Analysis of Fourier Transform Valuation Formulas and Applications

Transcription

1 Analysis of Fourier Transform Formulas and Applications Ernst Eberlein Freiburg Institute for Advanced Studies (FRIAS) and Center for Data Analysis and Modeling (FDM) University of Freiburg (joint work with Kathrin Glau and Antonis Papapantoleon) 6th World Congress of the Bachelier Finance Society Toronto June 22 26, 21 c Eberlein, Uni Freiburg, 1

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

3 Carr and Madan (1999) Raible (2) Fourier and Laplace based valuation formulas Borovkov and Novikov (22): exotic options Hubalek, Kallsen, and Krawczyk (26): hedging Lee (24): discretization error in fast Fourier transform Hubalek and Kallsen (25): options on several assets Biagini, Bregman, and Meyer-Brandis (28): indices Hurd and Zhou (29): spread options Eberlein and Kluge (26): interest rate Eberlein and Koval (26): cross currency Eberlein, Kluge, and Schönbucher (26): credit default swaptions Harmonic analysis (Parseval s formula) c Eberlein, Uni Freiburg, 2

4 Exponential semimartingale model B T = (Ω, F, F, P) stochastic basis, where F = F T and F = (F t) t T. Price process of a financial asset as exponential semimartingale S t = S e H t, t T. (1) H = (H t) t T semimartingale with canonical representation H = B + H c + h(x) (µ H ν) + (x h(x)) µ H. (2) For the processes 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 H. c Eberlein, Uni Freiburg, 3

5 Alternative model description E(X) = (E(X) t) t T stochastic exponential where Note S t = E( e H) t, t T 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) R E( H) e t = exp eht 1 2 e Y H c t <s t (1 + e H s) exp( e H s) Asset price positive only if e H > 1. c Eberlein, Uni Freiburg, 4

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

7 Supremum and infimum processes Let X = (X t) t T be a stochastic process. Denote by X t = sup X u and X t = inf u t 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 e H t = S e sup t T H t = S e H T. (4) t T t T Similarly S T = S e H T. (5) c Eberlein, Uni Freiburg, 6

8 formulas payoff functional We want to price an option with payoff Φ(S t, 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 e H T K ) + = `e H T +log S 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 = H or X = H). c Eberlein, Uni Freiburg, 7

9 formulas Consider the option price as a function of S or better of s = log S X driving process (X = H, H, H, etc.) Φ(S e H t, t T ) = f (X T s) Time- price of the option (assuming r ) V f (X; s) = EˆΦ(S t, t T ) = E[f (X T s)] formulas based on Fourier and Laplace transforms Carr and Madan (1999) Raible (2) plain vanilla options general payoffs, Lebesgue densities In these approaches: Some sort of continuity assumption (payoff or random variable) c Eberlein, Uni Freiburg, 8

10 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 bg L 1 (R) c Eberlein, Uni Freiburg, 9

11 formulas Theorem Assume that (C1) (C3) are in force. Then, the price V f (X; s) of an option on S = (S t) t T with payoff f (X T ) is given by Z V f (X; s) = e Rs e ius ϕ XT ( u ir) 2π b f (u + ir)du, (6) R where ϕ XT denotes the extended characteristic function of X T and b f denotes the Fourier transform of f. c Eberlein, Uni Freiburg, 1

12 Discussion of assumptions Alternative choice: (C1 ) g L 1 (R) (C3 ) er. P XT L 1 (R) (C3 ) = e R. P XT has a cont. bounded Lebesgue density Recall: (C3) bg L 1 (R) Sobolov space Lemma H 1 (R) = g L 2 (R) g exists and g L 2 (R) g H 1 (R) = bg L 1 (R) Similar for the Sobolev Slobodeckij space H S (R) (s > 1 ) 2 c Eberlein, Uni Freiburg, 11

