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A Hybrid Market ModelCalibration algorithm
A Hybrid Commodity and Interest Rate MarketModel
Kay Pilz and Erik Schlögl
University of Technology, Sydney
June 2010
Kay Pilz and Erik Schlögl A Hybrid Commodity and Interest Rate Market Model
A Hybrid Market ModelCalibration algorithm
Recall: The basic LIBOR Market ModelThe cross–currency LIBOR Market Model
Literature
LIBOR Market Model (LMM):Miltersen/Sandmann/Sondermann (1997),Brace/Gatarek/Musiela (1997), Jamshidian (1997),Musiela/Rutkowski (1997)Multicurrency LIBOR Market Model: Schlögl (2002)LMM calibration: Pedersen (1998)Integrating commodity risk, interest rate risk, andstochastic convenience yields: Gibson/Schwartz (1990),Miltersen/Schwartz (1998), Miltersen (2003)
Kay Pilz and Erik Schlögl A Hybrid Commodity and Interest Rate Market Model
A Hybrid Market ModelCalibration algorithm
Recall: The basic LIBOR Market ModelThe cross–currency LIBOR Market Model
A model of forward LIBOR
The forward LIBOR L(t ,T ) is defined in terms of zero couponbond prices by
L(t ,T ) :=1δ
(B(t ,T )
B(t ,T + δ)− 1)
Note that irrespective of the model we choose, L(t ,T ) is a mar-tingale under IPT+δ.
Therefore, assuming deterministic volatility for L(t ,T ) meansthat it is lognormally distributed under IPT+δ.
Kay Pilz and Erik Schlögl A Hybrid Commodity and Interest Rate Market Model
A Hybrid Market ModelCalibration algorithm
Recall: The basic LIBOR Market ModelThe cross–currency LIBOR Market Model
Setup
Discrete–tenor lognormal forward LIBOR model(as in Musiela/Rutkowski (1997))
Horizon date TN for some N ∈ IIN, finite number of maturities
Ti = TN − (N − i)δ, i ∈ {0, . . . ,N}
Dynamics of (domestic) forward LIBORs
dL(t ,Ti) = L(t ,Ti)λ(t ,Ti)dWTi+1(t)
whereλ(·, ·) is a deterministic function of its argumentsWTi+1(·) is a Brownian motion under the time Ti+1 forwardmeasure
Note that lognormality in this model is a measure–dependentproperty.
Kay Pilz and Erik Schlögl A Hybrid Commodity and Interest Rate Market Model
A Hybrid Market ModelCalibration algorithm
Recall: The basic LIBOR Market ModelThe cross–currency LIBOR Market Model
Links between domestic forward measures
By Ito’s lemma
d(
B(t ,T )
B(t ,T + δ)
)=
B(t ,T )
B(t ,T + δ)
δL(t ,T )
1 + δL(t ,T )λ(t ,T )dWT+δ(t)
Setting
γ(t ,T ,T + δ) =δL(t ,T )
1 + δL(t ,T )λ(t ,T )
we can write
dPTi
dPTi+1
∣∣∣∣Ft
=B(t ,Ti)
B(t ,Ti+1)
B(0,Ti+1)
B(0,Ti)= Et
(∫ ·0γ(u,Ti ,Ti+1) · dWTi+1(u)
)Thus
dWTi (t) = dWTi+1(t)− γ(t ,Ti ,Ti+1)dt
Kay Pilz and Erik Schlögl A Hybrid Commodity and Interest Rate Market Model
A Hybrid Market ModelCalibration algorithm
Recall: The basic LIBOR Market ModelThe cross–currency LIBOR Market Model
Adding a foreign economy, case 1
Assume lognormal forward LIBOR dynamics in the foreigneconomy as well
dL̃(t ,Ti) = L̃(t ,Ti)λ̃(t ,Ti)dW̃Ti+1(t)
Then the foreign forward measures are linked in a manneranalogous to the domestic forward measures.
This leaves us with the freedom of specifying one further link(only) between a domestic and a foreign forward measure.
