QuantMinds 2018 International Conference, Lisbon, Portugal. Swaptions, equity and bonds in HJM models.

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Swaptions, Bonds and Equities in HJM Models.
V.M. Belyaev
Allianz Life, Minneapolis, USA
QuantMinds, Lisbon, Portugal
May 15, 2018
V.M. Belyaev Swaptions, Bonds and Equities in HJM Models. May 15, 2018 1 / 60

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Introduction
Previous results were presented at Barcelona Global Derivative
Conference 2017.
It was demonstrated that the approximation works well and can
be used in calibration of all ATM swaptions.

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Introduction
Conference 2017.
New results:

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Introduction
Conference 2017.
New results:
New Calibration Procedure

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Introduction
Conference 2017.
New results:
Accuracy and Stability

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Introduction
Conference 2017.
New results:
Implied Volatility Smile

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Introduction
Conference 2017.
New results:
Eurodollar Futures

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Introduction
Conference 2017.
New results:
Eurodollar Futures
Equity and Bond Indices in HJM Model

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HJM Model
Heath, Jarrow, and Morton[2] model is deﬁned in terms of forward
rates f (t, τ):
B(t, T) = e−
∫ T
t f (t,τ)dτ
(1)
where B(t, T) is a zero coupon bond value.
Model dynamics has the following form:
df (t, T) = α(t, T)dt + σ(t, T)dW (t); (2)
where σ(t, T) is a deterministic forward rate volatility; dW (t) is a
Brownian motion; and
α(t, T) = σ(t, T)
∫ T
t
σ(t, τ)dτ; (3)
is a drift. This drift can not be arbitrary chosen but it depends on
volatility to satisfy arbitrage free conditions for bonds. This drift can
be calculated exactly in ﬁnite time step models.

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Swaps
Payer Swap Contract value at time t = 0 can be presented in the
following form
val(t = 0) = B(0, T) − B(0, TN) −
rX
ν
N∑
n=1
B(0, Tn); (4)
where rX - swap contract rate; ν - payment frequency; Tn - times of
N-payments; T - start time contract;
B(t, T) = e−
∫ T
t f (t,τ)dτ
; (5)
is a Z-bond price at initial time t, f (t, τ) is a risk free forward rate.
The initial swap value is equal to zero if
rX = rS =
B(0, T) − B(0, TN)
1
ν
∑N
n=1 B(0, Tn)
; (6)
where rS is a swap rate.

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Swaption Prices
Distribution of discounted swap contract values at time T is
PV (val(T)) = (7)
= e−
∫ T
0 r(t)dt
(
B(T, T) − B(T, TN) −
rX
ν
N∑
n=1
B(T, Tn)
)
.
Where discounted bond values distribution in HJM model has the
following form:
e−
∫ t
0 r(τ)dτ
B(t, T) = (8)
= B(0, T)e−
∫ t
0 dτ
∫ T
τ α(τ,t1)dt1−
∫ t
0 dW (τ)
∫ T
τ σ(τ,t1)dt1
;

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Swaption Prices
So, the discounted swap value distribution at time T has the
following form:
e−
∫ T
0 r(τ)dτ
(
B(T, T) − B(T, TN) −
rX
ν
N∑
n=1
B(T, Tn)
)
=
= B(0, T)e−
∫ T
0 dτ
∫ T
τ α(τ,t)dt+
∫ T
0 dW (τ)
∫ T
τ σ(τ,t)dt
−
−B(0, TN)e−
∫ T
0 dτ
∫ TN
τ α(τ,t)dt+
∫ T
0 dW (τ)
∫ TN
τ σ(τ,t)dt
−
−
rX
ν
N∑
n=1
B(0, Tn)e−
∫ T
0 dτ
∫ Tn
τ α(τ,t)dt+
∫ T
0 dW (τ)
∫ Tn
τ σ(τ,t)dt
; (9)
Martingale condition is satisﬁed
B(0, T) =
⟨
e−
∫ t
0 r(τ)dτ
B(t, T)
⟩
.

