the bonds of all maturities are
martingales for the numeraire pair (N,N).FUTURES & CIE. 3
Lemma 2. Let 0 ≤ u ≤ v. In the HJM one factor model, we have
NuN−1
v = exp

−
Z v
u
rsds

= P(u,v)exp
Z v
u
ν(s,v)dWs −
1
2
Z v
u
ν2(s,v)ds

.
4. Futures
The future ﬁxing date is denoted t0. The ﬁxing is on the rate between t1 and t2. The ﬁxing
rate is denoted Lt0. If the accrual factor for the period t1–t2 is δ, the ﬁxing is linked to the yield
curve by
1 + δLt =
P(t,t1)
P(t,t2)
.
The futures price is Φt. On the ﬁxing date, the relation between the price and the rate is
Φt0 = 1 − Lt0.
The futures margining is done on the futures price (multiply by the notional and divided by 4).
Theorem 1. Let 0 ≤ t ≤ t0 ≤ t1 ≤ t2. In the HJM one-factor model, the price of the futures
ﬁxing on t0 for the period t1–t2 with accrual factor δ is given by
Φt = 1 −
1
δ

P(t,t1)
P(t,t2)
γ(t) − 1

where
γ(t) = exp
Z t0
t
ν(s,t2)(ν(s,t2) − ν(s,t1))ds

.
Proof. Using the generic pricing future price process theorem [4, Theorem 12.6],
Φt = EN [1 − Lt0|Ft].
In Lt0, the only non constant part is the ratio of discount factors. Using Lemma 1 twice, we obtain
P(t0,t1)
P(t0,t2)
=
P(t,t1)
P(t,t2)
exp

−
1
2
Z t0
t
ν2(s,t1) − ν2(s,t2)ds +
Z t0
t
ν(s,t1) − ν(s,t2)dWs

.
Only the second integral contains a stochastic part. This integral is normally distributed of variance R t0
t (ν(s,t1) − ν(s,t2))2ds. So the expected value of the ratio of discount factors is reduced to
P(t,t1)
P(t,t2)
exp

−
1
2
Z t0
t
ν2(s,t1) − ν2(s,t2)ds +
Z t0
t
(ν(s,t1) − ν(s,t2))2ds

and we have the announced result. 
In the case of the extended Vasicek model, the adjustment factor can be written explicitly
lnγ =
σ2
2a3(exp(−at1) − exp(−at2))(exp(at0) − exp(at))(2 − exp(−a(t2 − t0)) − exp(−a(t2 − t))).
When the start date of the underlying rate is equal to the ﬁxing date (t0 = t1) like in the Sterling
market, this last formula is equivalent to the one of Kirikos and Novak [5].
In the time-dependent case, if we take t = 0 and sn = t0, we obtain
lnγ =
1
2a3(exp(−at1)−exp(−at2))
n X
i=1
σ2
i (exp(asi)−exp(asi−1))(2−exp(−a(t2−si))−exp(−a(t2−si−1))).
The impact on diﬀerentiating between the ﬁxing date and the start date is the integral in the
gamma between t0 and t1. For normal volatility level this is below 0.01 basis points. The date
impact is obviously larger for the options treated in the next section.4 M. HENRARD
5. Options on futures
We use the notation X for the strike rate, K = 1 − X for the strike price.
Theorem 2. Let 0 ≤ θ ≤ t0 ≤ t2 ≤ t3, X,
σ2
1 =
Z θ
0
(ν(s,t2) − ν(s,t1))
2 ds, σ12 =
Z θ
0
ν(s,θ)(ν(s,t2) − ν(s,t1))ds, σ2
2 =
Z θ
0
ν2(s,θ)ds
and Σ be the matrix deﬁned by
Σ =

σ2
1 σ12
σ12 σ2
2

.
In the a HJM one-factor model with deterministic volatility, if the matrix Σ is invertible, the price
of the call option with expiry t0 and strike rate X on the future with ﬁxing and payment date t0
on the rate between t2 and t3 is given in 0 by
C0 = P(0,θ)