Kay Pilz and Erik Schlögl A Hybrid Commodity and Interest Rate Market Model
A Hybrid Market ModelCalibration algorithm
Recall: The basic LIBOR Market ModelThe cross–currency LIBOR Market Model
Measure Links 1
Domestic Foreign
T0 forward measure T0 forward measure
T2 forward measure
T1 forward measure
T2 forward measure
T1 forward measure
Ti forward measureTi forward measure
TN forward measureTN forward measure
Measure Links 2
Domestic Foreign
T0 forward measure T0 forward measure
T2 forward measure
T1 forward measure
T2 forward measure
T1 forward measure
Ti forward measureTi forward measure
TN forward measureTN forward measure
Kay Pilz and Erik Schlögl A Hybrid Commodity and Interest Rate Market Model
A Hybrid Market ModelCalibration algorithm
Recall: The basic LIBOR Market ModelThe cross–currency LIBOR Market Model
Linking domestic & foreign forward measures
X (t): spot exchange rate in units of domestic currency per unitof foreign currency
Time Ti forward exchange rate:
X (t ,Ti) =B̃(t ,Ti)X (t)
B(t ,Ti)
This is a martingale under PTi . Conversely
1X (t ,Ti)
=B(t ,Ti)
1X(t)
B̃(t ,Ti)
is a martingale under P̃Ti .
So we can write
dX (t ,TN) = X (t ,TN)σX (t ,TN) · dWTN (t)
Kay Pilz and Erik Schlögl A Hybrid Commodity and Interest Rate Market Model
A Hybrid Market ModelCalibration algorithm
Recall: The basic LIBOR Market ModelThe cross–currency LIBOR Market Model
Domestic vs. foreign forward measures
dP̃TN
dPTN
=X (TN)B̃(TN ,TN)B(0,TN)
X (0)B̃(0,TN)B(TN ,TN)=
X (TN ,TN)
X (0,TN)
resp. restricting PTN , P̃TN to the information given at time t :
dP̃TN
dPTN
∣∣∣∣∣Ft
=X (t ,TN)
X (0,TN)
By the dynamics assumed forX (t ,TN),
dP̃TN
dPTN
= Et
(∫ ·0σX (u,TN)dWTN (u)
)PTN –a.s.
ThusdW̃TN (t) = dWTN (t)− σX (t ,TN)dt
Kay Pilz and Erik Schlögl A Hybrid Commodity and Interest Rate Market Model
A Hybrid Market ModelCalibration algorithm
Recall: The basic LIBOR Market ModelThe cross–currency LIBOR Market Model
Forward exchange rate volatilities
Note that all measure relationships and therefore all volatilitiesare now fixed.
To determine the remaining forward exchange rate volatilities,inductively make use of the relationship
X (t ,Ti)
X (t ,Ti+1)=
B(t ,Ti+1)
B(t ,Ti)
B̃(t ,Ti)
B̃(t ,Ti+1)
Kay Pilz and Erik Schlögl A Hybrid Commodity and Interest Rate Market Model
A Hybrid Market ModelCalibration algorithm
Recall: The basic LIBOR Market ModelThe cross–currency LIBOR Market Model
For ease of notation, consider just the first step of the induction.Writing all processes under the domestic time TN−1 forwardmeasure and applying Ito’s lemma then yields
dX (t ,TN−1) = X (t ,TN−1)((γ̃(t ,TN−1,TN)− γ(t ,TN−1,TN) + σX (t ,TN)) · dWTN−1(t)
)Thus we must set
σX (t ,TN−1) = γ̃(t ,TN−1,TN)− γ(t ,TN−1,TN) + σX (t ,TN)
i.e. we can choose only one σX (t ,Ti) to be a deterministicfunction of its arguments.
So for FX options we can have a Black/Scholes–type formulafor only one maturity, as all other forward exchange rates arenot lognormal.
Kay Pilz and Erik Schlögl A Hybrid Commodity and Interest Rate Market Model
A Hybrid Market ModelCalibration algorithm
Recall: The basic LIBOR Market ModelThe cross–currency LIBOR Market Model
Adding a foreign economy, case 2
Assume lognormal forward LIBOR dynamics in the domesticeconomy only; assume lognormal forward exchange rates
dX (t ,Ti) = X (t ,Ti)σX (t ,Ti)dWTi (t)
for σX a deterministic function of its arguments.