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Approximation
In the limit of small volatility σ → 0 the leading term of distribution
(9) is
rS − rX
ν
N∑
n=1
B(0, Tn) +
+B(0, T)
∫ T
0
dW (τ)
∫ T
τ
σ(τ, t)dt −
−
rX
ν
N∑
n=1
B(0, Tn)
∫ T
0
dW (τ)
∫ Tn
τ
σ(τ, t)dt −
−B(0, TN)
∫ T
0
dW (τ)
∫ TN
τ
σ(τ, t)dt. (10)

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Approximation
Sum of normal distributed values is a normal distribution, so
discounted swap contract values at time T is:
d =
rS − rX
ν
N∑
n=1
B(0, Tn) + Σ(T, N)ξ
√
T; (11)
where ξ is a normal distributed stochastic variable
(< ξ >= 0; < ξ2
>= 1) and
Σ2
(T, N) =
1
T
∫ T
0
v2
(t, N)dt;
v(t, N) = B(0, T)
∫ T
t
σ(t, τ)dτ − B(0, TN)
∫ TN
t
σ(t, τ)dτ −
−
rX
ν
N∑
n=1
B(0, Tn)
∫ Tn
t
σ(t, τ)dτ. (12)

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Approximation
So, proxy price of ATM swaption with time to expiration T can be
presented in the following form:
∫
[d0]+e− 1
2
ξ2 dξ
√
2π
= Σ(T, N)
√
T
2π
. (13)

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Approximation
Swap values can be improved by using corrected day count for ﬂoat
legs. For USD swap ﬂoat leg is calculated as Real number of days
360
. It can
be approximated by factor A = 365.25/360:
v(t, N) =
= A
(
B(0, T)
∫ T
t
σ(t, τ)dτ − B(0, TN)
∫ TN
t
σ(t, τ)dτ
)
−
−
rX
ν
N∑
n=1
B(0, Tn)
∫ Tn
t
σ(t, τ)dτ. (14)

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Approximation
For OIS discounting we can assume that Libor-OIS spread
s(t) = rLIBOR(t) − rOIS (t) (15)
is a deterministic function as it was done in [4] and [5].
Then the discounted swap values distribution at expiration time T is:
PV (val(T)) =
= e−
∫ T
0 r(t)dt
(
A
N∑
n=1
(
B(T, Tn−1)S(Tn−1)
S(Tn)
− B(T, Tn)
)
−
−
rX
ν
N∑
n=1
B(T, Tn)
)
; (16)
where S(t) = e−
∫ t
0 s(τ)dτ
.

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Approximation
Swaption forward volatility:
v(t, N) =
(
A +
rATM
ν
) N∑
n=1
B(0, Tn)
∫ Tn
t
σ(t, τ)dτ −
−A
N∑
n=1
B(0, Tn−1)
S(Tn−1)
S(Tn)
∫ Tn−1
t
σ(t, τ)dτ. (17)

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Non-Parametric Fits
Input Data:
USD OIS and Swap Rates, ATM Swaption Volatilities.
March 12, 2018, 3:55PM
Scenario Generation Procedure:
20,000 Scenarios, Time Step = 0.5 year, Horizon Time = 60 years

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Non-Parametric Fits
Input Data:
March 12, 2018, 3:55PM
Fitting procedures:

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Non-Parametric Fits
Input Data:
March 12, 2018, 3:55PM
Fitting procedures:
Set non-zero forward volatility for available expiration dates only.

