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A Hybrid Market Model Calibration algorithm A Hybrid Commodity and Interest Rate Market Model 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 Commodity and Interest Rate Market Model Hybrid Market Model Calibration algorithm Recall: The basic LIBOR Market Model The cross–currency LIBOR Market Model Setup Discrete–tenor

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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