13 Examples of payoff functions Example (Call and put option) Call payoff f (x) = (e x K ) +, K R +, b f (u + ir) = K 1+iu R (iu R)(1 + iu R), R I 1 = (1, ). (7) Similarly, if f (x) = (K e x ) +, K R +, b f (u + ir) = K 1+iu R (iu R)(1 + iu R), R I 1 = (, ). (8) c Eberlein, Uni Freiburg, 12

14 Example (Digital option) Call payoff 1 {e x >B}, B R +. b f (u + ir) = B iu R 1 iu R, R I 1 = (, ). (9) Similarly, for the payoff f (x) = 1 {e x <B}, B R +, b f (u + ir) = B iu R 1 iu R, R I 1 = (, ). (1) Example (Double digital option) The payoff of a double digital option is 1 {B<e x <B}, B, B R+. b 1 f (u + ir) = B iu R B iu R, R I 1 = R\{}. (11) iu R c Eberlein, Uni Freiburg, 13

15 Example (Asset-or-nothing digital) Payoff Similarly f (x) = e x 1 {e x >B} b f (u + ir) = 1 + iu R, R I 1 = (1, ) f (x) = e x 1 {e x <B} 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 ) + b f (u + ir) = K 2+iu R (1 + iu R)(2 + iu R), R I 1 = (2, ) c Eberlein, Uni Freiburg, 14

16 Non-path-dependent options European option on an asset with price process S t = e H t Examples: call, put, digitals, asset-or-nothing, double digitals, self-quanto options X T H T, i.e. we need ϕ HT Generalized hyperbolic model (GH model): Eberlein, Keller (1995), Eberlein, Keller, Prause (1998), Eberlein (21) 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 I 2 = ( α β, α β) ϕ HT (u) = (ϕ H1 (u)) T similar: NIG, CGMY, Meixner c Eberlein, Uni Freiburg, 15

17 Non-path-dependent options II Stochastic volatility Lévy models: Carr, Geman, Madan, Yor (23) Eberlein, Kallsen, Kristen (23) Stochastic clock Y t = e.g. CIR process Z t y sds (y s > ) dy t = K (η y t)dt + λy 1/2 t dw t Define for a pure jump Lévy process X = (X t) t Then H t = X Yt ( t T ) ϕ Ht (u) = ϕ Yt ( iϕ Xt (u)) (ϕ Yt ( iuϕ Xt ( i))) iu c Eberlein, Uni Freiburg, 16

18 Lévy model Classification of option types S t = S e H t payoff payoff function distributional properties (S T K ) + call 1 {ST >B} digital f (x) = (e x K ) + P HT usually has a density f (x) = 1 {e x >B} ` ST K + lookback 1 {ST >B} digital barrier = one touch f (x) = (e x K ) + density of P HT? f (x) = 1 {e x >B} c Eberlein, Uni Freiburg, 17

19 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) M XT (R) exists Assume (D1) (D2) then e Rs Z A V f (X; s) = lim e ius ϕ XT (u ir) A 2π b f (ir u) du (12) A if V f (X; ) is of bounded variation in a neighborhood of s and V f (X; ) is continuous at s. c Eberlein, Uni Freiburg, 18

20 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 X Modelling: St i = S i exp(ht i ) (1 i d) X T = Ψ(H t t T ) i=1 1 {L<STi <H} f : R d R + g(x) = e R,x f (x) (x R d ) Assumptions: (A1) g L 1 (R d ) (A2) M XT (R) exists (A3) bϱ L 1 (R d ) where ϱ(dx) = e R,x P XT (dx) c Eberlein, Uni Freiburg, 19

21 Theorem Options on multiple assets (cont.) If the asset price processes are modeled as exponential semimartingale processes such that S i M loc(p) (1 i d) and conditions (A1) (A3) are in force, then V f (X; s) = e R,s (2π) d Z R d e i u,s M XT (R + iu) b f (ir u)du Remark When the payoff function is discontinuous and the driving process does not possess a Lebesgue density L 2 -limit result c Eberlein, Uni Freiburg, 2

