SABR
Bridge representation
Edgeworth
Small time approximations
Probability density of lognormal
fractional SABR model
Tai-Ho Wang
IAQF/Thalesians Seminar Series
New York, January 24, 2017
(joint work with Jiro Akahori and Xiaoming Song)
Heuristic LDP
SABR
Bridge representation
Edgeworth
Small time approximations
Heuristic LDP
Outline
Review of SABR model and SABR formula
Lognormal fractional SABR (fSABR) model
A bridge representation for probability density of lognormal
fSABR
Edgeworth type of expansion
Heuristic derivation of sample path large deviation principle
from path-integral perspective
Approximations of implied volatility in small time
Conclusion
SABR
Bridge representation
Edgeworth
Small time approximations
Heuristic LDP
Stochastic αβρ (SABR) model
Stochastic αβρ (SABR) model was suggested and investigated by
Hagan-Lesniewski-Woodward as
dSt = Stβ αt (ρdBt + ρ̄dWt ),
dαt = ναt dBt ,
S0 = s;
α0 = α
where
pBt and Wt are independent Brownian motions,
ρ̄ = 1 − ρ2 .
SABR model is market standard for quoting cap and swaption
volatilities using the SABR formula for implied volatility.
Nowadays also used in FX and equity markets.
β = 0 is referred to as normal SABR
β = 1 is referred to as lognormal SABR
SABR
Bridge representation
Edgeworth
Small time approximations
Heuristic LDP
SABR formula
The SABR formula is a small time asymptotic expansion up to first
order for the implied volatilities of call/put option induced by the
SABR model.
σBS (K , τ ) = ν
log(s/K )
{1 + O(τ )}
D(ζ)
as the time to expiry τ approaches 0. D and ζ are defined
respectively as
!
p
1 − 2ρζ + ζ 2 + ζ − ρ
D(ζ) = log
1−ρ
and
ζ=



ν s 1−β −K 1−β
α
1−β
if β 6= 1;


ν
α
if β = 1.
log
s
K
SABR
Bridge representation
Edgeworth
Small time approximations
Plot of SABR formula for implied volatilities
0.22
α = 0.13927, ρ = −0.06867, ν = 0.5778, τ = 1.
s = 0.065
0.20
s = 0.076
s = 0.088
0.18
0.16
0.14
0.12
Implied Volatility
backbone
0.04
0.05
0.06
0.07
0.08
strike
0.09
0.10
0.11
Heuristic LDP
SABR
Bridge representation
Edgeworth
Small time approximations
Brief review on derivation of SABR formula
Take ν = 1 for simplicity. Change of variable
Z
ds
1
− ρα
x =
β
ρ̄
s0 s

