Valuation results for several cash settled Interest Rate Swap future, including the convexity correction due to margining.

1. Cash Settled Interest Rate Swap Futures
Convexity corrections in HJM one factor model
Gary J. Kennedy
ClarusFT Consulting
October 18, 2012

2. CME Cash Settled Interest Rate Swap Future
The value of the contract (on a notional of 1) on the last trading
day, θ, is
[0.04/R + (1 − 0.04/R)(1 + R/2)−n ]
Vθ (R) =
P D (θ, t0 )
where,
m
j=1 β k (τj−1 , τj )P D (θ, τj−1 ) − P D (θ, τj )
Rθ = n D
i=1 ai P (θ, ti )
P D (t, T ) is the discount factor at time t for a payment at time T ,
t0 = τ0 is the settlement day, {ti : i = 1, ..., n} are the payment
dates of the fixed leg of the reference swap, whilst ai denotes the
accrual period length of the fixed coupon paying at ti . The
floating leg is characterised by the payment dates
{τj : j = 1, ..., m}, whilst β k (t, T ) captures the spread between the
forecast curve and discount curve between t and T .

3. CME Cash Settled Interest Rate Swap Future
A more transparent, although less efficient, representation1 is;
4%× 1 4%× 1 1
4%× 2 (1+4%× 1 )2
2
+ 2
2 + ... + n−1 + n
(1+ 1 R )
2 (1+ 1 R )
2 (1+ 1 R )
2
(1+ 1 R )
2
Vθ (R) =
P(θ, t0 )
This is similar to the par curve cash settlement method of a
European swaption.
1
The two representations are linked by recognising the sum of a geometric
n
series, a + ax + ax 2 + ... + ax n−1 = a 1−x .
1−x

4. ASX Cash Settled Swap Future
The future is quoted directly in terms of the swap rate R, as 1 − R,
similar to a Eurodollar future. Yet the final settlement amount and
variation margin are determined by the price of a notional bond
future. The value of the future contract on the expiry date is;
1 6.5% × (1 − (1 + R/2)−n )
Vθ (R) = + (1 + R/2)−n
P D (θ, t0 ) R
This swap rate to price function has the same structure as the the
CME future (excepting the coupon rate).

5. NYSE Liffe SwapNote Future
Liffe’s SwapNote future is also cash-settled, but based on
discounting from a swap curve2 . Specifically, the price is the value
of the fixed leg of a swap with a 6% coupon paid annually on a
30/360 basis. In this case, θ again is the last trading day, t0 is the
delivery date, and {ti : i = 1, ..., n} are the notional cashflow dates.
n
P D (θ, ti ) P D (θ, tn )
Vθ = 6% × ai +
P D (θ, t0 ) P D (θ, t0 )
i=1
This is similar to the zero curve cash settlement method in
European swaptions.
2
The contract was designed before the two curve pricing world, the swap
curve assumes the same discount and forecst curve

6. 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].

7. Preliminaries
In the separable HJM one factor model, the following integrals of
the model parameters are identifiable in the convexity corrections;
w
H(v , w ) := h(s)ds
v
t
G (t) := g (s)ds
0
u
K (u, v ) := ν(s, v )g (s)ds
0

8. Preliminaries
Lemma 2
Let 0 ≤ t ≤ u ≤ v ≤ ti for i = 1, ..., n. In the separable HJM one
factor model the ratio of two discount bonds is given by;
P(u, ti ) P(t, ti ) 1
= ξi (t) exp αi X − αi2
P(u, v ) P(t, v ) 2
where,
αi2 = H(v , ti )2 [G (u) − G (t)]
and,
ln ξi (t) = H(v , ti ) [K (u, v ) − K (t, v )]
and, X is a N-normally distributed random variable and is the
same for all ti .
Proof.
See [Ken10].

9. Main Result
Theorem 3 (Generic Futures Valuation)
Let 0 ≤ t < θ ≤ t0 < ti for i = 1, ..., n. Suppose a futures contract
has a contract value, Vθ , that depends only on a finite number of
discount factors {P(θ, ti ) : i = 0, ..., n} on the expiry date. In a
separable HJM one factor model each discount factor can be
expressed as;
P(t, ti ) 1
P(θ, ti ) = ξi (t) exp αi X − αi2
P(t, θ) 2
where, αi2 = H 2 (ti ) [G (θ) − G (t)], ξi (t) = −H(θ, ti )K (θ, θ) and X
is a normally distributed variable. The future price is given by;
∞
1 1 2
Vt = √ e − 2 x Vθ (x)dx
−∞ 2π

10. Hull-White Parameterisation
For the Hull-White model, g (t) = σe at , h(s) = e −as ,
σ(t, u) = σe −a(u−t) and ν(t, u) = σ 1 − e −a(u−t) , in which case,
a
for a = 0,
t
σ 2 2at
G (t) = σ 2 e 2as ds = e −1
0 2a
T
1 −aθ
H(θ, T ) = e −as ds = e − e −aT
θ a
θ
σ2
K (θ, T ) = e as − e −a(T −2s) ds
a 0
σ2 1 1
= e aθ − 1 − e −a(T −2θ) + e −aT
a2 2 2

11. Hull-White Parameterisation
When a ≈ 0,
1
K (θ, T ) = σ 2 θ T − θ
2
G (θ) = σ 2 θ(1 + aθ)
1
H(θ, ti ) = (ti − θ) + (θ2 − ti2 )a
2
Notice that these approximations are exact at a = 0 (the Ho-Lee
model).

12. Bibliography I
ASX, ASX 3 and 10 year interest rate swap futures: Interest
rate markets fact sheet,
http://www.asx.com.au/documents/resources/asx_3_
10_year_interest_rate_swap_futures.pdf (2011).
, A guide to the pricing conventions of ASX interest
rate products, http://www.asx.com.au/documents/
professionals/pricing.pdf (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.

13. 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).