Hilbert transform approach for pricing Bermudan options in Lévy models

Transcription

1 Hilbert transform approach for pricing Bermudan options in Lévy models 1 1 Dept. of Industrial & Enterprise Systems Engineering University of Illinois at Urbana-Champaign Joint with Xiong Lin Spectral and cubature methods in finance and econometrics University of Leicester, 6/19/2009

2 Lévy models Introduction Lévy models Hilbert transform methods Better fit to empirical data, explain volatility smiles Finite activity jump diffusion models Merton 76 (normal jump diffusion), Kou 02 (double exponential jump diffusion) Infinite activity pure jump Lévy models Madan et al 90, 91, 98 (variance gamma), Barndorff-Nielsen 98 (normal inverse Gaussian), Carr, Geman, Madan & Yor 02 (CGMY), Eberlein, Keller & Prause 98 (Generalized hyperbolic) Books on Lévy processes Boyarchenko & Levendorskii 02, Cont & Tankov 04, Schoutens 03, Bertoin 96, Sato 99, Applebaum 04, Kyprianou 06

3 Lévy-Khintchine theorem Lévy models Hilbert transform methods Lévy process: independent and stationary increments Analytical tractability: characteristic functions available via Lévy-Khintchine Theorem φ t (ξ) = E[e iξxt ] = e tψ(ξ) where ψ is the characteristic exponent ψ(ξ) = 1 2 σ2 ξ 2 iµξ + (1 e iξx + iξx1 { x 1} )Π(dx) R

4 Fourier transform method Lévy models Hilbert transform methods Fourier transform method for European options: Carr & Madan 99 c(k) = 1 2π e αk e iξk Φ(ξ)dξ R where Φ(ξ) = e rt φ T (ξ i(α + 1)) (iξ + α)(iξ + α + 1) Extended in Lee 04: more general payoffs, exponentially decaying errors Bermudan options: vanilla, discrete barrier/lookback options, Hilbert transform method

5 Hilbert transform Introduction Lévy models Hilbert transform methods Hilbert transform of f L p (R), 1 p < Hf (x) = 1 π p.v. For any f L 1 (R) with ˆf L 1 (R) f (y) x y dy F(sgn f )(ξ) = ihˆf (ξ) F(1 (l, ) f )(ξ) = 1 2ˆf (ξ) + i 2 eiξl H(e iηl ˆf (η))(ξ)

6 Option pricing Introduction Lévy models Hilbert transform methods European vanilla options: E[(K S T ) + ] = KE[1 {ST <K}] E[S T 1 {ST <K}] P(X x) = 1 (,x) (y)p(y)dy R European style discrete barrier options: Feng & Linetsky 08 f j (x) = 1 (l,u) E tj,x[f j+1 (X tj+1 )] European style discrete lookbacks: Feng & Linetsky 09 M j X j = max(m j 1, X j ) X j = max(0, M j 1 X j 1 (X j X j 1 ))

7 Discrete Hilbert transform Lévy models Hilbert transform methods Discrete Hilbert transform with step size h > 0 H h f (x) = m= f (mh) 1 cos[π(x mh)/h], x R π(x mh)/h For f analytic in a horizontal strip {z C : I(z) < d} Hf H h f L (R) Ce πd/h πd(1 e πd/h ) Related to Whittaker cardinal series (sinc) expansion

8 Sinc expansion of analytic functions Lévy models Hilbert transform methods Whittaker cardinal series (sinc expansion) c(f, h)(x) = sin(π(x kh)/h) f (kh) π(x kh)/h For entire functions of exponential type π/h, sinc expansion is exact: c(f, h) = f For functions analytic in a strip {z C : I(z) < d}, Stenger 93 Ce πd/h f c(f, h) L πd(1 e 2πd/h )

9 Trapezoidal rule Introduction Lévy models Hilbert transform methods Take Hilbert transform on c(f, h), obtain discrete Hilbert transform Trapezoidal rule very accurate for f analytic in a strip {z C : I(z) < d} f (x)dx f (kh)h Ce 2πd/h 1 e 2πd/h R m= We use trapezoidal rule to compute Fourier inverse integral

10 Bermudan style vanilla options Backward induction in state space Discrete approximation Bermudan put: payoff G(S) = (K S) +, discrete monitoring T = {t 0, t 1,, t N } = {0,, 2,, N = T } Exponential Lévy model: X t a Lévy process in (Ω, F, F, P), start from ln(s 0 /K), equivalent martingale measure P is given Optimal stopping S t = Ke Xt V 0 (S 0 ) = sup E 0 [e rτ G(S τ )] τ

