Factor Gaussian Short Rate Model (HJM-G2++)

S. Acar, K. Natcheva-Acar A guide on the implementation of the Heath-Jarrow-Morton Two- Factor Gaussian Short Rate Model (HJM-G2++)

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A Guide on the Implementation of the Heath-Jarrow-Morton Two-Factor Gaussian Short Rate Model (HJM-G2++)

Dr. Sarp Kaya Acar, Dr. Kalina Natcheva-Acar

July 23, 2009

Kaiserslautern, Germany Contents

Contents

1 Introduction to the Heath-Jarrow-Morton Two-Factor Gaussian Model (HJM- G2++) 2

1.1 Motivation...... 2

1.2 DeﬁnitionoftheHJM-G2++model ...... 2

1.3 Closedformsolutions...... 4

1.3.1 ThepriceofaEuropeanoptiononazero-couponbond ...... 6

1.3.2 Thepriceofacaplet ...... 6

1.3.3 Thepriceofaswaption...... 7

1.3.4 Implementation of the swaption semi closed-form formula ...... 12

1.4 Degenerationtoaone-factormodel ...... 13

2 Numerical implementation 15

2.1 MonteCarlosimulation...... 15

2.2 Treeconstruction ...... 16

3 Calibration to market data 20

4 Conclusion 25

5 Appendix 26

1 Introduction to the Heath-Jarrow-Morton Two-Factor Gaussian Model (HJM-G2++)

1.1 Motivation

In the literature, there are at least two equivalent two-factor Gaussian models for the instantaneous short rate. These are the original two-factor Hull White model (see [3]) and the G2++ one by Brigo and Mercurio (see [1]). Both these models ﬁrst specify a time homogeneous two-factor short rate dynamics and then by adding a deterministic shift function ϕ( ) ﬁt exactly the initial term structure of interest rates. However, the obtained results are rather· clumsy and not intuitive which means that a special care has to be taken for their correct numerical implementation.

On the other side, as noticed by Heath-Jarrow-Morton (1992) (HJM), virtually any exogenous short rate model can be derived within the HJM framework by choosing the class of the forward rate volatilities. By starting within the HJM framework and limiting the volatility of the forward rate to a deterministic, exponentially decaying one, we derive another two-factor Gaussian short rate model which has a very simple and intuitive form. Moreover, no ﬁtting to the initial term structure is required as this is contained in the model by its construction. Additionally, the dynamics of the underlying factors under the forward risk-neutral measure is a very simple one, which facilitates the derivation of closed-form solutions. We call the offered construction HJM-G2++ model due to its similarity to the G2++ model of Brigo and Mercuirio.

In this report, we investigate the offered HJM-G2++ model and derive closed-form solutions to the standard options needed for calibration purposes (caps, ﬂoors, bond options and swaptions). Further, we calibrate the model parameters to at-the-monety (ATM) Caps market volatilities and ATM swaption volatility surfaces and make comments on the ability of the model to ﬁt real market data.

Finally, we deal with different methods for the numerical implementation of HJM-G2++ for pricing complex claims with no closed-form solution. For that purpose, we offer a Monte Carlo simulation scheme and a lattice approximation with a construction similar to the one of Li, Ritchken and Sankarasubramanian [4] (for the Cheyette model [2]).

1.2 Deﬁnition of the HJM-G2++ model

Let us assume that the instantaneous forward rate follows a two-factor HJM model and its volatility is deterministic and given by

κ1(T t) κ2(T t) σ(t, T ) = σ1e− − + σ2e− −

2 Introduction to the Heath-Jarrow-Morton Two-Factor Gaussian Model (HJM-G2++)

+ + with ki R , σi R , i = 1, 2. The forward rate equation for the so-deﬁned volatility can be witten∈ as ∈

2 2 κi(T t) f(t, T ) = f(0,T ) + e− − Xi(t) + bk(t, T )Zik(0, t) (1) i " k # X=1 X=1 for

κ1(T t) 1 e− − b1(t, T ) = − , κ1 = 0 κ1 6 b (t, T ) = T t, κ = 0; 1 − 1 κ2(T t) 1 e− − b2(t, T ) = − , κ2 = 0 κ2 6 b (t, T ) = T t, κ = 0; 2 − 2 and cumulative quadrative variation that is deﬁned as

t t σ2 e 2κ1(t u) 2 2κ1(t s) 1(1 − − ) Z11(u, t) = cov(X1(s),X1(s))ds = σ1e− − ds = − u u 2κ1 Z t Z Z12(u, t) = cov(X1(s),X2(s))ds u Z t (κ1+κ2)(t s) = Z21(u, t) = ρ1,2 σ1σ2e− − ds u ρσ σ Z 1 2 (κ1+κ2)(t u) = 1 e− − κ1 + κ2 − t t σ2 e κ2(t u) 2 2κ2(t s) 2(1 − − ) Z (u, t) = cov(X (s),X (s))ds = σ e− − ds = − 22 2 2 2 2κ Zu Zu 2 Using r(t) = f(t, t) we can ﬁnd from (1) the short rate to be

r(t) = f(0, t) + X1(t) + X2(t) (2)

where following Cheyette’s model (see [2]) the state variables X1(t) and X2(t) are respectively deﬁned under an Equivalent Martingale Measure Q as

2 Q dX (t) = κ X (t) + Z ,k(0, t) dt + σ dW˜ (t),X (0) = 0 (3) 1 − 1 1 1 1 1 1 k ! X=1 2 Q dX (t) = κ X (t) + Z ,k(0, t) dt + σ dW˜ (t),X (0) = 0 (4) 2 − 2 2 2 2 2 2 k ! X=1 ˜1 ˜2 with correlation d W , W t = ρdt. h i

3 Introduction to the Heath-Jarrow-Morton Two-Factor Gaussian Model (HJM-G2++)

The price of a zero coupon bond P (t, T ) at time t can easily be calculated using integration of the forward rate to be

T r(s)ds T f(t,u)du P (t, T ) = E e− Rt t = e− Rt |F P (0 ,T ) = exp b (t, T )X (t) b (t, T )X (t) 0P ( , t) − 1 1 − 2 2 2 1 bi(t, T )bj(t, T )Zij(0, t) . −2 i,j X=1

1.3 Closed form solutions

Using the form of the analytical strong solution of a generalized linear stochastic differential equation, we can write for s t ≤ t t κi(t s) κi(u s) κi(u s) Xi(t) = e− − Xi(s) + Zi1(s, u)e − du + Zi2(s, u)e − du s s t Z Z κi(u s) ˜ Q + σie − dWi (u) s Z t t κi(t s) κi(t u) κi(t u) = Xi(s)e− − + Zi1(s, u)e− − du + Zi2(s, u)e− − du s s t Z Z κi(t u) ˜ Q + σie− − dWi (u) Zs In speciﬁc for i = 1 we have t t κ1(t s) κ1(t u) κ1(t u) X1(t) = X1(s)e− − + Z11(s, u)e− − du + Z12(s, u)e− − du s s t Z Z κ1(t u) ˜ Q + σ1e− − dW1 (u) Zs t σ2 κ1(t s) 1 2κ1(u s) κ1(t u) = X1(s)e− − + 1 e− − e− − du s 2κ1 − t Z t ρ , σ σ 1 2 1 2 (κ1+κ2)(u s) κ1(t u) κ1(t u) ˜ Q + 1 e− − e− − du + σ1e− − dW1 (u) s κ1 + κ2 − s Z 2 Z 2 κ1(t s) σ 1 e− − = X (s)e κ1(t s) + 1 − 1 − − κ2 2 1 ρ σ σ 1,2 1 2 κ1(t s) (κ1+κ2)(t s) + κ2 (κ1 + κ2)e− − + κ1e− − (κ1 + κ2)κ1κ2 − t h i κ1(t u) ˜ Q + σ1e− − dW1 (u). Zs

4 Introduction to the Heath-Jarrow-Morton Two-Factor Gaussian Model (HJM-G2++)

Thus, we have for the short rate for s t ≤ r(t) = f(s, t) + X1(t) + X2(t) 2 2 σ2 1 e κ1(t s) σ2 1 e κ2(t s) κ1(t s) κ2(t s) 1 − − 2 − − = X (s)e− − + X (s)e− − + − + − 1 2 κ2 κ2 2 1 2 2 ρ σ σ 1,2 1 2 κ1(t s) κ2(t s) + 1 e− − 1 e− − κ1κ2 − − t t κ1(t u) ˜ Q κ2(t u) ˜ Q + σ1e− − dW1 (u) + σ2e− − dW2 (u) Zs Zs where

Q κ1(t s) κ2(t s) E r(t) Fs = f(s, t) + X (s)e− − + X (s)e− − | 1 2 σ2 2 σ2 2 1 κ1(t s) 2 κ2(t s) + 2 1 e− − + 2 1 e− − 2κ1 − 2κ2 − ρσ σ 1 2 κ1(t s) κ2(t s) + 1 e− − 1 e− − κ1κ2 − − σ2 1 e 2κ1(t s) ρσ σ σ2 1 e 2κ2(t s) Q 1 − − 1 2 (κ1+κ2)(t s) 2 − − Var r(t) Fs = − + 2 1 e− − + − . | 2κ1 κ1 + κ2 − 2κ2 where we notice that the short rate process r(t) is conditionally Gaussian.