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Non-Parametric Fits
Input Data:
March 12, 2018, 3:55PM
Fitting procedures:
Spline interpolation of Normal Volatilities

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Non-Parametric Fits
Input Data:
March 12, 2018, 3:55PM
Fitting procedures:
Spline interpolation of Normal Volatilities
Calibration Procedure

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Non-Parametric Fits
Setting not-zero forward volatility only for available expiration dates
is not realistic assumption but it leads to a good calibrated model
Figure: Tenor 1, No Interpolation

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Non-Parametric Fits
Forward bond volatilities:
Figure: No interpolation

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Non-Parametric Fits
Using spline interpolation is more realistic but it can not to reproduce
all swaption correctly
Figure: Tenor 1, Spline Interpolation

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Non-Parametric Fits
Figure: Spline interpolation

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Forward swaption volatility has the following form
v(t, N) =
(
A +
rATM
ν
) N∑
n=1
B(0, Tn)
∫ Tn
t
σ(t, τ)dτ −
−A
N∑
n=1
B(0, Tn−1)
S(Tn−1)
S(Tn)
∫ Tn−1
t
σ(t, τ)dτ. (18)
In case of the ﬁrst swaption tenor 0.5 (time step = 0.5) we have
t = 0 and N = 1:
v(0, 1) =
(
A +
rATM
ν
)
B(0, 2dt)(σ(0, 0) + σ(0, dt))dt −
−A × B(0, dt)
S(dt)
S(2dt)
σ(0, 0)dt. (19)

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0 1 2 3
0
1
2
3
E
Te + Tenor
T
Te

f f f f f f
v v v v v
f f f f
v v v
Σ(1, 1)

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0 1 2 3
0
1
2
3
E
Te + Tenor
T
Te

f f f f f f
v v v v v
f f f f
v v v
Σ(1, 1)
σ(0, 0)

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0 1 2 3
0
1
2
3
E
Te + Tenor
T
Te

f f f f f f
v v v v v
f f f f
v v v
Σ(1, 1)
σ(0, 0) σ(0, 0)

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0 1 2 3
0
1
2
3
E
Te + Tenor
T
Te

f f f f f f
v v v v v
f f f f
v v v
Σ(1, 1)
σ(0, 0) σ(0, 0)
Σ(1, 2)

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0 1 2 3
0
1
2
3
E
Te + Tenor
T
Te

f f f f f f
v v v v v
f f f f
v v v
Σ(1, 1)
σ(0, 0) σ(0, 0)
Σ(1, 2)
σ(0, 2)

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0 1 2 3
0
1
2
3
E
Te + Tenor
T
Te

f f f f f f
v v v v v
f f f f
v v v
Σ(1, 1)
σ(0, 0) σ(0, 0)
Σ(1, 2)
σ(0, 2) σ(0, 3) σ(0, 4) σ(0, 5)
Σ(1, 3) Σ(1, 4) Σ(1, 5)

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0 1 2 3
0
1
2
3
E
Te + Tenor
T
Te

f f f f f f
v v v v v
f f f f
v v v
Σ(3, 3)
σ(0, 0) σ(0, 0) σ(0, 2) σ(0, 3) σ(0, 4)

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0 1 2 3
0
1
2
3
E
Te + Tenor
T
Te

f f f f f f
v v v v v
f f f f
v v v
Σ(3, 3)
σ(0, 0) σ(0, 0) σ(0, 2) σ(0, 3) σ(0, 4)
σ(1, 1)

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0 1 2 3
0
1
2
3
E
Te + Tenor
T
Te

f f f f f f
v v v v v
f f f f
v v v
Σ(3, 3)
σ(0, 0) σ(0, 0) σ(0, 2) σ(0, 3) σ(0, 4)
σ(1, 1) σ(1, 1) σ(1, 1)

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0 1 2 3
0
1
2
3
E
Te + Tenor
T
Te

f f f f f f
v v v v v
f f f f
v v v
Σ(3, 3)
σ(0, 0) σ(0, 0) σ(0, 2) σ(0, 3) σ(0, 4)
σ(1, 1) σ(1, 1) σ(1, 1)
σ(1, 1) σ(1, 1)

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0 1 2 3
0
1
2
3
E
Te + Tenor
T
Te

f f f f f f
v v v v v
f f f f
v v v
Σ(3, 3)
σ(0, 0) σ(0, 0) σ(0, 2) σ(0, 3) σ(0, 4)
σ(1, 1) σ(1, 1) σ(1, 1)
σ(1, 1) σ(1, 1)
Σ(3, 4)