(X + 1/δ)N

−κ +
σ12
σ1

−
γ(θ)
δ
P(0,t1)
δP(0,t2)
exp(α − σ12)N

−κ − σ1 −
σ12
σ1

where κ is deﬁned by
X + 1/δ =
P(0,t1)
δP(0,t2)
exp(−σ1κ −
1
2
σ2
1 + α),
and
α =
Z θ
0
ν(s,t2)(ν(s,t2) − ν(s,t1))ds
The price of the put is given by
P0 = P(0,θ)

γ(θ)
δ
P(0,t1)
P(0,t2)
exp(α − σ12)N

κ + σ1 −
σ12
σ1

− (X + 1/δ)N

κ −
σ12
σ1

.
Proof. Using the generic pricing theorem [4, Theorem 7.33-7.34] we have
C0 = N0 EN
 
(Φθ − K)+N
−1
θ

= EN
 
X −
γ(θ)
δ
P(t0,t1)
P(t0,t2)
+
1
δ
+
N
−1
θ
!
.
This expected value can be computed explicitly using standard decomposition and computation
of normal distribution. Here those computation are a little bit more involved and required some
extra notations. Let
X1 =
Z θ
0
ν(s,t2) − ν(s,t1)dWs, X2 =
Z θ
0
ν(s,θ)dWs.
The random variables X1 and X2 are jointly normally distributed ([8, Theorem 3.1, p. 60]) with
covariance matrix Σ. Using Lemma 1 and Lemma 2, we have
P(θ,t1)
P(θ,t2)
=
P(0,t1)
P(0,t2)
exp(−X1 −
1
2
σ2
1 + α)
and
N
−1
θ = P(0,θ)exp(X2 −
1
2
σ2
2).
In the expectation, the parenthesis is positive when
X + 1/δ >
P(0,t2)
δP(0,t3)
exp(−X1 −
1
2
σ2
1 + α)
or when X1 > σ1κ. With these results, the expected value becomes
(1)
A
Z
x1>σ1κ
P(0,θ)

X + 1/δ −
γ
δ
P(0,t1)
P(0,t2)
exp(−x1 −
1
2
σ2
1 + α)

exp(x2 −
1
2
σ2
2)exp(−
1
2
xTΣ−1x)dxFUTURES & CIE. 5
where A = 1
2π
√
|Σ|. Like in the proof of [3, Theorem 8], we have
1
√
2π
Z
R
exp

x2 −
1
2
xTΣ−1x

dx2 =
p
|Σ|
σ1
exp

−
1
2σ2
1
 
x2
1 − 2x1σ12 − |Σ|


.
Note that x2
1 −2x1σ12 −|Σ|+σ2
1σ2
2 can be written as (x1 −σ12)2 and x2
1 −2x1σ12 −|Σ|+2x1σ2
1 +
σ4
1 + σ2
1σ2
2 can be written as (x1 + σ2
1 − σ12)2 + 2σ12σ2
1. So the double integral in (1) is equal to
P(0,θ)

(X + 1/δ)
1
√
2π
Z
x1>σ1κ
exp(−
1
2σ2
1
(x1 − σ12)2)
1
σ1
dx1
−
γ
δ
P(0,t1)
P(0,t1)
exp(α − σ12)
1
√
2π
Z
x1>σ1κ
exp(−
1
2σ2
1
(x1 + σ2
1 − σ12)2)
1
σ1
dx1

Writing the integrals as normal cumulative distributions gives the result. 
For option on futures, it is very important to diﬀerentiate between the option expiry date and
the ﬁxing of the future, specialy for mid-curve options.
In the extended Vasicek model, the coeﬃcients can be written explicitly. When the volatility is
constant the results are
σ1
2 =
σ2
2a3 (exp(−at1) − exp(−at2))
2 (exp(2aθ) − 1),
σ2
2 =
σ2
2a3 (2aθ − 3 + 4exp(−aθ) − exp(−2aθ)),
σ12 =
σ2
2a3 (exp(−at2) − exp(−at1))(exp(−aθ) + exp(aθ) − 2) and
α =
σ2
2a3 (exp(−at1) − exp(−at2))(2exp(aθ) − exp(−a(t2 − 2θ)) − 2 + exp(−at2)).
When the volatility is piecewise constant the formulas are a little bit more involved and include a
sum but are similar. We write the formulas with 0 = s0 and θ = sn.
σ1
2 =
1
2a3 (exp(−at1) − exp(−at2))
2
n X
i=1
σ2
i (exp(2asi) − exp(2asi−1)),
σ2
2 =
1
2a3
n X
i=1
σ2
i