11 Backward induction in state space Backward induction in state space Discrete approximation Variable change x = ln(s/k), g(x) = G(Ke x ) = K(1 e x ) +, f 0 (x) = V 0 (Ke x ) Backward induction f N (x) = g(x) ( ) f j (x) = max g(x), e r E j,x [f j+1 (X (j+1) )], 0 j < N V 0 (S 0 ) = f 0 (ln(s 0 /K))

12 Direct implementation Backward induction in state space Discrete approximation Knowing the transition density p ( ) E j,x [f j+1 (X (j+1) )] = f j+1 (y)p (y x)dy When discretized, the convolution becomes a Toeplitz matrix vector multiplication, can be implemented using FFT (Eydeland 94 ) Density may not be available, polynomial convergence Double exponential fast Gauss transform (Broadie & Yamamoto 05 ), limited to the Black-Schoels-Merton model and Merton s model For general Lévy models: COS method Fang & Oosterlee 08 R

13 Dampening for integrability Backward induction in state space Discrete approximation For α > 0 (for puts), define f j α(x) = e αx f j (x), g α (x) = e αx g(x) Backward induction fα N (x) = g α (x) ( ) fα(x) j = max g α (x), e αx r E j,x [e αx (j+1) fα j+1 (X (j+1) )], V 0 (S 0 ) = e α ln(s 0/K) f 0 α (ln(s 0 /K))

14 Esscher transform Introduction Backward induction in state space Discrete approximation Define new measure P α via dp α dp F t = e αxt φ t (iα) Esscher transformed Lévy process is still a Lévy process φ α t (ξ) = φ t(ξ + iα) φ t (iα) Applying property E α s [Y t ] = E s [Y t Z t /Z s ] E j,x [e αx (j+1) fα j+1 (X (j+1) )] = e αx ψ(iα) E α j+1 j,x [fα (X (j+1) )]

15 Dampened backward induction Backward induction in state space Discrete approximation Dampened backward induction For 0 j < N f j α(x) = max f N α (x) = g α (x) ( ) g α (x), e (r+ψ(iα)) E α j+1 j,x [fα (X (j+1) )] = g α (x)1 (,x j )(x) +e (r+ψ(iα)) E α j,x [ f j+1 α (X (j+1) ) ] 1 [x j, )(x) where x j is the early exercise boundary at time j

16 Backward induction in state space Discrete approximation By convolution theorem, Fourier transform of E α [ j,x f j+1 α (X (j+1) ) ] = fα j+1 (y)p α (y x)dy is a product ˆf α j+1 (ξ)φ α ( ξ) R ˆf N α (ξ) = ĝ α (ξ) ( 1 ˆf α(ξ) j = F(g α 1 (,x j ))(ξ) + e (r+ψ(iα)) α 2ˆf j+1 (ξ)φ α ( ξ) + i ( ) ) 2 eiξx j H e iηx j ˆf α j+1 (η)φ α ( η) (ξ)

17 Early exercise boundary Backward induction in state space Discrete approximation Early exercise boundary solves g α (x) = e (r+ψ(iα)) E α j+1 j,x [fα (X (j+1) )] Using Fourier inverse representation g α (x) = 1 2π e (r+ψ(iα)) j+1 e iξxˆf α (ξ)φ α (ξ)dξ xn = K. To solve for x j, use Newton-Raphson, with starting point xj+1 R

18 Algorithm summarized Backward induction in state space Discrete approximation Start with Fourier transform of dampened payoff ˆf α N = ĝ α j+1 At time j, with ˆf α, compute early exercise boundary xj using Newton Raphson Compute ˆf α j from ˆf α j+1 and xj (Hilbert transform) With ˆf α 1, option value at time 0 ( fα 0 1 (x) = max g α (x), 2π e (r+ψ(iα)) e iξxˆf ) α 1 (ξ)φ α (ξ)dξ R

19 Discrete approximation Backward induction in state space Discrete approximation Need to repeatedly evaluate a Fourier inverse integral and Hϕ(ξ) Trapezoidal rule for Fourier inverse integral, truncate infinite series with truncation level M, computational cost O(M) Replace Hϕ by discrete Hilbert transform, truncate resulting infinite series Hϕ(ξ) M m= M 1 cos[π(ξ mh)/h] ϕ(mh) π(ξ mh)/h