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0 1 2 3
0
1
2
3
E
Te + Tenor
T
Te

f f f f f f
v v v v v
f f f f
v v v
Σ(3, 3)
σ(0, 0) σ(0, 0) σ(0, 2) σ(0, 3) σ(0, 4)
σ(1, 1) σ(1, 1) σ(1, 1)
σ(1, 1) σ(1, 1)
Σ(3, 4)
σ(1, 4)
σ(1, 4)

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The calibration procedure leads to more realistic bond forward
volatility surface and it is able to reproduce all ATM swaption prices
Figure: Tenor 1

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Figure: Tenor 30

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Figure: Alternative Calibration

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Forward bond volatilities versus Time to Expirations and Tenors:
Figure: Alternative Calibration

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Proxy Stability and Accuracy
Ratio of historical normal implied volatilities to March 2018
volatilities are shown below:
Figure: Ratio of Swaption Normal Implied Volatilities to March 2018
Volatilities

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Figure: October 10, 2008 Tenor 1 Black-Scholes ATM Swaption
Volatilities.

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Figure: October 10, 2008 Tenor 20 Black-Scholes ATM Swaption
Volatilities.

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Figure: October 10, 2008 Forward Bond Volatilities.

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Volatility Smile
We can try to use calibrated HJM model to calculate OTM
swaptions.
Figure: 6Mx6M Swaption Implied Volatility.

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Volatility Smile
To be able to reproduce swaptions prices with shortest times to
expirations (in our case 0.5 year) we need to change normal
distribution. Here we consider the following combination of two
normal distribution:
ζ =
{
µ1 + w1ξ; with probability p
µ2 + w2ξ; with probability (1 − p)
. (20)
where to keep average value equals to zero and standard deviation to
1 we impose the following constraints:
pµ1 + (1 − p)µ2 = 0; pw2
1 + (1 − p)w2
2 = 1. (21)

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Volatility Smile
Assuming that µ1 = µ2 = 0 we obtained:
B(0, 2dt) = B(0, 2dt)e−α(0,dt)dt2
⟨
eσ(0,dt)ζdt
√
dt
⟩
. (22)
It means that to keep martingale condition the drift term must be
α(0, dt)dt2
= ln
(
p
⟨
eσ(0,dt)w1ξdt
√
dt
⟩
+
+(1 − p)
⟨
eσ(0,dt)w2ξdt
√
dt
⟩)
=
= ln
(
pe
1
2
σ2(0,dt)w2
1 dt3
+ (1 − p)e
1
2
σ2(0,dt)w2
2 dt3
)
. (23)

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Volatility Smile
General form of drift is:
α(kdt, k + Ndt)dt2
= ln
(
pe
1
2 (
∑N
n=1 σ(kdt,(n+k)dt))
2
w2
1 dt3
+
+(1 − p)e
1
2 (
∑N
2
w2
2 dt3
)
−
− ln
(
pe
1
2 (
∑N−1
2
w2
1 dt3
+
+(1 − p)e
1
2 (
∑N−1
2
w2
2 dt3
)
. (24)

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Volatility Smile
Let us select the following parameters:
p = 0.8; 1 − p = 0.2;
w1 =
√
0.6 = 0.775 . . . ; w2 =
√
1 − w1p
1 − p
= 1.612 . . . ;(25)

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Volatility Smile
Figure: Normal and Alternative Distributions.

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Volatility Smile
Generating scenarios that we can modify implied volatility smile of
the ﬁrst few expiration dates keeping good calibrated ATM swaption
prices. New distributions leads to the following volatility smile:
Figure: 6Mx6M Swaption Implied Volatility

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Volatility Smile
Short term swaption volatilities are little bit lower:
Figure: Tenor 1, New Distributions

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Volatility Smile
It happens because
R =
Modiﬁes Price
Original Price
= w1p + w2(1 − p) ≃ 0.94. (26)
To correct prices of ﬁrst time to expiration we need to increase initial
volatility by the following factor R−1
.