Denoting

2 2 σ 2 σ 2 Q F 1 κ1t 2 κ2t µr := E r(t) 0 = f(0, t) + 2 1 e− + 2 1 e− | 2κ1 − 2κ2 − ρσ σ 1 2 κ1t κ2t + 1 e− 1 e− κ1κ2 − − 2 2κ1t 2 2κ2t σ 1 e− ρσ σ σ 1 e− σ := VarQ r(t) F = 1 − + 2 1 2 1 e (κ1+κ2)t + 2 − r 0 κ κ κ − κ | 2 1 1 + 2 − 2 2

and using that the short rate is normally distributed with mean µr and variance σr we can easily estimate the risk-neutral probability for the short rate to become negative to be

µr Q(r(t) < 0) = EQ 11 r(t)<0 = EQ 11 r(t)−µr µr = ϕ > 0 { } σ < σ −σ r − r r where ϕ( ) denotes the cumulative distribution function of the standard normal distribution.· We remark here that although the upper probability is in most cases negligibly small, this is a deﬁnite drawback of the model.

Additionally, notice that the mean and the variance of the short rate in the HJM-G2++ and in the G2++ are the same (see [1], p.147) which means that the two models produce

5 Introduction to the Heath-Jarrow-Morton Two-Factor Gaussian Model (HJM-G2++)

equivalent dynamics of the short rate. This will be used later in the proof of the derivation of the closed-from solutions of caplets and bond options. However, the dynamics of the underlying factors X1(t) and X2(t) are slightly different which leads also to a slightly different closed-form solution for swaptions. For this reason, we will present in more details the derivation of the swaption formula.

1.3.1 The price of a European option on a zero-coupon bond

Theorem 1.1. Using the already speciﬁed short rate dynamics, the price ZBC(t, T, S, K) at time t of a European zero-bond call option with maturity T > t and strike K on a zero-coupon bond with maturity S > T is calculated to

P (t,S) 1 2 ln KP,T (t ) + 2 Σp(t,T,S) ZBC(t, T, S, K) = P (t, S)Φ  Σ,T,Sp(t ) 

 P (t,S) 1  2 ln KP,T (t ) 2 Σp(t,T,S) P (t, T )K Φ − −  Σ,T,Sp(t )    where Φ( ) denotes the cumulative standard normal distribution function and · 2 2 2 Σp(t, T, S) := b1(T,S) Z11(t, T ) + b2(T,S) Z22(t, T ) + 2b1(T,S)b2(T,S)Z12(t, T ).

The price ZBP(t, T, S, K) of a European bond put option is given by

P (t,S) 1 2 ln KP,T (t ) + 2 Σp(t,T,S) ZBP(t, T, S, K) = P (t, S)Φ − − Σ,T,Sp(t ) 

 P (t,S) 1 2 ln KP,T (t ) 2 Σp(t,T,S) +P (t, T )K Φ − . − Σ,T,Sp(t )    Proof: Due to the equivalence of the distribution of the short rate process r(t) in the 2FHW and in the G2++ model, we refer for the proof to [1].

1.3.2 The price of a caplet

Theorem 1.2. Using the already speciﬁed short rate dynamics, the price Caplet(t, T, S, K) at time t of a caplet resetting at time T > t, with payoff at time S > T

6 Introduction to the Heath-Jarrow-Morton Two-Factor Gaussian Model (HJM-G2++)

and strike K is calculated to

∗ K P (t,T ) 1 2 ln P,S(t ) + 2 Σp(t,T,S) Caplet(t, T, S, K) = P (t, T )Φ  Σ,T,Sp(t ) 

∗  K P (t,T ) 1 2  1 ln P,S(t ) 2 Σp(t,T,S) P (t,S)Φ − −K∗  Σ,T,Sp(t )    with K = 1 , Φ( ) denoting the cumulative standard normal distribution function ∗ 1+K(S T ) · and −

2 2 2 Σp(t, T, S) := b1(T,S) Z11(t, T ) + b2(T,S) Z22(t, T ) + 2b1(T,S)b2(T,S)Z12(t, T ).

Proof: see [1].

1.3.3 The price of a swaption

Consider an European option giving the right at time t0 = T (maturity) to enter an interest rate swap with payment times = t1, . . . , tn (reset times t0, . . . , tn 1 ), ﬁxed leg rate T { } { − } X and notional N. The year fraction from ti 1 to ti is denoted as usually by τi. Theorem 1.3. The arbitrage-free price at time− 0 of a European payer/receiver swaption is given by numerically computing the following one-dimensional integral

ESp(0,T, ,N,K) T − 2 1 x1 µx1 + 2 “ σ ” n e− x1 ∞ βi(x1) = NP (0,T ) Φ ( h (x )) λi(x )e Φ( h (x )) dx √ − 1 1 − 1 − 2 1 1 σx1 2π " i # Z−∞ X=1 (5)

and

ESr(0,T, ,N,K) T − 2 1 x1 µx1 + 2 “ σ ” n e− x1 ∞ βi(x1) = NP (0,T ) Φ (h (x )) λi(x )e Φ(h (x )) dx − √ 1 1 − 1 2 1 1 σx1 2π " i # Z−∞ X=1 (6)

7 Introduction to the Heath-Jarrow-Morton Two-Factor Gaussian Model (HJM-G2++)

where

x¯ µx ρx x (x µx ) h (x ) := 2 − 2 1 2 1 − 1 1 1 2 2 σx 1 ρ − σx 1 ρ 2 − x1x2 1 − x1x2 2 h (x ) := h (xp) + b (T, ti)σx p1 ρ 2 1 1 1 2 2 − x1x2 b1(T,ti)x1 q λi(x1) := ciA(T, ti)e−

1 2 2 x1 µx1 βi(x ) := b (T, ti) µx 1 ρ σ b (T, ti) + ρx x σx − 1 − 2 2 − 2 − x1x2 x2 2 1 2 2 σ x1 ti 1 2 f(0,u)du bk(T,ti)bj (T,ti)Zkj(0,T ) A(T, ti) := e− RT − 2 Pk,j=1 .

and c := c , . . . , cn such that ci := Xτi for i = 1, . . . , n 1 and cn := 1 + Xτn. { 1 } −

In addition x¯2 =x ¯2(x1) is the unique solution of the following equation (found via numerical methods)

n b1(T,ti)x1 b2(T,ti)x2 ciA(T, ti)e− − = 1 (7) i X=1 and

µx1 = µx2 = 0 T 2 2 2κ1(T u) σx1 := σ1e− − du = Z11(0,T ) 0 Z T 2 2 2κ2(T u) σx2 := σ2e− − du = Z22(0,T ) Z0 T (κ1+κ2)(T u) ρ1,2 σ1σ2e− − du Z (0,T ) ρ := 0 = 12 x1x2 σ σ R x1 x2 Z11(0,T )Z22(0,T ) Proof: p We will consider only the case of a European payer swaption and the one for the receiver will follow by analogy.

By deﬁnition of the payer swaption, we can write its discounted until time zero payoff as

n T r(u)du + Ne− R0 P (T, ti)τi (F (T, ti 1, ti) X) − − i ! X=1 where the forward Libor rate is deﬁned as

1 P (T, ti 1) P (T, ti) F (T, ti 1, ti) := − − , i = 1, . . . , n. − τ P (T,t ) i i

8 Introduction to the Heath-Jarrow-Morton Two-Factor Gaussian Model (HJM-G2++)

Using the deﬁnition of the forward Libor rate we can rewrite the payoff of the payer swaption as

n n n + T r(u)du Ne− R0 P (T, ti 1) P (T, ti) P (T, ti)τiX − − − i i i ! X=1 X=1 X=1 n + T r(u)du = Ne− R0 1 X τiP (T, ti) P (T, tn) − − i ! X=1 n + T r(u)du = Ne− R0 1 ciP (T, ti) . − i ! X=1 Taking expectation under the forward QT measure we obtain that the price of the payer swaption is given as

n + QT ESp(0,T, ,N,K) = NP (0,T )E 1 ciP (T, ti) . T − i ! ! X=1 where

t i f(0,u)du P (T, ti) = e− RT exp b1(T, ti)X1(T ) b2(T, ti)X2(T ) − − 2 1 bi(T, ti)bj(T, ti)Zij(0,T ) −2 i,j ! X=1 T where the joint density of X1(T ) and X2(T ) under measure Q is denoted by f(x1, x2). Further, due to Result (1) in the Appendix, the dynamics of X1(t) and X2(t) under the forward QT measure is given by

