Affine Processes. Martin Keller-Ressel TU Berlin

Transcription

1 TU Berlin Workshop on Interest Rates and Credit Risk 2011 TU Chemnitz 23. November 2011

2 Outline Introduction to Affine Jump-Diffusions The Moment Formula Bond & Option Pricing in Affine Models Extensions & Further Topics

3 Part I Introduction to

4 are a class of stochastic processes... with good analytic tractability (= explicit calculations and/or efficient numerical methods often available) that can be found in every corner of finance (stock price modeling, interest rates, commodities, credit risk,... ) efficient methods for pricing bonds, options,... dynamics and (some) distributional properties are well-understood They include models with mean-reversion (important e.g. for interest rates) jumps in asset prices (may represent shocks, crashes) correlation and more sophisticated dependency effects (stochastic volatility, simultaneous jumps, self-excitement... )

5 The mathematical tools used are characteristic functions (Fourier transforms) stochastic calculus (with jumps) ordinary differential equations Markov processes

6 Recommended Literature Transform Analysis and Asset Pricing for Affine Jump-Diffusions, Darrell Duffie, Jun Pan, and Kenneth Singleton, Econometrica, Vol. 68, No. 6, 2000 and Applications in Finance, DarrellDuffie, Damir Filipovic and Walter Schachermayer, The Annals of Applied Probability, Vol. 13, No. 3, 2003 A didactic note on affine stochastic volatility models, JanKallsen, In: From Stochastic Calculus to Mathematical Finance, pages Springer, Berlin, Affine Diffusion Processes: Theory and Applications, Damir Filipovic and Eberhard Mayerhofer, Radon Series Comp. Appl. Math 8, 1-40, 2009.

7 We start by looking at the Ornstein-Uhlenbeck process and the Feller Diffusion. The simplest (continuous-time) stochastic models for mean-reverting processes Used for modeling of interest rates, stochastic volatility, default intensity, commodity (spot) prices, etc. Also the simplest examples of affine processes!

8 Ornstein-Uhlenbeck process and Feller Diffusion Ornstein-Uhlenbeck (OU)-process dx t = λ(x t θ) dt + σdw t, X 0 R Feller Diffusion dx t = λ(x t θ) dt + σ X t dw t, X 0 R 0 θ... long-term mean λ > 0... rate of mean-reversion σ 0... volatility parameter σ for the OU-process We define σ(x t ):= σ X t for the Feller diffusion.

9 An important difference: The OU-process has support R, while the Feller diffusion stays non-negative What can be said about the distribution of X t? We will try to understand the distribution of X t through its characteristic function Φ Xt (y) =E e iyxt

10 Characteristic Function Characteristic Function For y R, the characteristic function Φ X (y) of a random variable X is defined as Φ X (y) :=E e iyx = e iyx df (x). Properties: Φ X (0) = 1, Φ X ( y) =Φ X (y), and Φ X (y) 1 for all y R. Φ X (y) =Φ Y (y) forally R, if and only if X d = Y. Let X and Y be independent random variables. Then Φ X +Y (y) =Φ X (y) Φ Y (y).

11 Let k N. IfE[ X k ] <, then E[X k ]=i k k y k Φ X (y). y=0 If the characteristic function Φ X (y) of a random variable X with density f (x) isknown,thenf (x) can be recovered by an inverse Fourier transform: f (x) = 1 2π e iyx Φ X (y) dy.

12 Back to the OU and CIR processes: We write u = iy and make the ansatz that the characteristic function of X t is of exponentially-affine form: Exponentially-Affine characteristic function E e iyxt = E e uxt =exp(φ(t, u)+ψ(t, u)x 0 ) (1) More precisely, if we can find functions φ(t, u), ψ(t, u) with φ(t, u) =0andψ(t, u) =u, suchthat M t = f (t, X t )=exp(φ(t t, u)+ψ(t t, u)x t ) is a martingale then we have E e ux T = E [M T ]=M 0 = exp (φ(t, u)+ψ(t, u)x 0 ), and (1) indeed gives the characteristic function.