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Volatility Smile
Figure: Tenor 1, Adjusted Distributions

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Volatility Smile
Adjusted distribution leads to the following volatility smile:
Figure: 6Mx6M Swaption

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Volatility Smile
:
Figure: 6Mx5Y Swaption

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Volatility Smile
:
Figure: 1Yx5Y Swaption

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Volatility Smile
:
Figure: 30Yx5Y Swaption

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Volatility Smile
To be able to calibrate all OTM swaptions we need to add
dependence of forward volatilities on forward rates.

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Eurodollar Futures
Recently, Mercurio [5] calculated Eurodollar futures in Shifted Libor
Market Model. Here we can calculate these futures in HJM model
from calibrated HJM model directly.
Eurodollar futures can be calculated as:
F(T) =
1
dt
⟨
1
B(T, T + dt)
− 1
⟩
(27)
where B(t, T) is a Libor Bond with Maturity Time T at time t
B(t, T) = e−
∫ T
t (S(t,τ)+f (t,τ))dτ
; (28)
and dt is a Maturity of Future Bonds:
dt = 0.25; dt = 0.5. (29)

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Eurodollar Futures.
In HJM model martingale condition can be used to calculate the drift
B(0, T) =
= B(0, T)edrift(T−dt)
⟨
e−
∫ T−dt
0 dW (τ)
∫ T−dt
τ σ(τ,t)dt
⟩
=
= B(0, T)edrift(T−dt)
e
1
2
Σ2(T−dt)(T−dt)
; (30)
where
Σ2
(T − dt) =
1
T − dt
∫ T−dt
0
v2
(t, T)dt;
v(t) =
∫ T−dt
t
σ(τ, T)dτ. (31)
Then from (30) we have:
drift(T − dt) = −
1
2
Σ2
(T − dt)(T − dt). (32)

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Eurodollar Futures.
Then we can calculate (27):
⟨
1
B(T, T + dt)
⟩
=
=
B(0, T)
B(0, T + dt)
e
1
2
Σ2(T)dt
⟨
e
∫ T
0 dW (τ)
∫ T
τ σ(τ,t)dt
⟩
=
=
B(0, T)
B(0, T + dt)
eΣ2(T)dt
. (33)
Eurodollar future value is:
F(T) =
1
dt
(
B(0, T)
B(0, T + dt)
eΣ2(T)dt
− 1
)
. (34)

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Eurodollar Futures
Taken into account that Eurodollar payments are calculated as
act days
360
we obtain:
Figure: 6 Months Eurodollar Futures.

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European Options and Stochastic Rates
Black-Scholes model in the presence of stochastic rates has the
following form:
d ln(S(t)) =
(
r(t) −
1
2
v2
)
dt + vdW (t); (35)
where r(t) = f (t, t) is a short term interest rate.

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In case of HJM model Call option price is
Call(X, T) =
⟨
e−
∫ T
0 r(t)dt
[
S(0)e
∫ T
0 r(t)dt−1
2
v2
eq(T)T+
∫ T
0 v(t)dWeq(t)
−X]+
⟩
=
=
⟨[
S(0)e−1
2
v2
eq(T)T+
∫ T
0 v(t)dWeq(t)
−
−Xe−1
2
v2
r (T)T−
∫ T
0 (T−t)dt
∫ T
t σ(τ,t)dWr (τ)
]
+
⟩
= (36)
=
⟨[
S(0)e−1
2
v2
eq(T)T+veq(T)
√
Tξeq
− Xe−1
2
v2
r (T)T−vr (T)
√
Tξr
]
+
⟩
;

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Volatilities can be calculated from their forward volatilities:
v2
eq(T) =
1
T
∫ T
0
v2
(t)dt;
v2
r (T) =
1
T
∫ T
0
dt1
∫ t1
0
dt2
∫ t2
0
σ(t, t1)σ(t, t2)dt. (37)

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In the case of constant interest rate volatility without mean reversion
process
σ(t, T) = σ; (38)
we have
vr (T) =
1
√
3
σT; (39)
and implied volatility is
IV 2
(T) = v2
eq(T) + v2
r (T). (40)

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Figure: Constant (σ = 1%, Formula and MC) and Actual Monte-Carlo
Implied Volatilities of Equity with Forward Constant Volatility 10%.

