Evaluation of the HJM equation by cubature methods for SPDE

Transcription

1 Evaluation of the HJM equation by cubature methods for SPDEs TU Wien, Institute for mathematical methods in Economics Kyoto, September 2008

2 Motivation Arbitrage-free simulation of non-gaussian bond markets with (jump-)events, for instance for risk management.

3 Motivation Arbitrage-free simulation of non-gaussian bond markets with (jump-)events, for instance for risk management. Pricing of all exotic derivatives in one model, for prototyping and pricing.

4 Motivation Arbitrage-free simulation of non-gaussian bond markets with (jump-)events, for instance for risk management. Pricing of all exotic derivatives in one model, for prototyping and pricing. conceptual progresses for term structure problems.

5 Numerics for HJM equation Finite factor approaches (works only in special cases).

6 Numerics for HJM equation Finite factor approaches (works only in special cases). Andreasen s (strong) Euler-Maruyama scheme as described in Glasserman s book.

7 Numerics for HJM equation Finite factor approaches (works only in special cases). Andreasen s (strong) Euler-Maruyama scheme as described in Glasserman s book. Björk-Szepessi-Tempone FE and Euler-scheme for the HJM equation.

8 Research program describe weak and strong high order schemes for (semi-linear) SPDEs.

9 Research program describe weak and strong high order schemes for (semi-linear) SPDEs. apply weak and strong schemes for the HJM equation (simulation, pricing).

10 Research program describe weak and strong high order schemes for (semi-linear) SPDEs. apply weak and strong schemes for the HJM equation (simulation, pricing). calibration of the HJM equation to market data? Short-time asymptotics?

11 SDEs in infinite dimension SPDEs with group-generator The Szekőfalvi-Nagy theorem SPDEs Existence and Uniqueness for SPDEs the moving frame SDEs in infinite dimension SPDEs with group-generator The Szekőfalvi-Nagy theorem SPDEs Rough Paths Lip(γ)-vector fields The universal limit theorem RPDEs Stochastic Taylor Expansion Theorem the cubature method Comparison of methods Cubature on Wiener space by moving frame

12 SDEs in infinite dimension SPDEs with group-generator The Szekőfalvi-Nagy theorem SPDEs Stochastic partial differential equations appear in many contexts as natural model (stochastic Navier-Stokes equation, stochastic Burger s equation, Zakai equation of filtering, Brownian motions on infinite dimensional manifolds, HJM equation, Bühler s equation on forward variance curves, etc). Usually one considers a non-perturbed PDE of semilinear type dx t = (AX t + V(X t ))dt, X 0 H and starts to add noise to this equation (where the noise-volatilities only have Lipschitz restrictions). Semigroup theory is the tool for solving those equations.

13 SDEs in infinite dimension SPDEs with group-generator The Szekőfalvi-Nagy theorem SPDEs When we consider an SPDE of the type dx t = (AX t + V(X t ))dt + d V i (X t )dbt i for a strongly continuous semigroup S with generator A we usually consider the concept of a mild solution as leading solution concept, i.e. t X t = S t X 0 + S t s V(X s )ds + 0 i=1 d i=1 t 0 S t s V i (X s )db i s up to some time horizon under obvious integrability conditions.

14 SDEs in infinite dimension SPDEs with group-generator The Szekőfalvi-Nagy theorem SPDEs SDEs in infinite dimension Let B be a d-dimensional Brownian motion on a filtered probability space with time horizon T. Let H be a separable Hilbert space. By standard methods we can prove that dy t = σ(t, Y t )dt + d σ i (t, Y t )dbt i has a unique strong solution for globally Lipschitz-bounded vector fields with at most linear growth (with uniform bound in t [0, T ]), i.e. i=1 σ(t, x) σ(t, y) D x y, σ(t, x) D(1 + x ) for x, y H, 0 t T and some constant D > 0, analogously for σ i, i = 1,..., d.

15 SDEs in infinite dimension SPDEs with group-generator The Szekőfalvi-Nagy theorem SPDEs SPDEs with group-generator Assume that we are given a strongly continuous group S on H with generator A. By jumping onto the moving frame x S t x we can solve the SPDE d dx t = (AX t + V(X t ))dt + V i (X t )dbt i in the mild sense under global Lipschitz assumptions on the vector fields. Indeed, define the pulled back vector fields i=1 σ(t, y) = S t V(S t y), σ i (t, y) = S t V i (S t y), for y H, 0 t T and i = 1,..., d, and solve the standard equation, then X t = S t Y t is a mild solution of the SPDE.