13 Assume φ, ψ are sufficiently differentiable and apply the Ito-formula to f (t, X t )=exp(φ(t t, u)+x t ψ(t t, u)). The relevant derivatives are t f (t, X t)= φ(t t, u)+x t ψ(t t, u) f (t, X t ) x f (t, X t)=ψ(t t, u)f (t, X t ) 2 x 2 f (t, X t)=ψ(t t, u) 2 f (t, X t )

14 We get: df (t, X t ) f (t, X t ) = φt t + X t ψ T t dt + ψ T t dx t ψ2 T t σ2 X t dt = = φt t + X t ψ T t dt + ψ T t λ(x t θ) dt+ + ψ T t σ(x t ) dw t ψ2 T t σ(x t) 2 dt f (t, X t ) is local martingale, if ( φ T t + X t ψ T t )= ψ T t λ(x t θ)+ 1 2 ψ2 T t σ(x t) 2 for all possible states X t. Note that both sides are affine in X t,since σ(x t ) 2 σ 2 for the OU-process = σ 2 X t for the CIR process

15 We can collect coefficients : For the OU-process this yields For the CIR process we get φ(s, u) =θλψ(s, u)+ σ2 ψ(s, u) 2 ψ(s, u) = λψ(s, u) φ(s, u) =θλψ(s, u) ψ(s, u) = λψ(s, u)+ σ2 ψ(s, u) 2 These are ordinary differential equations. We also know the initial conditions φ(0, u) =0, ψ(0, u) =u.

16 If φ(t, u) andψ(t, u) solve the ODEs on the preceding slide, then M t is a local martingale. It is easy to check that in both cases M is also bounded, hence a true martingale. If M t is a martingale, then E e iyxt =exp(φ(t, iy)+x o ψ(t, iy)) is the characteristic function of X t.

17 The OU process For the OU-process we solve and get φ(s, u) =θλψ(s, u)+ σ2 2 ψ(s, u)2, φ(0, u) =0 ψ(s, u) = λψ(s, u), ψ(0, u) =u ψ(t, u) =e λt u φ(t, u) =θu(1 e λt )+ σ2 4λ u2 (1 e 2λt )

18 Thus the characteristic function of the OU-process is given by E e iyxt =exp iy e λt X 0 + θ(1 e λt ) y 2 σ 2 2 2λ (1 e 2λt ) and we get the following: Distributional Properties of OU-process Let X be an Ornstein-Uhlenbeck process. Then X t is normally distributed, with EX t = θ + e λt (X 0 θ), VarX t = σ2 1 e 2λt, 2λ Q: Can you think of a simpler way to obtain the above result?

19 The CIR process For the CIR-process we solve φ(s, u) =θλψ(s, u), φ(0, u) =0 and get ψ(s, u) = λψ(s, u)+ σ2 2 ψ(s, u)2, ψ(0, u) =u. ue λt ψ(t, u) = 1 σ2 2λ u(1 e λt ) φ(t, u) = 2λθ 1 σ 2 log σ2 2λ u(1 e λt ) (2) (3) The differential equation for ψ is called a Riccati equation. Q: How was the solution of the Riccati equation determined?

20 Thus the characteristic function of the CIR-process is given by 2λθ E e iyxt = 1 σ2 2λ (1 σ 2 e λt iy e λt )iy exp 1 σ2 2λ (1 e λt )iy and we get the following: Distributional Properties of the Feller Diffusion Let X be an Feller-diffusion, and define b(t) = σ2 4λ (1 e λt ). Then X t b(t) has non-central χ2 -distribution, with parameters k = 4λθ σ 2, α = e λt b(t), Q: Does there exist a limiting distribution? What is it?

21 Summary The key assumption was that the characteristic function of X t is of exponentially-affine form E e iyxt =exp(φ(t, iy)+x 0 ψ(t, iy)) We derived that φ(t, u) andψ(t, u) satisfyordinary differential equations of the form φ(t, u) =F (ψ(t, u)), φ(0, u) =0 ψ(t, u) =R(ψ(t, u)), ψ(0, u) =u Solving the differential equation gave φ(t, u) and ψ(t, u) in explicit form. The same approach works if the coefficients of the SDEs are time-dependent; ODEs become time-dependent too.