2 2 dX (t) = κ X (t) + Z ,k(0, t) b (t, T )σ ρ , b (t, T )σ σ dt 1 − 1 1 1 − 1 1 − 1 2 2 1 2 k ! X=1 QT +σ1dW1 (t),X1(0) = 0 2 2 dX (t) = κ X (t) + Z ,k(0, t) b (t, T )σ ρ , b (t, T )σ σ dt 2 − 2 2 2 − 2 2 − 1 2 1 1 2 k ! X=1 T T Q 2 Q +ρ , σ dW (t) + 1 ρ σ dW (t),X (0) = 0 1 2 2 1 − 1,2 2 2 2 q and using Result (2) of the Appendix, we can ﬁnd conditioning on F0 T T κ1(T u) Q X1(T ) = b1(T,T )Z11(0,T ) b2(T,T )Z12(0,T ) + σ1e− − dW1 (u) − − 0 T Z T κ1(T u) Q = σ1e− − dW1 (u) Z0

9 Introduction to the Heath-Jarrow-Morton Two-Factor Gaussian Model (HJM-G2++)

and T T T T κ2(T u) Q 2 κ2(T u) Q X2(T ) = ρ1,2 σ2e− − dW1 (u) + 1 ρ1,2 σ2e− − dW2 (u). 0 − 0 Z q Z Thus,

QT E (X (T ) ) := µx = 0 1 |F0 1 QT E (X (T ) ) := µx = 0 2 |F0 2 QT 2 V ar (X1(T ) 0) := σx1 = Z11(0,T ) T |F V arQ (X (T ) ) := σ2 = Z (0,T ) 2 |F0 x2 22 T (κ1+κ2)(T u) ρ1,2 σ1(u)σ2(u)e− − du Z (0,T ) corr(X (T ),X (T )) := ρ = 0 = 12 1 2 x1x2 σ σ R x1 x2 Z11(0,T )Z22(0,T ) and p

2 2 x1 µx (x1 µx )(x2 µx ) x2 µx exp 1 − 1 2ρ − 1 − 2 + − 2 2(1 ρ2 ) σx x1x2 σx σx σx − x1x2 1 − 1 2 2 f(x , x ) = − 1 2 2 2σπ x σx 1 ρ 1 2 − x1x2 p Hereon, the proof follows the same steps as in [1], p.163 but for completeness we will present it in more details here.

n + QT ESp(0,T, ,N,K) = NP (0,T )E 1 ciP (T, ti) T − i ! ! X=1 R2 n t i f(0,u)du = NP (0,T ) 1 cie− RT − i Z X=1 2 + 1 exp b (T, ti)x b (T, ti)x bi(T, ti)bj(T, ti)Zij(0,T ) f(x , x )dx dx − 1 1 − 2 2 − 2 1 2 1 2 i,j ! X=1 R2 n + b1(T,ti)x1 b2(T,ti)x2 = NP (0,T ) 1 ciA(T, ti)e− − f(x , x )dx dx − 1 2 1 2 i ! Z X=1

Next, freezing x1 and integrating over x2 we choose numerically x¯2 such that n b1(T,ti)x1 b2(T,ti)¯x2 ciA(T, ti)e− − = 1. i X=1

10 Introduction to the Heath-Jarrow-Morton Two-Factor Gaussian Model (HJM-G2++)

Then, denoting

b1(T,ti)x1 1 λi(x1) := ciA(T, ti)e− , γ := 2σπ x1 σx2 1 ρx x2 − 1 2 2 q 1 x1 µx1 ρx1x2 x1 µx1 E(x1) := 2 − , F (x1) := 2 − −2(1 ρ ) σx (1 ρ ) σx σx − x1x2 1 − x1x2 1 2 1 G(x ) := 1 2(1 ρ2 )σ2 − x1x2 x2 we have that

+ + n ∞ ∞ b2(T,ti)x2 ESp(0,T, ,N,K) = NP (0,T ) 1 λi(x1)e− T x¯2 − Z−∞ Z i=1 ! 2 X E(x1)+F (x1)(x2 µx ) G(x1)(x2 µx ) γe − 2 − − 2 dx2dx1 + + 2 ∞ E(x1) ∞ F (x1)(x2 µx ) G(x1)(x2 µx ) = NP (0,T ) γe e − 2 − − 2 dx2dx1 x¯2 Z−∞ Z + n + ∞ E(x1) b2(T,ti)µx ∞ (F (x1) b2(T,ti))(x2 µx ) NP (0,T ) γe − 2 λi(x1) e − − 2 − x¯2 Z−∞ i=1 Z 2 X G(x1)(x2 µx ) e− − 2 dx2dx1.

Using that

b 2 Ax2+Bx √π B B B e− = e 4A ϕ b 2A ϕ a 2A a √A − √2A − − √2A Z " s ! s !# where ϕ( ) denotes the cumulative standard distribution function, we obtain ·

ESp(0,T, ,N,K) T + F (x )2 ∞ E x √π 1 F (x1) ( 1) 4G(x1) = NP (0,T ) γe e 1 ϕ (x¯2 µx2 ) 2G(x1) dx1 G(x1) − − − 2G(x1) Z−∞ " !# − p2 + (F (x1) b2(T,ti)) ∞ E(xp) √π b2(T,ti)µx + p NP (0,T ) γe 1 e− 2 fG − G(x1) Z−∞ n p F (x1) b2(T, ti) λi(x ) 1 ϕ (¯x µx ) 2G(x ) − . 1 − 2 − 2 1 − i=1 " 2G(x1) !# X p p

11 Introduction to the Heath-Jarrow-Morton Two-Factor Gaussian Model (HJM-G2++)

1.3.4 Implementation of the swaption semi closed-form formula

In this section, we shall devote some attention to the implementation of (semi-)closed formula for the payer (resp. receiver) swaption price given by equation (5) (resp.(6) ), since it requires numerical procedures of root ﬁnding and integration.

Let us consider the pricing formula of the payer swaption, given by equation (5). According to authors best knowledge; one can compute the integral only numerically.

After intensive tests, see Table 1, we have chosen the 20 point Gaussian quadrature,

19 ∞ g(x)dx wjg(xj ) ≈ j Z−∞ X=0 where the weights and abscissas are set according to the Gauss-Legendre polynomials, see [6].

The limits of the integration are set according to the fact that the integrand in formula (5) is actually a bounded function against a normal distribution. See Figure 1. Hence, one can truncate the domain of integration to the interval [µx Nσx , µx + Nσx ], where N N. 1 − 1 1 1 ∈ Since the precision of the integral approximation is affected by the integral bounds, by keeping in mind that the integrand is monotonic at the left and right wings, we have chosen the constant N adaptively, according to the following algorithm.

1. Compute lower and upper bounds for N = 1: l = Nσx1 , u = Nσx1 2. Compute the integrand at l and u − 8 3. If Integrand(l) < 10− , set lower bound as l. Else increase N and go to step 1. 8 4. If Integrand(u) < 10− , set lower bound as u. Else increase N and go to step 1.

Now, let us consider the equation (7). We shall find the root of this equation. We bracket l r the root in an interval [x2, x2] and use a hybrid root finding algorithm, which is a combination of Newton-Raphson and bisection algorithms. The numerical procedures that we used can be found in [6]1. The precision for the root finding algorithm is set to be 10E 7., see Table 1. − In the view of Table 1, one observes a trade-off between the precision of the results and the computing time. As a reasonable trade-off, we have chosen the 20 point Gauss-Legendre and the precision 10e 07 for the root finding algorithm. − 1Bracketing algorithm (zbrac) is in page 356 and root finding algorithm (rtsafe) is in page 370

12 Introduction to the Heath-Jarrow-Morton Two-Factor Gaussian Model (HJM-G2++)

0.2 (0.8, 0.5, 0.2, 0.1, −0.73) 0.18 (0.5, 0.3, 0.2, 0.2, −0.5 ) 0.16

0.14

0.12

0.1

0.08

0.06

0.04

0.02

0 −1 −0.5 0 0.5 1

Figure 1 Plot of the integrand in formula (5) for a 1Y-swaption with an underlying 1Y swap and the underlying process has the

parameters (κ1 = 0.8, κ2 = 0.5,σ1 = 0.2, σ2 = 0.1, ρ = 0.73), (κ1 = 0.5, κ2 = 0.3,σ1 = 0.2, σ2 = 0.2, ρ = 0.5) − − 10e-05 10e-07 10e-09 10e-11 0.023053435634 0.023053435640 0.023053435634 0.023053435634 10 00:00:01.79 00:00:02.34 00:00:02.90 00:00:03.46 0.023073833332 0.023073766462 0.023073766455 0.023073766455 20 00:00:02.48 00:00:03.26 00:00:04.04 00:00:04.82 0.023073835520 0.023073766434 0.023073766427 0.023073766427 30 00:00:03.17 00:00:04.20 00:00:05.21 00:00:06.25 0.023073834632 0.023073766434 0.023073766427 0.023073766427 40 00:00:03.87 00:00:05.15 00:00:06.39 00:00:08.18

Table 1 Comparison of the sum of squared errors of the swaption surface calculated by different number of Gauss-Legendre polynomials(on the rows) and different precisions of the root ﬁnding algorithm (on the columns).