1−β −s 1−β
1s

0

− ρ̄ρ α if β 6= 1;
 ρ̄ 1−β
=


 1 log s − ρ α
if β = 1,
ρ̄
s0
ρ̄
y
= α
Heuristic LDP
SABR
Bridge representation
Edgeworth
Small time approximations
Heuristic LDP
Ito’s formula implies
dXt = Yt dWt −
β
Yt2
dt
2ρ̄ (1 − β)(ρ̄Xt − ρYt )
dYt = Yt dBt
Infinitestimal generator
L = y 2 (∂x2 + ∂y2 ) −
β
y2
∂x
2ρ̄ (1 − β)(ρ̄x − ρy )
The principle part is the Laplace-Beltrami operator on
2-dimensional hyperbolic space. The corresponding diffusion
process in called the hyperbolic Brownian motion.
The transition density of 2-dimensional hyperbolic Brownian
motion is given by the McKean kernel.
Drift part does not play a role in the large deviation regime.
SABR
Bridge representation
Edgeworth
Small time approximations
Heuristic LDP
SABR density in small time
The transition density of the process (Xt , Yt ) from (x, y ) to (ξ, η)
in time τ is asymptotically given by
pτ (ξ, η|x, y ) ≈
1 − d 2 (ξ,η|x,y )
2τ
e
,
2πτ
where d is defined as
−1
d(ξ, η|x, y ) = cosh
with
(ξ − x)2 + η 2 + y 2
2y η
p
cosh−1 z = log z + z 2 − 1 .
d is the geodesic distance between the initial point (x, y ) and
terminal point (ξ, η).
SABR
Bridge representation
Edgeworth
Small time approximations
Heuristic LDP
From density to option price
For out-of-money calls,
ZZ
C (K , T ) =
(sT − K )+ pT (sT , αT |s0 , α0 )dsT dαT
Z ∞Z ∞
d 2 (sT ,αT |s0 ,α0 )
1
2T
dsT dαT
≈
(sT − K )e −
2πT 0
K
By applying the Laplace asymptotic formula, we have, as T → 0,
C (K , T ) ≈ e −
2 (s ,α )
d∗
0 0
2T
where d∗ is the minimal distance from the initial point (s0 , α0 ) to
the half plane {(s, α) : s ≥ K }, i.e.,
d∗ (s0 , α0 ) = min {d(s, α|s0 , α0 ) : s ≥ K } .
SABR
Bridge representation
Edgeworth
Small time approximations
Heuristic LDP
Matching with Black-Scholes price
The zeroth order SABR formula is thus obtained by matching the
exponent with the corresponding term in Black-Scholes price.
e
−
2 (s ,α )
d∗
0 0
2T
≈ C (K , T ) = CBS (K , T ) ≈ e
−
(log s0 −log K )2
2σ 2 T
BS
We end up with
d∗2 (s0 , α0 ) ≈
(log s0 − log K )2
.
2
σBS
Thus,
σBS (K , T ) ≈
|log s0 − log K |
.
d∗ (s0 , α0 )
SABR
Bridge representation
Edgeworth
Small time approximations
Heuristic LDP
Why fractional process?
Gatheral-Jaisson-Rosenbaum observed from empirical data that
Log-volatility behaves as a fractional Brownian Motion with
Hurst exponent H of order 0.1 at any reasonable time scale.
Indeed, they fitted the empirical qth moments m(q, ∆) in
various lags ∆ to
E [|log σt+∆ − log σt |q ] = Kq ∆ζq
proxied by daily realized variance estimates. Kq denotes the
qth moment of standard normal.
At-the-money volatility skew is well approximated by a power
law function of time to expiry
SABR
Bridge representation
Edgeworth
Small time approximations
Heuristic LDP
Gatheral-Jaisson-Rosenbaum
Log-volatility behaves as a fractional Brownian Motion with Hurst
exponent H of order 0.1 at any reasonable time scale
SABR
Bridge representation
Edgeworth
Small time approximations
Gatheral-Jaisson-Rosenbaum
Log-log plot of m(q, ∆) versus ∆ for various q.