Thus for all maturities Ti
dW̃Ti (t) = dWTi (t)− σX (t ,Ti)dt
Since the derivation of the links between forward exchange ratevolatilities did not depend on the lognormality assumptions, it isvalid in the present context as well and therefore
γ̃(t ,Ti−1,Ti) = σX (t ,Ti−1)− σX (t ,Ti) + γ(t ,Ti−1,Ti)
Kay Pilz and Erik Schlögl A Hybrid Commodity and Interest Rate Market Model
A Hybrid Market ModelCalibration algorithm
Recall: The basic LIBOR Market ModelThe cross–currency LIBOR Market Model
A commodity as “foreign currency”
The commodity market can naturally be considered as a“foreign interest rate market.”The currency is the physical commodity itself.The “zero coupon bond prices” C(t ,T ) quote (as seen attime t) the amount of the commodity that has to beinvested at time t to physically receive one unit of thecommodity at time T .Thus the yield of C(t ,T ) is the convenience yield (adjustedfor storage costs, if applicable).Since convenience yields are implicit rather than explicitlyquoted in the market, “Case 2” of the multicurrency modelis applicable.
Kay Pilz and Erik Schlögl A Hybrid Commodity and Interest Rate Market Model
A Hybrid Market ModelCalibration algorithm
The Pedersen approachFutures/Forward Relation & Convexity CorrectionMerging Interest Rate & Commodity Calibrations
Pedersen (1998) Calibration
Calibration to market prices of caps (or caplets) andswaptions.Calibration of a non-parametric volatility function λ(·, ·),piecewise constant on a discretisation of both time tomaturity and calendar time.Unconstrained non-linear optimisation of weighted sum ofquality–of–fit and smoothness criteria.Correlation is exogenous to the calibration procedure:Assumed to be constant in time and estimated fromhistorical data.Reduction of dimension via principal components analysis.
Kay Pilz and Erik Schlögl A Hybrid Commodity and Interest Rate Market Model
A Hybrid Market ModelCalibration algorithm
The Pedersen approachFutures/Forward Relation & Convexity CorrectionMerging Interest Rate & Commodity Calibrations
The nonparametric approach
Suppose we have nfac factors (the dimension of the drivingBrownian motion) and discretise process time into ncalsegments, and forward time (maturities) into nfwd segments.
The i-th component of (1 ≤ i ≤ nfac) of the volatility functionλ(t ,T ) will be given by
λi(t , x) = λijk , t ∈ [tj−1, tj) , x ∈ [xk−1, xk )
where x = T − t is the forward tenor, tj , j > 0, and xk , k > 0,are the chosen process and forward times, respectively.
For convenience set t0 = x0 = 0.
Kay Pilz and Erik Schlögl A Hybrid Commodity and Interest Rate Market Model
A Hybrid Market ModelCalibration algorithm
The Pedersen approachFutures/Forward Relation & Convexity CorrectionMerging Interest Rate & Commodity Calibrations
Objective function
wcapsQOFcaps + wswaptionsQOFswaptions + smooth
Quality of fit QOF =1N
N∑i=1
(PVi
PVi− 1
)2
smooth = scalefwd · smoothfwd + scalecal · smoothcal
smoothfwd =
ncal∑j=1
nfwd∑j=2
(voli,j
voli,j−1− 1)2
smoothcal =
ncal∑j=2
nfwd∑j=1
(voli,j
voli−1,j− 1)2
Kay Pilz and Erik Schlögl A Hybrid Commodity and Interest Rate Market Model
A Hybrid Market ModelCalibration algorithm
The Pedersen approachFutures/Forward Relation & Convexity CorrectionMerging Interest Rate & Commodity Calibrations
Reducing the dimesionality of the problem
Original dimensionality: nfac × ncal × nfwd
Separate volatility levels and correlation:
Volatility levels given by volatility grid
voli,j , 1 ≤ i ≤ ncal , 1 ≤ j ≤ nfwd
where voli,j is the volatility as seen at time ti−1 (assumedconstant until ti ) of the basic period rate L(·, ti−1 + xj) for theforward period beginning at time ti−1 + xj .
