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Convexity Adjustment Calculation
Interest Rate Futures
Convexity is an attribute of an interest rate instrument which measures how the
instruments' duration or sensitivity to rate movements (i.e. DV01) changes as
rate levels change. Convexity is often referred to as the second derivative of
price with respect to rate level. A Convexity adjustment is a general concept
important in many Fixed-Income derivative models whereby the rate of the
instrument is adjusted to reflect the applicable convexity for that instrument. The
definition of this adjustment varies depending on the context of the derivative
instrument and there is no one standard model.
Most interest rate futures, such as Eurodollars, trade at a fixed value per basis
point and thus have zero convexity. In contrast, a forward rate agreement or a
swap containing a fixed-rate leg exhibits positive convexity similar to that of a
bullet bond. Therefore, the rates implied by futures contracts are considered
“expected rates” whereas FRA’s and swaps are consistent with executable
forwards, which encompass convexity.
The purpose of this document is to describe convexity adjustment whereby the
interest rate future is converted into an equivalent forward rate agreement (FRA).
This adjustment generates a lower rate for the futures contract, and becomes
more pronounced the longer the expiry date of the futures contract.
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Convexity Correction Formulas
From a quantitative point of view the origin of the convexity correction can be
expressed as the fact that futures and forwards are expected values of the
forward rate but under different measures. The futures rate, due to the daily
margining feature is the expected value of the forward rate under the risk neutral
measure. This differs from the forward rate and is a model dependent quantity,
that is, it depends on the dynamics of the interest rate. A common approach to
compute convexity corrections is to use so called short rate models for the
interest rate evolution. A popular choice corresponds to the use of the Hull-White
model which postulates that the short rate under the risk neutral measure follows
a normal diffusion process with volatility σ and speed of mean reversion a.
The relation between (simple compounded) forward rate 𝑓𝑤𝑑0 (𝑇1 , 𝑇2 ) from 𝑇1 to
𝑇2 and the observable (simple compounded) future rate 𝑓𝑢𝑡0 (𝑇1 , 𝑇2 ) over the
same period is:
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𝑓𝑤𝑑0 (𝑇1 , 𝑇2 ) ≈ 𝑓𝑢𝑡0 (𝑇1 , 𝑇2 ) − [1 − 𝑒𝑥𝑝(−X(𝑇1 , 𝑇2 ))] [𝑓𝑢𝑡0 (𝑇1 , 𝑇2 ) + ]
𝛿
where
𝛿 = 𝑇2 − 𝑇1
X(𝑇1 , 𝑇2 ) =
𝐵(𝑡, 𝑇) =
σ2
B(𝑇1 , 𝑇2 )(B(𝑇1 , 𝑇2 )(1 − e−2a𝑇1 ) + 𝑎B2 (0, 𝑇1 ))
2a
1 − e−a(T−t)
𝑎
Derivation of this formula is provided in the appendix.
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Convexity Adjustment Settings
Mean Reversion Speed
The mean reversion speed a represents the speed with which the short term rate
reverts to its long term mean. The typical range of values for mean reversion is
0.001 for negligible effects to 0.1, which is a relatively high mean reversion. Its
inverse 1/a is the time scale on which mean reversion takes place.
Rate Volatility
The value used as default for the short rate volatility σ is chosen as the
approximate normal (Bachelier) 10 year cap volatility obtained by multiplying the
10 year Black volatility by the corresponding swap rate.
Rate volatility = (10-yr swap rate {USSWAP10}) * (10-yr cap volatility {USCV10}) /100
For each currency, different tenors may be used for rate volatility. The choice of
which tenors is based on the unique set of market conditions (i.e. liquidity) for each
swap curve and individual tenors.
One recent market condition impacting
determination of what market data to use is the continued existence of negative
rates. As a result, IR VOL market players have been migrating to normal volatilities
for trading; therefore, EUR and CHF was recently enhanced to use normal volatility.
For other currencies (besides EUR and CHF) if the swap rate becomes negative
then the forward swap rate and Swaption volatility is used for rate volatility.
Below we list several currencies and which tenors are used:
USD: USCV10 * USSWAP10
EUR: EUNS06 BVOL
GBP: BPNS06 BVOL
JPY: JYCV5 * JYSWAP5
CHF: SFNS03 BBIR
CAD: CDCV03* CDSW3
AUD: ADCV5* ADSWAP5
SEK: SKSN055 BBIR
SGD: SDCV2* SDSW2
HKD: HDCV2* HDSW2
NZD: NDSN011 BBIR
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Bloomberg Screens
SWDF saves the user default on Rate Volatility and Mean Rev Speed.