1.4 Degeneration to a one-factor model

We want to investigate the conditions under which we can derive the one factor Hull-White (extended Hull-White, Vasicek, from now on 1FHW) model as a special case of the HJM-G2++ model introduced in Subsection 1.1.

Let us deﬁne X(t) := X1(t) + X2(t).

13 Introduction to the Heath-Jarrow-Morton Two-Factor Gaussian Model (HJM-G2++)

Proposition 1. Let us assume that the coefﬁcients of the SDEs (3) and (4) satisfy

κ1 = κ2, ρ = 0.

Then, X(t) satisﬁes the following SDE:

e dX(t) = κX(t) + Z(0, t) dt + σdW (t) (8) − e 2 2 e e ˜ where κ = κ1 = κ2, σ = σ1 + σ2e, Z(0, t) = Z11(0, t) +eZ22(0, t), i.e., X is a 1FHW process. p Proof:eLet us considere the strong solutionse of SDEs (3) and (4) t t κ1(t s) κ1(t u) κ1(t u) ˜ Q X1(t) = X1(s)e− − + (Z11(s, u) + Z12(s, u)) e− − du + σ1e− − dW1 (u), Zs Zs

t t κ2(t s) κ2(t u) κ2(t u) ˜ Q X2(t) = X2(s)e− − + (Z21(s, u) + Z22(s, u)) e− − du + σ2e− − dW2 (u). Zs Zs Then,

κ1(t s) κ2(t s) X(t) := X1(s)e− − + X2(s)e− − t t κ1(t u) κ2(t u) e + (Z11(s, u) + Z12(s, u)) e− − du + (Z21(s, u) + Z22(s, u)) e− − du s s Z t t Z κ1(t u) ˜ Q κ2(t u) ˜ Q + σ1e− − dW1 (u) + σ2e− − dW2 (u). Zs Zs By substituting the assumptions in the above equation, we get

t κ1(t s) κ1(t u) X(t) = (X1(s) + X2(s)) e− − + (Z11(s, u) + Z22(s, u)) e− − du s t Z e 2 2 κ1(t u) ˜ Q + σ1 + σ2 e− − dW (u) s q Z t t κ(t s) κ(t u) κ(t u) Q = X(t)e− − + Z(s, u)e− − du + σ e− − dW˜ (u) e e e Zs Zs which is the solutione of SDE (8). e e

14 Numerical implementation

2 Numerical implementation

Since no closed solutions are available for the prices of complex structured notes, we have to use numerical methods for their approximation. For this purpose, we have chosen to use a Monte Carlo simulation and a lattice approximation. In this section, we shall shortly describe their implementation.

2.1 Monte Carlo simulation

For the Monte Carlo simulation we have chosen to use the classical Euler discretization scheme of the driving stochastic differential equations (3) and (4). Then, if π is an arbitrary discretization of [0,T ] such that 0 t < t < . . . < tn = T we have ≤ 0 1 2 π π π π π Q X (ti ) = X (ti) + κ X (ti) + Z ,k(0, ti) ∆t + σ ∆ W , 1 +1 1 − 1 1 1 i 1 i 1 k ! X=1 X1(0) = 0 2 π π π π π Q 2 π Q X (ti ) = X (ti) + κ X (ti) + Z ,k(0, ti) ∆t + σ ρ∆ W + 1 ρ ∆ W , 2 +1 2 − 2 2 2 i 2 i 1 − i 2 k ! X=1 p X2(0) = 0

π π Q Q π Q π with ∆i := ti ti 1, ∆i Wj := Wj (ti+1) Wj (ti) , i = 1, . . . , n, j = 1, 2. − − − Q Q Notice that we have decomposed the correlated Brownian motions W˜1 and W˜2 into a Q Q linear combination of independent Brownian motions W1 and W2 using Cholesky decomposition. In this form, it is straightforward to simulate the forward components using Monte Carlo simulation. If Z1 and Z2 are both standard normally distributed independent variables (i. e. Z1,Z2 (0, 1)) then we can rewrite the simulation scheme as ∼ N

2 π π π π π X (ti ) = X (ti) + κ X (ti) + Z ,k(0, ti) ∆t + σ Z ∆t , 1 +1 1 − 1 1 1 i 1 1 i k=1 ! X p X1(0) = 0 2 π π π π π 2 π X (ti ) = X (ti) + κ X (ti) + Z ,k(0, ti) ∆t + σ ρZ ∆t + 1 ρ Z ∆t , 2 +1 2 − 2 2 2 i 2 1 i − 2 i k=1 ! X p p p X2(0) = 0.

15 Numerical implementation

2.2 Treeconstruction

For shortness of the notations, let us denote the drifts of the two processes by ν1(t, X1) := κ1X1(t) + Z11(0, t) + Z12(0, t) and ν (t, X ) := −κ X (t) + Z (0, t) + Z (0, t). 2 2 − 2 2 21 22 a a Next, if we assume that at time t the tree is in state (x1, x2) then at time t + ∆t it can move to the following four states

a+ a+ x1 , x2 with probability puu

a+ a− x1 , x2 with probability pud

a− a+ x1 , x2 with probability pdu

a− a− x1 , x2 with probability pdd Deﬁnition 2.1. Let us deﬁne the up and down jumps of both process by

+ − xa := xa + J (t, xa) + 1 √∆σ t , xa := xa + J (t, xa) 1 √∆σ t 1 1 1 1 1 1 1 1 1 − 1 a+ a a a− a a x := x + J2(t, x ) + 1√∆σ t 2, x := x + J2(t, x ) 1√∆σ t 2 2 2 2 2 2 2 − where

a Z1 if Z1 even, J1(t, x1) := Z1 + 1 else. a Z2 if Z2 even, J2(t, x2) := Z2 + 1 else. ν t,xa √ t ν t,xa √ t for Z := ﬂoor 1( 1 ) ∆ and Z := ﬂoor 2( 2 ) ∆ . 1 σ1 2 σ2 Remark 2.1. Weh want to pointi out here thath we cannoti choose exactly a a a ν1(t, x1)√∆t a ν2(t, x2)√∆t J1(t, x1) = , J2(t, x1) = σ1 σ2 because of two reasons • If we want to construct a recombining tree we would like that the absolute value of the up-jump and the absolute value of the down-jump are integer multiples of the same a jump heights. Notice that this will not be the case for real values of J1(t, x1) and a a a J2(t, x2). Therefore, we need a construction of J1(t, x1) and J2(t, x2) that allows them to take only integer values;

• On the other side, if we choose simply

a a a ν1(t, x1)√∆t a ν2(t, x2)√∆t J1(t, x1) = ﬂoor , J2(t, x2) = ﬂoor " σ1 # " σ2 #

16 Numerical implementation

the tree will be allowed to expand endlessly, hence we will take no account of the mean-reversion in the underlying processes. Finally, if we set

a Z1 if Z1 even, J1(t, x1) := Z1 + 1 else. a Z2 if Z2 even, J2(t, x2) := Z2 + 1 else. ν t,xa √ t ν t,xa √ t for Z := ﬂoor 1( 1 ) ∆ and Z := ﬂoor 2( 2 ) ∆ we notice that in a "normal 1 σ1 2 σ2 case" of not tooh strong meani reversion, theh values of J1i(t, x1) and J2(t, x2) will most of the time be equal to zero. Thus, we will have a product of two binomial trees in the usual sense. But as soon as ν1(t, x1)√∆σ t > 1 or in other words as soon as the local mean dominates the local variance,| the branching| of the tree will be changed and thus J1(t, x1) and J2(t, x2) will take the values of 2 in the upper part of the tree and of 2 in the down part. Thus, we would have both required− effects, namely that the local variance will be approximately matched and the effect of mean reversion will be taken into account.