16 SDEs in infinite dimension SPDEs with group-generator The Szekőfalvi-Nagy theorem SPDEs The Szekőfalvi-Nagy theorem Let H be a separable Hilbert space and S a pseudo-contractive, strongly continuous semi-group, i.e. there is ω R such that S t exp(ωt) for t 0. The Szekőfalvi-Nagy theorem tells that there exists a Hilbert H containing H as a closed subspace and a strongly continuous group S on H extending S in the following sense: for h H we have S t h = π S t h, where π is the orthogonal projection of H on H.

17 SDEs in infinite dimension SPDEs with group-generator The Szekőfalvi-Nagy theorem SPDEs Steps of the proof Consider the real numbers R and Hilbert space H. A positive definite function T with values in the bounded operators of a Hilbert space H thereon is a map T : R L(H) such that for all t 1,..., t N R and all ψ 1,..., ψ N H we have holds true. N T tm tn ψ n, ψ m 0 n,m=1

18 SDEs in infinite dimension SPDEs with group-generator The Szekőfalvi-Nagy theorem SPDEs Steps of the proof Consider the real numbers R and Hilbert space H. A positive definite function T with values in the bounded operators of a Hilbert space H thereon is a map T : R L(H) such that for all t 1,..., t N R and all ψ 1,..., ψ N H we have holds true. N T tm tn ψ n, ψ m 0 n,m=1 If T is a positive definite function, then there exists a unitary representation U : R L(K) on a Hilbert space K, such that H is a closed subspace of K and such that πu t = T t holds true on H for all real t. If T is weakly continuous then U is strongly continuous.

19 SDEs in infinite dimension SPDEs with group-generator The Szekőfalvi-Nagy theorem SPDEs If S is a one-parameter contraction semigroup on H, then T t = S t and T t = S t for t 0 is a positive definite functions on R taking values in the bounded operators on H.

20 SDEs in infinite dimension SPDEs with group-generator The Szekőfalvi-Nagy theorem SPDEs If S is a one-parameter contraction semigroup on H, then T t = S t and T t = S t for t 0 is a positive definite functions on R taking values in the bounded operators on H. The unitary representation is uniquely defined up to isomorphisms on the closure of the span of t R U t H.

21 SDEs in infinite dimension SPDEs with group-generator The Szekőfalvi-Nagy theorem SPDEs SPDEs Let H be a separable Hilbert space and S a pseudo-contractive, strongly continuous semi-group with generator A. In order to construct a mild solution of dz t = (AZ t + V (Z t ))dt + d V i (Z t )dbt, i we consider V = V π and V i = V i π and construct a mild solution for SPDE with group generator A. The process Z t = π(x t ), for t 0, is then a mild solution of the SPDE. Uniqueness is shown in all cases by Gronwall s lemma. i=1

22 Rough Paths Lip(γ)-vector fields The universal limit theorem RPDEs Rough Paths GV m denotes the nilpotent Lie group of step m over V embedded into the nilpotent algebra A m V over V, which is often refered as truncated (at level m) tensor algebra over V. The p-variation norm (distance) d p is given through d p (X, Y ) = max sup 1 i [p] D [0,T ] ( D X i t l+1,t l Y i t l+1,t l p i ) i p on the respective set of multiplicative functionals on T with values in A [p] V.

23 Rough Paths Lip(γ)-vector fields The universal limit theorem RPDEs Rough Paths A 1-geometric rough path is a path with finite total variation which has unique lifts into GV m for every m 1.

24 Rough Paths Lip(γ)-vector fields The universal limit theorem RPDEs Rough Paths A 1-geometric rough path is a path with finite total variation which has unique lifts into GV m for every m 1. Fix p 1. We understand p-geometric rough paths X as closure of 1(-geometric) rough paths with respect to the p-variation norm.

25 Rough Paths Lip(γ)-vector fields The universal limit theorem RPDEs Rough Paths A 1-geometric rough path is a path with finite total variation which has unique lifts into GV m for every m 1. Fix p 1. We understand p-geometric rough paths X as closure of 1(-geometric) rough paths with respect to the p-variation norm. We denote the set of p-geometric rough paths with values in V by GΩ p (V ). Notice that p-geometric rough paths have liftings into G [p] V by construction.

