Basis risk with partial information
Two-dimensional Kalman filter
Indifference hedging with random drifts
Optimal hedging under partial information
Michael Monoyios
Mathematical Institute
University of Oxford
www.maths.ox.ac.uk/~monoyios
Further Developments in Quantitative Finance
ICMS, Edinburgh
July 2007
Michael Monoyios
Optimal hedging under partial information
Mathematical Institute University of Oxford www.maths.ox.ac.uk/~monoyios
Basis risk with partial information
Two-dimensional Kalman filter
Indifference hedging with random drifts
1 Basis risk with partial information
2 Two-dimensional Kalman filter
3 Indifference hedging with random drifts
Michael Monoyios
Optimal hedging under partial information
Mathematical Institute University of Oxford www.maths.ox.ac.uk/~monoyios
Basis risk with partial information
Two-dimensional Kalman filter
Indifference hedging with random drifts
Model
(Ω, F, F := (Ft )0≤t≤T , P), and Ft = σ((Bs , Bs⊥ ); 0 ≤ s ≤ t)
S := (St )0≤t≤T traded
dSt = rSt dt + σSt (λdt + dBt ) =: rSt dt + σSt dξt
Y := (Yt )0≤t≤T non-traded
dYt = rYt dt + βYt (θdt + dWt )
=: rYt dt + βYt dζt
Correlation ρ
d[B, W ]t = ρdt, W = ρB +
p
1 − ρ2 B ⊥ ,
ρ ∈ [−1, 1]
European claim pays h(YT )
Michael Monoyios
Optimal hedging under partial information
Mathematical Institute University of Oxford www.maths.ox.ac.uk/~monoyios
Basis risk with partial information
Two-dimensional Kalman filter
Indifference hedging with random drifts
Perfect correlation case
For ρ = 1, perfect hedging possible
No arbitrage requires θ = λ
(c)
Position in n claims hedged by ∆t
(c)
∆t
v (t, y )
units of S at t ∈ [0, T ]
β Yt ∂v
(t, Yt )
σ St ∂y
= BS(t, y ; β)
= −n
Perfect hedge does not require knowledge of λ, θ
Michael Monoyios
Optimal hedging under partial information
Mathematical Institute University of Oxford www.maths.ox.ac.uk/~monoyios
Basis risk with partial information
Two-dimensional Kalman filter
Indifference hedging with random drifts
Incomplete market case (ρ 6= 1)
Utility-based hedge, for U(x) = −e −αx , computable in analytic form
Position in n claims hedged by ∆t units of S at t ∈ [0, T ]
∆t
p(t, y )
= −nρ
= −
β Yt ∂p
(t, Yt )
σ St ∂y
M e −r (T −t)
log E Q exp −α(1 − ρ2 )nh(YT ) Yt = y
2
α(1 − ρ )n
Under Q M
dYt = rYt dt + βYt (θ − ρλ)dt + dWtM
Exponential hedge requires knowledge of λ, θ
Superior to BS hedge if drifts known (Monoyios, 2004)
Michael Monoyios
Optimal hedging under partial information
Mathematical Institute University of Oxford www.maths.ox.ac.uk/~monoyios
Basis risk with partial information
Two-dimensional Kalman filter
Indifference hedging with random drifts
Drift estimation impossible
Observe S over [0, t] and estimate λ
Z
1 t d(e −rs Ss )
Bt
λ̄(t) =
=λ+
∼ N(λ, 1/t)
t 0
σSs
t
√
(λ̄(t) − λ)/(1/ t) ∼ N(0, 1), and 95% confidence interval for λ is
1.96
1.96
[λ̄min (t), λ̄max (t)] = λ̄(t) − √ , λ̄(t) + √
t
t
Suppose true λ = 1. To be 95% sure of λ to within 5% of true value
(|λ̄(t) − λ| ≤ 0.05) requires
λ̄max (t) − λ̄min (t) = 0.1 ⇒ t ≈ 1537years
Drift mis-estimation ruins indifference hedging (Monoyios, 2007)
Can Bayesian learning help?
