Lévy Driven Financial Models

Transcription

1 Lévy Driven Models Ernst Eberlein Department of Mathematical Stochastics and Freiburg Center for Data Analysis and Modeling University of Freiburg Méthodes Statistiques et Applications en Actuariat et Finance Université Cadi Ayyad, Marrakech April 8 13, 2013 Reserve c Eberlein, Uni Freiburg, 1

2 Deutsche Bank densities Quantiles of Standard Normal Reserve c Eberlein, Uni Freiburg, 2

3 QQ plots for Deutsche Bank Deutsche Bank Quantiles of Standard Normal Reserve c Eberlein, Uni Freiburg, 3

4 density( x ) zero-bond log-returns ( ), 5 years to maturity empirical densities calculated from zero-yield data for Germany empirical normal log return x of zero-bond Reserve c Eberlein, Uni Freiburg, 4

5 log( density( x ) ) empirical normal log return x of zero-bond Reserve c Eberlein, Uni Freiburg, 5

6 log( density( x ) ) empirical NIG log return x of zero-bond Reserve c Eberlein, Uni Freiburg, 6

7 L = (L t) t 0 process with stationary and independent increments on a probability space (Ω, F, (F t) t 0, P) càdlàg paths: right-continuous with left limits canonical representation L t = bt + cw t + Z t + L s1 { Ls >1} s t b and c 0 real numbers, (W t) t 0 standard Brownian motion (Z t) t 0 purely discontinuous martingale independent of (W t) t 0 L s = L s L s jump at time s > 0 Reserve c Eberlein, Uni Freiburg, 7

8 Semimartingale representation (X t) t 0 semimartingale with X 0 = 0 X t s t X s1 { Xs >1} process with bounded jumps special semimartingale: unique decomposition into a local martingale and a predictable process with finite variation X t = V t + M t + s t X s1 { Xs >1} M t = M c t + M d t M c continuous, M d purely discontinuous For : V t = bt and M c = cw t (Z t) t 0 = (M d t ) t 0 the purely discontinuous local martingale (of a Lévy process) Reserve c Eberlein, Uni Freiburg, 8

9 The purely discontinuous martingale X s1 { Xs 1} does not converge in general compensating s t P lim ε 0 ( What is F? s t X s1 {ε Xs 1} t ) x1 {ε x 1} F(dx) µ X (ω; dt, dx) = s>0 1 { Xs(ω) 0}E (s, Xs(ω))(dt, dx) random measure of jumps of X = (X t) t 0 µ X (ω; [0, t] A) counts how many jumps of size within A occur for path ω from 0 to t E [ µ X ( ; [0, t] A) ] = tf(a), M d t = t 0 R ) x1 { x 1} (µ X (ds, dx) dsf(dx) cannot be separated in general F intensity measure or Lévy measure Reserve c Eberlein, Uni Freiburg, 9

10 Local characteristics of a Lévy process (1) Triplet of local characteristics of (L t) t 0 : (b, c, F) L t = bt + t cw t + ν = L(L 1 ) is infinitely divisible L(L 1 ) = L(L 1/n ) L(L 1/n ) 0 R t + x1 { x >1} µ L (ds, dx) 0 R x1 { x 1} ( µ L (ds, dx) dsf(dx) ) Reserve c Eberlein, Uni Freiburg, 10

11 Local characteristics of a Lévy process (2) Fourier transform in Lévy Khintchine form [ E[exp(iuL 1 )] = exp iub 1 ) ] 2 u2 c + (e iux 1 iux1 { x 1} F(dx) R = exp(ψ(u)) F Lévy measure: pricing of derivatives: E[f (L T )] R min(1, x 2 )F(dx) < uses E[exp(iuL T )] = exp(ψ(u)) T Reserve c Eberlein, Uni Freiburg, 11

12 Examples of Lévy measures The density of the Lévy measure of the normal inverse Gaussian (left) and the α-stable process. Reserve c Eberlein, Uni Freiburg, 12

13 Integrability properties of the Lévy measure Finiteness of the moments of the process depends on the frequency of the large jumps Proposition Let L be a Lévy process with triplet (b, c, F). 1 E[ L t p ] < for p R + if and only if 2 E[exp(pL t)] < for p R if and only if exp(px)f (dx) <. { x >1} { x >1} x p F(dx) <. Reserve c Eberlein, Uni Freiburg, 13

14 Consequences for the representation If L(L 1 ) has a finite expectation then { x >1} xf(dx) < add iux1 { x >1} F(dx) to the characteristic exponent [ E[exp(iuL 1 )] = exp iub 1 ( 2 u2 c + e iux 1 iux ) ] F (dx) R (different b) In the same way t added to (M d t ) t 0, consequently 0 R x1 { x >1} ( µ L (ds, dx) dsf (dx) ) L t = bt + t cw t + x ( µ L (ds, dx) dsf (dx) ) 0 R Martingale if b = E[X 1 ] = 0 Submartingale if b > 0, supermartingale if b < 0 can be Reserve c Eberlein, Uni Freiburg, 14

15 Example (Poisson process) Lévy measure F = λe 1, λ intensity parameter No Gaussian component: c = 0 jumps of size 1 occur with average rate λ per unit time Fourier transform E[exp(iuL t)] = exp[λt(e iu 1)] Canonical representation L t = λt + (L t λt) ( ) = λt + 1 {Tn t} λt n 1 Reserve c Eberlein, Uni Freiburg, 15

16 Generalized hyperbolic distributions (O.E. Barndorff-Nielsen (1977)) Density: d GH (x) = a(λ, α, β, δ) (δ 2 + (x µ) 2) (λ 1/2)/2 a(λ, α, β, δ) = K λ 1/2 ( a δ 2 + (x µ) 2 ) exp(β(x µ)) ( α 2 β 2) λ/2 2πα λ 1/2 δ λ K λ (δ α 2 β 2 ) K λ modified Bessel function of the third kind with index λ Parameters: λ R Class parameter µ R Location α > 0 Shape δ > 0 Scale parameter β with 0 β < α Skewness (Volatility) Reserve c Eberlein, Uni Freiburg, 16

17 y Normal NIG (0.02,0,0.5,0) NIG (6,-5.5,1,1) NIG (7,6,1,-1) x Reserve c Eberlein, Uni Freiburg, 17

18 y x Reserve c Eberlein, Uni Freiburg, 18

19 Special cases Hyperbolic Normal inverse Gaussian (NIG) Normal reciprocal inverse Gaussian (NRIG) Variance gamma Student t (limiting case) Cauchy (limiting case) Skewed Laplace Normal (limiting case) Generalized inverse Gaussian (limiting case) Reserve c Eberlein, Uni Freiburg, 19

20 Hyperbolic distribution (λ = 1) d H (x) = α2 β 2 ( ( 2αδK 1 δ ) exp α ) δ 2 + (x µ) 2 + β(x µ) α 2 β 2 Eb., Keller (1995); Eb., Keller, Prause (1998) Normal inverse Gaussian (NIG) (λ = 1/2) d NIG (x) = αδ ( π exp δ ) K1 (αg α 2 β 2 δ (x µ)) + β(x µ) g δ (x µ) where g δ (x) = δ 2 + x 2 O.E. Barndorff-Nielsen (1998) Reserve c Eberlein, Uni Freiburg, 20

21 y Normal NIG (0.02,0,0.5,0) NIG (6,-5.5,1,1) NIG (7,6,1,-1) x Reserve c Eberlein, Uni Freiburg, 21

22 y x Reserve c Eberlein, Uni Freiburg, 22

23 Credit profit and loss distribution 0.0 Reserve c Eberlein, Uni Freiburg, 23

24 Proposition Let L be a Lévy process with triplet (b, c, F). Activity and variation 1 If F(R) < then almost all paths of L have a finite number of jumps on every compact interval. In that case, the Lévy process has finite activity. 2 If F(R) = then almost all paths of L have an infinite number of jumps on every compact interval. In that case, the Lévy process has infinite activity. Proposition Let L be a Lévy process with triplet (b, c, F). 1 If c = 0 and x F (dx) < then almost all paths of L have x 1 finite variation. 2 If c 0 or x 1 x F(dx) = then almost all paths of L have infinite variation. Reserve c Eberlein, Uni Freiburg, 24

25 GH Levy process with marginal densities values of GH (-0.5,100,0,1,0.1) Levy process t Reserve c Eberlein, Uni Freiburg, 25

26 t 0 1 F (R) < 2 F (R) = 3 { x 1} Fine structure of the paths { x 1} { x 1} F(dx) < finite activity F(dx) = infinite activity x F(dx) < (and c = 0) finite variation the sum of the small jumps converges and R x ( µ L (ds, dx) dsf(dx) ) = t 0 R xµ L (ds, dx) t xf(dx) R Reserve c Eberlein, Uni Freiburg, 26

27 Stock prices and indices: geometric Brownian motion (Samuelson 1965) solution Log returns: ds t = µs t dt + σs tdw t ( S t = S 0 exp σw t + ) ) (µ σ2 t 2 ( ) log S t+1 log S t N µ σ2, 2 σ2 Correct return distributions: key ingredient Consistency of the model along different time grids Reserve c Eberlein, Uni Freiburg, 27

28 Deutsche Bank densities Quantiles of Standard Normal Reserve c Eberlein, Uni Freiburg, 28

29 density( x ) zero-bond log-returns ( ), 5 years to maturity empirical densities calculated from zero-yield data for Germany empirical normal NIG log return x of zero-bond Reserve c Eberlein, Uni Freiburg, 29

30 log( density( x ) ) empirical normal log return x of zero-bond Reserve c Eberlein, Uni Freiburg, 30

31 log( density( x ) ) empirical NIG log return x of zero-bond Reserve c Eberlein, Uni Freiburg, 31

32 y x Reserve c Eberlein, Uni Freiburg, 32

33 Exponential Lévy model L = (L t) t 0 S t = S 0 exp(l t) Lévy process with L(L 1 ) = ν Along a time grid with span 1: exact log returns Alternative description by a stochastic differential equation ds t = S t ( dl t + c 2 dt + can be written as R ) (e x 1 x)µ L (dt, dx) ds t = S t d L t where ( L t) t 0 is a Lévy process with jumps > 1 Reserve c Eberlein, Uni Freiburg, 33

34 GH Levy process with marginal densities values of GH (-0.5,100,0,1,0.1) Levy process t Reserve c Eberlein, Uni Freiburg, 34

35 stock price [DM] time [hours] Reserve c Eberlein, Uni Freiburg, 35

36 Consistency along different time grids Models are typically fitted (calibrated) on the basis of daily data (e.g. daily closing prices) Does this model describe the price movements for an intraday or weekly horizon? Classical Gaussian model: Log-returns are always normally distributed (selfsimilarity of Brownian motion) GH model: Empirical investigation shows that the model provides rather good distributions along other time grids as well Reserve c Eberlein, Uni Freiburg, 36

37 empirical densities Daily vs. one hour returns of Bayer data (Jan Aug. 1994) one hour returns and fitted hyperbolic density daily returns Reserve c Eberlein, Uni Freiburg, 37

38 empirical densities Daily vs. one hour returns of Bayer data (Jan Aug. 1994) one hour returns daily returns Reserve c Eberlein, Uni Freiburg, 38

39 Pricing of derivatives: Martingality of exponential Lévy models Necessary assumption: S t = S 0 exp(l t) martingale model E[S t] = S 0 E[exp(L t)] < This excludes a priori the class of stable processes in general E[exp(L t)] < E[L t] < consequently L t = bt + t cw t + x ( µ L (ds, dx) dsf(dx) ) 0 R S t = S 0 exp(l t) is a martingale if b = c 2 ( e x 1 x ) F(dx) Use either Itô s formula or verify that M t = exp(lt) is a martingale E[exp(L t)] R Reserve c Eberlein, Uni Freiburg, 39

40 Equivalent martingale measures (EMMs) ( e rt S t) t 0 has to be a martingale In general large set of EMMs: market is incomplete Characterization of the set of all EMMs (Eberlein and Jacod (1997)) Characterization of those EMMs under which L is again a Lévy process The range of call option prices under all EMMs spans the whole no-arbitrage interval (Eberlein and Jacod (1997)) ( (S0 Ke rt ) +, S 0 ) Criteria to choose an EMM: Esscher transform, minimal distance MM, minimal entropy MM, utility functions,... whole industry Reserve c Eberlein, Uni Freiburg, 40