26 Rough Paths Lip(γ)-vector fields The universal limit theorem RPDEs Lip(γ)-vector fields Let V and W be Banach spaces. Let k 0 be an integer and k < γ k + 1. Let F be a closed subset of V and f : F W a function and let furthermore f j : F L(V j, W ) for j = 1,..., k. We denote f 0 := f. The collection (f 0,..., f k ) belongs to Lip(γ)(F, W ) if there exists a constant M 0 such that sup f j (x) M x F and if there exists functions R j such that, for each x, u F and each v V j, we have f j (y)(v) = k j l=0 1 l! f j+l (x)(v (y x) l ) + R j (x, y)(v),

27 Rough Paths Lip(γ)-vector fields The universal limit theorem RPDEs and R j (x, y) M x y γ j holds for j = 0,..., k. We usually say that f is Lip(γ)(F, W ) without mentioning f 1,..., f k and we call the smallest constant M for which all described inequalities hold the Lip(γ)-norm of f and denote it by f Lip γ.

28 Rough Paths Lip(γ)-vector fields The universal limit theorem RPDEs The universal limit theorem Let p 1 and γ > p be fixed and f (t, y) L(V, W ) a family of endomorphisms for t [0, T ] and y W. Let s assume f Lip(γ)([0, T ] W, L(R V, R W )) for the extension f (s, y)(r, v) = (r, f (s, y)(v)), and let X denote the time-extended p-geometric rough path in GΩ p (R V ) extending X.

29 Rough Paths Lip(γ)-vector fields The universal limit theorem RPDEs The universal limit theorem Then for all X GΩ p (V ) and all ξ W the RDE dỹ t = f (Ỹ t )d X t, Y 0 = ξ admits a unique solution Z = ( X, Ỹ ) GΩ p (R V R W ) which depends continuously on (X, ξ), i.e. the map GΩ p (V ) W GΩ p (V W ) is continuous in the p-variation topology (and continuously extends the Itô map). The Picard iterations converge with an exponential rate with respect to p-variation norm.

30 Rough Paths Lip(γ)-vector fields The universal limit theorem RPDEs Let X be a p-geometric rough path for some p 1.

31 Rough Paths Lip(γ)-vector fields The universal limit theorem RPDEs Let X be a p-geometric rough path for some p 1. Let W be a Banach space together with a strongly continuous group P with generator A.

32 Rough Paths Lip(γ)-vector fields The universal limit theorem RPDEs Let X be a p-geometric rough path for some p 1. Let W be a Banach space together with a strongly continuous group P with generator A. Let f (y) L(V, W ) be a family of endomorphisms for y W.

33 Rough Paths Lip(γ)-vector fields The universal limit theorem RPDEs We would like to make sense of solutions of the RPDE dy t = AY t dt + f (Y t )dx t, Y 0 = ξ W. (1) In order to do this we consider the concept of a mild solution t Y t = P t ξ + P t P s (f (Y s )dx s ), Y 0 W. 0 The main idea (and concept) here is to consider instead of the defining equation for mild solutions the time-dependent RDE du t = P t (f (P t U t )dx t ), U 0 = ξ. (2) If we are able to solve this equation by a p-geometric rough path U, then Y t := P t U t is a mild solution of the equation by definition. This can be done under Lip(γ)-conditions.

34 Rough Paths Lip(γ)-vector fields The universal limit theorem RPDEs Main theorem Let p 1 and γ > p be fixed and f (y) L(V, W ) a family of endomorphisms for y W. Let s assume f Lip(γ)([0, T ] W, L(R V, R W )) for the extension f (s, y)(r, v) = (r, (P s f (P s y)(v))), then there is a unique mild solution for the RPDE dy t = (AY t + g(y t ))dt + f (Y t )dx t, Y 0 = ξ on compact intervals [0, T ].

35 Rough Paths Lip(γ)-vector fields The universal limit theorem RPDEs Corollary If we take a Brownian (or Gaussian) p-geometric rough path for X (extended by time) we can speak of mild solutions of Banach-space valued SPDEs of the type dy t = AY t dt + f (Y t )dx t, Y 0 W under Lip(γ) conditions on the vector field f.

36 Stochastic Taylor Expansion Theorem the cubature method Comparison of methods Cubature on Wiener space by moving frame We describe the methodology of Cubature on Wiener space in a bounded setting, since we assume smooth Lip(γ)-vector fields for some γ m + 1. Consider on a Hilbert space H. dy y t = d i=0 V i (Y y t ) db i t

37 Stochastic Taylor Expansion Theorem the cubature method Comparison of methods Cubature on Wiener space by moving frame Let B be the set of finite sequences (i 1,..., i k ) with degree deg(i 1,..., i k ) = k + #{j, i j = 0}, then for m 1 and for C -bounded f f (Y y T ) = V i1 V i k f (y) deg(i 1,...,i k ) m + R m (f, T, y) 0 t 1 t k T db i 1 t1 db i k tk + with remainder estimate R m (f, T, y) 2 = O(T m+1 2 ) as T 0.