Heuristic LDP
SABR
Bridge representation
Edgeworth
Small time approximations
Heuristic LDP
Gatheral-Jaisson-Rosenbaum
d
is well
At-the-money volatility skew ψ(τ ) = dk
σ
(k,
τ
)
BS
k=0
approximated by a power law function of time to expiry τ
SABR
Bridge representation
Edgeworth
Small time approximations
Heuristic LDP
Fractional volatility process
The observations suggest the following model for instantaneous
volatility
H
σt = σ0 e νWt ,
where W H is a fractional Brownian motion with Hurst exponent H.
As stationarity of σt is concerned, GJR suggested the model for
instantaenous volatility as σt = σ0 e Xt where
dXt = α(m − Xt )dt + νdWtH
is a fractional Ornstein-Uhlenbeck process. Again, drift term plays
no role in large deviation regime.
SABR
Bridge representation
Edgeworth
Small time approximations
Heuristic LDP
Review: fractional Brownian motion
A mean-zero Gaussian process BtH is called a fractional Brownian
motion with Hurst exponent H ∈ [0, 1] if its autocovariance
function R(t, s), for t, s > 0, satisfies
i 1
h
t 2H + s 2H − |t − s|2H .
R(t, s) := E BtH BsH =
2
d
H = aH B H for a > 0
B H is self-similar, indeed, Bat
t
B H has stationary increments
BtH is a standard Brownian motion when H =
1
2
SABR
Bridge representation
Edgeworth
Small time approximations
Heuristic LDP
Lognormal fSABR model
Consider the following lognormal fSABR model
dSt
= αt (ρdBt + ρ̄dWt ),
St
H
αt = α0 e νBt ,
where
pBt and Wt are independent Brownian motions,
ρ̄ = 1 − ρ2 . BtH is a fractional Brownian motion with Hurst
exponent H driven by Bt :
Z t
H
Bt =
KH (t, s)dBs .
0
KH is the Molchan-Golosov kernel.
Goal: to obtain an easy to access expression for the joint
density of (St , αt ).
SABR
Bridge representation
Edgeworth
Small time approximations
Heuristic LDP
Slightly more explicit form
Defining the new variables Xt = log St and Yt = αt , we may
rewrite the lognormal fSABR model in a slightly more explicit form
as
Z
Z t
Y02 t 2νBsH
νBsH
Xt − X0 = Y0
e
ds,
e
(ρdBs + ρ̄dWs ) −
2 0
0
H
Yt = Y0 e νBt .
We derive a bridge representation for the joint density of
(Xt , Yt ) in a “Fourier space”.
SABR
Bridge representation
Edgeworth
Small time approximations
Heuristic LDP
Bridge representation for joint density
The joint density of (Xt , Yt ) has the following bridge representation
p(t, xt , yt |x0 , y0 )
−
=
e
√
yt
Z
e
where i =
ηt2
2ν 2 t 2H
2πν 2 t 2H
i(xt −x0 )ξ
×
1
×
2π
"
R
H
y2
i −ρ 0t y0 e νBs dBs + 20 vt ξ
E e
e
−
ρ̄2 y02 vt
2
Rt
√
H
−1, vt = 0 e 2νBs ds and ηt = log yy0t .
#
H
νBt = ηt dξ,
ξ2 SABR
Bridge representation
Edgeworth
Small time approximations
Heuristic LDP
Bridge representation in uncorrelated case
The bridge representation for the joint density of (Xt , Yt ) reads
simpler when ρ = 0:
p(t, xt , yt |x0 , y0 )
−
=
e
√
ηt2
2ν 2 t 2H
yt 2πν 2 t 2H
where i =
×
1
2π
Z
i
h 1
2 e i(xt −x0 )ξ E e − 2 (ξ−i)ξy0 vt νBtH = ηt dξ,
Rt
√
H
−1, vt = 0 e 2νBs ds and ηt = log yy0t .
SABR
Bridge representation
Edgeworth
Small time approximations
Heuristic LDP
McKean kernel