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EDSF and EDS always assumes 0.03 as default for Mean Rev Speed, and use
cap volatility for the currency.
EDS<go> select 1
For more information, please press the <HELP> key twice on the BLOOMBERG
PROFESSIONAL® service.
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Appendix: Derivation of Convexity Correction Formulas
In the Hull-White model with mean reversion a and constant volatility σ,
𝑑𝑟(𝑡) = (𝜃(𝑡) − 𝑎 𝑟(𝑡)) 𝑑𝑡 + 𝜎 𝑑𝑊(𝑡)
where 𝜃(𝑡) is chosen to fit the interest rate term structure.
The solution to this SDE is:
t
𝑟(𝑡) = α(t) + σ ∫ e−a(t−u) dW(u)
0
where
α(t) = f M (0, t) +
σ2
(1 − e−a t )2
2a2
and 𝑓 𝑀 (0, 𝑡) is the market instantaneous forward rate at time 0 for the maturity T.
Some notation before proceeding to the derivation of convexity correction.
Denote the coverage period 𝛿 of the forward rate from 𝑇1 to 𝑇2 and the function
𝐵(𝑡, 𝑇) as:
𝛿 ≝ 𝑇2 − 𝑇1
1 − e−a(T−t)
𝑎
𝐵(𝑡, 𝑇) ≝
The time 0 value of a zero coupon bond maturity at T can be expressed as
following under the risk neutral measure Q:
𝑇
𝑃(0, 𝑇) = 𝐸 𝑄 [𝑒𝑥𝑝 (− ∫ 𝑟(𝑡)𝑑𝑡)]
0
𝑇
𝑇
t
= 𝐸 𝑄 [𝑒𝑥𝑝 (− ∫ α(t)dt − σ ∫ ∫ e−a(t−u) dWu 𝑑𝑡)]
𝑇
0
0
𝑄
0
𝑇
t
= 𝑒𝑥𝑝 (− ∫ α(t)dt) 𝐸 [𝑒𝑥𝑝 (−σ ∫ ∫ e−a(t−u) dWu 𝑑𝑡)]
0
𝑇
0 0
𝑇 T
0
𝑇
0 u
𝑇
= 𝑒𝑥𝑝 (− ∫ α(t)dt) 𝐸 𝑄 [𝑒𝑥𝑝 (−σ ∫ ∫ e−a(t−u) dt dWu )]
= 𝑒𝑥𝑝 (− ∫ α(t)dt) 𝐸 𝑄 [𝑒𝑥𝑝 (−σ ∫ B(u, T) dWu )]
0
𝑇
= 𝑒𝑥𝑝 (− ∫ α(t)dt +
0
2
𝑇
0
σ
∫ B2 (t, T) 𝑑𝑡)
2 0
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The time zero (simple compounded) forward rate is related to the bond price as:
1 + 𝛿 ∗ 𝑓𝑤𝑑0 (𝑇1 , 𝑇2 ) =
𝑃(0, 𝑇1 )
𝑃(0, 𝑇2 )
𝑇2
σ2 𝑇1 2
σ2 𝑇2 2
= 𝑒𝑥𝑝 (∫ α(t)dt + ∫ B (t, 𝑇1 ) 𝑑𝑡 − ∫ B (t, 𝑇2 ) 𝑑𝑡)
2 0
2 0
𝑇1
As described above, the time zero (simple compounded) future rate under the
risk neutral measure is given by:
= 1 + 𝛿 ∗ 𝑓𝑢𝑡0 (𝑇1 , 𝑇2 )
𝑇2
𝑄
= 𝐸 [𝑒𝑥𝑝 (∫ 𝑟(𝑡)𝑑𝑡)]
𝑇1
𝑇2
𝑇2
t
= 𝐸 𝑄 [𝑒𝑥𝑝 (∫ α(t)dt + σ ∫ ∫ e−a(t−u) dWu 𝑑𝑡)]