38 Stochastic Taylor Expansion Theorem the cubature method Comparison of methods Cubature on Wiener space by moving frame Fix an order m 2.

39 Stochastic Taylor Expansion Theorem the cubature method Comparison of methods Cubature on Wiener space by moving frame Fix an order m 2. There are H 1 -trajectories ω 1,..., ω r : [0, T ] R d and weights λ 1,..., λ r > 0. The trajectories ω j and their number r depend on d and m, but not on the specific vector fields V 0,..., V d.

40 Stochastic Taylor Expansion Theorem the cubature method Comparison of methods Cubature on Wiener space by moving frame Fix an order m 2. There are H 1 -trajectories ω 1,..., ω r : [0, T ] R d and weights λ 1,..., λ r > 0. The trajectories ω j and their number r depend on d and m, but not on the specific vector fields V 0,..., V d. Solve r non-autonomous ODEs (with a numerical scheme of order m+1 2 ) dy y t (ω j ) = V 0 (Y y t (ω j ))dt + Y y 0 (ω j) = y, then the estimate E(f (Y y T )) = holds true as T 0. r j=1 d i=1 V i (Y y t (ω j ))dω i j (t), λ j f (Y y T (ω j)) + O(T m+1 2 )

41 Stochastic Taylor Expansion Theorem the cubature method Comparison of methods Cubature on Wiener space by moving frame Define a linear operator on Lip(m + 1) functions f, Q s f (y) := r j=1 and obtain by the Markov property E(f (Y y t )) Q t n where K depends on f Lip(m+1). Q t n λ j f (Y y s (ω j )). f (y) K 1 n m 1 2,

42 Stochastic Taylor Expansion Theorem the cubature method Comparison of methods Cubature on Wiener space by moving frame For FD or FE numerical approaches one discretizes first the operator A in order to obtain an SDE and works then on this SDE by more classical methods.

43 Stochastic Taylor Expansion Theorem the cubature method Comparison of methods Cubature on Wiener space by moving frame For FD or FE numerical approaches one discretizes first the operator A in order to obtain an SDE and works then on this SDE by more classical methods. For the Euler-Maruyama scheme, or higher-order Taylor schemes, one discretizes first the Brownian motion (strongly or weakly). One obtains expressions involving random differential operators and iterates their expectations.

44 Stochastic Taylor Expansion Theorem the cubature method Comparison of methods Cubature on Wiener space by moving frame For FD or FE numerical approaches one discretizes first the operator A in order to obtain an SDE and works then on this SDE by more classical methods. For the Euler-Maruyama scheme, or higher-order Taylor schemes, one discretizes first the Brownian motion (strongly or weakly). One obtains expressions involving random differential operators and iterates their expectations. For cubature methods one discretizes first the Brownian motion by (finitely many) trajectories. Next one solves deterministic PDEs of the type t X t = S t X 0 + S t s α 0 (X s )ds + 0 along cubature trajectories ω 1,..., ω r. d i=1 t 0 S t s α i (X s )dω i s

45 Stochastic Taylor Expansion Theorem the cubature method Comparison of methods Cubature on Wiener space by moving frame Cubature on Wiener space by moving frame the Theorem Given A + α, α 1,..., α d and m 2, then under Lip(γ)-conditions on σ, σ 1,..., σ d for γ m + 1 we obtain convergence of the cubature method of order m 1 2 for certain classes of functions on H. The vector fields σ i are constructing by passing to the time-dependent system through the pull-back σ i (t, y) = S t α i (S t (y)), σ(t, y) = S t α(s t (y)) for i = 0,..., d, y H and 0 t T, where S is the strongly continuous group associated to A, and T some time horizon. The proof is performed by jumping to the moving frame, picking up the convergence rate and jumping back. The ladder is then not needed anymore.

