Sketched valuation results for the new CME Deliverable Interest Rate Swap Future, including the convexity correction due to margining.

1. CME Deliverable Interest Rate Swap Future
Convexity corrections in HJM one factor model
Gary J. Kennedy
ClarusFT Consulting
October 7, 2012

2. CME Deliverable Interest Rate Swap Future
The contract specifications are defined in [CME12].
Key highlights are;
Delivers a OTC swap which is to be cleared with CME
The fixed rate of the swap is set by CME when the contract is
listed
The price is 100 points + NPV of swap (each contract has a
notional of $100,000)
Futures style margining
The margining will introduce a convexity correction which is worth
investigating; problems have arisen recently on swap futures when
the effect of margining is not well understood [CMY11, Pen11].
Results similar to those found on existing cash settled swap futures
can be established, although we need to account for the
multi-curve pricing.

3. CME Deliverable Interest Rate Swap Future
The value of the contract (on a notional of 1) on the last trading
day, θ, is
n m
P D (θ, ti ) P D (θ, τj ) P k (θ, τj−1 )
Vθ = 1 + K ai − −1
P D (θ, t0 ) P D (θ, t0 ) P k (θ, τj )
i=1 j=1
where, P D (t, T ) is the discount factor at time t for a payment at
T , t0 denotes the settlement day, {ti : 1 = 1, ..., n} are the
payment dates on the fixed leg of the swap, whilst ai denotes the
fixed rate coupon accrual lengths paying at ti , and K is the fixed
rate of the swap. {τj : 1 = 1, ..., m} are the payment dates on the
float leg of the swap, and τ0 := t0 for convenience.

4. Connection to Bond Futures
P k (t,u) k
D
P (t,u)
Using the assumptions from [Hen10], P k (t,v )
= βt (u, v ) P D (t,v ) ,
k k
and βt (u, v ) = β0 (u, v )
n m
P D (θ, ti ) P D (θ, τj ) P k (θ, τj−1 )
Vθ =1 + K ai − −1
P D (θ, t0 ) P D (θ, t0 ) P k (θ, τj )
i=1 j=1
n
P D (θ, ti) P D (θ, τm)
=K ai D (θ, t )
+ D (θ, t )
+ 1 − β k (τ0 , τ1 )
P 0 P 0
i=1
m
P D (θ, τj−1 )
+ 1 − β k (τj−1 , τj )
P D (θ, τ0 )
j=2
The first two terms are the value of a notional bond future (and
NYSE Liffe’s SwapNote product). The last two terms are small but
necessary adjustments to account for the two curve pricing of the
underlying swap.

5. Preliminaries
Lemma 1
Let 0 ≤ t ≤ u ≤ v and u ≤ w . In the HJM one factor model, the
ratio of two discount bonds is given by;
u u
P D (u, w ) P D (t, w ) 1
= D ξ(t) exp µ(s)dWs − µ2 (s)ds
P D (u, v ) P (t, v ) t 2 t
where,
u
ln ξ(t) = ln ξ(t, u, v , w ) = ν(s, v ) (ν(s, v ) − ν(s, w )) ds
t
and µ(s) = ν(s, w ) − ν(s, v )
Proof.
See [Ken10, Hen10].

6. Main Result1
Theorem 2
Let 0 ≤ t < θ ≤ t0 < ti for i = 1, ..., n and t0 = τ0 < τj for
j = 1, ..., m. In the HJM one-factor model, the price of CME
deliverable swap future is given by;
n
P D (t, ti )
1+K ai ξi (t)
P D (t, t0 )
i=1
m
P D (t, τj−1 ) P D (t, τj )
− β k (τj−1 , τj ) ξj−1 (t) − D ξj (t)
P D (t, t0 ) P (t, t0 )
j=1
θ
where, ln ξi (t) = ln ξ(t, θ, t0 , ti ) = t ν(s, t0 ) (ν(s, t0 ) − ν(s, ti )) ds
Proof.
[HK04] and Lemma 1
1
Similar result was independently established in [Hen12]

7. Bibliography I
CME, Deliverable interest rate swap future,
http://www.cmegroup.com/trading/interest-rates/
deliverable-interest-rate-swap-futures.html (2012).
R. Cont, R. Mondescu, and Y. Yu, Central clearing of interest
rate swaps: A comparison of offerings, Available at SSRN:
http://papers.ssrn.com/sol3/papers.cfm?abstract_
id=1783798 (2011).
Marc Henrard, Bonds futures and their options: More than the
cheapest-to-deliver; quality option and margining, Journal of
Fixed Income 16 (2006), no. 2, 62–75.
, The irony in derivatives discounting part ii: The
crisis, Wilmott Journal 2 (2010), no. 6, 301–316.

8. Bibliography II
, Deliverable interest rate swap futures: Pricing in
Gaussian HJM model, Available at SSRN: http://papers.
ssrn.com/sol3/papers.cfm?abstract_id=2154429
(2012).
Phil Hunt and Joanne Kennedy, Financial derivatives in theory
and practice, vol. 555, Wiley, 2004.
Gary J. Kennedy, Swap futures in HJM one-factor model,
Available at SSRN: http://ssrn.com/abstract=1648419
(2010).
Mark Pengelly, Margin minutiae at issue in Jefferies v IDCG
suit, Risk Magazine (Nov) (2011).