41 f (S T ) payoff of the option at maturity T f (x) = (x K ) + f (x) = (K x) + Pricing of derivatives European call option European put option Similarly: digitals, quantos, asset-or-nothing, power options,... Given a specific martingale measure (calibration to market data) Explicit formula for European call V = S 0 γ V = E [ e rt f (S T ) ] d T GH(x; θ+1) dx e rt K where γ = ln(k /S 0 ) and d T GH γ d T GH(x; θ) dx GH-density under risk-neutral measure Reserve c Eberlein, Uni Freiburg, 41

42 Raible s method Numerical evaluation based on bilateral Laplace transforms (Raible (2000)). We want to price a European call option; V = e rt IE[(S T K ) + ] = e rt (S T K ) + dp Ω = e rt (S 0 e x K ) + dp LT (x) = e rt (S 0 e x K ) + ρ(x)dx R if P LT is absolutely continuous with respect to the Lebesgue measure λ\ with density ρ. Define g(x) = (e x K ) + and ζ = log S 0, then V = e rt g(ζ x)ρ(x)dx = e rt (g ρ)(ζ) R R Reserve c Eberlein, Uni Freiburg, 42

43 Raible s method (cont.) is a convolution at point ζ. Passing to bilateral Laplace transforms L h (z), z C L V (z) = e rt e zx (g ρ)(x)dx R = e rt e zx g(x)dx e zx ρ(x)dx = e rt L g(z)l ρ(z). R L g can be calculated explicitly; L ρ can be expressed in terms of the characteristic function ϕ LT (Lévy Khintchine formula!). By numerically inverting the Laplace transform, we recover the option price. The method applies to any European hence path-independent payoff, such as call, put, digital, self-quanto and power options. The Lévy motions we are interested in, e.g. generalized hyperbolic, have a (known) Lebesgue density. R Reserve c Eberlein, Uni Freiburg, 43

44 Supremum and infimum processes Let X = (X t) 0 t T be a stochastic process. Denote by X t = sup X u and X t = inf 0 u t 0 u t Xu the supremum and infimum process of X respectively. Since the exponential function is monotone and increasing ( S T = sup S t = sup S 0 e L ) t = S 0 e sup 0 t T L t = S0 e L T. (1) 0 t T 0 t T Similarly S T = S 0 e L T. (2) Reserve c Eberlein, Uni Freiburg, 44

45 formulas payoff functional We want to price an option with payoff Φ(S t, 0 t T ), where Φ is a measurable, non-negative functional. Separation of payoff function from the underlying process: Example Fixed strike lookback option (S T K ) + = (S 0 e L T K ) + = ( e L T +log S 0 K ) + 1 The payoff function is an arbitrary function f : R R +; for example f (x) = (e x K ) + or f (x) = 1 {e x >B}, for K, B R +. 2 The underlying process denoted by X, can be the log-asset price process or the supremum/infimum or an average of the log-asset price process (e.g. X = L or X = L). Reserve c Eberlein, Uni Freiburg, 45

46 formulas Consider the option price as a function of S 0 or better of s = log S 0 X driving process (X = L, L, L, etc.) Φ(S 0 e L t, 0 t T ) = f (X T s) Time-0 price of the option (assuming r 0) V f (X; s) = E [ Φ(S t, 0 t T ) ] = E[f (X T s)] formulas based on Fourier and Laplace transforms Carr and Madan (1999) Raible (2000) Borovkov and Novikov (2002) plain vanilla options general payoffs, Lebesgue densities plain vanilla and lookback options In these approaches: Some sort of continuity assumption (payoff or random variable) Reserve c Eberlein, Uni Freiburg, 46

47 formulas assumptions M XT moment generating function of X T g(x) = e Rx f (x) (for some R R) dampened payoff function L 1 bc(r) bounded, continuous functions in L 1 (R) Assumptions (C1) (C2) (C3) g L 1 bc(r) M XT (R) exists ĝ L 1 (R) Reserve c Eberlein, Uni Freiburg, 47

48 Theorem formulas Assume that (C1) (C3) are in force. Then, the price V f (X; s) of an option on S = (S t) 0 t T with payoff f (X T ) is given by V f (X; s) = e Rs e ius ϕ XT ( u ir) f (u + ir)du, (3) 2π R where ϕ XT denotes the extended characteristic function of X T and f denotes the Fourier transform of f. Proof V f (X; s) = Ω f (X T s)dp = e Rs R e Rx g(x s)p XT (dx). (4) cont. next page Reserve c Eberlein, Uni Freiburg, 48

49 Proof (cont.) Under assumption (C1), g L 1 (R) and ĝ is well-defined. With (C3) ĝ L 1 bc(r). g(x) = 1 e ixu ĝ(u)du. (5) 2π R Returning to the valuation problem (4) we get ( ) V f (X; s) = e Rs e Rx 1 e i(x s)u ĝ(u)du P XT (dx) R 2π R ( ) = e Rs e ius e i( u ir)x P XT (dx) ĝ(u)du 2π R R = e Rs e ius ϕ XT ( u ir) f (u + ir)du. (6) 2π R Reserve c Eberlein, Uni Freiburg, 49

50 Discussion of assumptions Alternative choice: (C1 ) g L 1 (R) L (R) (C3 ) er. P XT L 1 (R) (C3 ) = e R. P XT has a cont. bounded Lebesgue density Recall: (C3) ĝ L 1 (R) Sobolov space H 1 (R) = { g L 2 (R) g exists and g L 2 (R) } Lemma g H 1 (R) = ĝ L 1 (R) Similar for the Sobolev Slobodeckij space H s (R) (s > 1 ) 2 Reserve c Eberlein, Uni Freiburg, 50

51 Examples of payoff functions Example (Call and put option) Call payoff f (x) = (e x K ) +, K R +, f (u + ir) = Similarly, if f (x) = (K e x ) +, K R +, f (u + ir) = K 1+iu R (iu R)(1 + iu R), R I 1 = (1, ). (7) K 1+iu R (iu R)(1 + iu R), R I 1 = (, 0). (8) Reserve c Eberlein, Uni Freiburg, 51

52 Example (Digital option) Call payoff 1 {e x >B}, B R +. f (u + ir) = B iu R 1 iu R, R I 1 = (0, ). (9) Similarly, for the payoff f (x) = 1 {e x <B}, B R +, f (u + ir) = B iu R 1 iu R, R I 1 = (, 0). (10) Example (Double digital option) The payoff of a double digital call option is 1 {B<e x <B}, B, B R+. 1 ( f (u + ir) = B iu R B iu R), R I 1 = R\{0}. (11) iu R Reserve c Eberlein, Uni Freiburg, 52

53 Example (Asset-or-nothing digital) Call payoff Put payoff f (x) = e x 1 {e x >B} f (u + ir) = 1 + iu R, R I 1 = (1, ) f (x) = e x 1 {e x <B} f (u + ir) = B 1+iu R B 1+iu R 1 + iu R, R I 1 = (, 1) Example (Self-quanto option) Call payoff f (x) = e x (e x K ) + f (u + ir) = K 2+iu R (1 + iu R)(2 + iu R), R I 1 = (2, ) Reserve c Eberlein, Uni Freiburg, 53

54 Non-path-dependent options European option on an asset with price process S t = e L t Examples: call, put, digitals, asset-or-nothing, double digitals, self-quanto options X T L T, i.e. we need ϕ LT Generalized hyperbolic model (GH model): Eberlein, Keller (1995), Eberlein, Keller, Prause (1998), Eberlein (2001) ϕ L1 (u) = e iuµ( α 2 β 2 ) ( λ/2 K λ δ α2 (β + iu) ) 2 α 2 (β + iu) 2 ( K λ δ α2 β 2) similar: NIG, CGMY, Meixner ϕ LT (u) = (ϕ L1 (u)) T I 2 = ( α β, α β) Reserve c Eberlein, Uni Freiburg, 54

55 Computation of ϕ L1 Let (L t) t 0 be a gamma process, then E[e ul 1 ] = e ux c γ Γ(γ) x γ 1 e cx dx c γ = Γ(γ) x γ 1 e (c u)x dx c γ (c u) γ = x γ 1 e (c u)x dx (c u) γ Γ(γ) ( ) γ c = for u < c c u ϕ L1 (u) = E[e iul 1 ] = ( c ) γ c iu Reserve c Eberlein, Uni Freiburg, 55

56 Non-path-dependent options II Stochastic volatility Lévy models: Carr, Geman, Madan, Yor (2003) Eberlein, Kallsen, Kristen (2003) Stochastic clock Y t = e.g. CIR process t 0 y s ds (y s > 0) dy t = K (η y t) dt + λy 1/2 t dw t Define for a pure jump Lévy process X = (X t) t 0 Then H t = X Yt (0 t T ) ϕ Ht (u) = ϕ Yt ( iϕ Xt (u)) (ϕ Yt ( iuϕ Xt ( i))) iu Reserve c Eberlein, Uni Freiburg, 56

57 Classification of option types Lévy model S t = S 0 e L t payoff payoff function distributional properties (S T K ) + call 1 {ST >B} digital ( ST K ) + lookback 1 {ST >B} digital barrier = one touch f (x) = (e x K ) + P LT usually has a density f (x) = 1 {e x >B} f (x) = (e x K ) + density of P LT? f (x) = 1 {e x >B} Reserve c Eberlein, Uni Freiburg, 57

58 formula for the last case Payoff function f maybe discontinuous P XT does not necessarily possess a Lebesgue density Assumption (D1) (D2) Theorem g L 1 (R) L (R) M XT (R) exists Assume (D1) (D2) then e Rs A V f (X; s) = lim e ius ϕ XT (u ir) f (ir u) du (12) A 2π A if V f (X; ) is of bounded variation in a neighborhood of s and V f (X; ) is continuous at s. Reserve c Eberlein, Uni Freiburg, 58

59 Fixed strike lookback call: (S T K ) + Lookback options (analogous for lookback put). We get C T (S; K ) = 1 S R iu K 1+iu R 0 ϕ 2π LT ( u ir) du (13) R (iu R)(1 + iu R) where ϕ LT ( u ir) = lim A 1 A 2π A for R (1, M) and Y > α (M). T (Y +iv) e Y + iv κ(y + iv, 0) dv (14) κ(y + iv, iu R) The floating strike lookback option, ( S T S T ) +, is treated by a duality formula (Eb., Papapantoleon (2005)). Reserve c Eberlein, Uni Freiburg, 59

60 One-touch call option: 1 {ST >B}. One-touch options Driving Lévy process L is assumed to have infinite variation or has infinite activity and is regular upwards. L satisfies assumption (EM), then DC T (S; B) = lim A for R (0, M), ϕ = α (M) and ϕ LT (u ir) = lim N 1 A S R+iu 2π A = P(L T > log(b/s 0 )) 1 N 2π N T (Y +iv) e Y + iv 0 ϕ LT (u ir) B R iu du (15) R + iu κ(y + iv, 0) dv. (16) κ(y + iv, R iu) Reserve c Eberlein, Uni Freiburg, 60

61 Equity default swap (EDS) Fixed premium exchanged for payment at default default: drop of stock price by 30 % or 50 % of S 0 first passage time fixed leg pays premium K at times T 1,..., T N, if T i τ B if τ B T : protection payment C, paid at time τ B premium of the EDS chosen such that initial value equals 0; hence K = CE [ ] e rτ B 1 {τb T } N i=1 E [. (17) e rt i 1 {τb >T i }] Calculations similar to touch options, since 1 {τb T } = 1 {ST B}. Reserve c Eberlein, Uni Freiburg, 61

62 Options on multiple assets Basket options Options on the minimum: (ST 1 ST d K ) + Multiple functionals of one asset Barrier options: (S T K ) + 1 {ST >B} Slide-in or corridor options: (S T K ) + N i=1 1 {L<STi <H} Modelling: St i = S0 i exp(l i t) (1 i d) f : R d R + g(x) = e R,x f (x) (x R d ) Assumptions: (A1) g L 1 (R d ) L (R d ) (A2) M XT (R) exists (A3) ϱ L 1 (R d ) where ϱ(dx) = e R,x P XT (dx) Reserve c Eberlein, Uni Freiburg, 62

63 Sensitivities Greeks V f (X; S 0 ) = 1 S R iu 0 M XT (R iu) f (u + ir) du 2π R Delta of an option f (X; S 0 ) = V(X; S 0) = 1 S R 1 iu 0 M XT (R iu) f (u + ir) S 0 2π R (R iu) Gamma of an option Γ f (X; S 0 ) = 2 V f (X; S 0 ) = 1 2 S 0 2π R S R 2 iu 0 M XT (R iu) f (u + ir) (R 1 iu) 1 (R iu) 1 du 1 du Reserve c Eberlein, Uni Freiburg, 63