The McKean kernel pH2 (t, xt , yt |x0 , y0 ) reads
√ −t/8 Z ∞
2
ξe −ξ /2t
2e
√
pH2 (t, xt , yt |x0 , y0 ) =
dξ,
cosh ξ − cosh d
(2πt)3/2 d
where d = d(xt , yt ; x0 , y0 ) is the geodesic distance from (xt , yt ) to
(x0 , y0 ).
Note that the McKean kernel is a density with respect to the
Riemannian volume form y12 dxt dyt .
t
The bridge representation can be regarded as a generalization
of the McKean kernel.
Indeed, in the case where H = 12 , ν = 1 and ρ = 0,
Ikeda-Matsumoto showed how to recover the McKean kernel.
SABR
Bridge representation
Edgeworth
Small time approximations
Heuristic LDP
Expanding around bs
We expand the conditional expectation in the bridge
representation
around the deterministic path bs . Let Eηt [·] = E ·|νBtH = ηt .
First, define the deterministic path bs by
h
i
H
bs = log Eηt e 2νBs .
Indeed,
H
bs = log Eηt [e 2νBs ] = 2νEηt [BsH ] + 2ν 2 var ηt [BsH ]
n
o
= 2R(1, u)ηt + 2ν 2 t 2H u 2H − R 2 (1, u) ,
s
t
and R(t, s) = E BtH BsH .
h
i
H
Note that e bs = Eηt e 2νBs . In other words, e bs is an
where u =
H
unbiased estimator of e 2νBs conditioned on νBtH = ηt .
SABR
Bridge representation
Edgeworth
Small time approximations
Heuristic LDP
Now expand the conditional expectation in the bridge
representation around the deterministic path bs as
R t 2 2νB H
1
Eηt e − 2 (ξ−i)ξ 0 y0 e s ds
R
H
R
− 21 (ξ−i)ξ 0t y02 e 2νBs −e bs ds
− 12 (ξ−i)ξ 0t y02 e bs ds
= e
Eηt e
Rt
1
2 bs
≈ e − 2 (ξ−i)ξ 0 y0 e ds
#!
"×
Z t n
k
X 1
1
bs
2
2νBsH
−e
ds
.
1+
Eη
− (ξ − i)ξ
y0 e
k! t
2
0
k=2
Note that, by definition of bs , the first order term in the last
expansion is automatically zero.
SABR
Bridge representation
Edgeworth
Small time approximations
Heuristic LDP
In principle each term in the expansion can be computed in closed
form since they are given by the integrals of moments of shifted
lognormal distributions. Denoting these terms by Hk , i.e.,
" Z
#
k
t
2νBsH
bs
e
−e
Hk (t, ηt ) := Eηt
ds
0
"
ZZ
=
Eηt
k Y
e
2νBsH
i
− e bsi
#
ds1 · · · dsk
i=1
[0,t]k
For instance,
"
ZZ
H2 (t, ηt ) =
Eηt
=
[0,t]2
where v̂t =
Rt
0
e bs ds.
e
2νBsH
i
−e
2bsi
#
ds1 ds2
i=1
[0,t]2
ZZ
2 Y
h
i
H
H
Eηt e 2νBs1 +2νBs2 ds1 ds2 − v̂t2 ,
SABR
Bridge representation
Edgeworth
Small time approximations
Heuristic LDP
Substituting the last expansion into bridge representation we
obtain the following expansion (in the Fourier space) in terms of
the Hk functions as
p(t, xt , yt |x0 , y0 )
ηt2
1
−
√
≈
e 2ν 2 t 2H ×
yt 2πν 2 t 2H
Z
1
1
e i(xt −x0 )ξ e − 2 (ξ−i)ξv̂t ×
2π
(
)
k
n
X
1
1
1+
− (ξ − i)ξ y02k Hk (t, ηt ) dξ,
k!
2
k=2
where v̂t =
Rt
0
y02 e bs ds.
Note that it gives a natural expansion in t via the Hk
functions.
SABR
Bridge representation
Edgeworth
Small time approximations
Heuristic LDP
Edgeworth type of expansion
Finally, integrating term by term, we obtain a full Edgeworth type
of expansion for the probability density p when ρ = 0 around the
deterministic path bs .
For example, we have up to second order
p(t, xt , yt |x0 , y0 )
−
≈
ηt2
wt2
− 2
1
e 2ν 2 t 2H
2y v̂t
0
×
e
q
√
2
2H
2
yt 2πν t
2πy0 v̂t