𝑇1
𝑇2
𝑇1
0
𝑇2
t
= 𝑒𝑥𝑝 (∫ α(t)dt) 𝐸 𝑄 [𝑒𝑥𝑝 (σ ∫ ∫ e−a(t−u) dWu 𝑑𝑡)]
𝑇1
𝑇1
𝑇2
𝑇1
0
𝑇2
𝑇2
𝑇2
= 𝑒𝑥𝑝 (∫ α(t)dt) 𝐸 𝑄 [𝑒𝑥𝑝 (σ ∫ ∫ e−a(t−u) dt 𝑑dWu + σ ∫ ∫ e−a(t−u) dt 𝑑dWu )]
𝑇1
0
𝑇2
𝑇1
𝑇1
𝑇1
u
𝑇2
= 𝑒𝑥𝑝 (∫ α(t)dt) 𝐸 𝑄 [𝑒𝑥𝑝 (σ ∫ B(u, 𝑇2 ) − B(u, 𝑇1 ) dWu + σ ∫ B(u, 𝑇2 )dWu )]
𝑇1
0
𝑇2
= 𝑒𝑥𝑝 (∫ α(t)dt +
𝑇1
𝑇1
σ2 𝑇1
σ2 𝑇2
2
∫ (B(t, 𝑇2 ) − B(t, 𝑇1 )) 𝑑𝑡 + ∫ B 2 (t, 𝑇2 ) 𝑑𝑡)
2 0
2 𝑇1
The ratio of the future rate and forward rate expressions is:
=
1 + 𝛿 ∗ 𝑓𝑢𝑡0 (𝑇1 , 𝑇2 )
1 + 𝛿 ∗ 𝑓𝑤𝑑0 (𝑇1 , 𝑇2 )
= 𝑒𝑥𝑝 (
𝑇1
𝑇2
𝑇1
𝑇2
σ2
2
(∫ (B(t, 𝑇2 ) − B(t, 𝑇1 )) 𝑑𝑡 + ∫ B 2 (t, 𝑇2 ) 𝑑𝑡 − ∫ B 2 (t, 𝑇1 ) 𝑑𝑡 + ∫ B 2 (t, 𝑇2 ) 𝑑𝑡))
2 0
𝑇1
0
0
𝑇1
𝑇2
= 𝑒𝑥𝑝 (σ2 ∫ B 2 (t, 𝑇2 ) − B(t, 𝑇1 )B(t, 𝑇2 ) 𝑑𝑡 ) 𝑒𝑥𝑝 (σ2 ∫ B 2 (t, 𝑇2 ) 𝑑𝑡)
0
𝑇1
𝑇
We approximate this by dropping the second exponential 𝑒𝑥𝑝 (σ2 ∫𝑇 2 B2 (t, 𝑇2 ) 𝑑𝑡).
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In the case of large 𝑇1 , this term would be small compare to the term in the first
𝑇
exponential 𝑒𝑥𝑝 (σ2 ∫0 1 B2 (t, 𝑇2 ) 𝑑𝑡). In the case of small 𝑇1 and 𝑇2 , the overall
convexity adjustment would be small anyway and the effect of omitting this term
would be negligible. Thus, we have the following approximation:
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=
𝑇1
1 + 𝛿 ∗ 𝑓𝑢𝑡0 (𝑇1 , 𝑇2 )
≈ 𝑒𝑥𝑝 (σ2 ∫ B 2 (t, 𝑇2 ) − B(t, 𝑇1 )B(t, 𝑇2 ) 𝑑𝑡)
1 + 𝛿 ∗ 𝑓𝑤𝑑0 (𝑇1 , 𝑇2 )
0
The expression inside this exponential function is equivalent to the following
function:
𝑇1
X(𝑇1 , 𝑇2 ) ≝ σ2 ∫ B 2 (t, 𝑇2 ) − B(u, 𝑇1 )B(u, 𝑇2 ) 𝑑𝑡
0
σ2
= B(𝑇1 , 𝑇2 )(B(𝑇1 , 𝑇2 )(1 − e−2a𝑇1 ) + 𝑎B 2 (0, 𝑇1 ))
2a
With further arrangement of above equation, we have the following relationship
between the (simple compounded) forward rate and the observable (simple
compounded) future rate:
1 + 𝛿 ∗ 𝑓𝑢𝑡0 (𝑇1 , 𝑇2 )
≈ 𝑒𝑥𝑝(X(𝑇1 , 𝑇2 ))
1 + 𝛿 ∗ 𝑓𝑤𝑑0 (𝑇1 , 𝑇2 )
1
𝑓𝑤𝑑0 (𝑇1 , 𝑇2 ) ≈ 𝑓𝑢𝑡0 (𝑇1 , 𝑇2 ) − [1 − 𝑒𝑥𝑝(−X(𝑇1 , 𝑇2 ))] [𝑓𝑢𝑡0 (𝑇1 , 𝑇2 ) + ]
𝛿