64 Numerical examples Option prices in the 2d Black-Scholes model with negative correlation Option prices in the 2d stochastic volatility model Option prices in the 2d GH model with positive (left) and negative (right) correlation. c Eberlein, Uni Freiburg, 64 Reserve

65 Consequences for risk management More precise quantification of market risk The stochastic uncertainty of a book or portfolio corresponding to a specified time horizon is given by its P&L-distribution Risk measures (e.g. VaR, volatility, shortfall measure) simple functions of the P&L-distribution Chance measures (e.g. expected return) also functions of the P&L-distribution portfolio management Reserve c Eberlein, Uni Freiburg, 65

66 Probability P&L-distribution 1 0,9 0,8 Confidence Interval 0,7 0,6 0,5 0,4 0,3 0,2 0, Loss Value at Risk (in $ 1000) Profit Reserve c Eberlein, Uni Freiburg, 66

67 Standard risk measure: Value at Risk P[X t < u α] = α α-quantile of return distribution VaR(α) = S 0 S 0 exp(u α) Functional value at risk α VaR(α) Improvement: Shortfall measure Shortfall(α, t) = E[S 0 S 0 exp(x t) X t < u α] Reserve c Eberlein, Uni Freiburg, 67

68 Profit-and-Loss Distribution Density 1%-Quantile Value at Risk 0 Reserve c Eberlein, Uni Freiburg, 68

69 Value at Risk Value at Risk empirical normal hyperbolic NIG GH level of probability c Eberlein, Uni Freiburg, 69 Reserve

70 Stochastic volatility Basic model S t = S 0 exp(x t) where X t = µt + σl t (L t) t 0 standardized Lévy process: E[L 1 ] = 0 and Var(L 1 ) = 1 σ (σ t) t 0 volatility process Dynamic version: dx t = σ tdl t Discrete version: X t = σ t L t Various models for (σ t) t 0 : historic volatility Ornstein Uhlenbeck process d(log σt 2 ) = a(log σt 2 c) dt + bdb t GARCH model implicit volatility Reserve c Eberlein, Uni Freiburg, 70

71 VaR and actual losses implied volatility times of excessive losses trading day Reserve c Eberlein, Uni Freiburg, 71

72 models models should be able to reproduce the observable term structures of interest rates, market prices of interest rate derivatives (caps, floors, swaptions) but they should also be analytically tractable. Idea: Use an HJM-type model driven by a (possibly non-homogeneous) Lévy process Reserve c Eberlein, Uni Freiburg, 72

73 interest rates in percent Real and estimated interest rates of the USA Svensson parameters: b0 = b1 = b2 = b3 = tau1 = tau2 = real estimated time to maturity Termstructure, February 17, 2004 c Eberlein, Uni Freiburg, 73 Reserve

74 Comparison of estimated interest rates (least squares Svensson) interest rate in percent Euroland Japan Switzerland USA time to maturity Reserve Termstructure, February 17, 2004 c Eberlein, Uni Freiburg, 74

75 Nelson Siegel (1987) curves m = maturity, parameters β 0, β 1, β 2, τ 1 )) ( ) 1 s(m) = β 0 + β 1 (1 exp ( mτ1 mτ1 ( ( ( + β 2 1 exp m )) ( ) 1 ) m exp ( ) mτ1 τ 1 τ 1 Improved curves: Svensson (1994) parameters β 0, β 1, β 2, β 3, τ 1, τ 2 )) ( ) 1 s(m) = β 0 + β 1 (1 exp ( mτ1 mτ1 ( ( ( + β 2 1 exp m )) ( ) 1 ) m exp ( ) mτ1 τ 1 τ 1 ( ( ( + β 3 1 exp m )) ( ) 1 ) m exp ( ) mτ2 τ 2 τ 2 Reserve c Eberlein, Uni Freiburg, 75

76 1-month rate in % time One month rate at the German market, March 01, 1967 March 31, 1997 Reserve!" #$#%& ' ()!" #*+,!.- " #/ !879+ ;:8 c Eberlein, Uni Freiburg, - 76

77 Merton (1970) dr t = θ dt + σ db t Short rate dynamics Vasiček (1977) dr t = k(θ r t) dt + σ db t Dothan (1978) dr t = ar tdt + σr t db t Brennan-Schwartz (1979) Constantinides-Ingersoll (1984) Cox-Ingersoll-Ross (1985) dr t = (θ(t) ar t) dt + σr t db t dr t = σr 3/2 t db t dr t = k(θ r t) dt + σ r t db t Ho-Lee (1986) dr t = θ(t) dt + σ db t Black-Derman-Toy (1990) Hull-White (1990) (extended CIR) Sandmann-Sondermann (1993) dr t = dr t = r t ( θ(t) a ln rt σ2 (t) ) dt + σ(t)r t db t k (θ(t) r t) dt + σ(t) r t db t dr t = (1 e r t ) [( θ(t) 1 2 (1 e r t )σ 2) dt +σ db t ] c Eberlein, Uni Freiburg, 77 Reserve

78 Classical of the dynamics of term structures B(t, T ) price at time t [0, T ] of a default-free zero coupon bond with maturity T [0, T ] (B(T, T ) = 1) ( T ) f (t, T ) instantaneous forward rate: B(t, T ) = exp f (t, u) du t Heath, Jarrow, Morton (HJM) framework df (t, T ) = α(t, T ) dt + σ(t, T ) dw t (W t) t 0 d-dimensional Brownian motion σ(t, T ) volatility structure (e.g. Vasiček) Under the risk-neutral measure [ t B(t, T ) = B(0, T ) exp r(s) ds where r(t) = f (t, t) short rate t 0 σ (s, T ) 2 ds + t 0 σ (s, T ) dw s ] Reserve c Eberlein, Uni Freiburg, 78

79 density( x ) zero-bond log-returns ( ), 10 years to maturity empirical densities calculated from zero-yield data for Germany empirical normal log return x of zero-bond Reserve c Eberlein, Uni Freiburg, 79

80 density( x ) zero-bond log-returns ( ), 5 years to maturity empirical densities calculated from zero-yield data for Germany empirical normal NIG log return x of zero-bond Reserve c Eberlein, Uni Freiburg, 80

81 The driving process L = (L 1,..., L d ) is a d-dimensional time-inhomogeneous Lévy process, i.e. L has independent increments and the law of L t is given by the characteristic function IE[exp(i u, L t )] = exp θ s(z) = z, b s z, csz + t R d 0 θ s(iu) ds with ( ) e z,x 1 z, x F s(dx) where b t R d, c t is a symmetric nonnegative-definite d d-matrix and F t is a Lévy measure Integrability: T 0 T 0 ( ) b s + c s + x 2 F s(dx) ds < { x 1} { x >1} exp(ux)f s(dx) ds < for u M Reserve c Eberlein, Uni Freiburg, 81

82 Description in terms of modern stochastic analysis L = (L t) is a special semimartingale with canonical representation t t t L t = b s ds + c 1/2 s dw s + x(µ L ν)(ds, dx) R d and characteristics A t = t 0 b s ds, C t = t 0 c s ds, ν(ds, dx) = F s(dx) ds W = (W t) is a standard d-dimensional Brownian motion, µ L the random measure of jumps of L and ν is the compensator of µ L L is also called a process with independent increments and absolutely continuous characteristics (PIIAC) Reserve c Eberlein, Uni Freiburg, 82

83 Simulation of a GH Lévy motion NIG Levy process with marginal densities values of NIG(100,0,1,0) Levy process t Reserve c Eberlein, Uni Freiburg, 83

84 Simulation of a Lévy process NIG(10,0,0.100,0) on [0,1] NIG(10,0,0.025,0) on [1,3] t Reserve c Eberlein, Uni Freiburg, 84

85 Lévy forward rate approach Eberlein, Raible (1999), Eberlein, Özkan (2003), Eberlein, Jacod, Raible (2005), Eberlein, Kluge (2006) df (t, T ) = 2 A(t, T ) dt 2 Σ(t, T ) dl t (0 t T T ) Σ and A are deterministic functions with values in R d and R respectively whose paths are continuously differentiable in the second variable. The volatility structure is bounded 0 Σ i (t, T ) M (i {1,..., d}). Furthermore, Σ(t, T ) 0 for t < T and Σ(T, T ) = 0 for T [0, T ]. The drift condition A(t, T ) = θ s(σ(t, T )) holds. Reserve c Eberlein, Uni Freiburg, 85

86 Implications Savings account and default-free zero coupon bond prices are given by ( 1 t t ) B t = B(0, t) exp θ s(σ(s, t)) ds Σ(s, t) dl s and 0 ( t t ) B(t, T ) = B(0, T )B t exp θ s(σ(s, T )) ds + Σ(s, T ) dl s. 0 0 Bond prices, once discounted by the savings account, are martingales. In case d = 1, the martingale measure is unique (see Eberlein, Jacod, and Raible (2004)). 0 Reserve c Eberlein, Uni Freiburg, 86

87 L = (L 1,..., L d ) Key tool d-dimensional time-inhomogeneous Lévy process t IE[exp(i u, L t )] = exp θ s(z) = z, b s z, csz + 0 R d θ s(iu) ds where ( ) e z,x 1 z, x F s(dx) in case L is a (time-homogeneous) Lévy process, θ s = θ is the cumulant (log-moment generating function) of L 1. Proposition Eberlein, Raible (1999) Suppose f : R + C d is a continuous function such that R(f i (x)) M for all i {1,..., d} and x R +, then [ ( t )] ( t ) IE exp f (s)dl s = exp θ s(f (s))ds Take f (s) = (s, T ) for some T [0, T ] 0 0 Reserve c Eberlein, Uni Freiburg, 87

88 Pricing of European options [ t B(t, T ) = B(0, T ) exp (r(s) + θ s(σ(s, T ))) ds + 0 where r(t) = f (t, t) short rate t 0 Σ(s, T )dl s ] V (0, t, T, w) time-0-price of a European option with maturity t and payoff w(b(t, T ), K ) V (0, t, T, w) = IE P [B 1 t w(b(t, T ), K )] Volatility structures Σ(t, T ) = σ (1 exp( a(t t))) a (Vasiček) Σ(t, T ) = σ(t t) (Ho Lee) Fast algorithms for Caps, Floors, Swaptions, Digitals, Reserve c Eberlein, Uni Freiburg, 88

89 Forward measure associated with data T T Density dp T dp = 1 B T B(0, T ) or IE P [ dpt dp Ft ] = B(t, T ) B tb(0, T ) For the case of the Lévy term structure model this equals ( t t ) exp Σ(s, T ) dl s θ s(σ(s, T )) ds Compensator of µ L under P T : Standard Brownian motion under P T : 0 0 ν T (dt, dx) = e Σ(t,T ),x ν(dt, dx) W T t = W t t 0 c 1/2 s Σ(s, T ) ds Reserve c Eberlein, Uni Freiburg, 89

90 Pricing formula for caps (Eberlein, Kluge (2006)) w(b(t, T ), K ) = (B(t, T ) K ) + Call with strike K and maturity t on a bond that matures at T Write X = t 0 C(0, t, T, K ) = IE P [B 1 t (B(t, T ) K ) + ] (Σ(s, T ) Σ(s, t))dl s, then C(0, t, T, K ) = 1 KB(0, t) exp(rξ) 2π where ξ is a constant and R < 1. = B(0, t)ie Pt [(B(t, T ) K ) + ] e iuξ (R + iu) 1 (R iu) 1 M X t ( R iu) du Analogous for the corresponding put and for swaptions Reserve c Eberlein, Uni Freiburg, 90

91 to market data Eberlein Kluge (2006) performed for a driving homogeneous as well as for a time-inhomogeneous Lévy process Time-inhomogeneous case: piecewise Lévy process (maturities up to 1 year, 1 to 5 years, greater than 5 years) Minimize the sum of ( model price market price ATM market price for the respective maturity ) 2 Reserve c Eberlein, Uni Freiburg, 91

92 Caplet market data Strike rates Maturity (years) Euro caplet implied volatility surface on February 19, 2002 Reserve c Eberlein, Uni Freiburg, 92

93 results Strike rates Maturity (years) Absolute differences between implied volatility of model and market price 1 Reserve c Eberlein, Uni Freiburg, 93

94 Implied volatility curve for 2 years Market price Model price (homogeneous) Model price (non homogeneous) Implied volatility (in %) Strike rate (in %) Reserve c Eberlein, Uni Freiburg, 94

95 Implied volatility curve for 5 years Market price Model price (homogeneous) Model price (non homogeneous) Implied volatility (in %) Strike rate (in %) c Eberlein, Uni Freiburg, 95 Reserve