22 Part II Affine Jump-Diffusions

23 Jump Diffusions We consider a jump-diffusion on D = R m 0 Rn Jump-Diffusion where dx t = µ(x t ) dt + σ(x t ) dw t + dz t diffusion part jump part W t is a Brownian motion in R d ; µ : D R d, σ : D R d d,and Z is a right-continuous pure jump process, whose jump heights have a fixed distribution ν(dx) and arrive with intensity λ(x t ), for some λ : D [0, ). The Brownian motion W, the jump heights of Z, andthe jump times of Z are assumed to be independent. (4)

24 Jump Diffusions (2)

25 Some elementary properties and notation for the jump process Z t : Z t is RCLL (right continuous with left limits) Z t := lim s t,s t Z s and Z t := Z t Z t. Z t = Z t if and only Z t = 0 if and only a jump occurs at time t. Let τ(i) be the time of the i-th jump of Z t.letf be a function such that f (0) = 0. Then 0 s t f ( Z s ):= 0 τ(i) t f ( Z s ) is a well-defined sum, that runs only over finitely many values (a.s.)

26 Ito formula for jump-diffusions Ito formula for jump diffusions Let X be a jump-diffusion with diffusion part D t and jump part Z t. Assume that f : R d R is a C 1,2 -function and that Z t is a pure jump process of finite variation. Then t f (t, X t )=f(0, X 0 )+ t + 1 tr f (s, X s ). 0 s t Here f x = f x 1,..., f 2 f x 2 = 0 f t t (s, X f s ) ds f x 2 (s, X s )σ(x s )σ(x s ) x d x (s, X s ) dd s + ds+ denotes the gradient of f,and 2 f x i x j is the Hessian matrix of the second derivatives of f.

27 Affine Jump-Diffusion Affine Jump-Diffusion We call the jump diffusion X (defined in (4)) affine, ifthedrift µ(x t ), the diffusion matrix σ(x t )σ(x t ) and the jump intensity λ(x t )areaffine functions of X t. More precisely, assume that µ(x) =b + β 1 x β d x d σ(x)σ(x) = a + α 1 x α d x d λ(x) =m + µ 1 x 1 + µ d x d where b, β i R d ; a, α i R d d and m,µ i [0, ). Note: (d + 1) 3 parameters for a d-dimensional process.

28 We want to show that an affine jump-diffusion has a (conditional) characteristic function of exponentially-affine form: Characteristic function of Affine Jump Diffusion Let X be an affine jump-diffusion on D = R m 0 Rn.Then E e u X T F t =exp(φ(t t, u)+x t ψ(t t, u)) for all u = iz ir d and 0 t T,whereφ and ψ solve the system of differential equations with... φ(t, u) =F (ψ(t, u)), φ(0, u) = 0 (5) ψ(t, u) =R(ψ(t, u)), ψ(0, u) =u (6)

29 (continued) κ(u) = R d (e u x 1) ν(dx), and F (u) =b u u au + mκ(u) R 1 (u) =β 1 u u α 1 u + µ 1 κ(u),. R d (u) =β d u u α d u + µ d κ(u). The differential equations satisfied by φ(t, u) and ψ(t, u) are called generalized Riccati equations. The functions F (u), R 1 (u),...,r d (u) are of Lévy-Khintchine form.

