Efficient valuation of exotic derivatives in Lévy models

Transcription

1 Efficient valuation of exotic derivatives in models Ernst Eberlein and Antonis Papapantoleon Department of Mathematical Stochastics and Center for Data Analysis and Modeling (FDM) University of Freiburg Exotic Conference on Stochastic Processes: Theory and Applications on occasion of the 65th birthday of Wolfgang Runggaldier Bressanone (Italy), July 16 20,

2 Volatility smile and surface implied vol (%) delta (%) or strike maturity Strike rate (in %) Maturity (in years) 0 Exotic Volatility surfaces of foreign exchange and interest rate Volatilities vary in strike (smile) Volatilities vary in time to maturity (term structure) Volatility clustering 1

3 Exponential semimartingale model Let B T = (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. (1) H = (H t) 0 t T is a semimartingale with canonical representation H = H 0 + B + H c + h(x) (µ H ν) + (x h(x)) µ H. (2) Exotic For the B, C = H c, and the measure ν we use the notation T(H P) = (B, C, ν) which is called the triplet of predictable characteristics of the semimartingale H. 2

4 Alternative model description E(X) = (E(X) t) 0 t T stochastic exponential S t = E( e H) t, 0 t T where ds t = S t d e H t eh t = H t + 1 Z t Z 2 Hc t + (e x 1 x)µ H (ds, dx) 0 R Exotic Note E( H) e t = exp( H e t 1 2 e H c Y t) (1 + H e s) exp( H e s) Asset price positive only if e H > 1. 0<s t 3

5 Let M loc(p) be the class of local martingales. Assumption (ES) Martingale modeling The process 1 {x>1} e x ν has bounded variation. Then S = e H M loc(p) B + C 2 + (ex 1 h(x)) ν = 0. (3) Exotic Throughout, we assume that P is a (local) martingale measure for S. By the Fundamental Theorem of Asset Pricing, the value of an option on S equals the discounted expected under a martingale measure. We assume zero interest rates. 4

6 Supremum and infimum Let X = (X t) 0 t T be a stochastic process. We 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 0 t T S t = sup 0 t T e H t = e sup 0 t T H t = e H T. (4) Exotic Similarly S T = e H T. (5) 5

7 formulae functional We want to price an option with f (X T ), where X T = p(h t, 0 t T ) is an F T -measurable functional. The functionals we consider are European style, and consist of two parts: 1 The 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 can be the asset price or the supremum/infimum or an average of the asset price process (e.g. X = H or X = H). Exotic Exotic 6

8 Assumptions: formulae assumptions (R1) Assume that R R e Rx f (x)dx < for all R I 1 R. (R2) Assume that M XT (z) = E[e zx T ] <, for all z I 2 R. (R3) Assume that I 1 I 2. formulae based on Fourier transforms; similar to Raible (2000), but no need for Lebesgue density. Consider the Fourier transform of the function like Borovkov and Novikov (2002); also Hubalek et al. (2006) and Černý (2007), for hedging. Carr and Madan (1999) and Raible (2000) transform the option price. Exotic 7

9 Theorem 1 formulae Assume that (R1) (R3) are in force. Then, the price V f (X) of an option on S = (S t) 0 t T with f (X T ) is given by V f (X) = 1 Z ϕ XT ( u ir)f f (u + ir)du, (6) 2π R where ϕ XT denotes the extended characteristic function of X T and F f denotes the Fourier transform of f. Proof Introduce the dampened function g(x) = e Rx f (x), R I 1. Then Z V f (X) = E[f (X T )] = E[e RX T g(x T )] = e Rx g(x)p XT (dx). (7) R Exotic cont. next page 8

10 Proof (cont.) Under assumption (R1), g has a Fourier transform F g; inverting it, we get a representation as g(x) = 1 Z e ixu F g(u)du. (8) 2π R Returning to the valuation problem (7) we get Z V f (X) = R = 1 2π = 1 2π Z! e Rx 1 e ixu F g(u)du P XT (dx) 2π R Z Z! Z R R R e i( u ir)x P XT (dx) F g(u)du ϕ XT ( u ir)f f (u + ir)du. (9) Exotic 9

11 formulae II formulae for that depend on two functionals of the driving process. Examples: barrier, slide-in or corridor and two-asset correlation option (S T K ) + 1 {ST >B} ; (S T K ) + N X i=1 1 {L<STi <H}; Exotic (S 1 T K ) + 1 {S 2 T >B}. 10

12 Theorem 2 formulae II The price V f,g (X, Y ) of an option on S = (S t) 0 t T with function f (X T )g(y T ) is given by V f,g (X, Y ) = 1 Z Z ϕ 4π 2 XT,Y T ( u ir 1, v ir 2 ) R R F g(v + ir 2 )F f (u + ir 1 )dvdu, (10) where ϕ XT,Y T denotes the extended characteristic function of the random vector (X T, Y T ). Exotic Proof. Assumptions and proof are similar to Theorem 1. 11