This is the object which will be calibrated.
Kay Pilz and Erik Schlögl A Hybrid Commodity and Interest Rate Market Model
A Hybrid Market ModelCalibration algorithm
The Pedersen approachFutures/Forward Relation & Convexity CorrectionMerging Interest Rate & Commodity Calibrations
Covariance and correlation — Principal componentsrepresentation
Let vol be the vector of basic period forward rate volatilities asseen on time tj−1. Let Corr be the corresponding correlationmatrix. The covariance matrix is then computed as
Cov = volT Corr vol
Let Γ be the diagonal matrix containing the eigenvalues of Covand V be the corresponding matrix of eigenvectors, i.e. wehave the eigenvalue/eigenvector decomposition of Cov
Cov = V T ΓV
Kay Pilz and Erik Schlögl A Hybrid Commodity and Interest Rate Market Model
A Hybrid Market ModelCalibration algorithm
The Pedersen approachFutures/Forward Relation & Convexity CorrectionMerging Interest Rate & Commodity Calibrations
As Cov is positive semidefinite, all entries γk on the diagonal ofΓ will be non-negative and we have
Cov = W T W
wherewik =
√γkvik
We can then extract the stepwise constant volatility function forforward LIBORs as
λijk = wik
W will provide values for as many factors as the rank of thecovariance matrix. For a given nfac, we only use the rows of Wcorresponding to the nfac largest eigenvalues.
Kay Pilz and Erik Schlögl A Hybrid Commodity and Interest Rate Market Model
A Hybrid Market ModelCalibration algorithm
The Pedersen approachFutures/Forward Relation & Convexity CorrectionMerging Interest Rate & Commodity Calibrations
Spot measure dynamics
Brownian motion under the rolling spot LIBOR measure Q isrelated to BM under the Ti forward measure by
dWTi (t) = φ(t ,Ti)dt + dWQ(t)
with φ(·,Ti) defined recursively as
φ(t ,Ti)− φ(t ,Ti−1) = γ(t ,Ti−1,Ti) =δL(t ,Ti−1)
1 + δL(t ,Ti−1)λ(t ,Ti−1)
Under an appropriate extension of the discrete–tenor LMM tocontinuous tenor, Q coincides with the spot risk–neutralmeasure and the futures price corresponds to the expectedfuture spot price under this measure.
Kay Pilz and Erik Schlögl A Hybrid Commodity and Interest Rate Market Model
A Hybrid Market ModelCalibration algorithm
The Pedersen approachFutures/Forward Relation & Convexity CorrectionMerging Interest Rate & Commodity Calibrations
Futures vs. forward
Thus for the futures price G(t ,T ) observed at time t for maturityT , we have
G(t ,T ) = EQ[X (T ,T )|Ft ]
= X (t ,T )EQ
[exp
{∫ T
tσX (u,T )dWQ(u)
− 12
∫ T
tσ2
X (u,T )du +
∫ T
tσX (u,T )φ(u,T )du
}∣∣∣∣∣Ft
]
≈ X (t ,T ) exp
{∫ T
tσX (u,T )φ(u,T )du
}
where φ is the “frozen coefficient” approximation for φ.
Kay Pilz and Erik Schlögl A Hybrid Commodity and Interest Rate Market Model
A Hybrid Market ModelCalibration algorithm
The Pedersen approachFutures/Forward Relation & Convexity CorrectionMerging Interest Rate & Commodity Calibrations
Merging Interest Rate & Commodity Calibrations
Step 1:
Calibrate LMM for interest rates using Pedersen approach. Anoutput of this is the matrix W (I).
Step 2:
Calibrate the volatility of forward commodity prices to themarket using an appropriately modified Pedersen approach. Anoutput of this is the matrix W (C).
Kay Pilz and Erik Schlögl A Hybrid Commodity and Interest Rate Market Model
A Hybrid Market ModelCalibration algorithm
The Pedersen approachFutures/Forward Relation & Convexity CorrectionMerging Interest Rate & Commodity Calibrations
Step 3:
Suppose we have an exogenously given covariance matrix ΣCIof all forward LIBORSs and commodity prices.