30 Proof (sketch:) Show that the generalized Riccati equations have unique global solutions φ, ψ (This is the hard part, and here the assumption that D = R m 0 Rn enters!) Fix T 0, define M t = f (t, X t )=exp(φ(t t, u)+ψ(t t, u) X t ) and show that M t remains bounded. Apply Ito s formula to M t :

31 The relevant quantities for Ito s formula are t f (t, X t ) = φ(t t, u)+x t ψ(t t, u) f (t, X t ) x f (t, X t ) =ψ(t t, u)f (t, X t ) 2 x 2 f (t, X t ) =ψ(t t, u) ψ(t t, u) f (t, X t ) f (t, X t )= e ψ(t t,u) Xt 1 f (t, X t ) Also define the cumulant generating function of the jump measure: κ(u) = (e u x 1)ν(dx). R d

32 We can write f (t, X t )as... f (t, X t ) = local martingale t φ(t s, u)+x s ψ(t s, u) f (s, X s ) ds t 0 t 0 t 0 ψ(t s, u) µ(x s )f (s, X s ) ds+ ψ(t s, u) σ(x s )σ(x s ) ψ(t s, u)f (s, X s ) ds+ κ ψ(t s, u) λ(x s )f (s, X s ) ds Inserting the definitions of µ(x s ), σ(x s )σ(x s ) and λ(x s ) and using the generalized Riccati equations we obtain the local martingale property of M.

33 Since M is bounded it is a true martingale and it holds that E e ux T F t = E [M T F t ]= = M t = exp (φ(t t, u)+ψ(t t, u) X t ), showing desired form of the conditional characteristic function.

34 Example: The Heston model Heston proposes the following model for a stock S t and its (mean-reverting) stochastic variance V t (under the risk-neutral measure Q) 1 : Heston model ds t = V t S t dwt 1 dv t = λ(v t θ) dt + η V t ρ dwt ρ 2 dwt 2 where W t =(W 1 t, W 2 t ) is two-dimensional Brownian motion. 1 We assume here that the interest rate r =0

35 The Heston model (2) The parameters have the following interpretation: λ... mean-reversion rate of the variance process θ... long-term average of V t η... vol-of-var : the volatility of the variance process ρ... leverage : correlation bet. moves in stock price and in variance.

36 The Heston model (3) Transforming to the log-price L t = log(s t ) we get dl t = V t 2 dt + V t dwt 1 dv t = λ(x t θ) dt + η V t ρ dwt ρ 2 dwt 2 which is a two dimensional affine diffusion! Writing X t =(L t, V t )wefind 0 1/2 µ(x t )= + λθ 0 L t + λ β 1 b σ(x t )σ(x t ) = 0 a 1 L t + α β 2 ηρ ηρ η 2 α 2 V t V t

37 The Heston model (4) Thus, the characteristic function of log-price L t and stochastic variance V t of the Heston model can be calculated from φ(t, u) =λθψ 2 (t, u) ψ 2 (t, u) = 1 2 u 2 1 u 1 λψ2 (t, u)+ η2 2 ψ2 2(t, u)+ηρu 1 ψ 2 (t, u) with initial conditions φ(0, u) = 0, ψ 2 (t, u) =u 2. Note that ψ 1 (t, u) =0andthusψ 1 (t, u) =u 1 for all t 0.

38 Duffie-Garleanu default intensity process Duffie and Garleanu propose to use the following process (taking values in D = R 0 )asamodelfordefaultintensities: Duffie-Garleanu model dx t = λ(x t θ) dt + σ X t dw t + dz t where Z t is a pure jump process with constant intensity c, whose jumps are exponentially distributed with parameter α. The above process is an affine jump diffusion, whose characteristic function can be calculated from the generalized Riccati equations where φ(t, u) =F (ψ(t, u)), F (u) =λθu + cu α u, ψ(t, u) =R(ψ(t, u)) u2 R(u) = λu + 2 σ2

39 Parameter Restrictions Revisit the Feller Diffusion Feller Diffusion dx t = λ(x t θ) dt + σ X t dw t, X 0 R 0 Can we allow θ < 0? When X t = 0, then X t+ t λθ < 0 and X t+ t is not well-defined. = Parameter restrictions are necessary. Ideally, we can find necessary & sufficient parameter restrictions.