96 Implied volatility curve for 10 years Market price Model price (homogeneous) Model price (non homogeneous) Implied volatility (in %) Strike rate (in %) c Eberlein, Uni Freiburg, 96 Reserve

97 Basic interest rates B(t,T ): price at time t [0, T ] of a default-free zero coupon bond f (t,t ): instantaneous forward rate B(t,T ) = exp ( ) T f (t,u) du t L(t,T ): default-free forward Libor rate for the interval T to T + δ ( ) L(t,T ) := 1 B(t,T ) 1 δ B(t,T +δ) F B (t,t,u): forward price process for the two maturities T and U F B (t,t,u) := B(t,T ) B(t,U) = 1 + δl(t,t ) = B(t,T ) B(t,T + δ) = F B(t,T,T + δ) Reserve c Eberlein, Uni Freiburg, 97

98 LIBOR market model T 0 T1 T2 T3 T M+1 =T * with δ = T n+1 T n (fixed accrual period) L(t, T ) forward LIBOR rate for the interval T to T + δ as of time t T δ-forward LIBOR rate L(t, T ) = 1 ( ) B(t, T ) δ B(t, T + δ) 1 For two maturities T, U define the forward process F B (t, T, U) = B(t, T ) B(t, U) = 1 + δl(t, T ) = F B (t, T, T + δ) Sandmann, Sondermann, Miltersen (1995); Miltersen, Sandmann, Sondermann (1997); Brace, Gatarek, Musiela (1997); Jamshidian (1997) Reserve c Eberlein, Uni Freiburg, 98

99 The Lévy Libor model (Eberlein, Özkan (2005)) Tenor structure T 0 < T 1 < < T M < T M+1 = T with T i+1 T i = δ, set Ti = T iδ for i = 1,..., M T M * T* M 1 T * 2 T * 1 T 0 T 1 T 2 T 3 Assumptions T M 1 T M T * (LR.1): For any maturity T i there is a bounded deterministic function λ(, T i ), which represents the volatility of the forward Libor rate process L(, T i ). (LR.2): We assume a strictly decreasing and strictly positive initial term structure B(0, T ) (T ]0, T ]). Consequently the initial term structure of forward Libor rates is given by L(0, T ) = 1 ( ) B(0, T ) δ B(0, T + δ) 1 Reserve c Eberlein, Uni Freiburg, 99

100 Backward Induction (1) Given a stochastic basis (Ω, F T, P T, (F t) 0 t T ) T* * M T * 2 T * 1 T M 1 T 0 T 1 T 2 T 3 T M 1 T M T * We postulate that under P T ( t L(t, T1 ) = L(0, T1 ) exp where t t L T t = b s ds c 1/2 s dw T s + ) λ(s, T1 )dl T s t 0 R x(µ L ν T,L )(ds, dx) is a non-homogeneous Lévy process with random measure of jumps µ L and P T -compensator ν T,L (ds, dx) = F s(dx) ds, F s({0}) = 0, where F s satisfies some integrability conditions Reserve c Eberlein, Uni Freiburg, 100

101 Backward Induction (2) In order to make L(t, T 1 ) a P T -martingale, choose the drift characteristic (b s) s.t. t 0 λ(s, T 1 )b s ds= 1 2 t 0 t c sλ 2 (s, T 1 ) ds Transform L(t, T 1 ) in a stochastic exponential where H(t, T 1 ) = t 0 0 R ( ) e λ(s,t 1 )x 1 λ(s, T1 )x ν T,L (ds, dx) L(t, T 1 ) = L(0, T 1 )E(H(t, T 1 )) λ(s, T1 )c 1/2 s dw T s + t 0 R ( ) e λ(s,t 1 )x 1 (µ L ν T,L )(ds, dx) c Eberlein, Uni Freiburg, 101 Reserve

102 Backward Induction (3) Equivalently ( dl(t, T1 ) = L(t, T1 ) λ(t, T1 )c 1/2 t dw T t ( ) ) + e λ(t,t 1 )x 1 (µ L ν T,L )(dt, dx) R with initial condition L(0, T 1 ) = 1 δ ( ) B(0, T 1 ) B(0, T ) 1 Reserve c Eberlein, Uni Freiburg, 102

103 Backward Induction (4) Recall F B (t, T 1, T ) = 1 + δl(t, T 1 ), therefore, df B (t, T1, T ) = δdl(t, T1 ) ( δl(t, T = F B (t, T1, T ) 1 ) + R 1 + δl(t, T 1 ) λ(t, T 1 ) }{{} = α(t,t 1,T ) c 1/2 t dw T t δl(t, T1 ) ( ) ) 1 + δl(t, T1 ) e λ(t,t 1 )x 1 (µ L ν T,L )(dt, dx) }{{} = β(t,x,t 1,T ) 1 Define the forward martingale measure associated with T 1 M 1 t = t dp T 1 dp T α(s, T1, T )c 1/2 dw T 0 s = E T 1 (M 1 ) where t s + 0 R (β(s, x, T1, T ) 1) (µ L ν T,L )(ds, dx) c Eberlein, Uni Freiburg, 103 Reserve

104 Then W T 1 t = W T t t 0 Backward Induction (5) α(s, T1, T )c 1/2 s ds is the forward Brownian motion for date T 1 and ν T 1,L (dt, dx) = β(t, x, T 1, T )ν T,L (dt, dx) is the P T 1 -compensator for µ L. Second step T 2 * T 1 * T 0 T 1 T 2 T M 1 T M T * We postulate that under P T 1 ( t L(t, T2 ) = L(0, T2 ) exp 0 λ(s, T 2 ) dl T 1 s ) where L T 1 t t = b T 1 t s ds + c 1/2 s dw T t 1 s + x(µ L ν T 1,L )(ds, dx) R Reserve c Eberlein, Uni Freiburg, 104

105 Backward Induction (6) L(t, T 2 ) is a P T 1 -martingale if (b T 1 s t 0 λ(s, T 2 )b T 1 s ds = 1 2 t 0 t 0 ) is chosen s.t. c sλ 2 (s, T 2 ) ds ( ) e λ(s,t 2 )x 1 λ(s, T2 )x ν T 1,L (ds, dx) R Reserve c Eberlein, Uni Freiburg, 105

106 Second measure change where t Mt 2 = 0 t + Backward Induction (7) dp T 2 dp T 1 = E T 2 (M 2 ) α(s, T2, T1 )c 1/2 s dw T 1 s 0 R ( β(s, x, T 2, T 1 ) 1 ) (µ L ν T 1,L )(ds, dx) This way we get for each time point Tj in the tenor structure a Libor rate process which is under the forward martingale measure P T j 1 of the form ( t ) L(t, Tj ) = L(0, Tj ) exp λ(s, Tj ) dl T j 1 s 0 Reserve c Eberlein, Uni Freiburg, 106

107 Comparison with Lévy forward rate approach F B (t, T 1, T ) = B(t, T 1 ) B(t, T ) dp T 1 dp T Ft = dp T 1 dp dp dp T Ft = B(t, T 1 ) B tb(0, T ) B tb(0, T1 ) B(t, T ) = B(0, T ) B(0, T 1 ) F B(t, T 1, T ) Reserve c Eberlein, Uni Freiburg, 107

108 Forward process model (1) Postulate ( t 1 + δl(t, T1 ) = (1 + δl(0, T1 )) exp equivalently ( t F B (t, T1, T ) = F B (0, T1, T ) exp In differential form ( df B (t, T1, T ) = F B (t, T1, T ) + R 0 0 λ(t, T1 )c 1/2 t ) ( e λ(t,t 1 )x 1 ) λ(s, T1 ) dl T s ) λ(s, T1 ) dl T s dw T t ) (µ L ν T,L )(dt, dx) Reserve c Eberlein, Uni Freiburg, 108

109 Forward process model (2) Define the forward martingale measure associated with T 1 where M 1 t = t 0 dp T 1 dp T = E T 1 ( M 1 ) t ( ) λ(s, T1 )c 1/2 s dw T s + e λ(s,t 1 )x 1 (µ L ν T,L )(ds, dx). 0 R Reserve c Eberlein, Uni Freiburg, 109

110 Then W T 1 t = W T t motion for date T 1 Forward process model (3) t and 0 λ(s, T1 )c 1/2 s ds is the forward Brownian ν T 1,L (dt, dx) = exp(λ(t, T 1 )x) ν T,L (dt, dx) is the P T 1 -compensator of µ L. Continuing this way we get for each time point Tj a Libor rate process under P T j 1 in the form in the tenor structure 1 + δl(t, Tj ) = ( 1 + δl(0, Tj ) ) ( t exp λ(s, Tj ) dl T s 0 with successive compensators ( j ) ν T j,l (dt, dx) = exp λ(t, Ti )x F t(dx) dt. i=1 Consequence of this alternative approach: negative Libor rates can occur j 1 ). Reserve c Eberlein, Uni Freiburg, 110

111 Pricing of caps and floors (1) Time-T j -payoff of a cap settled in arrears Nδ(L(T j 1, T j 1 ) K ) + N notional amount (set N = 1) K strike rate Time-t value C t = = [ n IE P j=1 B t B Tj δ(l(t j 1, T j 1 ) K ) + F t n [ B(t, T j )IE PTj δ(l(tj 1, T j 1 ) K ) + ] F t j=1 Analogous for floor Nδ(K L(T j 1, T j 1 )) + ] Reserve c Eberlein, Uni Freiburg, 111

112 Pricing of caps and floors (2) Dynamics of L(t, T j 1 ) under P Tj (purely discontinuous case) ( ) ( dl(t, T j 1 ) = L(t, T j 1 ) e λ(t,t j 1 )x 1 µ L ν T,L) j (dt, dx) R Solution ( t ) L(t, T j 1 ) = L(0, T j 1 ) exp λ(s, T j 1 ) dl T j s Write 0 ( t = L(0, T j 1 ) exp b T j s λ(s, T j 1 ) ds 0 t ( + (xλ(s, T j 1 )) µ L ν T,L) ) j (ds, dx) 0 R X t = t 0 λ(s, T j 1) dl T j s then L(t, T j 1 ) = L(0, T j 1 ) exp(x t) is a martingale with respect to P Tj Reserve c Eberlein, Uni Freiburg, 112

113 Numerical evaluation Denote ζ j = ln(l(0, T j 1 )) and v K (x) = (e x K ) + Bilateral Laplace transform of v K : L[v K ](z) = + e zx v K (x) dx Characteristic function of X Tj 1 : χ(u) = IE PTj [exp(iux Tj 1 ] Assume mgf( R)<, then the time-0 price of the j-th caplet is given by V j (ζ j, K ) = δb(0, T j ) eζ j R 2π + e iuζ j L[v K ](R+iu)χ(iR u) du whenever the right-hand side exists χ(u) easy to compute for generalized hyperbolic Lévy motion Reserve c Eberlein, Uni Freiburg, 113

114 Representation as convolution V j (ζ j, K ) = δb(0, T j ) eζ j R 2π + e iuζ j V j (ζ j, K ) = δb(0, T j )IE PTj [(L(T j 1, T j 1 ) K ) + ] ] = δb(0, T j )IE PTj [v K (ζ j X Tj 1 ) = δb(0, T j ) v K (ζ j x) P X T j 1 T j (dx) R = δb(0, T j ) v K (ζ j x)ρ(x) dx = δb(0, T j ) (v K ρ) (ζ j ). R L[v K ](R+iu)χ(iR u) du And L[V j ](R + iu) = δb(0, T j )L[v K ](R + iu)l[ρ](r + iu) for u R. Reserve c Eberlein, Uni Freiburg, 114

115 Extensions of the basic Lévy market model Lévy market model (Eb Özkan (2005)) Multi-currency setting (Eb Koval (2006)) model (Eb Kluge Schönbucher (2006)) Swap rate model (Eb Liinev (2006)) Duality principle (Eb Kluge Papapantoleon (2006)) Reserve c Eberlein, Uni Freiburg, 115

116 Lévy market model Domestic Market Foreign Market P 0,T -forward measure P 0,T N -forward measure P i,t N -forward measure P 0,T N 1 -forward measure P i,t N 1 -forward measure P 0,Tj+1 -forward measure P i,tj+1 -forward measure P 0,Tj -forward measure P i,t -forward measure F X i (, T ) F B (, T j, T j+1 ) F B i (, T j, T j+1 ) F X i (, T j ) P 0,T 1 -forward measure P i,t 1 -forward measure Relationship between domestic and foreign fixed income markets in a discrete-tenor framework. P i,tj -forward measure Reserve c Eberlein, Uni Freiburg, 116