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If we have constant interest rate/equity correlations
⟨dWeq(t)dWr (t)⟩ = ρdt (41)
then in continuous limit (dt → 0) under constant volatility
assumptions we obtain:
⟨ξeqξr ⟩ = ρ
veqσ
∫ T
t
(T − τ)dτ
√
v2
eq
(∫ T
t
dτ
)
σ2
(∫ T
0
(T − τ)2dτ
) =
= ρ
∆2
2
√
3
∆2
=
ρ
2
√
3 = ρIV . (42)
Implied volatility is
IV 2
(T) = v2
eq(T) + 2ρIV veq(T)vr (T) + v2
r (T). (43)

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Bond Index and Stochastic Rates
Bond index model:
S(t + dt) =
B(t + dt, T)
B(t, T)
S(t); (44)
where S(t) is a Bond Index, B(t, T) is a price of Z-bond at time t
and T − t = ∆ = const is time to bond maturity, index duration.
In HJM model
B(t + dt, T)
B(t, T)
= e(
∫ T
t+dt σ(t,τ)dτ)dW (t)+drift
; (45)
where drift is a deterministic function.

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From eq.(45) we can see that the Bond Index Model is equivalent to
equity model which forward volatility is
v(t) =
∫ T
t
σ(t, τ)dτ; (46)
and equity-rate correlations = −100%.
It means that the Bond-Index volatility has the following form
v2
B(t) = v2
(t) −
√
3v(t)vr (t) + v2
r (t). (47)

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In case of a constant volatility we have
v = σ∆; vr =
t
√
3
σ;
v2
B(t) = σ2
(
∆2
− t∆ +
1
3
t2
)
. (48)
Then Bond Fund minimal implied volatility is:
vB
(
t =
3
2
∆
)
=
1
2
σ∆. (49)

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Figure: Constant (σ = 1%, Formula and MC) and Actual Monte-Carlo
Implied Volatilities of Bond Index.

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Main Results
Closed Form formula was derived for swaption prices. It can be
used for fast and accurate calculations of ATM swaptions.
It was shown that it is possible to ﬁt all ATM swaption prices.
It was checked that this approximation works well even in the
case of extreme high normal swaption volatility of 2008 year.
It was demonstrated that it is possible to ﬁt short term swaption
implied volatility smile.
Eurodollars futures are calculated. Reasonable agreement with
market data.
In this model bond index with constant duration has a minimum
volatility because of bond index value has a negative correlation
with short term interest rate.

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What needs to be done
To control errors we need to calculate a ﬁrst correction to this
approximation.
Add volatility dependence on forward rates to make possible to
calibrate all swaptions.

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Bibliography
Hull, J. and A. White (1990): ”Pricing interest-rate derivative
securities”, The Review of Financial Studies, 3(4): 573−592.
Heath, D., R. Jarrow, and A. Morton (1990): ”Bond Pricing and
the Term Structure of Interest Rates: A Discrete Time
Approximation”.Journal of Financial and Quantitative Analysis,
25: 419−440.
Martin Haugh: “The Heath-Jarrow-Morton Framework”, Term
Structure Models: IEOR E4710, (2010).
V.M. Belyaev : “Swaption Prices in HJM Model. Nonparametric
Fit”, arXiv:1697.01619, [ q-ﬁn.PR], (2016).
F. Mercurio : “The present of futures”,Risk Magazine, 3 2018:
78-83.

QuantMinds 2018 International Conference, Lisbon, Portugal. Swaptions, equity and bonds in HJM models.

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