40 Characterization of affine jump-diff. ond = R n R m 0 Duffie, Filipovic & Schachermayer (2003) derive the necessary & sufficient parameter restrictions ( admissibility conditions ) for all affine jump-diffusions on the state space D = R n R m 0 Rd.We write J := {1,...,n}, I := {n +1,...,n + m} for indices of the real-valued and the non-negative components. The following holds: Characterization of an affine jump-diffusion on R n R m 0 Let X be an affine jump-diffusion with state space D = R n R m 0. Then the parameters a, α k, b, β k, m,µ k, ν(dx) satisfy the following conditions:

41 (continued) a, α k are positive semi-definite matrices and α j =0forall j J. ae k =0forallk I α i e k =0forallk I and i I \{k} α j =0forallj J b D β i e k 0 for all k I and i I \{k} β j e k =0forallk I and j J µ j =0forallj J supp ν D. Conversely, if the parameters a, α k, b, β k, m,µ k, ν(dx) satisfythe above conditions, then an affine jump-diffusion X with state space D = R n R m 0 exists.

42 Illustration of the parameter conditions a = α j (j J) = 0 α i (i I ) = α i ii where α i ii 0 b =.. β j (j J) = β i (i I ) =.. β i i. where β i i R Stars denote arbitrary real numbers; the small -signs denote non-negative real numbers and the big -sign a positive semi-definite matrix.

43 We sketch a proof of the conditions necessity: σ(x)σ(x) = a + α 1 x 1 + α d x d has to be positive semidefinite for all x D = a, a i are positive semidefinite for i I and α j =0for j J. λ(x) =m + µ 1 x 1 + µ d x d has to be non-negative for all x D = µ j =0forj J. The process must not move outside D by jumping = supp ν D.

44 Assume that X t has reached the boundary of D, thatisx t = x with x k = 0 for some k I. The following conditions have to hold, such that X t does not cross the boundary: b + i=k β ix i inward pointing drift: 0 ek µ(x) =e k = b D, βi e k 0 for all i I \{i}, andβj e k =0 for all j J. diffusion parallel to the boundary: 0=ek σ(x) =e k a + i=k α ix i = ae k =0andα i e k =0foralli I \{k}. (e k denotes the k-th unit vector.)

45 Part III The Moment Formula

46 The Moment formula Let X be an affine jump-diffusion on D = R m 0 Rn.Wehave shown that E e u X T F t =exp(φ(t t, u)+x t ψ(t t, u)) for all u ir d where φ and ψ solve the generalized Riccati equations. What can be said about general u C d and in particular about the moment generating function θ E e θ X T with θ R d?

47 In general we should expect that The exponential moment E e u X T may be finite or infinite depending on the value of u C d and on the distribution of X T The generalized Riccati equations no longer have global solutions for arbitrary starting values u C d (blow-up of solutions may appear)

48 Moment formula Let X be an affine jump-diffusion on D = R m 0 Rn with X 0 D and assume that dom κ R d is open. Let φ(t, u) =F (ψ(t, u)), t φ(0, u) = 0 (7) ψ(t, u) =R(ψ(t, u)), t ψ(0, u) =u (8) be the associated generalized Riccati equations, with F and R analytically extended to S(dom κ) := u C d : Re u dom κ. Then the following holds...,

49 Moment formula (contd.) (a) Let u C d and suppose that E e u X T <. Then u S(dom κ) and there exists unique solutions φ, ψ of the gen. Riccati equations such that E e u X T F t =exp(φ(t t, u)+ψ(t t, u) X t ) (9) for all t [0, T ]. (b) Let u S(dom κ) and suppose that the gen. Riccati equations have solutions φ, ψ that start at u and exist up to T.Then E e u X T < and (9) holds for all t [0, T ]. Essentially: Solution to gen. Riccati equation exists Exponential Moment exists.

50 Sketch of the proof of (a) (for real arguments θ R d ): Show by analytic extension that there exist functions φ(t, θ) and ψ(t, θ) suchthat M t := E e θ X T F t =exp(φ(t t, θ)+ψ(t t, θ) X t ). By the assumption of (a) M is a martingale. Show that φ and ψ are differentiable in t (This is the hard part!) Use the Ito-formula to show that the martingale property of M implies that φ and ψ solve the generalized Riccati equations

51 Sketch of the proof of (b): Let θ dom κ. Define M t =exp(φ(t t, θ)+ψ(t t, θ) X t ) Use the Ito-formula and the generalized Riccati equations to show that M is a local martingale Since M is positive, it is a supermartingale and E e θ,x T = E [M T ] M 0 <. Apply part (a) of the theorem and use that the solutions of the gen. Riccati equations are unique.