117 Libor rates in a cross currency setting Discrete tenor structure Accrual periods T 0 < T 1 < < T n < T n+1 = T δ = T j+1 T j T T 1 T 2 T j 1 T j 0 (m + 1) markets i = 0,..., m 0 = domestic market T * N T Want to model the dynamics of the Libor rate L i (t,t j 1 ) which applies to time period [T j 1,T j ] in market i (i = 0,..., m) We target at the form ( t ) L i (t,t j 1 ) = L i (0,T j 1 ) exp λ i (s,t j 1 ) dl i,t j s 0 Reserve c Eberlein, Uni Freiburg, 117

118 The driving process L 0,T = (L 0,T 1,..., L 0,T d ) is a d-dimensional time-inhomogeneous Lévy process. The law of L 0,T t is given by t IE[exp(iu L 0,T t )] = exp 0 θ 0,T s (z) = z b 0,T s z C sz + R d θ 0,T s (iu) ds with ( e z x 1 z x where b 0,T t R d, C s is a symmetric nonnegative-definite d d-matrix and λ 0,T s is a Lévy measure. Integrability assumptions )λ 0,T (dx), s Reserve c Eberlein, Uni Freiburg, 118

119 Description in terms of modern stochastic analysis L 0,T = (L 0,T t ) is a special semimartingale with canonical representation t t t L 0,T t = b 0,T s ds + c s dw 0,T s + x(µ ν 0,T )(ds, dx) R d (W 0,T t ) is a P 0,T -standard Brownian motion with values in R d c t is a measurable version of the square root of C t µ the random measure of jumps of (L 0,T t ) ν 0,T (ds, dx) = λ 0,T s (dx) ds is the P 0,T -compensator of µ (L 0,T t ) is also called a process with independent increments and absolutely continuous characteristics (PIIAC). Reserve c Eberlein, Uni Freiburg, 119

120 The foreign forward exchange rate for date T (1) Assumptions (FXR.1): For every market i {0,..., m} there are strictly decreasing and strictly positive zero-coupon bond prices B i (0,T j )(j = 0,..., N + 1) and for every foreign economy i {1,..., m} there are spot exchange rates X i (0) given. Consequently the initial foreign forward exchange rate corresponding to T is F X i (0,T ) = Bi (0,T )X i (0) B 0 (0,T ) Reserve c Eberlein, Uni Freiburg, 120

121 The foreign forward exchange rate for date T (2) Assumptions (FXR.2): For every foreign market i {1,..., m} there is a continuous deterministic function ξ i (,T ) : [0,T ] R d +. For every coordinate 1 k d we assume where M < (ξ i (s,t )) k M (s [0,T ], 1 i m) M N + 2. Reserve c Eberlein, Uni Freiburg, 121

122 Assumptions The foreign forward exchange rate for date T (3) (FXR.3): For every i {1,..., m} the foreign forward exchange rate for date T is given by ( t t ) F X i (t,t ) = F X i (0,T ) exp γ i (s,t ) ds + ξ i (s,t ) dl 0,T s 0 0 where γ i (s,t ) = ξ i (s,t ) b 0,T s 1 ξ i (s,t ) c s 2 2 ( ) e ξi (s,t ) x 1 ξ i (s,t ) x λ 0,T s (dx) R d Equivalently ( F X i (t,t ) = F X i (0,T )E t ξ i (s,t ) c s dw 0,T s 0 ( + exp ( ξ i (s,t ) x ) ) ) 1 (µ ν 0,T )(ds, dx) 0 R d Reserve c Eberlein, Uni Freiburg, 122

123 Consequences: Define The foreign forward exchange rate for date T (4) F X i (, T ) is a P 0,T -martingale [ ] FX E P 0,T i (t,t ) F X i (0,T = 1 ) dp i,t dp 0,T = F X i (t,t ) Ft F X i (0,T ) By Girsanov s theorem we get a P i,t -standard Brownian motion W i,t t and a P i,t -compensator = W 0,T t t c sξ i (s,t ) ds 0 ν i,t (dt, dx) = exp(ξ i (t,t ) x)ν 0,T (dt, dx) Reserve c Eberlein, Uni Freiburg, 123

124 Tenor structure with T j+1 T j = δ, set T N * T0 T 1 T 2 T 3 Assumptions The Lévy Libor model as in Eberlein Özkan (2005) T 0 < T 1 < < T N < T N+1 = T T* N 1 T j = T jδ for j = 1,..., N T 2 * T 1 * T N 1 T * N T (CLM.1): For every market i and every maturity T j there is a bounded deterministic function λ i (,T j ), which represents the volatility of the forward Libor rate process L i (,T j ) in market i. (CLM.2): The initial term structure of forward Libor rates in market i is given by L i (0,T j ) = 1 ( B i ) (0,T j ) δ B i (0,T j + δ) 1 Reserve c Eberlein, Uni Freiburg, 124

125 Backward Induction Given a stochastic basis (Ω, F T, P 0,T, (F t) 0 t T ) T N * T* N 1 T 2 * T 1 * T0 T 1 T 2 T 3 We postulate that under P i,t where L i,t t = T N 1 T * N T ( t ) L i (t, T1 ) = L i (0, T1 ) exp λ i (s, T1 ) dl i,t s 0 t 0 t t b i,t s ds + c s dw i,t s + x(µ ν i,t )(ds, dx) 0 0 R d with W i,t and ν i,t as given before. Reserve c Eberlein, Uni Freiburg, 125

126 Lévy market model Domestic Market Foreign Market P 0,T -forward measure P 0,T N -forward measure P i,t N -forward measure P 0,T N 1 -forward measure P i,t N 1 -forward measure P 0,Tj+1 -forward measure P i,tj+1 -forward measure P 0,Tj -forward measure P i,t -forward measure F X i (, T ) F B (, T j, T j+1 ) F B i (, T j, T j+1 ) F X i (, T j ) P 0,T 1 -forward measure P i,t 1 -forward measure Relationship between domestic and foreign fixed income markets in a discrete-tenor framework. P i,tj -forward measure Reserve c Eberlein, Uni Freiburg, 126

127 Relationship between the domestic and the foreign market The forward exchange rates in the i-th foreign market are related by F X i (t,t j ) = F X i (t,t j+1 ) F B i (t,t j,t j+1 ) F B 0(t,T j,t j+1 ) From this one gets the dynamics of F X i (t,t j ) df X i (t,t j ) df X i (t,t j ) = ζi (t,t j,t j+1 ) dw 0,T j t + (ζ i (t, x,t j,t j+1 ) 1)(µ ν 0,Tj )(dt, dx) R d where the coefficients are given recursively ζ i (t,t j,t j+1 ) = α i (t,t j,t j+1 ) α 0 (t,t j,t j+1 ) + ζ i (t,t j+1,t j+2 ) ζ i (t, x,t j,t j+1 ) = βi (t, x,t j,t j+1 ) β 0 (t, x,t j,t j+1 ) ζi (t, x,t j+1,t j+2 ) Reserve c Eberlein, Uni Freiburg, 127

128 Foreign forward caps and floors Pricing cross-currency derivatives (1) δx[l i (T j 1,T j 1 ) K i ] + Time-0-value of a foreign T N -maturity cap N+1 [ ( FC i (0, T N ) = δ B i (0,T j )E i,t P j L i (T j 1,T j 1 ) K i) ] + j=1 Alternatively if we define K i = 1 + δk i (forward process approach) N+1 [ ( FC i (0, T N ) = B i (0,T j )E i,t P j 1 + δl i (T j 1,T j 1 ) K i) ] +, j=1 N+1 = C i (0,T j, K i ) j=1 Reserve c Eberlein, Uni Freiburg, 128

129 Numerical evaluation of the cap price Define X i T j 1 (t) = t 0 Pricing cross-currency derivatives (2) λ i (s,t j 1 ) dl i,t j s = ln 1 + δli (t,t j 1 ) 1 + δl i (0,T j 1 ) and let χ i,t j 1 (z) be its characteristic function, then C i (0,T j, K i ) = B i (0,T j ) K i exp( ξ i j R) 2π exp(iu ξ j i χ i,t j 1 (ir u) ) (R + iu)(1 + R + iu) du where ξ i j = ln( K i ) ln(1 + δl i (0,T j 1 )) and R is s.t. χ i,t j 1 (ir) <. Reserve c Eberlein, Uni Freiburg, 129

130 swaps Pricing cross-currency derivatives (3) Floating-for-floating cross-currency (i; l; 0) swap Libor rate L i (T j 1,T j 1 ) of currency i is received at each date T j Libor rate L l (T j 1,T j 1 ) of currency l is paid Payments are made in units of the domestic currency Thus the cashflow at time point T j is (notional = 1) δ(l i (T j 1,T j 1 ) L l (T j 1,T j 1 )) Reserve c Eberlein, Uni Freiburg, 130

131 Pricing cross-currency derivatives (4) The time-0-value of a floating-for-floating (i; l; 0) cross-currency forward swap in units of the domestic currency is N+1 CCFS [i;l;0] (0) = B 0 (0,T j ) B i (0,T j 1 ) exp (D i (0,T B i j 1,T j )) (0,T j ) j=1 N+1 B l (0,T j 1 ) exp (D l (0,T B l j 1,T j )) (0,T j ) where D i (0,T j 1,T j ) = Tj 1 0 Tj 1 0 j=1 λ i (s, T j 1 ) c sζ i (s,t j,t j+1 ) ds R d ( ( ) exp λ i (s,t j 1 ) (ζi x 1) (s, x,t j,t j+1 ) 1 ) ν 0,Tj (ds, dx) c Eberlein, Uni Freiburg, 131 Reserve

132 Pricing cross-currency derivatives (5) A quanto caplet with strike K i, which expires at time T j 1, pays at time T j QCpl i (T j,t j, K i ) = δx i (L i (T j 1,T j 1 ) K i ) + where X i is the preassigned foreign exchange rate Time-0-value QCpl i (0,T j, K i ) = B 0 (0,T j )IE 0,T P j [δx i (L i (T j 1,T j 1 ) K i ) + ] = B 0 (0,T j )X i IE 0,T P j [(1 + δl i (T j 1,T j 1 ) (1 + δk i )) + ] (forward process approach) Reserve c Eberlein, Uni Freiburg, 132

133 Pricing cross-currency derivatives (6) Numerical evaluation of quanto caplets. Write ( Tj δl i (T j 1,T j 1 ) = (1 + δl i (0,T j 1 )) exp then for v(x) = (e x 1) + Finally we get 0 λ i (s,t j 1 ) dl i,t j s = (1 + δl i (0,T j 1 )) exp ( M i (0,T j 1,T j ) + D i ) (0,T j 1,T j ) }{{}}{{} random non-random QCpl i (0,T j, K i ) = B 0 (0,T j )X i (1 + δk i )(v ϱ)(ξ j ) QCpl i (0,T j, K i ) = B 0 (0,T j )X i (1 + δk i ) exp(ξ jr) 2π ) χ Mi,T j 1 (ir u) exp(iuξ j ) (R + iu)(r iu) du c Eberlein, Uni Freiburg, 133 Reserve

134 Absolute errors of EUR caplet calibration 3.0% 2.0% 1.0% 0.0% 1.50% 1.75%2.00% 2.25% 2.50% Strike Rate Maturity (years) 2 1 Reserve c Eberlein, Uni Freiburg, 134

135 Absolute errors of USD caplet calibration 3.0% 2.0% 1.0% 0.0% 1.00% 1.25%1.50% 2.00% Strike Rate 3.00% Maturity (years) 2 1 Reserve c Eberlein, Uni Freiburg, 135

136 Basic interest rates P(t,T ): price at time t [0, T ] of a default-free zero coupon bond with maturity T [0,T ] (P(T,T ) = 1) f (t,t ): instantaneous forward rate P(t,T ) = exp ( ) T f (t,u) du t L(t,T ): default-free forward Libor rate for the interval T to T + δ as of time t T (δ-forward Libor rate) ( ) L(t,T ) := 1 P(t,T ) 1 δ P(t,T +δ) F P (t,t,u): forward price process for the two maturities T and U F P (t,t,u) := P(t,T ) P(t,U) = 1 + δl(t,t ) = P(t,T ) P(t,T + δ) = F P(t,T,T + δ) Digital options Range notes Reserve