20 Error estimate Introduction Backward induction in state space Discrete approximation Discretization error O(exp( πd/h) With φ t (ξ) exp( ct ξ ν ), truncation error is essentially Select h = h(m) according to O(exp( c(mh) ν )) h(m) = Total error: O(exp( CM ν 1+ν )) ( ) 1 πd 1+ν M ν 1+ν c

21 Toeplitz matrix vector multiplication Backward induction in state space Discrete approximation Evaluate Hψ(ξ) M m= M for ξ = Mh,, Mh 1 cos[π(ξ mh)/h] ψ(mh) π(ξ mh)/h Correspond to Toeplitz matrix vector multiplication FFT based method for such multiplications: O(M log(m)) Total computational cost of the method: O(NM log(m))

22 NIG model Introduction Backward induction in state space Discrete approximation Pricing Bermudan put option in NIG NIG process: time changed Brownian motion, with an inverse Gaussian process as the stochastic time Characteristic exponent ψ(ξ) = iµξ + δ NIG ( αnig 2 (β NIG + iξ) 2 αnig 2 β2 NIG ) φ t (ξ) has exponential tails with ν = 1, error estimate in M: O(e C M )

23 Bermudan put in the NIG model Backward induction in state space Discrete approximation Figure: T = 1, N = 252, S 0 = 100, K = 110, r = 0.1, q = 0, α NIG = 25, β NIG = 5, δ NIG = 0.5, implemented in Matlab using a Laptop with CPU 2GHz, RAM 1G, last node takes 2.9s

24 Bermudan barrier options Barrier options Lookback options Bermudan down-and-out put with lower barrier L, l = ln(l/k) Backward induction f N (x) = g(x)1 (l, ) (x) For 0 j < N, f j (x) = 1 (l, ) (x) max(g(x), e r E j,x [f j+1 (X (j+1) )]) = g(x)1 (l, ) (x)1 (,x j )(x) +e r E j,x [f j+1 (X (j+1) )]1 (l, ) (x)1 (x j,,)(x)

25 CGMY model Introduction Barrier options Lookback options Pricing Bermudan down-and-out put in CGMY Levy density Characteristic exponent π(x) = CeGx x 1+Y 1 {x<0} + Ce Mx x 1+Y 1 {x>0} ψ(ξ) = iµξ CΓ( Y )((M iξ) Y M Y + (G + iξ) Y G Y ) Characteristic function has exponential tails with ν = Y, error estimate in M: O(exp( CM Y /(1+Y ) ))

26 Barrier options Lookback options Bermudan DOP in the CGMY model Figure: T = 1, N = 252, S 0 = 100, K = 110, r = 0.1, q = 0, C = 3, G = 22, M = 28, Y = 0.8, implemented in Matlab using a Laptop with CPU 2GHz, RAM 1G, last node takes 2.0s

27 Barrier options Lookback options Bermudan floating strike lookbacks Standard backward induction involves two state variables: asset price, maximum asset price Can be reduced to one state variable, maximum asset price/asset price f j (y) = max(e y 1, e q E j,y [f j+1 (e Y (j+1) )] Double exponential fast Gauss transform method (Yamamoto 05 ), limited to BSM and Merton s models

28 Kou s model Introduction Barrier options Lookback options Pricing Bermudan floating strike lookback put Log asset price follows N t X t = µt+σb t + Z n, Z n pη 1 e η1x 1 {x>0} +(1 p)η 2 e η2x 1 {x<0} n=1 Characteristic exponent ψ(ξ) = iµξ σ2 ξ 2 + λ ( 1 pη 1 η 1 iξ (1 p)η ) 2 η 2 + iξ Characteristic function has exponential tails with ν = 2, error estimate O(exp( CM 2/3 ))

29 Barrier options Lookback options Bermudan floating strike lookback put in Kou s model Figure: T = 1, N = 252, S 0 = 100, r = 0.1, q = 0, σ = 0.25, λ = 3, p = 0.3, η 1 = 17, η 2 = 12, implemented in Matlab using a Laptop with CPU 2GHz, RAM 1G, last node takes 1.1s

30 Summary Introduction Barrier options Lookback options Hilbert transform method for pricing Bermudan style options in Lévy process models Very accurate with exponentially decay errors Fast with computational cost O(NM log(m)) European vanilla, barrier, lookback, defaultable bonds,, barrier, floating strike lookback, monte carlo simulation etc.