52 Some consequences (we still assume that dom κ is open) Exponential Martingales: t e θ Xt is a martingale if and only if θ dom κ and F (θ) =R(θ) = 0. Exponential Measure Change: Let X be an affine jump diffusion and θ dom κ. ThenthereexistsameasureP θ P such that X is an affine jump-diffusion under P θ with F θ (u) =F (u + θ) F (θ) R θ (u) =R(u + θ) R(θ). Exponential Family: The measures (P θ ) θ dom κ form a curved exponential family with likelihood process t L θ t = dpθ dp =exp θ X t F (θ)t R(θ) X s ds. 0

53 Proof: Extension of state-space approach Consider the process (X t, Y t = t 0 X s). The process (X, Y )is again an affine jump-diffusion (note: dy t = X t dt) Define L θ t =exp(θ X t F (θ)t R(θ) Y t ) Applying the moment formula to find the exponential moment of order (θ, R(θ)) of the extended process (X, Y ) we get E L θ T F t = =exp(p(t t)+q(t t) X t ) exp ( F (θ)t R(θ) Y t ) where p(t) =F (q(t)), p(0) = 0 t q(t) =R(q(t)) R(θ), q(0) = θ. t

54 θ is a stationary point of the second Riccati equation. Hence, the (global) solutions are q(t) = θ and p(t) = tf (θ) for all t 0 Inserting the solution yields E F t =exp(θ X t F (θ)t R(θ) Y t )=L θ t, L θ T and hence t L θ t is a martingale. Define the measure P θ by dp θ dp = L t. Ft A similar calculation yields F θ (u) andr θ (u) fortheprocessx under P θ.

55 Part IV Bond and Option Pricing in Affine Models

56 Pricing of Derivatives We consider the following setup: The goal is to price a European claim on some underlying asset S t,whichhaspayoff f (S T )attimet. We denote the value of the claim at time t by V t. As numeraire asset, we use the money market account t M t =exp 0 R(X s) ds determined by the short rate process R(X s ). Under the assumption of no-arbitrage, there exists a martingale measure Q for the discounted asset price process S t,suchthat M 1 t V t = M t E Q M 1 T f (S T ) F t.

57 To allow for analytical calculations we make the following assumption: Both the short rate process R(X t )andtheassets t are modelled under the risk-neutral measure Q through an affine jump-diffusion process X t in the following way: R(X t )=r + ρ X t, S t = e ϑ X t for some fixed parameters r, ρ 0 and ϑ dom κ. This setup includes the combination of many important short rate and stock price models: Vasicek, Cox-Ingersoll-Ross, Black-Scholes, Heston, Heston with jumps,...

58 Extension-of-state-space-approach and moment formula yield the following: Discounted moment generating function Let u S(dom κ) andφ(t, u) =M t E Q M 1 T eu X T F t. Suppose the differential equations with φ (t, u) =F (ψ (t, u)), φ (0, u) = 0 (10) ψ (t, u) =R (ψ (t, u)), ψ (0, u) =u (11) F (u) =F (u) r, and R (u) =R(u) ρ, or more precisely...

59 (continued) F (u) =b u u au + mκ(u) r R1 (u) =β1 u u α 1 u + µ 1 κ(u) ρ 1,. Rd (u) =β d u u α d u + µ d κ(u) ρ d. have solutions t φ (t, u) andt ψ (t, u) uptotimet,then Φ(t, u) =exp(φ (T t, u)+ψ (T t, u) X t ) for all t T.