137 P(t,T ): underlying Pricing of options on bonds w(p(t,t ), K ): payoff of a European option with maturity t and strike K V (0, t,t, w): time-0-price of the option V (0, t,t, w) = IE P [B 1 t w(p(t,t ), K )] Caps, Floors, Swaptions, Digitals, Turnbull (1995): floating range notes in 1-factor Gaussian HJM Navatte and Quittard-Pinon (1999): delayed digital options Nunes (2004): multifactor Gaussian HJM Digital options Range notes Reserve

138 Forward measure and adjusted forward measure Forward martingale measure for settlement day T dp T dp := 1 ( T T ) B T P(0,T ) = exp A(s,T ) ds + Σ(s,T ) dl s 0 0 Adjusted forward measure P T,T for T < T For 0 t T : dp T,T dp T := F(T,T,T ) F(0,T,T ) = P(0,T ) P(0,T )P(T,T ) dp T,T dp = dp T Ft dp Ft Digital options Range notes Reserve

139 Digital options (1) Standard European interest rate digital call (put) with strike r K SD(Θ) T [r n(t, T + δ); r k ; T ] := 1 {Θrn(T,T +δ)>θr k }, where r n(t, T + δ) is the reference rate (Libor) r n(t, T + δ) = 1 [ ] 1 δ P(T, T + δ) 1 and Θ = 1 for a digital call, Θ = 1 for a digital put Delayed digital option for maturity T and payment date T 1 DD(Θ) T1 [r n(t, T + δ); r k ; T 1 ] := 1 {Θrn(T,T +δ)>θr k } Digital options Range notes Reserve

140 Digital options (2) Delayed range digital options (T T 1 ) DRD T1 [r n(t, T + δ); r l ; r u; T 1 ] := 1 {rn(t,t +δ) [r l,r u]} Obvious relationship for time-t prices DRD t[r n(t, T + δ); r l ; r u; T 1 ] = P(t, T 1 ) DD(1) t[r n(t, T + δ); r u; T 1 ] DD( 1) t[r n(t, T + δ); r l ; T 1 ]. Call-put parity only when L(P(T, T + δ)) without point masses DD(1) t[r n(t, T + δ); r k ; T 1 ] = P(t, T 1 ) DD( 1) t[r n(t, T + δ); r k ; T 1 ] Digital options Range notes Reserve

141 Pricing formulae for delayed digital options (1) D t := DD(1) t[r n(t, T + δ); r k ; T 1 ] [ ] 1 = B tie P 1 {rn(t,t +δ)>r B k } F t T1 ] = P(t,T 1 )IE T1 [1 {rn(t,t +δ)>r k } Ft = P(t,T 1 )IE T1 [1 ] { } P(T,T +δ)< δr 1 Ft k +1 ( ) P(t,T + δ) = P(t,T 1 )h P(t,T ) ] where h(y) = IE T1 [1 { y exp[ T t A(s,T,T +δ) ds+ } T t Σ(s,T,T +δ) dl s]< δr 1 k +1 and A(s, T, T + δ) = A(s,T + δ) A(s,T ), Σ(s, T,T + δ) = Σ(s,T + δ) Σ(s,T ) Digital options Range notes Reserve

142 Pricing formulae for delayed digital options (2) T Denote X := Σ(s,T,T + δ) dl s K := t ( 1 T ) δr K + 1 exp A(s,T,T + δ) ds P X T 1 = distribution of X under P T1 then h(y) = 1 { } dp X e x < K T1 (x) y = f y( x)ϕ(x) dx = (f y ϕ)(0) = V (0) for f y(x) = 1 { e x < K y } and V (ζ) = (fy ϕ)(ζ) Denote by M X T 1 the moment generating function of X w.r.t. P T1 t Digital options Range notes Reserve

143 Theorem Pricing formulae for delayed digital options (3) Suppose the distribution of X possesses a Lebesgue density. Choose an R > 0 such that M X T 1 ( R) <. Then D t = 1 π P(t,T 1) 0 ( ( ) R+iu ) P(t,T ) R P(t,T + δ) K 1 R + iu MX T 1 ( R iu) du Proof: L[V ](z) = L[f y](z)l[ϕ](z) V (0) = 1 lim 2πi Y L[f y](r + iu) = 1 R+iu R+iY L[V ](z) dz R iy ( ) R+iu K y Digital options Range notes Reserve

144 Pricing range notes (1) n n + n T 0 t T 1 T 2 T N T j = coupon payment dates n j = number of days between T j and T j+1 based on some day count convention δ j = number of years between T j and T j+1 T j,i = T j + i δ j,i = length (in years) of the compounding period starting at T j,i Digital options Range notes Reserve

145 Pricing range notes (2) For a floating range note, the coupon at time T j+1 is ν j+1 (T j+1 ) := rn(t j,t j + δ j ) + s j D j H(T j,t j+1 ) where s j is the spread over the reference rate D j number of days for the (j +1)-th compounding period n j H(T j,t j+1 ) = i=1 Time-t value of a flaoting range note 1 {rl (T j,i ) r n(t j,i,t j,i +δ j,i ) r u(t j,i )} N 1 FlRN(t) := P(t,T N ) + ν j+1 (t) j=0 Digital options Range notes Reserve

146 of FlRN coupons (1) [ ] 1 r n(t 0,T 0 + δ 0 ) + s 0 ν 1 (t) = B tie H(T 0,T 1 ) F t B T1 D 0 = rn(t 0,T 0 + δ 0 ) + s [ 0 P(t,T 1 )IE T1 H(T 0,T 1 ) ] Ft D 0 = rn(t ( 0,T 0 + δ 0 ) + s 0 P(t,T 1 )H(T 0, t) D 0 n 0 [ ] ) + P(t, T 1 ) IE T1 1 {rl (T 0,i ) r n(t 0,i,T 0,i +δ 0,i ) r u(t 0,i )} F t i=n 0 +1 }{{} =DRD t[r n(t 0,i,T 0,i +δ 0,i );r l (T 0,i );r u(t 0,i );T 1] Digital options Range notes Reserve

147 of FlRN coupons (2) [ rn(t j,t j+1 ) + s ] j ν j+1 (t) = P(t,T j+1 )IE Tj+1 H(T j,t j+1 ) F t D j ( sj = 1 ) n j ] P(t,T j+1 ) IE Tj+1 [1 {rl (T D j δ j D j,i ) r n(t j,i,t j,i +δ j,i ) r u(t j,i )} F t j + P(t,T j+1) δ j D j i=1 =: ν 1 j+1(t) + ν 2 j+1(t). i=1 n j [ ] 1 IE Tj+1 P(T j,t j+1 ) 1 {r l (T j,i ) r n(t j,i,t j,i +δ j,i ) r u(t j,i )} F t Note that ( νj+1(t) 1 sj = 1 ) n j DRD t[r n(t j,i,t j,i + δ j,i ); r l (T j,i ); r u(t j,i );T j+1 ] D j δ j D j and ν 2 j+1(t) = n j i=1 i=1 P(t,T j ) [ ] IE Tj,T 1{rl δ j D j+1 (T j,i ) r n(t j,i,t j,i +δ j,i ) r u(t j,i )} F t j }{{} =:D j,i t Digital options Range notes Reserve

148 of FlRN coupons (3) Neglect now the indices i, j Theorem Suppose the distribution of X possesses a Lebesgue density. Choose an R > 0 such that M X ( R) <. Then D t = 1 ( ( ) ) R+iu P(t,T ) R π 0 P(t,T + δ) K 1 R + iu MX ( R iu) du 1 ( ( ) ) R+iu P(t,T ) R π P(t,T + δ) K 1 R + iu MX ( R iu) du with 0 K := K := ( 1 T δr l (T ) + 1 exp t ( T 1 δr u(t ) + 1 exp t ) A(s,T, T + δ) ds ) A(s,T, T + δ) ds Digital options Range notes Reserve

149 Lévy credit risk model Maturities in years Government Bond Caa B3 B2 B1 Ba3 Ba2 Ba1 Baa3 Baa1 A1 Aaa c Eberlein, Uni Freiburg, 149 Credit derivatives Reserve

150 Basic interest rates B(t,T ): price at time t [0, T ] of a default-free zero coupon bond with maturity T [0, T ] (B(T,T ) = 1) f (t,t ): instantaneous forward rate B(t,T ) = exp ( ) T f (t,u) du t L(t,T ): default-free forward Libor rate for the interval T to T + δ as of time t T (δ-forward Libor rate) ( ) L(t,T ) := 1 B(t,T ) 1 δ B(t,T +δ) F B (t,t,u): forward price process for the two maturities T < U F B (t,t,u) := B(t,T ) B(t,U) = 1 + δl(t,t ) = B(t,T ) B(t,T + δ) = F B(t,T,T + δ) Credit derivatives Reserve c Eberlein, Uni Freiburg, 150

151 The Lévy Libor model with default risk (Eberlein, Kluge, Schönbucher 2006) B 0 (t,t k ): time-t price of a defaultable zero coupon bond with zero recovery and maturity T k τ: time of default B(t,T k ): pre-default value of the defaultable bond = B 0 (t,t k ) = 1 {τ>t} B(t,T k ), B(T k,t k ) = 1 (k = 1,..., n) Terminal value of the defaultable bond B 0 (T k,t k ) = 1 {τ>tk }B(T k,t k ) = 1 {τ>tk } Credit derivatives Reserve c Eberlein, Uni Freiburg, 151

152 The Lévy Libor model with default risk (2) The defaultable forward Libor rates for the interval [T k, T k+1 ] are given by L(t,T k ) := 1 ( ) B(t,Tk ) δ k B(t,T k+1 ) 1. The forward Libor spreads are given by S(t,T k ) := L(t,T k ) L(t,T k ). The default risk factors or forward survival processes are given by D(t,T k ) := B(t,T k) B(t,T k ). The discrete-tenor forward default intensities are given by H(t,T k ) := 1 ( ) D(t,Tk ) δ k D(t,T k+1 ) 1 = S(t,T k) 1 + δl(t,t k ). c Eberlein, Uni Freiburg, 152 Credit derivatives Reserve

153 Canonical construction of the time of default Let Γ = (Γ t) t 0 be an ( F t)-adapted, right-continuous, increasing process on ( Ω, F, P T ), Γ 0 = 0, lim t Γ t =. Let η be a random variable on ( Ω, F, P) uniformly distributed on [0, 1]. Define Ω := Ω Ω, G := F F, Q T := P T P (F t) trivial extension of ( F t) to (Ω, G, Q T ) τ := inf{t R + : e Γ t η} Denote H t := σ ( 1 {τ u} 0 u t ), G t := F t H t = τ is a (G t)-stopping time Q T {τ > s F T } = Q T {τ > s F s} = e Γs (0 s T ) Credit derivatives = (Γ t) is the (F t)-hazard process of τ under Q T (and also under all Q Tk ) c Eberlein, Uni Freiburg, 153 Reserve

154 Consequences for the price of a defaultable bond Payoff at maturity: B 0 (T k,t k ) = 1 {τ>tk } Therefore, define = B 0 (t,t k ) = B(t,T k )IE QTk [1 {τ>tk } G t] = B(t,T k )1 {τ>t} IE QTk [1 {τ>tk } F t] e Γ t B(t,T k ) := B(t,T k ) IE Q Tk [1 {τ>tk } F t] e Γ t = H(t,T k ) = 1 δ k ( IEQTk [e Γ Tk Ft] IE QTk+1 [e Γ T k+1 F t] 1 (Γ Tk ) k=1,...,n can be chosen such that H(t,T k ) has the form ( t H(t,T k )=H(0, T k ) exp + 0 t 0 t b H (s,t k,t k+1 ) ds + 0 ) c 1/2 s γ(s,t k ) dw T k+1 s ) γ(s, T k ), x (µ ν T k+1 )(ds, dx). R d c Eberlein, Uni Freiburg, 154 Credit derivatives Reserve

155 Defaultable forward measures The defaultable forward measure (or survival measure) Q Ti for the settlement day T i is defined on (Ω, G Ti ) by dq Ti dq Ti := B(0,T i) B 0 (0,T i ) B0 (T i,t i ) = B(0,T i) B(0,T i ) 1 {τ>t i }. = Q Ti (A) = Q Ti (A {τ > T i }) (A G Ti ), forward measure conditioned on survival until T i Denote P Ti := Q Ti FTi survival measure The restricted defaultable forward measure P Ti for the settlement day T i is defined on (Ω, F Ti ) by dp Ti = B(0,T i) dp Ti B(0,T i ) Q T i ({τ > T i } F Ti ) = B(0,T i) i 1 1 B(0,T i ) 1 + δ k H(T k,t k ). k=0 Credit derivatives Reserve c Eberlein, Uni Freiburg, 155