In order to achieve an approximate fit to this covariance matrix,we exploit the property that multivariate normally distributedrandom variables are invariant under orthonormal rotations.
We seek a square matrix Q, which minimises
‖ΣCI −W (C)Q(W (I))>‖
and‖QQ> − I‖
We then replace W (C) by W (C)Q when determining thevolatility functions for forward commodity prices.
Kay Pilz and Erik Schlögl A Hybrid Commodity and Interest Rate Market Model
A Hybrid Market ModelCalibration algorithm
The Pedersen approachFutures/Forward Relation & Convexity CorrectionMerging Interest Rate & Commodity Calibrations
Notes
The dimension of Q is the total number of factors, whichmay be greater than or equal to the greater of the numberof factors in W (I) and W (C).W (I) and W (C) are padded with zeroes where needed.Due to the dependence of the convexity adjustment oninterest rate volatilities, steps 2 and 3 need to be repeatediteratively.
Kay Pilz and Erik Schlögl A Hybrid Commodity and Interest Rate Market Model
A Hybrid Market ModelCalibration algorithm
The Pedersen approachFutures/Forward Relation & Convexity CorrectionMerging Interest Rate & Commodity Calibrations
The commodity and interest rate market on thecalibration date 5 May 2008
2005 2006 2007 2008 200940
60
80
100
120
140
160
Years
US
D
Crude Oil Nearest Futures
0 1 2 3 4 5110
112
114
116
118
120Crude Oil Futures Curve
Maturity Times
US
D
0 1 2 3 4 50.025
0.03
0.035
0.04
0.045
0.053M Forward Rates
Reset Times
Dec
imal
Po
ints
Left: The WTI Crude Oil nearest futures between 2005 and end of 2008. The circle indicates the calibration date.
Middle: The futures curve as seen at calibration date with maturities up to five years. Right: The 3–month USD
forward rates for reset dates (expiries) between 3 months and 4 years and 9 months.
Kay Pilz and Erik Schlögl A Hybrid Commodity and Interest Rate Market Model
A Hybrid Market ModelCalibration algorithm
The Pedersen approachFutures/Forward Relation & Convexity CorrectionMerging Interest Rate & Commodity Calibrations
Historically estimated interest rate correlation matrix
01
23
45
6
0
1
2
3
4
5
60.4
0.5
0.6
0.7
0.8
0.9
1
Forward Time
Interest Forward Rate Correlation Matrix
Forward Time
Kay Pilz and Erik Schlögl A Hybrid Commodity and Interest Rate Market Model
A Hybrid Market ModelCalibration algorithm
The Pedersen approachFutures/Forward Relation & Convexity CorrectionMerging Interest Rate & Commodity Calibrations
Calibrated interest rate volatility matrix
0
0.5
1
1.5
2
2.5
30 1 2 3 4 5 6
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Interest Forward Rate Volatility
Forward Time
CalendarTime
Kay Pilz and Erik Schlögl A Hybrid Commodity and Interest Rate Market Model
A Hybrid Market ModelCalibration algorithm
The Pedersen approachFutures/Forward Relation & Convexity CorrectionMerging Interest Rate & Commodity Calibrations
Market prices vs. model prices
0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
1
1.2x 10
−3
Caplet Expiries
Caplets
1 1.5 21
2
3
4
5
6
7
8x 10
−3
Last Caplet Expiries
Caps
0 1 2 30
0.005
0.01
0.015
0.02
Swaption Expiries
Swaptions
Kay Pilz and Erik Schlögl A Hybrid Commodity and Interest Rate Market Model
A Hybrid Market ModelCalibration algorithm
The Pedersen approachFutures/Forward Relation & Convexity CorrectionMerging Interest Rate & Commodity Calibrations
Historically estimated commodity correlation matrix
00.5
11.5
22.5
3
0
0.5
1
1.5
2
2.5
30.88
0.9
0.92
0.94
0.96
0.98
1
Future Maturity