Michael Monoyios
Optimal hedging under partial information
Mathematical Institute University of Oxford www.maths.ox.ac.uk/~monoyios
Basis risk with partial information
Two-dimensional Kalman filter
Indifference hedging with random drifts
Partial information
Require strategies to be adapted to observation filtration generated
by asset returns
Incorporate learning via filtering
Use prior distribution for (λ, θ), now considered as random variables
Suppose (λ, θ) is bivariate normal
λ ∼ N(λ0 , v0 ),
√
θ ∼ N(θ0 , γ0 ), cov(λ, θ) = κ0 = ρ v0 γ0
Infer prior by observing S and Y over [−τ, 0]
Filter (update) estimates of λ, θ from subsequent observations of
ξt :=
1
σ
Z
0
t
d(e −rs Ss )
= λt + Bt ,
Ss
ζt :=
1
β
Z
0
t
d(e −rs Ys )
= θt + Wt
Ys
over hedging interval [0, T ]
Michael Monoyios
Optimal hedging under partial information
Mathematical Institute University of Oxford www.maths.ox.ac.uk/~monoyios
Basis risk with partial information
Two-dimensional Kalman filter
Indifference hedging with random drifts
Observation and signal
Observation filtration
F̂ := (F̂t )0≤t≤T ,
F̂t = σ(ξs , ζs ; 0 ≤ s ≤ t)
Observation process O, (unobservable) signal process U
ξt
λ
,
U=
O :=
θ
ζt 0≤t≤T
Observation and signal SDEs
dOt = Udt + DdMt ,
1 p 0
D=
,
1 − ρ2
ρ
Michael Monoyios
Optimal hedging under partial information
0
dU =
0
dBt
Mt =
dBt⊥
Mathematical Institute University of Oxford www.maths.ox.ac.uk/~monoyios
Basis risk with partial information
Two-dimensional Kalman filter
Indifference hedging with random drifts
Kalman-Bucy equations
Optimal filter Ût := E [U|F̂t ], 0 ≤ t ≤ T
E [λ|F̂t ]
λ̂t
Ût :=
=:
,
θ̂t
E [θ|F̂t ]
λ̂0
θ̂0
satisfies Kalman-Bucy filtering equation
−1
(dOt − Ût dt)
d Ût = Vt DD T
=
λ0
θ0
=: Vt dNt
(Nt )0≤t≤T is innovations process
Z t
Nt := Ot −
Ûs ds
0
Rt
ξt − 0 λ̂s ds
B̂t
Rt
=
=:
Ŵ
ζt − 0 θ̂s ds
t
and B̂, Ŵ are F̂-BMs with correlation ρ
Michael Monoyios
Optimal hedging under partial information
Mathematical Institute University of Oxford www.maths.ox.ac.uk/~monoyios
Basis risk with partial information
Two-dimensional Kalman filter
Indifference hedging with random drifts
Riccati and Lyapunov equations
Conditional variance-covariance matrix
i
h
Vt := E (U − Ût )(U − Ût )T F̂t
satisfies deterministic matrix Riccati equation
−1
dVt
= −Vt DD T
Vt
dt
Equivalently, Rt := Vt−1 satisfies Lyapunov equation
−1
dRt
= DD T
dt
Michael Monoyios
Optimal hedging under partial information
Mathematical Institute University of Oxford www.maths.ox.ac.uk/~monoyios
Basis risk with partial information
Two-dimensional Kalman filter
Indifference hedging with random drifts
Expanded covariance matrix
Written in full, conditional covariance is
E [(λ − λ̂t )2 |F̂t ]
E [(λ − λ̂t )(θ − θ̂t )|F̂t ]
=
E [(λ − λ̂t )(θ − θ̂t )|F̂t ]
E [(θ − θ̂t )2 |F̂t ]
vt κt
=:
κt γt
Vt
Michael Monoyios
Optimal hedging under partial information
Mathematical Institute University of Oxford www.maths.ox.ac.uk/~monoyios
Basis risk with partial information
Two-dimensional Kalman filter
Indifference hedging with random drifts
Expanded filtering equation
Equation for optimal filter is
d λ̂t
d θ̂t
Michael Monoyios
Optimal hedging under partial information
=
1
1 − ρ2
vt − ρκt
κt − ργt
1
1 − ρ2
=
vt − ρκt
κt − ργt
dξt − λ̂t dt
dζt − θ̂t dt
d B̂t
κt − ρvt
γt − ρκt
d Ŵt
κt − ρvt
γt − ρκt
Mathematical Institute University of Oxford www.maths.ox.ac.uk/~monoyios
Basis risk with partial information
Two-dimensional Kalman filter
Indifference hedging with random drifts
Solution of Lyapunov equation
Lyapunov equation is
d
dt
γt
vt γt −κ2t
t
− vt γκt −κ
2
t
t
− vt γκt −κ
2
vt
vt γt −κ2t
t
!