wt
2y0
wt
1
1  y2
wt
 4
+ q He3 √
+
1 +  0 He2 √
He4 √
y0 H2 (t, ηt ) ,

2


8 v̂t
v̂t
v̂t
v̂t
v̂t


v̂ 3
t
y 2 v̂
where wt = xt − x0 + 02 t and Hen (·) denotes the nth Hermite
Rt
polynomial. Recall that v̂t = 0 e bs ds and ηt = log yy0t .
SABR
Bridge representation
Edgeworth
Small time approximations
Heuristic LDP
Small time asymptotics - uncorrelated
To the lowest order as t → 0, the density p has the following small
time asymptotic behaviour
−
p(t, xt , yt |x0 , y0 ) =
where recall that v̂t =
e
√
ηt2
2ν 2 t 2H
yt 2πν 2 t 2H
Rt
0
e bs ds.
−
(xt −x0 )2
2y 2 v̂t
0
xt −x0
e
q
e 2 {1 + o(1)} ,
2πy02 v̂t
SABR
Bridge representation
Edgeworth
Small time approximations
Heuristic LDP
Probability density in small time - correlated case
The Edgeworth type of expansion in the correlated case is more
involved because of the appearance of the stochastic integral.
However, the lowest order term is manageable. Define the
functions CRK and CeR by
Z 1
Z 1
R(1,u)η
CRK (η) :=
e
KH (1, u)du, CeR (η) :=
e 2R(1,u)η du.
0
0
Then to the lowest order we have
p(t, xt , yt |x0 , y0 )
η2
− 2 t 2H
2ν t
1
2y 2 ṽt
0
1
xt −x0 −ρy0 CRK (ηt ) ηνt t 2 −H
1
1
1
× √
e
× √ e
2π yt ν 2 t 2H
y0 ṽt
2 (η ) t.
where ṽt = tψ(ηt ) := (1 + ρ2 )CeR (ηt ) − ρ2 CRK
t
≈
−
2
SABR
Bridge representation
Edgeworth
Small time approximations
Heuristic LDP
Approximate distance function
Rewrite the joint density p as
p(t, xt , yt |x0 , y0 )
d˜2 (xt ,yt |x0 ,y0 )
1
1
1
√
√ e − 2t 2H
≈
2π yt ν 2 t 2H y0 ṽt
where
2
1
˜ t , yt |x0 , y0 ) := ηt +
d(x
2
2
ν
y0 ψ(ηt )
xt − x0
1
t 2 −H
ηt
− ρy0 CRK (ηt )
ν
is regarded as the approximate “distance function”.
2
SABR
Bridge representation
Edgeworth
Small time approximations
Heuristic LDP
Convexity of approximate distance function
1.0
0.5
η
0.0
−0.5
−1.0
−1.5
−1.5
−1.0
−0.5
η
0.0
0.5
1.0
1.5
Contour plot of approximate distance H = 0.75
1.5
Contour plot of approximate distance function
−1.5
−1.0
−0.5
0.0
x
0.5
1.0
1.5
−1.5
−1.0
−0.5
0.0
0.5
1.0
1.5
x
Figure : The contour plots. Parameters ρ = −0.7, ν = 1, y0 = 1,
t = 0.5. H = 0.75 on the right; H = 0.25, on the left.
SABR
Bridge representation
Edgeworth
Small time approximations
Heuristic LDP
Implied volatility approximation by bridge representation
By matching with the Black-Scholes price to the lowest order, we
obtain a small time approximation of the implied volatility as
follows. Let α = 12 − H and k = log sK0 .
Implied volatility approximation
2
σBS
k2
≈ 2α
T
(
η∗2
1
+ 2
2
ν
y0 ψ(η∗ )
k
η∗
− ρy0 CRK (η∗ )
α
T
ν
2 )−1
where η∗ is the minimizer
(
)
η2
1
k
η 2
− ρy0 CRK (η)
.
η∗ = argmin η ∈ R : 2 + 2
ν
ν
y0 ψ(η) T α
Note that η∗ = η∗
k
Tα
SABR
Bridge representation
Edgeworth
Small time approximations
Heuristic LDP
Approximate implied volatility plots
0.50
σ
0.45
0.40
0.40
0.45
σ
0.50
0.55
fSABR implied volatility, t = 1
0.55
fSABR implied volatility, t = 0.01
−1.0
−0.5
0.0
logmoneyness
0.5
1.0
−1.0
−0.5
0.0
0.5
1.0
logmoneyness
Figure : The implied volatility curves. t = 0.01 on the left, t = 1 on the
right. Parameters are set as ρ = −0.4, ν = 0.58, α0 = 0.38. H = 0.1 in
red, H = 0.3 in orange, H = 21 in green, H = 0.7 in blue, H = 0.9 in