13 Example (Call and put option) Call f (x) = (e x K ) +, K R +, F f (u + ir) = Similarly, if f (x) = (K e x ) +, K R +, K 1+iu R (iu R)(1 + iu R), R I 1 = (1, ). (11) Exotic F f (u + ir) = K 1+iu R (iu R)(1 + iu R), R I 1 = (, 0). (12) 12

14 Example (Digital option) Call 1 {e x >B}, B R +. F f (u + ir) = B iu R 1 iu R, R I 1 = (0, ). (13) Similarly, for the f (x) = 1 {e x <B}, B R +, F f (u + ir) = B iu R 1 iu R, R I 1 = (, 0). (14) Exotic Example (Double digital option) The of a double digital call option is 1 {B<e x <B}, B, B R+. F f (u + ir) = 1 B iu R B iu R, R I 1 = R\{0}. (15) iu R 13

15 Example (Asset-or-nothing digital) Call Put f (x) = e x 1 {e x >B} F f (u + ir) = 1 + iu R, R I 1 = (1, ) f (x) = e x 1 {e x <B} F f (u + ir) = B1+iu R B1+iu R 1 + iu R, R I 1 = (, 1) Exotic Example (Self-quanto option) Call f (x) = e x (e x K ) + F f (u + ir) = K 2+iu R (1 + iu R)(2 + iu R), R I 1 = (2, ) 14

16 Let L = (L t) 0 t T be a process with triplet of local characteristics (b, c, λ), i.e. B t(ω) = bt, C t(ω) = ct, ν(ω; dt, dx) = dtλ(dx), λ measure. Assumption (EM) There exists a constant M > 1 such that Z e ux λ(dx) <, { x >1} u [ M, M]. Exotic Using (EM) and Theorems 25.3 and in Sato (1999), we get that for all u [ M, M]. Eˆe ul t <, Eˆe ul t < and Eˆe ul t < 15

17 Lemma 3 On the characteristic function of the supremum I Let L = (L t) 0 t T be a process that satisfies assumption (EM). Then, the moment generating function of L t is defined for all u (, M] and t [0, T ]. Lemma 4 Let L = (L t) 0 t T be a process that satisfies assumption (EM). Then, the characteristic function ϕ Lt of L t is holomorphic in the half plane {z C : M < Iz < } and can be represented as a Fourier integral in the complex domain ϕ Lt (z) = Eˆe izl Z t = e izx P Lt (dx). R Exotic 16

18 Fluctuation theory for Theorem 5 (Wiener Hopf factorization) Let L be a process. The Laplace transform of L at an independent and exponentially distributed time θ can be identified from the Wiener Hopf factorization of L via Eˆe βl θ = κ(q, 0) κ(q, β) (16) where κ(α, β), α 0, β 0, is given by Z Z κ(α, β) = k exp (e t e αt βx ) 1 «t P L t (dx) dt. (17) 0 0 Exotic Moreover, κ can be analytically extended to α, β C with Rα 0 and Rβ M. Proof. Theorem 6.16 in Kyprianou (2006). 17

19 Lemma 6 Linking fixed and exponential times Let L = (L t) 0 t T be a process that satisfies assumption (EM) and consider β C with Rβ [ M, ). The Laplace transforms of L t, t [0, T ] and L θ, θ Exp(q), are related via Eˆe βl θ = q Z 0 e qt Eˆe βl t dt. (18) Moreover, the Laplace transform of L θ is finite for β C with Rβ [ M, ). Exotic Proof. An application of Fubini s theorem yields Eˆe βl h Z i θ = E qe qt e βl t dt = q Z 0 0 e qt Eˆe βl t dt. 18

20 Theorem 7 On the characteristic function of the supremum II Let L = (L t) 0 t T be a process. The Laplace transform of L t at a fixed time t, t [0, T ], is given by Eˆe βl t = 1 Z t(y +iv) e κ(y + iv, 0) dv, (19) 2π Y + iv κ(y + iv, β) R for Y > 0. Moreover, the Laplace transform can be extended to the complex plane for β C with Rβ [ M, ). Exotic Proof. Combining eqs. (16) and (18) we get q Z 0 e qt E[e βl t ] dt = κ(q, 0) κ(q, β). (20) Applying Doetsch (1950), we invert the Laplace transform and the claim follows. 19

21 Non-path-dependent European option on an asset with price process S t = e H t Examples: call, put, digitals, asset-or-nothing, double digitals, self-quanto X t H T, i.e. we need ϕ HT Generalized hyperbolic model (GH model): ϕ H1 (u) = e iuµ α 2 β 2 λ/2 K λ `δp α2 (β + iu) 2 α 2 (β + iu) 2 K λ`δp α2 β 2 Exotic I 2 = ( α β, α β) ϕ HT (u) = (ϕ H1 (u)) T similar: NIG, CGMY, Meixner 20