1
=
1 − ρ2
1
−ρ
−ρ
1
Obtiain 3 equations for γt , κt , vt
γt
γ0
−
2
vt γt − κt
v0 γ0 − κ20
κt
κ0
−
vt γt − κ2t
v0 γ0 − κ20
v0
vt
−
2
vt γt − κt
v0 γ0 − κ20
Michael Monoyios
Optimal hedging under partial information
=
=
=
t
1 − ρ2
ρt
1 − ρ2
t
1 − ρ2
Mathematical Institute University of Oxford www.maths.ox.ac.uk/~monoyios
Basis risk with partial information
Two-dimensional Kalman filter
Indifference hedging with random drifts
Simplification
Assume γ0 = v0
Corresponds to having past observations over same time interval
[−τ, 0] for both S and Y
Then κ0 = ρv0 , and we obtain
vt =
Michael Monoyios
Optimal hedging under partial information
v0
,
1 + v0 t
γt = vt ,
κt = ρvt
Mathematical Institute University of Oxford www.maths.ox.ac.uk/~monoyios
Basis risk with partial information
Two-dimensional Kalman filter
Indifference hedging with random drifts
Solution of filtering problem
Equation for optimal filter simplifies to
so
Michael Monoyios
Optimal hedging under partial information
d λ̂t
d θ̂t
= vt
dξt − λ̂t dt
dζt − θ̂t dt
= vt
d B̂t
d Ŵt
Z t λ̂t
d B̂s
λ0
=
+
vs
θ0
θ̂t
d Ŵs
0
Mathematical Institute University of Oxford www.maths.ox.ac.uk/~monoyios
Basis risk with partial information
Two-dimensional Kalman filter
Indifference hedging with random drifts
Equivalently
λ̂(t, ξt )
λ̂(t, St )
λ̂t
≡
≡
θ̂t
θ̂(t, ζt )
θ̂(t, Yt )
given by
λ̂t =
λ0 + v0 ξt
,
1 + v0 t
θ̂t =
θ0 + v0 ζt
1 + v0 t
ζt =
1
log
β
with
ξt =
1
log
σ
Michael Monoyios
Optimal hedging under partial information
e −rt St
S0
1
+ σt,
2
e −rt Yt
Y0
1
+ βt
2
Mathematical Institute University of Oxford www.maths.ox.ac.uk/~monoyios
Basis risk with partial information
Two-dimensional Kalman filter
Indifference hedging with random drifts
Effective full information model
Asset price SDEs under F̂ become
dSt = rSt dt + σSt dξt
dYt = rYt dt + βYt dζt
= rSt dt + σSt (λ̂t dt + d B̂t )
= rYt dt + βYt (θ̂t dt + d Ŵt )
Full information model with random drift parameters (λ̂t , θ̂t )
Same result obtained by treating filtering of λ and θ as two
one-dimensional Kalman filters with same prior variance
Correlated asset Y gives no additional information about drift of S
(and vice versa) under continuous observation unless past
observations of S, Y are over different time intervals
Michael Monoyios
Optimal hedging under partial information
Mathematical Institute University of Oxford www.maths.ox.ac.uk/~monoyios
Basis risk with partial information
Two-dimensional Kalman filter
Indifference hedging with random drifts
Portfolio and wealth processes
Trading strategy π := (πt )0≤t≤T , F̂-adapted,
Wealth process X π ≡ X := (Xt )0≤t≤T
RT
0
πt2 dt < ∞ a.s.