WTI Crude Oil Futures Correlation
Future Maturity
Kay Pilz and Erik Schlögl A Hybrid Commodity and Interest Rate Market Model
A Hybrid Market ModelCalibration algorithm
The Pedersen approachFutures/Forward Relation & Convexity CorrectionMerging Interest Rate & Commodity Calibrations
Calibrated commodity volatility matrix
0
0.5
1
1.5
2
2.5
3 00.5
11.5
22.5
3
0.2
0.25
0.3
0.35
0.4
0.45
Calendar Time
WTI Crude Oil Forward Volatility
Forward Time
Kay Pilz and Erik Schlögl A Hybrid Commodity and Interest Rate Market Model
A Hybrid Market ModelCalibration algorithm
The Pedersen approachFutures/Forward Relation & Convexity CorrectionMerging Interest Rate & Commodity Calibrations
Commodity futures vs. forwards & call option prices
0 0.5 1 1.5 2 2.5 3109
110
111
112
113
114
115
116
117
118
119
120Futures and Forwards
Time to Maturity0 0.5 1 1.5 2 2.5 3
2
4
6
8
10
12
14
16
18Fit of Call Prices
Time to Maturity
Kay Pilz and Erik Schlögl A Hybrid Commodity and Interest Rate Market Model
A Hybrid Market ModelCalibration algorithm
The Pedersen approachFutures/Forward Relation & Convexity CorrectionMerging Interest Rate & Commodity Calibrations
Target & model cross correlations
02
46
01
23
0.1
0.15
0.2
0.25
0.3
0.35
Interest Forwards
Cross−Correlations
Commodity Forwards 0
2
4
6
01
23
−0.15
−0.1
−0.05
0
0.05
0.1
Interest Forwards
Absolute Error of Cross−Correlation Fit
Commodity Forwards
Kay Pilz and Erik Schlögl A Hybrid Commodity and Interest Rate Market Model
A Hybrid Market ModelCalibration algorithm
The Pedersen approachFutures/Forward Relation & Convexity CorrectionMerging Interest Rate & Commodity Calibrations
Six-factor correlation fitting errors
02
46
01
23
0.1
0.15
0.2
0.25
0.3
0.35
Interest Forwards
Cross−Correlations
Commodity Forwards 0
2
4
6
01
23
−0.15
−0.1
−0.05
0
0.05
0.1
Interest Forwards
Absolute Error of Cross−Correlation Fit
Commodity Forwards
Kay Pilz and Erik Schlögl A Hybrid Commodity and Interest Rate Market Model
A Hybrid Market ModelCalibration algorithm
The Pedersen approachFutures/Forward Relation & Convexity CorrectionMerging Interest Rate & Commodity Calibrations
Factors before & after rotation
0 0.5 1 1.5 2 2.5 3−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6Originally Calibrated Volatility Factors
Forward Time0 0.5 1 1.5 2 2.5 3
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35Cross−Transformed Volatility Factors
Forward Time
Kay Pilz and Erik Schlögl A Hybrid Commodity and Interest Rate Market Model
A Hybrid Market ModelCalibration algorithm
The Pedersen approachFutures/Forward Relation & Convexity CorrectionMerging Interest Rate & Commodity Calibrations
Target & model cross correlations
02
46
01
23
0.1
0.15
0.2
0.25
0.3
0.35
Interest Forwards
Cross−Correlations
Commodity Forwards
0
2
4
6
0
1
2
3−0.06
−0.04
−0.02
0
0.02
0.04
Interest Forwards
Absolute Error of Cross−Correlation Fit
Commodity Forwards
Kay Pilz and Erik Schlögl A Hybrid Commodity and Interest Rate Market Model
A Hybrid Market ModelCalibration algorithm
The Pedersen approachFutures/Forward Relation & Convexity CorrectionMerging Interest Rate & Commodity Calibrations
Twelve-factor correlation fitting errors
02
46
01
23
0.1
0.15
0.2
0.25
0.3
0.35
Interest Forwards
Cross−Correlations
Commodity Forwards
0
2
4
6
0
1
2
3−0.06
−0.04
−0.02
0
0.02
0.04
Interest Forwards
Absolute Error of Cross−Correlation Fit
Commodity Forwards
Kay Pilz and Erik Schlögl A Hybrid Commodity and Interest Rate Market Model