156 Successive restricted defaultable forward measures The defaultable Libor rate (L(t, T i )) 0 t Ti turns out to be a P Ti+1 -martingale and dp Ti dp Ti+1 Ft = B(0, T i+1) B(0, T i ) (1 + δ il(t, T i )) = 1 + δ il(t, T i ) 1 + δ i L(0, T i ) Credit derivatives Reserve c Eberlein, Uni Freiburg, 156

157 Pricing contingent claims with defaultable forward measures X promised payoff at day T i with zero recovery upon default π X t its price at time t [0, T i ] π X t = 1 {τ>t} B(t,T i )IE QTi [X1 {τ>ti } G t] (t [0, T i ]) The defaultable forward measures Q Ti and P Ti are the appropriate tools. If X is G Ti -measurable π X t If X is F Ti -measurable π X t = 1 {τ>t} B(t,T i )IE QTi [X G t] = B 0 (t,t i )IE QTi [X G t]. = 1 {τ>t} B(t,T i )IE PTi [X F t] = B 0 (t,t i )IE PTi [X F t]. Credit derivatives Reserve c Eberlein, Uni Freiburg, 157

158 Recovery rules and bond prices Defaultable zero coupon bonds fractional recovery of treasury value scheme At maturity of the bond B π (T, T ) = 1 {τ>t } + π1 {τ T } = π + (1 π)1 {τ>t } Time-t value (t [0, T ]) B π (t,t ) = πb(t,t ) + (1 π)1 {τ>t} B(t,T ) Defaultable coupon bonds recovery of par scheme Recovery of par: If default occurs in the time interval (T k, T k+1 ], recovery is given by the recovery rate π times the sum of the notional and the accrued interest over (T k, T k+1 ]. It is paid at T k+1. Corresponding cashflow pattern at T k+1 (k = 0,..., m 1): c1 {τ>tk+1 } + π(1 + c)1 {Tk <τ T k+1 } at T m: 1 {τ>tm} c Eberlein, Uni Freiburg, 158 Credit derivatives Reserve

159 Default payments Denote by e X k (t) the time-t value of receiving an amount of X at T k+1 default occured in period (T k, T k+1 ] Lemma If X is F Tk -measurable, then for t T k e X k (t) = 1 {τ>t} B(t, T k+1 )δ k IE PTk+1 [XH(T k, T k ) F t] Credit derivatives Reserve c Eberlein, Uni Freiburg, 159

160 Pricing of defaultable coupon bonds Fixed coupon of c to be paid at dates T 1,..., T m m 1 ( ) Bfixed(0)=B(0,T π m) + B(0,T k+1 ) c + π(1 + c)δ k IE PTk+1 [H(T k,t k )]. k=0 Floating coupon bond that pays Libor plus a constant spread x Promised payoff at the date T k+1 : δ k (L(T k,t k ) + x) m 1 ( Bfloating(0) π = B(0,T m) + δ k B(0,T k+1 ) x + IE PTk+1 [L(T k,t k )] k=0 + π(1 + δ k x)ie PTk+1 [H(T k,t k )] ) + πδ k IE PTk+1 [H(T k,t k )L(T k,t k )]. Credit derivatives Reserve c Eberlein, Uni Freiburg, 160

161 Numerical aspects ( t t H(t, T k ) = H(0, T k ) exp b H (s, T k, T k+1 ) ds c 1/2 s γ(s, T k ) dw T k+1 s t + γ(s, T k ), x (µ ν T k+1 (ds, dx) 0 R d Drift coefficient b H (s, T k, T k+1 ) to be approximated IE PTk+1 [H(T k, T k )L(T k, T k )] = 1 ) (L(0, T k ) IE δ PTk+1 [L(T k, T k )] IE PTk+1 [H(T k, T k )] k ) Credit derivatives Reserve c Eberlein, Uni Freiburg, 161

162 Credit default swaps (CDS) Standard default swap: Default of a coupon bond A receives: 1 π(1 + c) (fixed coupon) 1 π(1 + δ k (L(T k,t k ) + x)) (floating coupon) Time-0 value of the fee payments: s default swap rate s fixed = 1 π(1 + c) m k=1 B(0,T k 1) m m k=1 s m k=1 B(0,T k 1) ( ) B(0,T k )δ k 1 IE PTk [H(T k 1,T k 1 )] (B(0,T k )δ k 1 ( (1 π(1 + δ k 1 x)) 1 s floating = m k=1 B(0,T k 1) k=1 ) IE PTk [H(T k 1,T k 1 )] πδ k 1 IE PTk [H(T k 1,T k 1 )L(T k 1,T k 1 )] c Eberlein, Uni Freiburg, 162 Credit derivatives Reserve

163 Credit default swaptions (1) Assumption: The volatility structures factorize in the following way: λ(s, T i ) = λ i σ(s) and γ(s, T i ) = γ i σ(s) (0 s T i ). Payoff of a credit default swaption that is knocked out at default with strike S and maturity T i on a CDS that terminates at T m: ( ) m 1 (s(t i ; T i, T m) S) + B(T i, T k ) 1 {τ>ti } where s(t i ; T i, T m) denotes the default swap rate at T i. Price at time 0: [( (1 π(1 + c))δ m 1 C i,m 1 H(T i, T m 1 ) 0 = B(0, T i )IE PTi m 1 l=i (1 + δ l L(T i, T l ))(1 + δ l H(T i, T l )) π CDS k=i ) m 2 + (1 π(1 + c))δ k C i,k H(T i, T k ) S + k ]. l=i (1 + δ ll(t i, T l ))(1 + δ l H(T i, T l )) S k=i c Eberlein, Uni Freiburg, 163 Credit derivatives Reserve

164 Credit default swaptions (2) Forward Libor rates and default intensities can be written as ( λl ) L(T i, T l ) = L(0, T l ) exp X Ti + Bl L, ( σ sum γl ) H(T i, T l ) = H(0, T l ) exp X Ti + Bl H σ sum with σ sum := m 1 l=i Bl H. (λ l +γ l ), X Ti :=σ sum Ti 0 σ(s) dlt s and constants B L l, Assume the distribution of X Ti w.r.t. P Ti has a Lebesgue-density ϕ, then π0 CDS = B(0, T i ) g( x)ϕ(x) dx = B(0, T i )(g ϕ)(0) for some (explicitly given) function g. R Performing Laplace and inverse Laplace transformations and denoting by M X T i T i the P Ti -moment generating function of X Ti yields π0 CDS = B(0, T i ) 1 ( ) R L[g](R + iu)m X T i T π i ( R iu) du. 0 c Eberlein, Uni Freiburg, 164 Credit derivatives Reserve

165 Options on defaultable bonds (1) Payoff of a call with maturity T i and strike K (0, 1) on a defaultable zero coupon bond with maturity T m (i < m) which is knocked out at default πt CO i (K, T i, T m) = 1 {τ>ti }(B π (T i, T m) K ) + Price at time 0: π CO 0 = B(0, T m)ie PTm [(π m 1 l=i K ( 1 + δl H(T i, T l ) ) + (1 π) m 1 l=i ( (1 + δl L(T i, T l ) )( 1 + δ l H(T i, T l ) ))) +] Credit derivatives Reserve c Eberlein, Uni Freiburg, 165

166 Options on defaultable bonds (2) Using again a convolution representation π0 CO = B(0, T m) g( x)ϕ(x) dx = B(0, T m)(g ϕ)(0) R one gets for an R > 0 such that M X T i T m ( R) < the following (approximate) formula π CO 0 (K, T i, T m) = B(0, T m) 1 π 0 R ( L[g](R + iu) M X T i T m ( R iu) ) du Credit derivatives Reserve c Eberlein, Uni Freiburg, 166

167 Further credit derivatives Total rate of return swaps Asset package swaps Credit spread options Credit derivatives Reserve c Eberlein, Uni Freiburg, 167

168 Call option in FX market: Euro/Dollar The Theme Gives you the right to buy Euros paying in Dollars. At the same time a right to sell Dollars getting Euros. Payoff (S T K ) + (S t) exchange rate, K strike duality principle (S T K ) + = KS T ( 1 K 1 S T ) + ( = KS T K S T ) + Dollar/Euro rate Call price = K put price (in the dual market) Exponential semimartingale models Call-Put Duality Multiasset setting Reserve

169 Brief literature survey Carr (1994) put-call duality in BS-setting and for diffusions Chesney and Gibson (1995) two-factor diffusion model Bates (1997) diffusions and jump-diffusions Schroder (1999) various payoffs in diffusions and jump-diffusions Carr, Ellis, and Gupta (1998) static hedging strategies for exotic derivatives Carr and Chesney (1996) put-call for American options Detemple (2001) American options with general payoffs Henderson and Wojakowski (2002) Asian options Eberlein and Papapantoleon (2005a,b) Exotic options for Lévy and time-inhomogeneous Lévy models Vanmaele, Deelstra, Liinev, Dhaene, Goovaerts (2006) Forward start Asian options Fajardo and Mordecki (2006a,b) Lévy models Vecer (2002), Vecer and Xu (2004) Asian options (PIDE) Eberlein, Kluge, and Papapantoleon (2006) options Eberlein, Papapantoleon, Shiryaev (2006) Semimartingales Exponential semimartingale models Call-Put Duality Multiasset setting Reserve

170 Exponential semimartingale models Let B = (Ω, F, F, P) be a stochastic basis, where F = F T and F = (F t) 0 t T. We model the price process of a financial asset as an exponential semimartingale S t = e H t, 0 t T. H = (H t) 0 t T is a semimartingale with canonical representation or, in detail H = H 0 + B + H c + h(x) (µ H ν) + (x h(x)) µ H H t = H 0 + B t + H c t + where t 0 R t h(x)d(µ H ν) + (x h(x))dµ H, 0 R Exponential semimartingale models Call-Put Duality Multiasset setting Reserve

171 h = h(x) is a truncation function; canonical choice h(x) = x1 { x 1} ; B = (B t) 0 t T is a predictable process of bounded variation; H c = (H c t ) 0 t T is the continuous martingale part of H; ν = ν(ω; dt, dx) is the predictable compensator of the random measure of jumps µ H = µ H (ω; dt, dx) of H. For the processes B, C = H c, and the measure ν we use the notation T(H P) = (B, C, ν) which will be called the triplet of predictable characteristics of the semimartingale H with respect to the measure P. Assumption: The truncation function satisfies the antisymmetry property h( x) = h(x). Exponential semimartingale models Call-Put Duality Multiasset setting Reserve

172 Alternative model description E(X) = (E(X) t) 0 t T stochastic exponential S t = E( H) t, 0 t T ds t = S t d H t where Note H t = H t + 1 t 2 Hc t + (e x 1 x)µ H (ds, dx) 0 R E( H) t = exp( H t 1 2 H c t) (1 + H s) exp( H s) 0<s t Exponential semimartingale models Call-Put Duality Asset price positive only if H > 1. Multiasset setting Reserve

173 Martingale and dual martingale measures Assumption (ES) The process 1 {x>1} e x ν has bounded variation. Then, H is exponentially special and S = e H M loc(p) B + C 2 + (ex 1 h(x)) ν H = 0. Moreover, we assume that S M(P), therefore ES T = 1. Define on (Ω, F, (F t) 0 t T ) a new probability measure P with dp dp = S T. Since S > 0 (P-a.s.), we have P P and dp dp = 1 S T. Exponential semimartingale models Call-Put Duality Multiasset setting Reserve

174 Introduce the process S = 1 S. Then, denoting by H the dual of the semimartingale H, i.e. H = H, we have S = e H. Proposition Suppose S = e H M(P) i.e. S is a P-martingale. Then the process S M(P ) i.e. S is a P -martingale. Lemma Let f be a predictable, bounded process. The triplet of predictable characteristics of the stochastic integral process J = f dh, denoted by 0 T(J P) = (B J, C J, ν J ), is B J C J = f B + [h(fx) fh(x)] ν = f 2 C 1 A (x) ν J = 1 A (fx) ν, A B(R). Exponential semimartingale models Call-Put Duality Multiasset setting Reserve

175 Theorem The triplet T(H P ) = (B, C, ν ) can be expressed via the triplet T(H P) = (B, C, ν) by the following formulae: B = B C h(x)(e x 1) ν C = C 1 A (x) ν = 1 A ( x)e x ν, A B(R). Structure of the proof: (G) T(H P ) ( ) T(H P) (a) (c) T(H P ) Exponential (b) (d) semimartingale models Call-Put Duality ( ) T(H P) (G) Multiasset setting (G) : Girsanov s theorem, ( ) : dual of a semimartingale. Reserve