a a Theorem 2.1. If the subsequent probabilities for the (x1, x2) node are given by

a a 1 ρσ1σ2 + (J1 1)(J2 1)σ1σ2 ν1(t, x1)√∆) t(J2 1 σ2 ν2(t, x2)√∆) t(J1 1 σ1 puu = − − − − − − 4 " σ1σ2 # a a 1 ρσ1σ2 + (J1 + 1)(J2 + 1)σ1σ2 ν1(t, x1)√∆ t(J2 + 1)σ2 ν2(t, x2)√∆ t(J1 + 1)σ1 pdd = − − 4 " σ1σ2 # a a 1 ρσ1σ2 (J1 1)(J2 + 1)σ1σ2 + ν1(t, x1)√∆ t(J2 + 1)σ2 + ν2(t, x2)√∆) t(J1 1 σ1 pud = − − − − 4 " σ1σ2 # a a 1 ρσ1σ2 (J1 + 1)(J2 1)σ1σ2 + ν1(t, x1)√∆) t(J2 1 σ2 + ν2(t, x2)√∆t(J1 + 1)σ1 pdu = − − − − 4 " σ1σ2 #

then with the choice of jump heights given in Definition 2.1, the tree matches locally perfectly the first and approximately the second moments of the underlying processes and in addition, up to terms of order ∆t2 it matches the local covariance between the underlying processes. Proof: Denoting the approximated (with the lattice) local mean and variance respectively with E(∆xi) and var(∆xi), for i = 1, 2 and the true local mean and variance of the underlying process by E(∆xi) and var(∆xi), for i = 1, 2 we notice that matching the first b c

17 Numerical implementation

two moments implies that the risk-neutral tree probabilities have to solve

a+ a a− a ! a E(∆x ) = (puu + pud)(x x ) + (pdu + pdd)(x x ) = E(∆x ) = ν (t, x )∆t 1 1 − 1 1 − 1 1 1 1 (9) b 2 a+ a 2 a− a 2 E(∆x ) = (puu + pud)(x x ) + (pdu + pdd)(x x ) 1 1 − 1 1 − 1 ! = E(∆x2) = σ2∆t + ν (t, xa)2∆t2 (10) b 1 1 1 1 a+ a a− a ! a E(∆x ) = (puu + pdu)(x x ) + (pud + pdd)(x x ) = E(∆x ) = ν (t, x )∆t 2 2 − 2 2 − 2 2 2 2 (11) b 2 a+ a 2 a− a 2 E(∆x ) = (puu + pdu)(x x ) + (pdu + pdd)(x x ) 2 2 − 2 2 − 2 =! E(∆x2) = σ2∆t + ν (t, xa)2∆t2 (12) b 2 2 2 2 E(∆x1∆x2) = cov(∆x1∆x2) + E(∆x1)E(∆x2) a+ a a+ a a+ a a− a = puu(x x )(x x ) + pud(x x )(x x ) 1 − 1 2 − 2 1 − 1 2 − 2 b d a− a ba+ ab a− a a− a +pdu(x x )(x x ) + pdd(x x )(x x ) 1 − 1 2 − 2 1 − 1 2 − 2 ! a a 2 = E(∆x1∆x2) = σ1σ2ρ∆t + ν1(t, x1)ν(t, x2)∆t (13) 1 = puu + pud + pdu + pdd (14)

where ∆x1 and ∆x2 denote the changes in the respective processes x1 and x2 from time t to time t + ∆t. Ignoring the terms of order ∆t2 in the matching of the correlation and a± a± substituting with the deﬁnitions of x1 and x2 we solve equations (9), (11), (13) and (14) for the four unknown probabilities and obtain

a a 1 ρσ1σ2 + (J1 1)(J2 1)σ1σ2 ν1(t, x1)√∆) t(J2 1 σ2 ν2(t, x2)√∆) t(J1 1 σ1 puu = − − − − − − 4 " σ1σ2 # a a 1 ρσ1σ2 + (J1 + 1)(J2 + 1)σ1σ2 ν1(t, x1)√∆ t(J2 + 1)σ2 ν2(t, x2)√∆ t(J1 + 1)σ1 pdd = − − 4 " σ1σ2 # a a 1 ρσ1σ2 (J1 1)(J2 + 1)σ1σ2 + ν1(t, x1)√∆ t(J2 + 1)σ2 + ν2(t, x2)√∆t(J1 1)σ1 pud = − − − − 4 " σ1σ2 # a a 1 ρσ1σ2 (J1 + 1)(J2 1)σ1σ2 + ν1(t, x1)√∆) t(J2 1 σ2 + ν2(t, x2)√∆t(J1 + 1)σ1 pdu = − − − − 4 " σ1σ2 #

Next, let us calculate the local conditional variances for the already found probabilities ﬁrst

18 Numerical implementation

for the x1 process: var (∆x ) = E ∆x2 E (∆x )2 1 1 − 1 a+ a 2 a− a 2 a 2 2 = (puu + pud)(x x ) + (pdu + pdd)(x x ) ν1(t, x ) ∆t b b 1 − 1 1 − 1 − 1 c a ν1(t, x1)√∆t (J1 1)σ1 2 2 = − − (1 + J1) ∆tσ1 2σ1 ! a ν1(t, x1)√∆t (J1 1)σ1 2 2 a 2 + 1 − − (1 J1) ∆tσ1 ν1(t, x1) ∆t − 2σ1 ! − − = J2∆tσ2 + 2ν (t, xa)∆t ∆σ t 2J ν (t, xa)2∆t + σ2∆t − 1 1 1 1 1 1 − 1 1 1 = (J √∆σ t ν (t, xa)∆qt)2 + σ2∆t − 1 1 − 1 1 1 2 and since var (∆x1) = σ1∆t we have as in the rotated tree that the local variance is approximately matched due to the right deﬁnition of the J1 process. By symmetry, the same follows for the x2 process.

Notice that although the sum of the probabilities is one, they are not necessarily bounded between 0 and 1. It can however be shown that when we increase the reﬁnement, the probabilities converge to values between 0 and 1 (see [5]). A similar problem is encountered also in the Hull and White [3] 2-factor trinomial tree construction and also by Brigo and Mercurio [1] in their G2++ quadrinomial tree construction. There are (at least) three possible ways to proceed.

The first approach is to change the correlation coefficient at every node where the probabilities are not bounded in [0, 1], until we have well-defined probabilities. Brigo and Mercurio claim that this procedure although theoretically not consistent is practically applicable as the correlation has a very negligible contribution to the result.

A second way, (considered by Zvan, Forsyth and Vetzal [7]) is to treat the probabilities of the tree as weights and neglect that they are not properly defined. In this respect, the authors show for a variety of finite difference constructions (although for rather regular parameters) that the meshes allowing for negative coefficients satisfy discrete maximum and minimum principles as the mesh size parameter approaches zero and show no obvious oscillations in the level curves of the priced option values.

Here we follow the second standard approach in the construction of the quadrinomial tree, since at least theoretically we can increase the reﬁnement of the tree until we have well-deﬁned probabilities.

A third way to proceed is to deﬁne two new processes, constructed as a linear combination of the original ones and orthogonal to each other (see [5]). The approximating tree for the new orthogonal processes will have then well deﬁned probabilities.

19 Calibration to market data

3 Calibration to market data

In this section we will ﬁt the HJM-G2++ model to market ATM caps/ﬂoors volatilities and ATM swaption volatility surfaces.

We will calibrate the model parameters by minimizing the squared differences between the market and the model prices. For that reason, let us assume we have n market prices Market price1,..., Market pricen and with the same speciﬁcations for a parameter vector Θ which we want to calibrate, we have n model prices Model price1,..., Model pricen. The objective function is the sum of the squared differences and is deﬁned as

n (Market price Model price )2 . i − i i X=1

Since most of the market prices are cited in BS volatilities, we deﬁne also the absolute error between a one market BS volatility and its calibrated model correspondence as

absolute errori = Market volatility Model volatility , i = 1, . . . , n. | i − i|

In the implementation of the calibration of the HJM-G2++ we have the following inputs and outputs:

Input: Output:

Initial yield curve Calibrated parameter values of κ1, κ2, σ1, σ2, ρ

Market Caps/Floors or ATM Swaption Reports of the calibration results volatilities or prices

Type of the used Solver - Local (Simplex) or Global (Hybrid Asa) optimizer

The results of the calibration are delivered in Table 2 until Table 9.