purple.
SABR
Bridge representation
Edgeworth
Small time approximations
Heuristic LDP
Recovery of SABR formula?
Q: Does it recover the SABR formula to the lowest order
when H = 21 ?
SABR formula
σBS (k) ≈
where
−νk
,
D(ζ)
p
D(ζ) = log
ζ=−
ν
k,
α0
1 − 2ρζ + ζ 2 + ζ − ρ
.
1−ρ
SABR
Bridge representation
Edgeworth
Small time approximations
Heuristic LDP
Recovery of SABR formula?
Q: Does it recover the SABR formula to the lowest order
when H = 21 ?
A: NO!
SABR formula
σBS (k) ≈
where
−νk
,
D(ζ)
p
D(ζ) = log
ζ=−
ν
k,
α0
1 − 2ρζ + ζ 2 + ζ − ρ
.
1−ρ
SABR
Bridge representation
Edgeworth
Small time approximations
Heuristic LDP
Graphic comparison with SABR formula
Difference between implied volatilities
0.015
0.40
Comparison of implied volatilities
fSABR with H = 0.5
iv_sabr − vol
0.36
σ
0.34
0.30
0.000
0.32
0.005
0.010
0.38
SABR formula
−1.0
−0.5
0.0
Logmoneyness
0.5
1.0
−1.0
−0.5
0.0
0.5
1.0
Logmoneyness
Figure : The implied volatility curves from SABR and fSABR formula.
Parameters are set as τ = 1, ρ = −0.06867, ν = 0.58, α0 = 0.13927.
SABR
Bridge representation
Edgeworth
Small time approximations
Heuristic LDP
Recovery of SABR formula?
Q: Does it recover the SABR formula to the lowest order
when H = 21 ?
A: NO!
Q: Maybe a smarter choice of bs might work?
SABR
Bridge representation
Edgeworth
Small time approximations
Heuristic LDP
Recovery of SABR formula?
Q: Does it recover the SABR formula to the lowest order
when H = 21 ?
A: NO!
Q: Maybe a smarter choice of bs might work?
A: Unfortunately, doesn’t really work that way either.
SABR
Bridge representation
Edgeworth
Small time approximations
Heuristic LDP
Recovery of SABR formula?
Q: Does it recover the SABR formula to the lowest order
when H = 21 ?
A: NO!
Q: Maybe a smarter choice of bs might work?
A: Unfortunately, doesn’t really work that way either.
Q: Is it even possible to recover the SABR formula from the
bridge representation?
SABR
Bridge representation
Edgeworth
Small time approximations
Heuristic LDP
Recovery of SABR formula?
Q: Does it recover the SABR formula to the lowest order
when H = 21 ?
A: NO!
Q: Maybe a smarter choice of bs might work?
A: Unfortunately, doesn’t really work that way either.
Q: Is it even possible to recover the SABR formula from the
bridge representation?
A: Most-likely-path from bridge representation
SABR
Bridge representation
Edgeworth
Small time approximations
Heuristic LDP
Bridge representation for multiperiod joint density
Notations
t = (t1 , · · · , tn ),
BH
t
=
(BtH1 , · · ·
, BtHn ),
ξ t = (ξt1 , · · · , ξtn ),
x t = (xt1 , · · · , xtn ),
y t = (yt1 , · · · , ytn ),
X t = (Xt1 , · · · , Xtn ), Y t = (Yt1 , · · · , Ytn ),
η t = (ηt1 , · · · , ηtn ),
ζ t = (ζt1 , · · · , ζtn ).
For 0 < t1 < · · · < tn = T , we are interested in obtaining a bridge
expression for the multiperiod joint density
p(xt1 , yt1 , · · · , xtn , ytn )
:= P [(Xt1 , Yt1 ) = (xt1 , yt1 ), · · · , (Xtn , Ytn ) = (xtn , ytn )] .
SABR
Bridge representation
Edgeworth
Small time approximations
Heuristic LDP
The multiperiod joint density p has the following bridge
representation
p(x1 , y1 , · · · , xn , yn )