dXt = rXt dt + σπt (λ̂t dt + d B̂t )
U(x) = − exp(−αx), x ∈ R, α > 0
Primal value function u ≡ u (n)
u(t, x, s, y ) = E [U(XT + nh(YT ))|Xt = x, St = s, Yt = y ]
May be expressed as u(t, x, y ; λ̂t , θ̂t )
Optimal strategy π ∗ ≡ π ∗,n
Optimal wealth process by X ∗ ≡ X ∗,n
Random endowment nh(YT ) is bounded below
Michael Monoyios
Optimal hedging under partial information
Mathematical Institute University of Oxford www.maths.ox.ac.uk/~monoyios
Basis risk with partial information
Two-dimensional Kalman filter
Indifference hedging with random drifts
Indifference price and hedge
Indifference price per claim at t ∈ [0, T ], given
Xt = x, St = s, Yt = y , is p ≡ p (n) given by
u (n) (t, x − np (n) (t, x, s, y ), s, y ) = u (0) (t, x, s)
Optimal hedging strategy π H := πtH 0≤t≤T
(H)
πt
Michael Monoyios
Optimal hedging under partial information
:= πt∗,n − πt∗,0 , 0 ≤ t ≤ T
Mathematical Institute University of Oxford www.maths.ox.ac.uk/~monoyios
Basis risk with partial information
Two-dimensional Kalman filter
Indifference hedging with random drifts
Problem without claim, n = 0
Dual value function for n = 0, v (t, η) ≡ v (t, η; λ̂t )
"
! #
Z̃T v (t, η) = E V η
F̂t
Z̃t where
Z̃t = e −rt Zt ,
Zt = E(−λ̂ · B̂)t
This gives
h
i
η
v (t, η) = e −r (T −t) V (η) +
−r (T − t) + H (0) (t, λ̂t )
α
where
H (0) (t, λ̂t ) = E
ZT
log
Zt
ZT F̂t
Zt is entropic term
Michael Monoyios
Optimal hedging under partial information
Mathematical Institute University of Oxford www.maths.ox.ac.uk/~monoyios
Basis risk with partial information
Two-dimensional Kalman filter
Indifference hedging with random drifts
Entropy computation
Apply Itô to Zt =: g (t, λ̂t ) and use
dZt = −λ̂t Zt d B̂t ,
to obtain
ZT
=
Zt
vt
vT
1/2
d λ̂t = vt d B̂t ,
"
1
exp −
2
λ̂2T
λ̂2
− t
vT
vt
!#
Conditional on F̂t , λ̂T ∼ N(λ̂t , vt vT (T − t)), so
" #
1
vt
λ̂2t
(0)
H (t, λ̂t ) =
log
− vT (T − t) +
2
vT
vt
Michael Monoyios
Optimal hedging under partial information
Mathematical Institute University of Oxford www.maths.ox.ac.uk/~monoyios
Basis risk with partial information
Two-dimensional Kalman filter
Indifference hedging with random drifts
Primal value function for n = 0
Recover u(t, x) ≡ u(t, x; λ̂t ) from bidual relation
u(t, x) = inf [v (t, η) + xη]
η>0
so for n = 0, u ≡ u (0) given by
n
o
u(t, x; λ̂t ) = − exp −αxe r (T −t) − H (0) (t, λ̂t )
Recall λ̂t ≡ λ̂(t, St ) so, conditional on St = s, u(t, x; λ̂t ) ≡ u(t, x, s)
Michael Monoyios
Optimal hedging under partial information
Mathematical Institute University of Oxford www.maths.ox.ac.uk/~monoyios
Basis risk with partial information
Two-dimensional Kalman filter
Indifference hedging with random drifts
Optimal trading strategy (n = 0)
Given X0 = x, and writing u(x) ≡ u(0, x)
U 0 (XT∗ ) = ηe −rT ZT ,
η = u 0 (x)
Combine with Q-martingale property of (e −rt Xt∗ )0≤t≤T , and
dQ/dP = ZT ,
ZT ∗ ∗
−r (T −t)
Xt = e
E
X F̂t
Zt T e −r (T −t) (0)
H (0, λ0 ) − H (0) (t, λ̂t ) − log Zt
= xe rt +
α
Compute dXt∗ via Itô, recall dXt∗ = rXt∗ dt + σπt∗ (λ̂t dt + d B̂t ):
πt∗ = e −r (T −t)
Michael Monoyios
Optimal hedging under partial information
λ̂t vT
σα vt
Mathematical Institute University of Oxford www.maths.ox.ac.uk/~monoyios
Basis risk with partial information
Two-dimensional Kalman filter
Indifference hedging with random drifts
Problem with claim, n 6= 0
Dual value function now v (t, η, y ) ≡ v (t, η, y ; λ̂t , θ̂t ) ≡ v (t, η, s, y )
#
"
!