22 Delta of an option Sensitivities Greeks V f (X; S ) = 1 Z S R iu M XT (R iu) 2π b f (u + ir)du R f (X; S ) = V(X; S ) = 1 Z S R 1 iu M XT (R iu) b f (u + ir) S 2π R (R iu) 1 du Gamma of an option Γ f (X; S ) = 2 V f (X; S ) = 1 Z 2 S 2π R S R 2 iu M XT (R iu) b f (u + ir) (R 1 iu) 1 (R iu) 1 du c Eberlein, Uni Freiburg, 21

23 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, 22

24 Lévy processes Let L = (L t) t T be a Lévy 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 Lévy measure. Assumption (EM) There exists a constant M > 1 such that Z e ux λ(dx) <, { x >1} u [ M, M]. 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 < c Eberlein, Uni Freiburg, 23

25 On the characteristic function of the supremum I Proposition Let L = (L t) t T be a Lévy process that satisfies assumption (EM). Then, the characteristic function ϕ Lt of L t has an analytic extension to 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 c Eberlein, Uni Freiburg, 24

26 Fluctuation theory for Lévy processes Theorem (Extension of Wiener Hopf to the complex plane) Let L be a Lévy process. The Laplace transform of L at an independent and exponentially distributed time θ, θ Exp(q), can be identified from the Wiener Hopf factorization of L via Eˆe βl θ = Z qe[e βl t ]e qt dt = κ(q, ) κ(q, β) (13) for q > α (M) and β {β C R(β) > M} where κ(q, β), is given by Z Z κ(q, β) = k exp (e t e qt βx ) 1 «t P L t (dx) dt. (14) c Eberlein, Uni Freiburg, 25

27 On the characteristic function of the supremum II Theorem Let L = (L t) t T be a Lévy process satisfying assumption (EM). The Laplace transform of L t at a fixed time t, t [, T ], is given by Eˆe βl 1 t = lim A 2π Z A A t(y +iv) e κ(y + iv, ) dv, (15) Y + iv κ(y + iv, β) for Y > α (M) and β C with Rβ ( M, ). Remark Note that β = iz provides the characteristic function. c Eberlein, Uni Freiburg, 26

28 Application to lookback options Fixed strike lookback call: (S T K ) + (analogous for lookback put). Combining the results, we get C T (S; K ) = 1 Z S R iu K 1+iu R ϕ 2π LT ( u ir) du (16) R (iu R)(1 + iu R) where ϕ LT ( u ir) = lim A Z 1 A 2π A for R (1, M) and Y > α (M). T (Y +iv) e Y + iv κ(y + iv, ) dv (17) κ(y + iv, iu R) The floating strike lookback option, ( S T S T ) +, is treated by a duality formula (Eb., Papapantoleon (25)). c Eberlein, Uni Freiburg, 27

29 One-touch options One-touch call option: 1 {ST >B}. 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 Z 1 A S R+iu 2π A ϕ LT (u ir) B R iu du (18) R + iu = P(L T > log(b/s )) for R (, M). c Eberlein, Uni Freiburg, 28

30 Equity default swap (EDS) Fixed premium exchanged for payment at default default: drop of stock price by 3 % or 5 % of S 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 ; hence K = CE ˆe rτ B 1 {τb T } P N i=1 E ˆe. (19) rt i 1 {τb >T i } Calculations similar to touch options, since 1 {τb T } = 1 {ST B}. c Eberlein, Uni Freiburg, 29

31 Basic interest rates B(t,T ): price at time t [, T ] of a default-free zero coupon bond with maturity T [, T ] (B(T,T ) = 1) f (t,t ): instantaneous forward rate B(t,T ) = exp R 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) c Eberlein, Uni Freiburg, 3

32 Dynamics of the forward rates (Eb Raible (1999), Eb Özkan (23), Eb Jacod Raible (25), Eb Kluge (26) df (t, T ) = α(t, T ) dt σ(t, T ) dl t ( t T T ) α(t, T ) and σ(t, T ) satisfy measurability and boundedness conditions and α(s, T ) = σ(s, T ) = for s > T Define A(s, T ) = Z T s T α(s, u) du and Σ(s, T ) = Assume Σ i (s, T ) M (1 i d) Z T s T σ(s, u) du For most purposes we can consider deterministic α and σ c Eberlein, Uni Freiburg, 31