176 Symmetry of markets If the original market (S, P) and the dual market (S, P ) satisfy Law(S P) = Law(S P ) then we say these markets are symmetric. In cases where the triplets T(H P) and T(H P ) determine these laws completely (e.g. for H and H ) symmetry holds iff ν = ν The equation in the Theorem is then 1 A (x) ν = 1 A ( x)e x ν, A B(R) Exponential semimartingale models Call-Put Duality Multiasset setting Reserve

177 Example 1: Diffusion models ds t = S tσ(t, S t) dw t, S 0 = 1 local volatility models (Dupire (1994), Skiadopoulos (2001)) H t = t 0 B = 1 2 σ(u, e Hu ) dw u 1 2 Theorem B = B C = t 0 σ 2 (u, e Hu ) du σ 2 (u, e Hu ) du, C = 0 σ 2 (u, e Hu ) du, ν 0 0 σ2 (u, e Hu ) du, C = C, ν 0 Exponential semimartingale models Call-Put Duality Multiasset setting Reserve

178 Example 2: Purely discontinuous Lévy models S t = e H t, T(H, P) = (B, 0, ν) local characteristics: B t(ω) = bt, ν(ω; dt, dx) = dtf (dx), F Lévy measure S M loc(p) b = (e X 1 h(x))f(dx) R Actually: S M(P) Parametric models: F(dx) = e ϑx f (x) dx f even Generalized hyperbolic (includes hyperbolic, NIG, VG,... ) CGMY, Meixner Dual process H : 1 A (x)f (dx) = 1 A ( x)e (1+ϑ)x f (x) dx b = (e x 1 h(x))f (dx) R Exponential semimartingale models Call-Put Duality Multiasset setting Reserve

179 Theorem European options (1) The prices of standard call and put options satisfy the following duality relations: C T (S; K ) = K P T (K ; S ) and Proof: Using the dual measure (S T K ) C T (S; K ) = E [S + ] T S T where K = 1 K. = KE [( 1 K S T P T (K ; S) = K C T (S ; K ). = E [ (S T K ) + S T ] = E [(1 KS T ) + ] ) + ] = KE [(K S T ) + ], Exponential semimartingale models Call-Put Duality Multiasset setting Reserve

180 European options (2) Corollary Call and put prices in a dual pair of markets (S, P) and (S, P ) satisfy a call-call parity C T (S; K ) = K C T (S ; K ) + 1 K and a put-put parity P T (K ; S) = K P T (K ; S ) + K 1 Proof: Combine with classical call-put parity C T (S; K ) = P T (K ; S) + 1 K Exponential semimartingale models Call-Put Duality Multiasset setting Reserve

181 Floating strike lookback options (1) Payoff of a call: ( ) + S T α inf 0 t T St for an α 1 Assume H = (H t ) 0 t T satisfies the reflection principle ( Law sup H t H T P ) = Law( inf t T t T H t P ) (holds for ), then C T (S; α inf S) = αp T Value of a floating strike lookback call value of a fixed strike lookback put ( 1 ; inf S ) α Exponential semimartingale models Call-Put Duality Multiasset setting Reserve

182 Payoff of a put: Floating strike lookback options (2) ( ) + β sup S t S T for a 0 < β 1 0 t T Assume H = (H t ) 0 t T satisfies ( Law H T inf t T H t P ) ( = Law sup H t P ) t T (holds for ), then P T (β sup S; S) = βc T Value of a floating strike lookback put value of a fixed strike lookback call ( sup S ; 1 ) β Exponential semimartingale models Call-Put Duality Multiasset setting Reserve

183 Payoff of a call: Floating strike Asian options ( S T 1 T ) + S t dt T 0 Assume H = (H t ) 0 t T satisfies Law(H T H (T t) ; 0 t < T P ) = Law(H t ; 0 t < T P ) (holds for ), then C T (S; 1 ( S) = P T 1; 1 T T S ) Value of a floating strike Asian call value of a fixed strike Asian put ( 1 ) ( Similarly P T S; S = C 1 ) T S, 1 T T Exponential semimartingale models Call-Put Duality Multiasset setting Reserve

184 Forward-start options Payoff of a call: (S T S t) + Payoff of a put: (S t S T ) + Assume H = (H t ) 0 t T satisfies Law(H T H (T t) ; 0 t < T P ) = Law(H t ; 0 t < T P ) then and C t,t (S; S) = P T t(1; S ) P t,t (S; S) = C T t(s ; 1) Exponential semimartingale models Call-Put Duality Value of a forward-start call value of a plain vanilla put Multiasset setting Reserve

185 Multiasset price model Each component S i of the vector of asset price processes S = (S 1,..., S d ) is an exponential time-inhomogeneous Lévy process S i t = S i 0 exp L i t, 0 t T. The driving process L = (L t) 0 t T is an R d -valued time-inhomogeneous Lévy process that satisfies Assumption (EM), with canonical decomposition t t t L t = b s ds + c 1/2 s dw s + x(µ L ν)(ds, dx) R d Exponential semimartingale models Call-Put Duality Multiasset setting Reserve

186 Theorem Multiasset price model (2) Let L = (L t) 0 t T be an R d -valued PIIAC that satisfies Assumption (EM), with characteristics T(L P) = (B, C, ν). Let u, v be vectors in R d such that v ( M, M) d and u + v [ M, M] d. Define the measure P dp dp = e v,lt E[e v,l T ]. Then, the process L u = (L u t ) 0 t T, where L u t := u, L t, is a 1-dimensional PIIAC with characteristics T(L u P ) = (B u, C u, ν u ) with bs u = u, b s + u, c sv + u, x ( e v,x 1 ) λ s(dx) R d cs u = u, c su λ u s(e) = λ s({x R d : u, x E}), E B(R), where λ s is a measure defined by λ s(a) = e v,x λ s(dx), A B(R d ). Application: Multiasset options A Exponential semimartingale models Call-Put Duality Multiasset setting Reserve

187 Theorem Example: Swap option (Margrabe) We can relate the value of a swap, with payoff (S 1 T S 2 T ) +, and a plain vanilla option via the following duality: M ( S 1 0, S 2 0; C, ν ) = S 1 0 P ( 1, S 2 0/S 1 0; C, ν ) where the characteristics (C, ν ) are given in the previous Theorem for v = (1, 0) and u = ( 1, 1). Proof: Using asset S 1 to form the Radon Nikodym derivative [ ( ) ] [ ( ) + M = E ST 1 ST 2 = S0E 1 ST 1 + ] 1 S2 T S0 1 ST 1 [ ( ) + ] [ ( ) + ] = S0E 1 e L1 T 1 S2 T = S0E 1 1 S2 T, ST 1 ST 1 Exponential semimartingale models Call-Put Duality Multiasset setting Reserve

188 where v = (1, 0). Now, note that St 2 St 1 = S2 0 e L2 t S0 1 e L1 t where u = ( 1, 1) and Then, we have that = S2 0 e u,l t, S0 1 0 t T e u,l M(P ) since e u,l e v,l = e L2 M(P). M = S 1 0E [ ( 1 S T ) + ] where S is an exponential PIIAC with characteristics C and ν. Exponential semimartingale models Call-Put Duality Multiasset setting Reserve

189 Model with interest rates Asset price processes St i = S0 i exp[(r δ i )t + L i t] where L = (L 1,..., L d ) is a PIIAC with triplet (B, C, ν) payoff of a Margrabe option: (ST 1 ST 2 ) + value M(S 1 0, S 2 0; r, δ, C, ν) = e rt E[(S 1 T S 2 T ) + ] then M(S 1 0, S 2 0; r, δ, C, ν) = E[S 1 T ]e C T P(K, S 2 0/S 1 0, δ 1, r, C, ν ) Exponential semimartingale models Call-Put Duality where K = e C T and C T is a constant. Multiasset setting Reserve

190 Duality in the Lévy forward rate model Denote the value of a call option on a zero coupon bond by ( ) B(0, U) V c B(0, T ); B(0, T ), K ; C, ν = IE and similarly for a put option ( ) B(0, U) V p B(0, T ); B(0, T ), K ; C, ν Theorem [ 1 B T (B(T, U) K ) + ], [ ] 1 = IE (K B(T, U)) +. B T Assume that bond prices are modeled according to the Lévy forward rate model. Then, the value of a call and a put option on a bond are related via: ( ) ( B(0, U) V c B(0, T ); B(0, T ), K ; C, ν = V p B(0, T ); K, where f (s, x) = exp ( (Σ(s, U) + Σ(s, T ))x ). ) B(0, U) ; C, f ν B(0, T ) Exponential semimartingale models Call-Put Duality Multiasset setting Reserve

191 [ Idea of proof: Define the constant D := IE B(T,U) via (B T ) 2 ] and the measure P d P B(T, U) := dp D (B T ) = η 2 T. [ ] P P B(T,U) and the density process (η t) t [0,T ] is η t = IE F D(B T ) 2 t. Using Girsanov s theorem for semimartingales we deduce the P-characteristics of the driving process L. Now, [ ] 1 V c = IE (B(T, U) K ) + B T [ ] B(T, U) = IE D (B T ) KDB 2 T (K 1 B(T, U) 1 ) + Exponential semimartingale models Call-Put Duality Multiasset setting Reserve

192 and changing measure from P to P, we get that This can be re-written as [ V c = Ẽ KDB T (K 1 B(T, U) 1 ) +]. V c = Ẽ [ 1 ) ] + ( K B(T, U), B T for ( B T ) 1 := B(0,T ) B(0,U) DB T, K := B(0,U) B(0,T ) and B(T, U) := K B(0,U) B(0,T ) B(T, U) 1. Showing that B T and B(T, U) have dynamics analogous to that of B T and B(T, U) concludes the proof. Exponential semimartingale models Call-Put Duality Multiasset setting Reserve

193 Equivalent formulation (1) time-t i+1 payoff of a caplet: Nδ(L(T i, T i ) K ) + Recall 1 + δl(t i, T i ) = B(T i,t i ) B(T i,t i+1 ) δ(l(t i, T i ) K ) + = (1 + δl(t i, T i ) (1 + δk )) ( 1 ) = B(T i, T i+1 ) K time-t i value of this payoff ( 1 ) + ( 1 B(T i, T i+1 ) B(T i, T i+1 ) K = K K B(T i, T i+1 ) payoff of a put option on a bond with strike Analogously for a floorlet δk ) + Exponential semimartingale models Call-Put Duality Multiasset setting Reserve

194 Equivalent formulation (2) Value of a floorlet with strike K maturing at time T i that settles in arrears at time T i+1 [ 1 FL(L(0, T i ), K ; C, ν) = E δ(k L(T i, T i )) +] B Ti+1 [ 1 = (1 + δk )IE (B(T i, T i+1 ) K) +] B Ti where L(0, T i ) = 1 δ Therefore where C = ( B(0,Ti ) B(0,T i+1 ) 1 ) initial forward Libor rate FL(L(0, T i ), K ; C, ν) = C CL(K, L(0, T i ); C, f ν) 1 + δk 1 + δl(0, T i ) Exponential semimartingale models Call-Put Duality Multiasset setting Reserve

195 Duality in the Lévy Libor model Value of a caplet with strike K maturing at time T i that settles in arrears at time T i+1 CL(L(0, T i ), K ; C, ν T i+1 ) = B(0, T i+1 )E PTi+1 [δ(l(t i, T i ) K ) + ] Duality result CL(L(0, T i ), K ; C, ν T i+1 ) = FL(K, L(0, T i ); C, f ν T i+1 ) where f (s, x) = exp(λ(s, T i )x) Exponential semimartingale models Call-Put Duality Multiasset setting Reserve

196 Duality in the Lévy forward process model Value of a call option on the forward process with strike K which is settled in arrears at time T i+1 C(F(0, T i, T i+1 ), K ; C, ν T i+1 ) = B(0, T i+1 )E PTi+1 [(F(T i, T i, T i+1 ) K ) + ] Duality for call and put options on the forward process C(F(0, T i, T i+1 ), K ; C, ν T i+1 ) = P(K, F (0, T i, T i+1 ); C, f ν T i+1 ) Exponential semimartingale models Call-Put Duality Multiasset setting Reserve

197 statement Deutsche Bank 1,623,811 2,202,423 1,378,011 1,925,003 Reserve c Eberlein, Uni Freiburg, 197

198 statement Deutsche Bank Reserve c Eberlein, Uni Freiburg, 198

199 processes Levy 1,333, ,085 LIBOR Levy 2,202,423 1,925,003 Reserve c Eberlein, Uni Freiburg, 199