20 Calibration to market data

0.06 ATM market volatility 18 0.058 ATM model volatility

0.056 17

0.054 16 0.052

Yield 0.05 15 Volatility in % 0.048 14

0.046 13 0.044

0.042 12 5 10 15 20 25 30 0 5 10 15 20 Time Maturity

(a) Initial term structure (b) At the money caps/ﬂoors market/model volatilities

Figure 2 Calibration to ATM Caps - data from 13.02.2001

0.056 21 ATM market volatility 20 ATM model volatility 0.054

19 0.052 18

0.05 17 Yield 0.048 16 Volatility in % 15 0.046 14

0.044 13

0.042 12 0 10 20 30 40 50 60 0 5 10 15 20 Time Maturity

(a) Initial term structure (b) At the money caps/ﬂoors market/model volatilities

Figure 3 Calibration to ATM Caps - data from 06.08.2008

21 Calibration to market data

0.04 70 ATM model volatility 65 ATM market volatility

0.035 60

55 0.03 50

45 0.025 Yield 40 Volatility in % 35 0.02 30

0.015 25 20

0.01 15 0 10 20 30 40 50 60 0 5 10 15 20 Time Maturity

(a) Initial term structure (b) At the money caps/ﬂoors market/model volatilities

Figure 4 Calibration to ATM Caps - data from 30.04.2009

4.5 80 ATM market volatility 4 ATM model volatility 70

3.5 60 3

2.5 50 Yield 2 40 Volatility in % 1.5 30 1

20 0.5

0 10 0 10 20 30 40 50 0 5 10 15 20 25 30 Time Maturity

(a) Initial term structure (b) At the money caps/ﬂoors market/model volatilities

Figure 5 Calibration to ATM Caps - data from 08.07.2009

22 Calibration to market data

18 5

16 4

14 3

12 2 Absolute error 10 1

Swaption BS implied volatility 8 0 10 10 10 10 8 8 5 6 5 6 4 4 2 2 0 0 0 0 Swaption maturity Tenor Swaption maturity Tenor

(a) At the money market swaption volatility surface (b) Calibration absolute error

Figure 6 Calibration to ATM Swaptions - data from 13.02.2001

25 6

5 20 4

15 3

2 10 Absolute error 1

5 0 Swaption BS implied volatility in % 30 30 30 30 20 20 20 20 10 10 10 10 0 0 0 0 Swaption maturity Tenor Swaption maturity Tenor

(a) At the money market swaption volatility surface (b) Calibration absolute error

Figure 7 Calibration to ATM Swaptions - data from 06.08.2008

23 Calibration to market data

60 60

50 50 40 40 30 30 20 Absolute error 20 10

10 0 Swaption BS implied volatility in % 30 30 30 30 20 20 20 20 10 10 10 10 0 0 0 0 Swaption maturity Tenor Swaption maturity Tenor

(a) At the money market swaption volatility surface (b) Calibration absolute error

Figure 8 Calibration to ATM Swaptions - data from 30.04.2009

60 15

50 10 40

30 5 Absolute error 20

10 0 Swaption BS implied volatility in % 30 30 30 30 20 20 20 20 10 10 10 10 0 0 0 0 Swaption maturity Tenor Swpation maturity Tenor

(a) At the money market swaption volatility surface (b) Calibration absolute error

Figure 9 Calibration to ATM Swaptions - data from 08.07.2009

24 Conclusion

4 Conclusion

We have presented the HJM-G2++ model which is a two-factor Gaussian short rate model, equivalent to the original two-factor Hull and White model and the G2++ model by Brigo and Mercurio. Although an equivalent representation, the model has (due to its derivation within the HJM framework) a very intuitive form which does not need additional ﬁtting to the initial term structure and is an easier one for numerical implementation.

We have calibrated the HJM-G2++ model to different sets of market data of ATM Caps/Floors volatilities and ATM Swaption volatility surfaces. The obtained results show that due to its gaussian nature, the model is not suitable of ﬁtting strongly humped short-term effects. However, the model delivers satisfactory ﬁtting to options with middle and long-term maturities and is able to reproduce humped volatility structures.

Finally, we have offered a Monte Carlo simulation scheme and a lattice approximation as numerical methods for pricing claims with no closed-form solution. As a more general method for pricing structured notes, we consider here the tree approximation method since it can be easily adapted for pricing derivatives with American features as well as strongly path-dependent options (see [5]). Further, the lattice approximation scheme has a faster convergence than the Monte Carlo methods. However, one can use the Monte Carlo simulation results as a back-testing for the tree approximation.

25 Appendix

5 Appendix

Result (1) The equivalent T -forward risk-adjusted measure QT (associated with the bond maturing at time T ) is deﬁned as dQT B(0)P (T,T ) = , dQ 0B(T )P ( ,T ) where B(t) denotes the money-market account, driven by dB(t) = B(t)r(t)dt, B(0) = 1. Notice that T T T dQ exp 0 r(u)du exp 0 [f(0, u) + X1(u) + X2(u)] du = − = − dQ 0PR( ,T ) R 0P ( ,T ) T = exp [X (u) + X (u)] du . − 1 2 Z0

Then we ﬁnd that T 1 T T X (u)du = σ2b (u,T )2du + b (u, T )σ dW˜ Q(u) 1 2 1 1 1 1 1 Z0 Z0 Z0 T κ (κ + κ )e κ1(T u) + κ e (κ1+κ2)(T u) +ρ σ σ 2 − 1 2 − − 1 − − du 1,2 1 2 κ κ (κ + κ ) Z0 1 2 1 2 and applying the same steps in the integration of the X2-process, we obtain T T T T 1 2 2 1 2 2 X1(u)du + X2(u)du = b1(x,T ) σ1dx + b2(x,T ) σ2dx 0 0 2 0 2 0 Z Z Z T Z +ρ1,2 b1(x, T )b2(x, T )σ1σ2dx 0 T Z T ˜ Q ˜ Q + b1(x, T )σ1dW1 (x) + b2(x, T )σ2dW2 (x). Z0 Z0 ˜ Q ˜ Q Further, we decompose the Brownian motions W1 (t) and W2 (t) into a sum of two Q Q independent Brownian motions W1 (t) and W2 (t) such that ˜ Q Q dW1 (t) = dW1 (t) Q Q 2 Q dW˜ (t) = ρ , dW (t) + 1 ρ dW (t) 2 1 2 1 − 1,2 2 q

26 Appendix

and then we can write

T T T 2 dQ 1 2 1 2 = exp (b1(x,T )σ1 + ρ1,2b2(x, T )σ2(x)) dx 1 ρ1,2b2(x,T )σ2 dx dQ − 2 0 − 2 0 − hT Z T Z q Q 2 Q (b1(x, T )σ1 + ρ1,2b2(x, T )σ2) dW1 (x) 1 ρ1,2b2(x,T )σ2 dW2 (x) . − 0 − 0 − Z Z q i Next, the Girsanov Theorem for the change of measure implies (referring to [1]) that the QT QT T Brownian motions W1 (t) and W2 (t), under the forward measure Q , are given as

QT Q dW1 (t) = dW1 (t) + (b1(t, T )σ1 + ρ1,2b2(t, T )σ2) dt T dW Q (t) = dW Q(t) + 1 ρ2 b (t,T )σ dt. 2 2 − 1,2 2 2 q Q Q QT Since W1 (t) and W2 (t) are independent Brownian motions, so are also W1 (t) and QT W2 (t). Then, recalling that the processes X1(t) and X2(t) are given under the measure Q as

2 Q dX1(t) = κ1X1(t) + Z1,k(0, t) dt + σ1dW1 (t),X1(0) = 0 − ! Xk=1 2 Q 2 Q dX2(t) = κ2X2(t) + Z2,k(0, t) dt + ρ1,2σ2dW1 (t) + 1 ρ1,2σ2dW2 (t),X2(0) = 0 − ! − Xk=1 q we can write their dynamics under the forward QT measure as

2 2 dX1(t) = κ1X1(t) + Z1,k(0, t) b1(t, T )σ1 ρ1,2b2(t, T )σ1σ2 dt − − − ! Xk=1 QT +σ1dW1 (t),X1(0) = 0 2 2 dX2(t) = κ2X2(t) + Z2,k(0, t) b2(t, T )σ2 ρ1,2b1(t, T )σ1σ2 dt − − − ! Xk=1 T T Q 2 Q +ρ , σ dW (t) + 1 ρ σ dW (t),X (0) = 0. 1 2 2 1 − 1,2 2 2 2 q

27 Appendix

Result (2): Under measure QT , we want to ﬁnd X (t) and X (t) for t T . We have that 1 2 ≤ t t t κ1(t u) κ1(t u) 2 κ1(t u) X1(t) = Z11(0, u)e− − du + Z12(0, u)e− − du σ1e− − b1(0,T )du 0 0 − 0 Z t Z t Z T (κ1+κ2)(t 0) κ1(t u) Q ρ12 σ1σ2e− − b2(u, T )du + σ1e− − dW1 (u) − 0 0 t Z t Z 2 κ1(t u) (κ1+κ2)(t 0) = σ1e− − b1(0, t)du + ρ12 σ1σ2e− − b2(u, t)du 0 0 Z t Z t 2 κ1(t u) (κ1+κ2)(t 0) σ1e− − b1(0,T )du ρ12 σ1σ2e− − b2(u, T )du − 0 − 0 Z t Z T κ1(t u) Q + σ1e− − dW1 (u) 0 Z t T κ1(t u) Q = b (t, T )Z (0, t) b (t, T )Z (0, t) + σ e− − dW (u) − 1 11 − 2 12 1 1 Z0 Thus,