22 Non-path-dependent II Stochastic volatility models: Carr, Geman, Madan, Yor (2003) Stochastic clock Y t = e.g. CIR process Z t 0 y sds (y s > 0) dy t = K (η y t)dt + λy 1/2 t dw t Define for a pure jump process X = (X t) t 0 Exotic Then H t = X Yt (0 t T ) ϕ Ht (u) = ϕ Yt ( iϕ Xt (u)) (ϕ Yt ( iuϕ Xt ( i))) iu 21

23 Lookback Fixed strike lookback option: (S T K ) +. Combining Theorem 1 and Theorem 7, we get C T (S; K ) = 1 Z K 1+iu R ϕ 2π LT ( u ir) du (21) R (iu R)(1 + iu R) where ϕ LT ( u ir) = 1 Z 2π R T (Y +iv) e Y + iv κ(y + iv, 0) dv. (22) κ(y + iv, iu R) Exotic The floating strike lookback option, ( S T S T ) +, is treated by a duality formula. 22

24 Payoff of a put: Floating strike lookback (1) + 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 sup S ; 1 β Exotic Value of a floating strike lookback put value of a fixed strike lookback call 23

25 Payoff of a call: Floating strike lookback (2) + 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 Exotic C T (S; α inf S) = αp T 1 ; inf S α Value of a floating strike lookback call value of a fixed strike lookback put 24

26 Proof Floating strike lookback (3) C T `S; α inf S = Eˆ`ST α inf St + = E»S T 1 α inf «+ t T S t t T S T = E h + i 1 αe inf t T H t H T = E h + i 1 αe H T sup t T H t The process 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 C T `S; α inf S h = αe 1 α einf t T H + i t = αe h 1 + i α inf t T S t = αp 1 T α ; inf S Exotic 25

27 One-touch call option: 1 {ST >B}. One-touch Combining Theorem 1, Theorem 7 and the example for digital, we get DC T (S; B) = 1 Z Z 4π 2 R R T (Y +iv) e Y + iv Similarly for the one-touch put option: 1 {ST B}. κ(y + iv, 0) dvdu. (23) κ(y + iv, iu R) R iu B iu R Exotic DP T (S; B) = 1 Z Z 4π 2 R R T (Y +iv) e Y + iv bκ(y + iv, 0) dvdu. (24) bκ(y + iv, iu R) iu R B iu R 26

28 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, paid at time τ B premium of the EDS chosen such that initial value equals 0; hence K = E ˆe rτ B 1 {τb T } P N i=1 E ˆe. (25) rt i 1 {τb >T i } Exotic Calculations similar to touch, since 1 {τb T } = 1 {ST B}. 27

29 Options on two assets Two-asset correlation : Payoff of a correlation call: (ST 1 K ) + 1 {S 2 T >B} Measurement asset S 2 in the money call on a payment asset S 1 Asset price S i t = exp(l i t) i = 1, 2 where L = (L 1, L 2 ) is a time-inhomogeneous process TAC T (S 1, S 2 ; K, B) = 1 Z Z ϕ 4π 2 LT ( u ir 1, v ir 2 ) R R K 1+iu R 1 B iu R 2 (iu R 1 )(1 + iu R 1 ) R 2 iu dvdu Exotic 28

30 Borovkov, K. and A. Novikov (2002). On a new approach to calculating expectations for option pricing. J. Appl. Probab. 39, Carr, P. and D. B. Madan (1999). Option valuation using the fast Fourier transform. J. Comput. Finance 2 (4), Černý, A. (2007). Optimal continuous-time hedging with leptokurtic returns. Math. Finance 17, Eberlein, E. and A. Papapantoleon (2007). of exotic and credit derivatives in models. FDM Preprint 97, University of Freiburg Eberlein, E., A. Papapantoleon, and A. N. Shiryaev (2006). On the duality principle in option pricing: semimartingale setting. FDM-Preprint Nr. 92, University of Freiburg Exotic 29

31 (cont.) Hubalek, F., J. Kallsen and L. Krawczyk (2006). Variance-optimal hedging for with stationary independent increments. Ann. Appl. Probab. 16, Kyprianou, A. E. (2006). Introductory lectures on fluctuations of with applications. Springer. A. Papapantoleon (2007). Applications of semimartingales and in finance: duality and valuation. Ph.D. thesis, University of Freiburg. Raible, S. (2000). in finance: theory, numerics, and empirical facts. Ph.D. thesis, University of Freiburg. Sato, K.-I. (1999). and infinitely divisible distributions. Cambridge University Press. Exotic 30