60 Bond Pricing in Affine Jump Diffusion models As an immediate application we derive the following formula for pricing of zero-coupon bonds: Bond Pricing Suppose the gen. Riccati equations for the discounted mgf have solutions up to time T for the initial value u = 0. Then the price at time t of a (unit-notional) zero-coupon bond P t (T )maturingat time T is given by P t (T )=exp(φ (T t, 0) + X t ψ (T t, 0)). Yields the well-known pricing formulas for the Vasicek and the CIR-Model as special cases.

61 No-arbitrage constraints on F and R : The martingale assumption E Q M 1 T S T F t = M 1 t leads to the following no-arbitrage constraints on F and R : No-arbitrage constraints S t F (ϑ) =F (ϑ) r =0 R (ϑ) =R(ϑ) ρ =0.

62 Pricing of European Options A European call option with strike K and time-to-maturity T pays (S T K) + at time T. We will parameterize the option by the log-strike y = log K and denote its value at time t by C t (y, T ). The goal is to derive a pricing formula based on our knowledge of the discounted moment generating function Φ(t, u) =M t E Q M 1 eu X T T F t

63 Idea: Calculate the Fourier transform of C t (y, T ) (regarded as a function of y), and hope that it is a nice expression involving Φ(T t, u). Problem: C t (y, T ) may not be integrable, and thus may have no Fourier transform. Solution 1: Use the exponentially dampened call price C t (y, T )=e yζ C t (y, T )whereζ > 0. Solution 2: Replace the call option by a covered call with payoff S T (S T K) + =min(s T, K). Several other (related) solutions can be found in the literature...

64 Fourier pricing formula for European call options: Let C t (y, T ) be the price of a European call option with log-strike y and maturity T.ThenC t (y, T ) is given by the inverse Fourier transform C t (y, T )= e ζy 2π iωy Φ(T t, (ζ +1+iω)ϑ) e dω (12) (ζ + iω)(ζ +1+iω) where ζ is chosen such that ζ > 0 and the generalized Riccati equations starting at (ζ + 1)θ have solutions up to time T. (This formula is obtained by exponential dampening) Note: the required ζ can always be found, since dom κ is open and contains 0 and θ.

65 Fourier pricing formula for European put options: Let P t (y, T ) be the price of a European put option with log-strike y and maturity T.ThenP t (y, T ) is given by the inverse Fourier transform P t (y, T )= e ζy 2π iωy Φ(T t, (ζ +1+iω)ϑ) e dω (13) (ζ + iω)(ζ +1+iω) where ζ is chosen such that ζ > 1 and the generalized Riccati equations starting at (ζ + 1)θ have solutions up to time T. (This formula is obtained by exponential dampening) Note: the required ζ can always be found, since dom κ is open and contains 0 and θ.

66 Part V Extensions and further topics

67 Extensions Allow jumps with infinite activity, superpositions of d + 1 different jump measures and killing. These are the affine processes in the sense of Duffie, Filipovic and Schachermayer (2003)) This definition includes all Lévy process and all so-called continuous-state branching processes with immigration. Consider other state spaces: Positive semidefinite matrices: Wishart process, etc. Polyhedral and symmetric cones Quadratic state spaces (level sets of quadratic polynomials) Time-inhomogeneous processes

68 Further Topics/Current Research Utility maximization and variance-optimal hedging in affine models (Jan Kallsen, Johannes Muhle-Karbe et al.) Distributional Properties of affine processes: (non-central) Wishart distributions, infinite divisibility of marginal laws (Eberhard Mayerhofer et al.) Feller property, path regularity, regularity of the characteristic function (Christa Cuchiero, Josef Teichmann et al.) Relation to branching processes and superprocesses, infinite-dimensional generalizations Large-deviations and stationary distributions of affine processes Interaction between state-space geometry and distributionalor path-properties

69 Further Topics/Current Research Statistical estimation, density approximations, spectral approximations State-space-independent classification and/or characterization results Affine processes as finite-dimensional realizations of HJM-type models Applications, applications, applications: Affine term structure models (ATSMs) Affine stochastic volatility models (ASVMs) Credit risk models,...

70 Thank you for your attention!