T EQ (X (t) ) := b (t, T )Z (0, t) b (t, T )Z (0, t) 1 |F0 − 1 11 − 2 12 QT E (X2(t) 0) := b2(t, T )Z21(0, t) b2(t, T )Z22(0, t) |F − t − T Q 2 2κ1(t u) V ar (X1(t) 0) := σ1e− − du = Z11(0, t) |F 0 Z t T Q 2 2κ2(t u) V ar (X (t) ) := σ e− − du = Z (0, t) 2 |F0 2 22 Z0 t (κ1+κ2)(t u) ρ1,2 0 σ1σ2e− − du Z12(0, t) corr(X1(t),X2(t)) := = R Z11(0, t)Z22(0, t) Z11(0, t)Z22(0, t) p p

28 References

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Nickel Modeling and simulation of the pressing Keywords: Stationary heat equation, effective ther- Dynamic transportation of patients in hos- section of a paper machine mal conductivity, explicit jump, discontinuous coeffi- pitals Keywords: paper machine, computational fluid dynam- cients, virtual material design, microstructure simula- Keywords: in-house hospital transportation, dial-a-ride, ics, porous media tion, EJ-HEAT dynamic mode, tabu search (41 pages, 2007) (21 pages, 2006) (37 pages, 2006) 114. R. Ciegis, O. Iliev, Z. Lakdawala 95. A. Naumovich 105. Th. Hanne On parallel numerical algorithms for simu- On a finite volume discretization of the Applying multiobjective evolutionary algo- lating industrial filtration problems three-dimensional Biot poroelasticity sys- rithms in industrial projects Keywords: Navier-Stokes-Brinkmann equations, finite tem in multilayered domains Keywords: multiobjective evolutionary algorithms, dis- volume discretization method, SIMPLE, parallel comput- Keywords: Biot poroelasticity system, interface problems, crete optimization, continuous optimization, electronic ing, data decomposition method finite volume discretization, finite difference method circuit design, semi-infinite programming, scheduling (24 pages, 2007) (21 pages, 2006) (18 pages, 2006) 115. N. Marheineke, R. Wegener 96. M. Krekel, J. Wenzel 106. J. Franke, S. Halim Dynamics of curved viscous fibers with sur- A unified approach to Credit Default Swap Wild bootstrap tests for comparing signals face tension tion and Constant Maturity Credit Default and images Keywords: Slender body theory, curved viscous bers Swap valuation Keywords: wild bootstrap test, texture classification, with surface tension, free boundary value problem Keywords: LIBOR market model, credit risk, Credit De- textile quality control, defect detection, kernel estimate, (25 pages, 2007) fault Swaption, Constant Maturity Credit Default Swap- nonparametric regression method (13 pages, 2007) 116. S. Feth, J. Franke, M. Speckert (43 pages, 2006) Resampling-Methoden zur mse-Korrektur 107. Z. Drezner, S. Nickel und Anwendungen in der Betriebsfestigkeit 97. A. Dreyer Solving the ordered one-median problem in Keywords: Weibull, Bootstrap, Maximum-Likelihood, Interval Methods for Analog Circiuts the plane Betriebsfestigkeit Keywords: interval arithmetic, analog circuits, tolerance Keywords: planar location, global optimization, ordered (16 pages, 2007) analysis, parametric linear systems, frequency response, median, big triangle small triangle method, bounds, symbolic analysis, CAD, computer algebra numerical experiments 117. H. Knaf (36 pages, 2006) (21 pages, 2007) Kernel Fisher discriminant functions – a con- cise and rigorous introduction 98. N. Weigel, S. Weihe, G. Bitsch, K. Dreßler 108. Th. Götz, A. Klar, A. Unterreiter, Keywords: wild bootstrap test, texture classification, Usage of Simulation for Design and Optimi- R. Wegener textile quality control, defect detection, kernel estimate, zation of Testing Numerical evidance for the non-existing of nonparametric regression Keywords: Vehicle test rigs, MBS, control, hydraulics, solutions of the equations desribing rota- (30 pages, 2007) testing philosophy tional fiber spinning (14 pages, 2006) Keywords: rotational fiber spinning, viscous fibers, 118. O. Iliev, I. Rybak boundary value problem, existence of solutions On numerical upscaling for flows in hetero- 99. H. Lang, G. Bitsch, K. Dreßler, M. Speckert (11 pages, 2007) geneous porous media Comparison of the solutions of the elastic and elastoplastic boundary value problems Keywords: numerical upscaling, heterogeneous porous 128. M. Krause, A. Scherrer 137. E. Savenkov, H. Andrä, O. Iliev∗ media, single phase flow, Darcy‘s law, multiscale prob- On the role of modeling parameters in IMRT An analysis of one regularization approach lem, effective permeability, multipoint flux approxima- plan optimization for solution of pure Neumann problem tion, anisotropy Keywords: intensity-modulated radiotherapy (IMRT), Keywords: pure Neumann problem, elasticity, regular- (17 pages, 2007) inverse IMRT planning, convex optimization, sensitiv- ization, finite element method, condition number ity analysis, elasticity, modeling parameters, equivalent (27 pages, 2008) 119. O. Iliev, I. Rybak uniform dose (EUD) (18 pages, 2007) On approximation property of multipoint 138. O. Berman, J. Kalcsics, D. Krass, S. Nickel flux approximation method The ordered gradual covering location Keywords: Multipoint flux approximation, finite volume 129. A. Wiegmann problem on a network method, elliptic equation, discontinuous tensor coeffi- Computation of the permeability of porous Keywords: gradual covering, ordered median function, cients, anisotropy materials from their microstructure by FFF- network location (15 pages, 2007) Stokes (32 pages, 2008) Keywords: permeability, numerical homogenization, 120. O. Iliev, I. Rybak, J. Willems fast Stokes solver 139. S. Gelareh, S. Nickel (24 pages, 2007) On upscaling heat conductivity for a class of Multi-period public transport design: A industrial problems novel model and solution approaches Keywords: Multiscale problems, effective heat conduc- 130. T. Melo, S. Nickel, F. Saldanha da Gama Keywords: Integer programming, hub location, public tivity, numerical upscaling, domain decomposition Facility Location and Supply Chain Manage- transport, multi-period planning, heuristics (21 pages, 2007) ment – A comprehensive review (31 pages, 2008) Keywords: facility location, supply chain management, 121. R. Ewing, O. Iliev, R. Lazarov, I. Rybak network design 140. T. Melo, S. Nickel, F. Saldanha-da-Gama (54 pages, 2007) On two-level preconditioners for flow in Network design decisions in supply chain porous media planning Keywords: Multiscale problem, Darcy‘s law, single 131. T. Hanne, T. Melo, S. Nickel Keywords: supply chain design, integer programming phase flow, anisotropic heterogeneous porous media, Bringing robustness to patient flow models, location models, heuristics numerical upscaling, multigrid, domain decomposition, management through optimized patient (20 pages, 2008) efficient preconditioner (18 pages, 2007) transports in hospitals Keywords: Dial-a-Ride problem, online problem, case 141. C. Lautensack, A. Särkkä, J. Freitag, study, tabu search, hospital logistics K. Schladitz 122. M. Brickenstein, A. Dreyer (23 pages, 2007) POLYBORI: A Gröbner basis framework Anisotropy analysis of pressed point processes for Boolean polynomials 132. R. Ewing, O. Iliev, R. Lazarov, I. Rybak, Keywords: Gröbner basis, formal verification, Boolean Keywords: estimation of compression, isotropy test, J. Willems polynomials, algebraic cryptoanalysis, satisfiability nearest neighbour distance, orientation analysis, polar (23 pages, 2007) An efficient approach for upscaling proper- ice, Ripley’s K function ties of composite materials with high con- (35 pages, 2008) trast of coefficients 123. O. Wirjadi Keywords: effective heat conductivity, permeability of 142. O. Iliev, R. Lazarov, J. Willems Survey of 3d image segmentation methods fractured porous media, numerical upscaling, fibrous A Graph-Laplacian approach for calculating Keywords: image processing, 3d, image segmentation, insulation materials, metal foams the effective thermal conductivity of com- binarization (16 pages, 2008) (20 pages, 2007) plicated fiber geometries Keywords: graph laplacian, effective heat conductivity, 133. S. Gelareh, S. Nickel numerical upscaling, fibrous materials 124. S. Zeytun, A. Gupta New approaches to hub location problems (14 pages, 2008) A Comparative Study of the Vasicek and the in public transport planning CIR Model of the Short Rate Keywords: integer programming, hub location, trans- 143. J. Linn, T. Stephan, J. Carlsson, R. Bohlin Keywords: interest rates, Vasicek model, CIR-model, portation, decomposition, heuristic Fast simulation of quasistatic rod deforma- calibration, parameter estimation (25 pages, 2008) (17 pages, 2007) tions for VR applications Keywords: quasistatic deformations, geometrically 134. G. Thömmes, J. Becker, M. Junk, A. K. Vai- exact rod models, variational formulation, energy min- 125. G. Hanselmann, A. Sarishvili kuntam, D. Kehrwald, A. Klar, K. Steiner, imization, finite differences, nonlinear conjugate gra- Heterogeneous redundancy in software A. Wiegmann dients quality prediction using a hybrid Bayesian A Lattice Boltzmann Method for immiscible (7 pages, 2008) approach multiphase flow simulations using theLevel Keywords: reliability prediction, fault prediction, non- Set Method 144. J. Linn, T. Stephan homogeneous poisson process, Bayesian model aver- Keywords: Lattice Boltzmann method, Level Set aging Simulation of quasistatic deformations us- method, free surface, multiphase flow (17 pages, 2007) ing discrete rod models (28 pages, 2008) Keywords: quasistatic deformations, geometrically exact rod models, variational formulation, energy min- 126. V. Maag, M. Berger, A. Winterfeld, K.