n
Y
e

= E
−
1
2y 2 ρ̄2 ∆vt
0
k
∆xtk −y0 ρ
k=1
R tk
H
tk−1
e νBs dBs +
q
2πy02 ρ̄2 ∆vtk
i
h
H
P y0 e νB t = y t ,
where ∆vtk = vtk − vtk−1 =
R tk
tk−1
H
e 2νBs ds.
y02
∆vtk
2
2 
H

νB t = η t  ×

SABR
Bridge representation
Edgeworth
Small time approximations
Heuristic LDP
Heuristic derivation of LDP
Consider the log likelyhood function
log p(xt1 , yt1 , · · · , xtn , ytn )

n
Y
e
= log E 

−
1
2y 2 ρ̄2 ∆vt
0
k
+ log P
νB H
t
R tk
tk−1
H
e νBs dBs +
q
2πy02 ρ̄2 ∆vtk
k=1
h
∆xtk −y0 ρ
i
= ηt −
n
X
y02
∆vtk
2
2 

H
νB t = η 

log ytk .
k=1
We calculate the first two terms in the limit as n → ∞ in the
following.
SABR
Bridge representation
Edgeworth
Small time approximations
Note that
h
i
1 0 −1
log P νB H
=
η
η R η,
t ≈−
t
2ν 2
where R = [R(ti , tj )] is the covariance matrix of B H .
We approximate R(ti , tj ) by a Riemann sum
h
i Z ti ∧tj
H H
R(ti , tj ) = E Bti Btj =
K (ti , s)K (tj , s)ds
0
≈
i∧j
X
K (ti , tk )K (tj , tk )∆t = K 0 K ∆t,
k=0
where K is the upper triangular matrix

 K (ti , tj ), if i ≥ j;
K ij =

0,
otherwise.
Heuristic LDP
SABR
Bridge representation
Edgeworth
Small time approximations
Heuristic LDP
1
Therefore, R −1 = K −1 (K 0 )−1 ∆t
.
Let b = (bt1 , · · · btn ) be the unique solution to the linear system
η
= K b∆t
ν
!
ηti = ν
i
X
K (ti , tk )btk ∆t.
k=0
Hence,
1 0 −1
1
η R η = ∆tb 0 K 0 R −1 K b∆t
2ν 2
2
n
X
1 0
1
=
b b∆t =
bt2k ∆t
2
2
k=1
Z T
1
b 2 dt as n → ∞.
−→
2 0 t
Rt
Also, in the limit n → ∞, ηt = ν 0 K (t, s)bs ds for t ∈ [0, T ].
SABR
Bridge representation
Edgeworth
As for the conditional expectation

n
Y
e
log E 

−
1
2y 2 ρ̄2 ∆vt
0
k
k=1
≈
n
X

E −
k=1
1
2y02 ρ̄2 ∆vtk
Small time approximations
Heuristic LDP
2 
tk−1

H
νB t = η 
q

2πy02 ρ̄2 ∆vtk

!2 Z tk
H
∆xtk − y0 ρ
e νBs dBs νB H = η 
tk−1
∆xtk −y0 ρ
R tk
H
e νBs dBs +
y02
∆vtk
2
Note that conditioned on νB H = η, we have
Z tk
Pk−1
H
∆vtk =
e 2νBs ds ≈ e 2ηtk−1 ∆t = e 2ν j=0 K (tk−1 ,tj )btj ∆t ∆t
tk−1
and
Z
∆xtk − y0 ρ
t
=
tk
H
e νBs dBs ≈ ∆xtk − y0 ρe ηtk−1 btk−1 ∆t
k−1
P
∆xtk
ν k−1
K
(t
,t
)b
∆t
t
j
k−1
j
− y0 ρe j=0
btk−1 ∆t
∆t
SABR
Bridge representation
Edgeworth
Small time approximations
Heuristic LDP
Thus,
n
X