46 Stochastic Taylor Expansion Theorem the cubature method Comparison of methods Cubature on Wiener space by moving frame The evaluation-simulation algorithms Solve t r t = S t r 0 + S t s α 0 (r s )ds + 0 along cubature trajectories ω 1,..., ω r r sn+1 = S n r sn + α 0 (r sn ) n + d i=1 t 0 S t s σ i (r s )dω i s d σ i (r sn ) ω i (s n ) for a grid 0 = s 0 < s 1 <... < s n < t, n = s n+1 s n in order to obtain a mild solution of the previous PDE. i=1

47 Stochastic Taylor Expansion Theorem the cubature method Comparison of methods Cubature on Wiener space by moving frame The evaluation-simulation algorithms Solve t r t = S t r 0 + S t s α 0 (r s )ds + 0 along cubature trajectories ω 1,..., ω r r sn+1 = S n r sn + α 0 (r sn ) n + d i=1 t 0 S t s σ i (r s )dω i s d σ i (r sn ) ω i (s n ) for a grid 0 = s 0 < s 1 <... < s n < t, n = s n+1 s n in order to obtain a mild solution of the previous PDE. Build the operator Q t and iterate the expression. i=1

48 Stochastic Taylor Expansion Theorem the cubature method Comparison of methods Cubature on Wiener space by moving frame The evaluation-simulation algorithms Solve t r t = S t r 0 + S t s α 0 (r s )ds + 0 along cubature trajectories ω 1,..., ω r r sn+1 = S n r sn + α 0 (r sn ) n + d i=1 t 0 S t s σ i (r s )dω i s d σ i (r sn ) ω i (s n ) for a grid 0 = s 0 < s 1 <... < s n < t, n = s n+1 s n in order to obtain a mild solution of the previous PDE. Build the operator Q t and iterate the expression. i=1 Evaluate-simulate the tree by Monte-Carlo methods.

49 Interest Rate mechanics 1 Prices of T -bonds are denoted by P(t, T ). Interest rates are governed by a market of (default free) zero-coupon bonds modelled by stochastic processes (P(t, T )) 0 t T for T 0. We assume the normalization P(T, T ) = 1. T denotes the maturity of the bond, P(t, T ) its price at a time t before maturity T.

50 Interest Rate mechanics 1 Prices of T -bonds are denoted by P(t, T ). Interest rates are governed by a market of (default free) zero-coupon bonds modelled by stochastic processes (P(t, T )) 0 t T for T 0. We assume the normalization P(T, T ) = 1. T denotes the maturity of the bond, P(t, T ) its price at a time t before maturity T. The yield Y (t, T ) = 1 log P(t, T ) T t describes the compound interest rate p. a. for maturity T.

51 Interest Rate mechanics 1 Prices of T -bonds are denoted by P(t, T ). Interest rates are governed by a market of (default free) zero-coupon bonds modelled by stochastic processes (P(t, T )) 0 t T for T 0. We assume the normalization P(T, T ) = 1. T denotes the maturity of the bond, P(t, T ) its price at a time t before maturity T. The yield Y (t, T ) = 1 log P(t, T ) T t describes the compound interest rate p. a. for maturity T. f is called the forward rate curve of the bond market for 0 t T. P(t, T ) = exp( T t f (t, s)ds)

52 Interest Rate mechanics 2 The short rate process is given through R t = f (t, t) for t 0 defining the bank account process t (exp( R s ds)) t 0. 0

53 Interest Rate mechanics 2 The short rate process is given through R t = f (t, t) for t 0 defining the bank account process t (exp( R s ds)) t 0. 0 No arbitrage is guaranteed if on the filtered probability space (Ω, F, Q) with filtration (F t ) t 0, T E(exp( R s ds) F t ) = P(t, T ) t holds true for 0 t T for some equivalent (martingale) measure P Q. We write E = E P.

54 HJM-drift condition The forward rates (f (t, T )) 0 t T are best parametrized through r(t, x) := f (t, t + x) for t, x 0 (Musiela parametrization). No-Arbitrage is guaranteed if the HJM-equation dr t = ( d dx r t + d σ i (r t ) i=1. 0 σ i (r t ))dt + d σ i (r t )dbt i describes the time-evolution of the term structure of interest rates with respect to a martingale measure P. i=1

55 If we are interested in simulation with respect to Q we have to add drift d α(r) = φ i (r)σ i (r) i=1 for some real-valued functions φ i, due to Girsanov s theorem.

56 Questions Let r 1,..., r K be a time series of forward rates. Can we calibrate HJM equations in a simple way to this time series?

57 Questions Let r 1,..., r K be a time series of forward rates. Can we calibrate HJM equations in a simple way to this time series? Does a simulation of a HJM equation calibratd in a simple way share the stylized facts of the time series?