-H. 135. J. Orlik imization, finite differences, nonlinear conjugate gra- Küfer Homogenization in elasto-plasticity dients (9 pages, 2008) A novel non-linear approach to minimal Keywords: multiscale structures, asymptotic homogeni- area rectangular packing zation, nonlinear energy Keywords: rectangular packing, non-overlapping con- (40 pages, 2008) 145. J. Marburger, N. Marheineke, R. Pinnau straints, non-linear optimization, regularization, relax- Adjoint based optimal control using mesh- ation (18 pages, 2007) 136. J. Almquist, H. Schmidt, P. Lang, J. Deitmer, less discretizations M. Jirstrand, D. Prätzel-Wolters, H. Becker Keywords: Mesh-less methods, particle methods, Eul- Determination of interaction between erian-Lagrangian formulation, optimization strategies, 127. M. Monz, K.-H. Küfer, T. Bortfeld, C. Thieke adjoint method, hyperbolic equations MCT1 and CAII via a mathematical and Pareto navigation – systematic multi-crite- (14 pages, 2008 physiological approach ria-based IMRT treatment plan determina- Keywords: mathematical modeling; model reduction; tion electrophysiology; pH-sensitive microelectrodes; pro- 146. S. Desmettre, J. Gould, A. Szimayer Keywords: convex, interactive multi-objective optimiza- ton antenna Own-company stockholding and work effort tion, intensity modulated radiotherapy planning (20 pages, 2008) preferences of an unconstrained executive (15 pages, 2007) Keywords: optimal portfolio choice, executive compen- sation (33 pages, 2008) 147. M. Berger, M. Schröder, K.-H. Küfer Keywords: Fiber-fluid interaction, slender-body theory, 166. J. I. Serna, M. Monz, K.-H. Küfer, C. Thieke A constraint programming approach for the turbulence modeling, model reduction, stochastic dif- Trade-off bounds and their effect in multi- two-dimensional rectangular packing prob- ferential equations, Fokker-Planck equation, asymptotic criteria IMRT planning expansions, parameter identification lem with orthogonal orientations Keywords: trade-off bounds, multi-criteria optimization, (21 pages, 2009) Keywords: rectangular packing, orthogonal orienta- IMRT, Pareto surface tions non-overlapping constraints, constraint propa- (15 pages, 2009) gation 157. E. Glatt, S. Rief, A. Wiegmann, M. Knefel, (13 pages, 2008) E. Wegenke 167. W. Arne, N. Marheineke, A. Meister, R. We- Structure and pressure drop of real and vir- gener 148. K. Schladitz, C. Redenbach, T. Sych, tual metal wire meshes Numerical analysis of Cosserat rod and M. Godehardt Keywords: metal wire mesh, structure simulation, string models for viscous jets in rotational model calibration, CFD simulation, pressure loss Microstructural characterisation of open spinning processes (7 pages, 2009) foams using 3d images Keywords: Rotational spinning process, curved viscous Keywords: virtual material design, image analysis, open fibers, asymptotic Cosserat models, boundary value foams 158. S. Kruse, M. Müller problem, existence of numerical solutions (30 pages, 2008) Pricing American call options under the as- (18 pages, 2009) sumption of stochastic dividends – An ap- 149. E. Fernández, J. Kalcsics, S. Nickel, plication of the Korn-Rogers model 168. T. Melo, S. Nickel, F. Saldanha-da-Gama R. Ríos-Mercado Keywords: option pricing, American options, dividends, An LP-rounding heuristic to solve a multi- A novel territory design model arising in dividend discount model, Black-Scholes model period facility relocation problem (22 pages, 2009) the implementation of the WEEE-Directive Keywords: supply chain design, heuristic, linear pro- Keywords: heuristics, optimization, logistics, recycling gramming, rounding (28 pages, 2008) 159. H. Lang, J. Linn, M. Arnold (37 pages, 2009) Multibody dynamics simulation of geomet- 150. H. Lang, J. Linn rically exact Cosserat rods 169. I. Correia, S. Nickel, F. Saldanha-da-Gama Lagrangian field theory in space-time for Keywords: flexible multibody dynamics, large deforma- Single-allocation hub location problems tions, finite rotations, constrained mechanical systems, geometrically exact Cosserat rods with capacity choices structural dynamics Keywords: Cosserat rods, geometrically exact rods, Keywords: hub location, capacity decisions, MILP for- (20 pages, 2009) small strain, large deformation, deformable bodies, mulations Lagrangian field theory, variational calculus (27 pages, 2009) (19 pages, 2009) 160. P. Jung, S. Leyendecker, J. Linn, M. Ortiz Discrete Lagrangian mechanics and geo- 170. S. Acar, K. Natcheva-Acar 151. K. Dreßler, M. Speckert, R. Müller, metrically exact Cosserat rods A guide on the implementation of the Ch. Weber Keywords: special Cosserat rods, Lagrangian mechanics, Heath-Jarrow-Morton Two-Factor Gaussian Noether’s theorem, discrete mechanics, frame-indiffer- Customer loads correlation in truck engi- Short Rate Model (HJM-G2++) ence, holonomic constraints Keywords: short rate model, two factor Gaussian, neering (14 pages, 2009) Keywords: Customer distribution, safety critical compo- G2++, option pricing, calibration nents, quantile estimation, Monte-Carlo methods (30 pages, 2009) (11 pages, 2009) 161. M. Burger, K. Dreßler, A. Marquardt, M. Speckert 152. H. Lang, K. Dreßler Calculating invariant loads for system simu- An improved multiaxial stress-strain correc- lation in vehicle engineering Keywords: iterative learning control, optimal control Status quo: July 2009 tion model for elastic FE postprocessing theory, differential algebraic equations(DAEs) Keywords: Jiang’s model of elastoplasticity, stress-strain (18 pages, 2009) correction, parameter identification, automatic differ- entiation, least-squares optimization, Coleman-Li algorithm 162. M. Speckert, N. Ruf, K. Dreßler (6 pages, 2009) Undesired drift of multibody models excit- ed by measured accelerations or forces 153. J. Kalcsics, S. Nickel, M. Schröder Keywords: multibody simulation, full vehicle model, A generic geometric approach to territory force-based simulation, drift due to noise (19 pages, 2009) design and districting Keywords: Territory design, districting, combinatorial optimization, heuristics, computational geometry 163. A. Streit, K. Dreßler, M. Speckert, J. Lichter, (32 pages, 2009) T. Zenner, P. Bach Anwendung statistischer Methoden zur 154. Th. Fütterer, A. Klar, R. Wegener Erstellung von Nutzungsprofilen für die An energy conserving numerical scheme for Auslegung von Mobilbaggern the dynamics of hyperelastic rods Keywords: Nutzungsvielfalt, Kundenbeanspruchung, Keywords: Cosserat rod, hyperealstic, energy conserva- Bemessungsgrundlagen tion, finite differences (13 pages, 2009) (16 pages, 2009) 164. I. Correia, S. Nickel, F. Saldanha-da-Gama 155. A. Wiegmann, L. Cheng, E. Glatt, O. Iliev, Anwendung statistischer Methoden zur Er- S. Rief stellung von Nutzungsprofilen für die Design of pleated filters by computer sim- Auslegung von Mobilbaggern ulations Keywords: Capacitated Hub Location, MIP formulations Keywords: Solid-gas separation, solid-liquid separation, (10 pages, 2009) pleated filter, design, simulation (21 pages, 2009) 165. F. Yaneva, T. Grebe, A. Scherrer An alternative view on global radiotherapy 156. A. Klar, N. Marheineke, R. Wegener optimization problems Hierarchy of mathematical models for pro- Keywords: radiotherapy planning, path-connected sub- duction processes of technical textiles levelsets, modified gradient projection method, improv- ing and feasible directions (14 pages, 2009)