Z̃TQ
Z̃TQ v (t, η, y ) = inf E V η Q + η Q F̂t
Q∈M
Z̃t
Z̃t where now Z̃tQ = e −rt ZtQ and
dQ = E(−λ̂ · B̂ − ψ · B̂ ⊥ )t
ZtQ =
dP F̂t
p
and under Q ∈ M, with := 1 − ρ2 ,
dSt
dYt
d λ̂t
d θ̂t
Michael Monoyios
Optimal hedging under partial information
=
=
=
=
rSt dt + σSt d B̂tQ
rYt dt + βYt [(θ̂t − ρλ̂t − ψt )dt + d ŴtQ ]
vt [−λ̂t dt + d B̂tQ ]
vt [−(ρλ̂t + ψt )dt + d ŴtQ ]
Mathematical Institute University of Oxford www.maths.ox.ac.uk/~monoyios
Basis risk with partial information
Two-dimensional Kalman filter
Indifference hedging with random drifts
Dual value function, n 6= 0
This gives
i
h
η
v (t, η, y ) = e −r (T −t) V (η) +
−r (T − t) + H (n) (t, y ; λ̂t , θ̂t )
α
where
"
H (n) (t, y ; λ̂t , θ̂t ) = inf E Q log
Q∈M
ZTQ
ZtQ
!
#
+ αnh(YT ) F̂t
is entropic term
Michael Monoyios
Optimal hedging under partial information
Mathematical Institute University of Oxford www.maths.ox.ac.uk/~monoyios
Basis risk with partial information
Two-dimensional Kalman filter
Indifference hedging with random drifts
Entropic term
H (n) decomposes according to
"
H (n) (t, y ; λ̂t , θ̂t )
= EQ
= H
(0)
1
2
Z
t
T
#
λ̂2u du F̂t + C (t, y ; λ̂t , θ̂t )
(t, λ̂t ) + C (t, y ; λ̂t , θ̂t )
where C (t, y ; λ̂t , θ̂t ) ≡ C (t, s, y ) is value function of stochastic
control problem
" Z
#
1 T 2
Q
C (t, s, y ) := inf E
ψu du + αnh(YT ) St = s, Yt = y
ψ
2 t
Michael Monoyios
Optimal hedging under partial information
Mathematical Institute University of Oxford www.maths.ox.ac.uk/~monoyios
Basis risk with partial information
Two-dimensional Kalman filter
Indifference hedging with random drifts
HJB equation and optimal control
HJB equation for C (t, s, y ) is
1 2
M
Ct + AS,Y C + min ψ − βψyCy = 0
ψ
2
C (T , s, y ) = αnh(y )
where AM
S,Y is generator of (S, Y ) under minimal measure
1 2 2
1 s
AM
S,Y C = rsCs + s Css +(r +β(θ̂ −ρλ̂))yCy + β y Cyy +ρσβsyCsy
2
2
Optimal control is ψt∗ ≡ ψ ∗ (t, St , Yt ) where
ψ ∗ (t, s, y ) = βyCy (t, s, y )
so HJB equation is semi-linear PDE
1
2
2 2 2
Ct + AM
S,Y C − (1 − ρ )β y Cy = 0
2
Michael Monoyios
Optimal hedging under partial information
Mathematical Institute University of Oxford www.maths.ox.ac.uk/~monoyios
Basis risk with partial information
Two-dimensional Kalman filter
Indifference hedging with random drifts
Primal value function and indifference price
Primal value function recovered via bidual relation
n
o
u(t, x, s, y ) = − exp −αxe r (T −t) − H (0) (t, s) − C (t, s, y )
Indifference price p ≡ p (n)
p (n) (t, s, y ) =
e −r (T −t)
C (t, s, y )
αn
satisfies
1
2 r (T −t) 2 2 2
pt + AM
β y py = 0, p(T , s, y ) = h(y )
S,Y p − rp − αn(1 − ρ )e
2
Michael Monoyios
Optimal hedging under partial information