58 Questions Let r 1,..., r K be a time series of forward rates. Can we calibrate HJM equations in a simple way to this time series? Does a simulation of a HJM equation calibratd in a simple way share the stylized facts of the time series? Can such a method be extended to more risk factors?

59 Questions Let r 1,..., r K be a time series of forward rates. Can we calibrate HJM equations in a simple way to this time series? Does a simulation of a HJM equation calibratd in a simple way share the stylized facts of the time series? Can such a method be extended to more risk factors? Can one improve the conceptual fundamentals of term structure theory in view of working numerical schemes?

60 The Euler scheme for the HJM equation Consider the HJM equation in the moving frame dr t = (Ar t + β(r t ))dt + d β i (r t )dbt i i=1 on a Hilbert space of forward rate curves H where the shift semigroup extends to a group.

61 The Euler scheme for the HJM equation Consider the HJM equation in the moving frame dr t = (Ar t + β(r t ))dt + d β i (r t )dbt i i=1 on a Hilbert space of forward rate curves H where the shift semigroup extends to a group. Consider the Euler scheme for this bounded equation for initial value f 0 = r 0. df t = σ(t, f t )dt + d σ i (t, f t )dbt i i=1

62 The Euler scheme with n 1 steps looks like f (n) (i+1)t n = σ ( it n, f (n) ) t d it n n + σ j( it n, f (n) it n j=1 ) t n Nj i for a finite i.i.d. sequence of standard normal random variables taking values in R d with f (n) 0 = r 0.

63 The Euler scheme with n 1 steps looks like f (n) (i+1)t n = σ ( it n, f (n) ) t d it n n + σ j( it n, f (n) it n j=1 ) t n Nj i for a finite i.i.d. sequence of standard normal random variables taking values in R d with f (n) 0 = r 0. Under differentiability conditions of type Lip(3) one obtains weak convergence of order 1 for the Euler scheme. The rate transfers via r (n) t written in terms of r, i.e. r (n) (i+1)t n = S t n = S t f (n) t r (n) it n. The scheme can be directly + β ( r (n) ) t it n n + d j=1 with a valid weak convergence rate of order 1. β j( r (n) ) t it n n Nj i

64 Calibration The implemented dynamics calibrated to the time series is of the following type dr t =( d dx r t + α(r t) 2 K 1 1 2(K 1) α(r i=1 i ) 2 (r i+1 r i ) + α(r K 1 t) K 1 i=1 (r i+1 r i ) dwt i, α(r i ). 0 (r i+1 r i ) + µ)dt+ where α(r) is a non-vanishing Lipschitz function on forward rates.

65 Simulated Example α(r)(x) = Y (x) = 1 x x 0 r(y)dy for x 0.

66 Simulated Example α(r)(x) = 1 x Y (x) = x 0 r(y)dy for x 0. Data from ECB for risk-free bonds measured by the Svensson family.

67 Simulated Example α(r)(x) = 1 x Y (x) = x 0 r(y)dy for x 0. Data from ECB for risk-free bonds measured by the Svensson family. Additional simulation of FX-rates.

68 Simulated Example α(r)(x) = 1 x Y (x) = x 0 r(y)dy for x 0. Data from ECB for risk-free bonds measured by the Svensson family. Additional simulation of FX-rates. Time series of length K = 50.

69 Simulated Example α(r)(x) = 1 x Y (x) = x 0 r(y)dy for x 0. Data from ECB for risk-free bonds measured by the Svensson family. Additional simulation of FX-rates. Time series of length K = 50. Time to maturity T = 100 days and simulation time t = 10 days.

70 Applications in financial mathematics No finite factor dynamics has to be assumed ( model-free approach ). direct MC valuation of exotic derivatives on interest rates (pricing).

71 Applications in financial mathematics No finite factor dynamics has to be assumed ( model-free approach ). direct MC valuation of exotic derivatives on interest rates (pricing). direct szenario generation from HJM equations (risk management).

72 Applications in financial mathematics No finite factor dynamics has to be assumed ( model-free approach ). direct MC valuation of exotic derivatives on interest rates (pricing). direct szenario generation from HJM equations (risk management). high order schemes allow for quick evaluation algorithms (implementation into Premia).

73 Applications in financial mathematics No finite factor dynamics has to be assumed ( model-free approach ). direct MC valuation of exotic derivatives on interest rates (pricing). direct szenario generation from HJM equations (risk management). high order schemes allow for quick evaluation algorithms (implementation into Premia). Events can be priced and simulated in an arbitrage-free way (risk management).

74 Links The Premia Project The Scilab Project ECB Yield Curve

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