2a(si − si−1) − (exp(−a(θ − si)) − exp(−a(θ − si−1)))
(4 − exp(−a(θ − si)) − exp(−a(θ − si−1)))

,
σ12 =
1
2a3 (exp(−at1) − exp(−at2))
n X
i=1
σ2
i (exp(asi) − exp(asi−1))(2 − exp(−a(θ − si)) − exp(−a(θ − 2si−1))) and
α =
1
2a3(exp(−at1) − exp(−at2))
n X
i=1
σ2
i (exp(asi) − exp(asi−1))(2 − exp(−a(t2 − si)) − exp(−a(t2 − 2si−1))).
6. Conclusion
We review the pricing of euro-dollar futures in the HJM framework with deterministic volatility.
Even if this note is not the ﬁrst one to deal with this convexity adjustment problem, it improves
existing results by taking care of all the relevant dates, including the spot-lag between the ﬁxing and
the underlying start date. The formula obtained is specialized for the case of the time-dependent
extended Vasicek model with piecewise constant volatility parameter.6 M. HENRARD
The note also provide explicit formulas for exchange traded options on futures. In a similar
way all the relevant date are taken into account, including the expiry, ﬁxing and start date. The
explicit results are also specialized for the extended Vasicek model. To our knowledge this is the
ﬁrst note that gives and explicit formula for those products. The resulting formula is somehow
involved in its formulation. this is due to the margining characteristic of the underlying future
that requires an convexity adjustment factor. On top of it, the option is on the rate, not on the
deposit which creates a second and diﬀerent convexity adjustment.
Disclaimer: The views expressed here are those of the author and not necessarily those of the
Bank for International Settlements.
References
[1] D. C. Brody and L. P. Hughston. Chaos and coherence: a new framework for interest-rate modelling. Proc. R.
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[4] P. J. Hunt and J. E. Kennedy. Financial Derivatives in Theory and Practice. Wiley series in probability and
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[5] G. Kirikos and D. Novak. Convexity conundrums. Risk, pages 60–61, March 1997. 1, 3
[6] M. Musiela and M. Rutkowski. Martingale Methods in Financial Modelling, volume 36. Springer, second edition,
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report, Bank of America, 2004. 1
[8] R. Rebonato. Modern pricing of interest-rate derivatives: the LIBOR market model and Beyond. Princeton
University Press, Princeton and Oxford, 2002. 4
Contents
1. Introduction 1
2. Model and hypothesis 2
3. Preliminary results 2
4. Futures 3
5. Options on futures 4
6. Conclusion 5
References 6
Derivatives Group, Banking Department, Bank for International Settlements, CH-4002 Basel (Switzer-
land)
E-mail address: Marc.Henrard@bis.org

A semi-explicit approach to canary swaptions in HJM one-factor model. Applied Mathematical Finance, To appear

Bond pricing and the term structure of interest rates: a new methodology for contingent claims valuation.

Chaos and coherence: a new framework for interest-rate modelling.

Convexity conundrums.

Eurodollar futures convexity adjustments in stochastic volatility models.

Financial Derivatives in Theory and Practice. Wiley series in probability and statistics.

Martingale Methods in Financial Modelling,

Modern pricing of interest-rate derivatives: the LIBOR market model and Beyond.

Eurodollar futures and options: convexity adjustment in HJM one- factor model

http://128.118.178.162/eps/fin/papers/0503/0503005.pdf

Eurodollar futures and options: convexity adjustment in HJM one- factor model

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