Mathematical Institute University of Oxford www.maths.ox.ac.uk/~monoyios
Basis risk with partial information
Two-dimensional Kalman filter
Indifference hedging with random drifts
Optimal hedge
HJB equation for primal value function gives π ∗ in terms of
derivatives of C (t, s, y )
Optimal hedge (∆H
t )0≤t≤T follows as
β Yt (n)
(n)
∆H
=
−n
p
(t,
S
,
Y
)
+
ρ
p
(t,
S
,
Y
)
t
t
t
t
t
s
σ St y
Michael Monoyios
Optimal hedging under partial information
Mathematical Institute University of Oxford www.maths.ox.ac.uk/~monoyios
Basis risk with partial information
Two-dimensional Kalman filter
Indifference hedging with random drifts
Residual risk
Trade n claims at time 0 for p (n) (0, Y0 , S0 ) per claim and hedge with
strategy (∆H
t )0≤t≤T
Overall position worth L = (Lt )0≤t≤T given by
Lt = XtH + np (n) (t, St , Yt ), L0 = 0
Itô and PDE satisfied by p ≡ p (n) gives
dLt
h = rLt dt + n β θ̂t − ρλ̂t − (θ − ρλ) Yt py (t, St , Yt )
1
αn(1 − ρ2 )e r (T −t) β 2 Yt2 (py )2 (t, St Yt ) dt
+
2
p
+ n 1 − ρ2 βYt py (t, St Yt )dBt⊥
Michael Monoyios
Optimal hedging under partial information
Mathematical Institute University of Oxford www.maths.ox.ac.uk/~monoyios
Basis risk with partial information
Two-dimensional Kalman filter
Indifference hedging with random drifts
Approximation for indifference price
Indifference price given St = s, Yt = y
" Z
#
e −r (T −t)
1 T 2
Q
inf E
ψu du + αnh(YT ) St = s, Yt = y
p(t, s, y ) =
ψ
αn
2 t
dSt
dYt
d λ̂t
d θ̂t
=
=
=
=
rSt dt + σSt d B̂tQ
rYt dt + βYt [(θ̂t − ρλ̂t − ψt )dt + d ŴtQ ]
vt [−λ̂t dt + d B̂tQ ]
vt [−(ρλ̂t + ψt )dt + d ŴtQ ]
For small approximate optimal control by ψ = 0, so trade claim at
marginal price p̂(0, S0 , Y0 ) where
p̂(t, s, y ) = e −r (T −t) E M [h(YT )|St = s, Yt = y ]
Michael Monoyios
Optimal hedging under partial information
Mathematical Institute University of Oxford www.maths.ox.ac.uk/~monoyios
Basis risk with partial information
Two-dimensional Kalman filter
Indifference hedging with random drifts
Distribution of YT under Q M
Marginal price p̂ computable in analytic form
Under Q M , conditional on St = s, Yt = y , log YT is Gaussian
∼ N(m(t, s, y )(T − t), Σ2 (t)(T − t))
1
m(t, s, y ) = log y + r + β(θ̂t − ρλ̂t − β)
2
2
2
2
Σ = β (1 + (1 − ρ )vt (T − t))
log YT
Allows fast simulation of L, which now satisfies
dLt = rLt dt + nβ θ̂t − ρλ̂t − (θ − ρλ) Yt py (t, St , Yt )dt
p
+ n 1 − ρ2 βYt py (t, St Yt )dBt⊥
Michael Monoyios
Optimal hedging under partial information
Mathematical Institute University of Oxford www.maths.ox.ac.uk/~monoyios
Basis risk with partial information
Two-dimensional Kalman filter
Indifference hedging with random drifts
How effective is optimal hedge with learning?