33 Implications Savings account and default-free zero coupon bond prices are given by Z 1 t Z t «B t = B(, t) exp A(s, T ) ds Σ(s, t) dl s and Z t Z t «B(t, T ) = B(, T )B t exp A(s, T ) ds + Σ(s, T ) dl s. If we choose A(s, T ) = θ s(σ(s, T )), then bond prices, discounted by the savings account, are martingales. In case d = 1, the martingale measure is unique (see Eberlein, Jacod, and Raible (24)). c Eberlein, Uni Freiburg, 32

34 L = (L 1,..., L d ) Key tool d-dimensional time-inhomogeneous Lévy process Z t E[exp(i u, L t )] = exp θ s(z) = z, b s z, csz + Z 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» Z t «Z t «E exp f (s)dl s = exp θ s(f (s))ds Take f (s) = P (s, T ) for some T [, T ] c Eberlein, Uni Freiburg, 33

35 Pricing of European options»z t B(t, T ) = B(, T ) exp (r(s) + θ s(σ(s, T ))) ds + where r(t) = f (t, t) short rate Z t Σ(s, T )dl s V (, t, T, w) time--price of a European option with maturity t and payoff w(b(t, T ), K ) V (, t, T, w) = E P [B 1 t w(b(t, T ), K )] Volatility structures Σ(t, T ) = bσ (1 exp( a(t t))) a (Vasiček) Σ(t, T ) = bσ(t t) (Ho Lee) Fast algorithms for Caps, Floors, Swaptions, Digitals, Range options c Eberlein, Uni Freiburg, 34

36 Pricing formula for caps (Eberlein, Kluge (26)) w(b(t, T ), K ) = (B(t, T ) K ) + Call with strike K and maturity t on a bond that matures at T Assume X = C(, t, T, K ) = E P [B 1 t (B(t, T ) K ) + ] Z t = B(, t)e Pt [(B(t, T ) K ) + ] (Σ(s, T ) Σ(s, t))dl s has a Lebesgue density, then C(, t, T, K ) = 1 KB(, t) exp(rξ) 2π Z where ξ is a constant and R < 1. e iuξ (R + iu) 1 (R iu) 1 M X t ( R iu)du Analogous for the corresponding put and for swaptions c Eberlein, Uni Freiburg, 35

37 Borovkov, K. and A. Novikov (22). 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), Eberlein, E., K. Glau, and A. Papapantoleon (29). Analysis of Fourier transform valuation formulas and applications. To appear in Applied Mathematical Finance. Eberlein, E., K. Glau, and A. Papapantoleon (29). Analyticity of the Wiener Hopf Factors and valuation of exotic options in Lévy models. Preprint, University of Freiburg. Eberlein, E. and A. Papapantoleon (25). Symmetries and pricing of exotic options in Lévy models. In Exotic Option Pricing and Advanced Lévy Models, A. Kyprianou, W. Schoutens, P. Wilmott (Eds.), Wiley, pp c Eberlein, Uni Freiburg, 36

38 (cont.) Eberlein, E., A. Papapantoleon, and A. N. Shiryaev (28). On the duality principle in option pricing: Semimartingale setting. Finance & Stochastics 12, Hubalek, F., J. Kallsen and L. Krawczyk (26). Variance-optimal hedging for processes with stationary independent increments. Ann. Appl. Probab. 16, Kyprianou, A. E. (26). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer. Papapantoleon, A. (27). Applications of semimartingales and Lévy processes in finance: Duality and valuation. Ph.D. thesis, University of Freiburg. Raible, S. (2). Lévy processes in finance: theory, numerics, and empirical facts. Ph.D. thesis, University of Freiburg. c Eberlein, Uni Freiburg, 37