E −
k=1
n
X
1
2y02 ρ̄2 ∆vtk
1

!2 H
∆xtk − y0 ρ
e νBs dBs νB H = η 
tk−1
Z
tk
×
2y02 ρ̄2 e 2ν j=0 K (tk−1 ,tj )btj ∆t
2
Pk−1
∆xtk
− y0 ρe ν j=0 K (tk−1 ,tj )btj ∆t btk−1 ∆t
∆t
Z T
2
Rt
1
1
Rt
ẋt − y0 ρe ν 0 K (t,s)bs ds bt dt
−→ −
2 0 y 2 ρ̄2 e 2ν 0 K (t,s)bs ds
0
Z
1
1 T
= −
(ẋt − ρy0 e ηt bt )2 dt
2
2
2 0 ρ̄ y0 e 2ηt
≈
−
Pk−1
k=0
as n → ∞.
SABR
Bridge representation
Edgeworth
Small time approximations
Large deviations principle for fSABR
We end up with
− log P [Xt = xt , Yt = yt for t ∈ [0, T ]]
= − lim log p(xt1 , yt1 , · · · , xtn , ytn )
n→∞
Z T
Z
1
1
1 T 2
2
ηt
=
(ẋt − ρy0 e bt ) dt +
b dt
2 0 ρ̄2 y02 e 2ηt
2 0 t
Z
Z
1 T 2
1 T 1
2
(ẋt − ρyt bt ) dt +
b dt
=
2 0 ρ̄2 yt2
2 0 t
where b ∈ L2 [0, T ] satisfying
Z
ηt = log yt − log y0 = ν
t
KH (t, s)bs ds
0
for t ∈ [0, T ].
This should be the rate function for sample path LDP.
Heuristic LDP
SABR
Bridge representation
Edgeworth
Small time approximations
Recovery of Freidlin-Wentzell when H =
Indeed,
1
2
1 −1
K [η](t).
ν H
is simply the usual differential operator, thus
bt =
When H = 12 , KH−1
Heuristic LDP
bt =
η̇t
1 ẏt
=
.
ν
ν yt
Therefore, the rate function reduces to
− log P [Xt = xt , Yt = yt for t ∈ [0, T ]]
Z
Z 1 T 1
η̇t 2
1 T η̇t 2
=
ẋt − ρyt
dt +
dt
2 0 ρ̄2 yt2
ν
2 0
ν
Z
1 T
1
=
ν 2 ẋt2 − 2ρν ẋt ẏt + ẏt2 dt
2
2
2
2 0 ρ̄ ν yt
which recovers the classical large deviations principle of
Freidlin-Wentzell
SABR
Bridge representation
Edgeworth
Small time approximations
Heuristic LDP
Implied volatility approximation by LDP
Again, by matching with the Black-Scholes price, we obtain
fSABR formula
2
σBS
k2
≈
T
Z
0
T
1
(ẋt∗ − ρyt∗ bt∗ )2 + bt∗ 2 dt
2
∗
2
ρ̄ yt
where (x ∗ , b ∗ ) is the minimizer
Z
∗ ∗
(x , b ) = argmin ẋ, b ∈ L2 [0, T ] :
0
T
−1
,
1
(ẋt − ρyt bt )2 + bt2 dt
2
ρ̄ yt2
with xT = k and yt∗ is given by, for t ∈ [0, T ],
Z t
∗
log yt − log y0 = ν
KH (t, s)bs∗ ds
0
This recovers the SABR formula when H = 12 !
SABR
Bridge representation
Edgeworth
Small time approximations
Heuristic LDP
Conclusion
We show a bridge representation for the (single and multi
period) joint density of the lognormal SABR model.
We obtained a full Edgeworth type of expansion in the
uncorrelated case.
Small time asymptotics to the lowest order are presented for
both correlated and uncorrelated cases.
A heuristic derivation of large deviations principle from
multiperiod bridge representation for the fSABR model is
shown, which recovers the classical Freidlin-Wentzell large
deviations principle when H = 12 .
Approximations of implied volatility in small time are obtained
accordingly by matching terms with the Black-Scholes price.
We emphasize that the fSABR formula also holds for general
fSABR model, not only for lognormal fSABR.
SABR
Bridge representation
Edgeworth
Small time approximations
THANK YOU FOR YOUR ATTENTION.
Heuristic LDP