Compare terminal residual risk with that from using by BS-style
hedge, given by
dLBS
t
= rLBS
t dt + nβYt (θ − λ)vy (t, Yt )dt
p
+ nβYt vy (t, Yt )[(ρ − 1)dBt + 1 − ρ2 dBt⊥ ]
For each simulation, obtain λ0 , θ0 as point estimates of λ, θ over past
data (use [−τ, 0], so v0 = 1/τ ), then update using filtering results
Michael Monoyios
Optimal hedging under partial information
Mathematical Institute University of Oxford www.maths.ox.ac.uk/~monoyios
Basis risk with partial information
Two-dimensional Kalman filter
Indifference hedging with random drifts
Typical result over 10,000 paths, short put
λ = 0.25, θ = 0.4, ρ = 0.95, K = 100
hS0 i = 82.9, hY0 i = 87.2
hp̂i = 16.0, hBSi = 17.2
Hedging Error Distributions
2000
Optimal Hedge
mean error = −0.53
Frequency
1500
sd error = 5.67
median error = 0.08
1000
500
0
−40
−30
−20
−10
0
Terminal Hedge Error
10
2000
Frequency
30
Naive Hedge
mean error = 1.72
1500
20
sd error = 4.16
median error = 1.24
1000
500
0
−40
Michael Monoyios
Optimal hedging under partial information
−30
−20
−10
0
Terminal Hedge Error
10
20
30
Mathematical Institute University of Oxford www.maths.ox.ac.uk/~monoyios
Basis risk with partial information
Two-dimensional Kalman filter
Indifference hedging with random drifts
λ = 0.25, θ = 0.4, ρ = 0.6, K = 100
hS0 i = 82.8, hY0 i = 87.1
hp̂i = 18.0, hBSi = 17.2
Hedging Error Distributions
2500
Optimal Hedge
mean error = −2.86
Frequency
2000
1500
sd error = 13.05
median error = 0.62
1000
500
0
−80
−60
−40
−20
0
Terminal Hedge Error
20
2500
2000
40
60
Naive Hedge
mean error = 1.68
Frequency
sd error = 11.59
1500
median error = 1.64
1000
500
0
−80
Michael Monoyios
Optimal hedging under partial information
−60
−40
−20
0
Terminal Hedge Error
20
40
60
Mathematical Institute University of Oxford www.maths.ox.ac.uk/~monoyios
Basis risk with partial information
Two-dimensional Kalman filter
Indifference hedging with random drifts
λ = 0.39, θ = 0.4, ρ = 0.75, K = 100
hS0 i = 84.3, hY0 i = 85.1
hp̂i = 16.9, hBSi = 15.9
Hedging Error Distributions
2500
2000
Optimal Hedge
mean error = −1.76
sd error = 9.89
Frequency
median error = 0.29
1500
1000
500
0
−50
2500
Frequency
2000
−40
−30
−20
−10
0
Terminal Hedge Error
10
20
mean error = 0.04
30
40
Naive Hedge
sd error = 8.07
median error = 0.33
1500
1000
500
0
−50
Michael Monoyios
Optimal hedging under partial information
−40
−30
−20
−10
0
Terminal Hedge Error
10
20
30
40
Mathematical Institute University of Oxford www.maths.ox.ac.uk/~monoyios
Basis risk with partial information
Two-dimensional Kalman filter
Indifference hedging with random drifts
Conclusions
BS-style hedging robust
Filtering and learning not sufficient to handle parameter uncertainty
in utility-based hedge
Exact numerical results seem unlikely to alter conclusion
Michael Monoyios
Optimal hedging under partial information
Mathematical Institute University of Oxford www.maths.ox.ac.uk/~monoyios