Hilbert transform approach for pricing Bermudan options in … · Introduction Bermudan vanilla...

IntroductionBermudan vanilla

Bermudan barrier/look

Hilbert transform approach for pricing Bermudanoptions in Levy models

Liming Feng1

1Dept. of Industrial & Enterprise Systems EngineeringUniversity of Illinois at Urbana-Champaign

Joint with Xiong Lin

Spectral and cubature methods in finance and econometricsUniversity of Leicester, 6/19/2009

Liming Feng Pricing Bermudan options in Levy models



Levy modelsHilbert transform methods

Levy models

Better fit to empirical data, explain volatility smiles

Finite activity jump diffusion modelsMerton 76 (normal jump diffusion), Kou 02 (double exponential

jump diffusion)

Infinite activity pure jump Levy modelsMadan 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 Levy processesBoyarchenko & Levendorskii 02, Cont & Tankov 04, Schoutens03, Bertoin 96, Sato 99, Applebaum 04, Kyprianou 06





Levy-Khintchine theorem

Levy process: independent and stationary increments

Analytical tractability: characteristic functions available viaLevy-Khintchine Theorem

φt(ξ) = E[e iξXt ] = e−tψ(ξ)

where ψ is the characteristic exponent

ψ(ξ) =1

2σ2ξ2 − iµξ +

∫R(1− e iξx + iξx1|x |≤1)Π(dx)





Fourier transform method

Fourier transform method for European options: Carr &Madan 99

c(k) =1

2πe−αk

∫R

e−iξkΦ(ξ)dξ

where

Φ(ξ) =e−rTφT (ξ − i(α+ 1))

(iξ + α)(iξ + α+ 1)

Extended in Lee 04: more general payoffs, exponentiallydecaying errors

Bermudan options: vanilla, discrete barrier/lookback options,Hilbert transform method





Hilbert transform

Hilbert transform of f ∈ Lp(R), 1 ≤ p <∞

Hf (x) =1

πp.v .

∫ ∞

−∞

f (y)

x − ydy

For any f ∈ L1(R) with f ∈ L1(R)

F(sgn · f )(ξ) = iHf (ξ)

F(1(l ,∞) · f )(ξ) =1

2f (ξ) +

i

2e iξlH(e−iηl f (η))(ξ)





Option pricing

European vanilla options:

E[(K − ST )+] = KE[1ST<K]− E[ST1ST<K]

P(X ≤ x) =

∫R

1(−∞,x)(y)p(y)dy

European style discrete barrier options: Feng & Linetsky 08

f j(x) = 1(l ,u) · Etj ,x [fj+1(Xtj+1)]

European style discrete lookbacks: Feng & Linetsky 09

Mj−Xj = max(Mj−1,Xj)−Xj = max(0,Mj−1−Xj−1−(Xj−Xj−1))





Discrete Hilbert transform

Discrete Hilbert transform with step size h > 0

Hhf (x) =∞∑

m=−∞f (mh)

1− cos[π(x −mh)/h]

π(x −mh)/h, x ∈ R

For f analytic in a horizontal strip z ∈ C : |=(z)| < d

||Hf −Hhf ||L∞(R) ≤Ce−πd/h

πd(1− e−πd/h)

Related to Whittaker cardinal series (sinc) expansion





Sinc expansion of analytic functions

Whittaker cardinal series (sinc expansion)

c(f , h)(x) =∞∑−∞

f (kh)sin(π(x − kh)/h)

π(x − kh)/h

For entire functions of exponential type π/h, sinc expansion isexact: c(f , h) = f

For functions analytic in a strip z ∈ C : |=(z)| < d, Stenger93

||f − c(f , h)||L∞ ≤ Ce−πd/h

πd(1− e−2πd/h)





Trapezoidal rule

Take Hilbert transform on c(f , h), obtain discrete Hilberttransform

Trapezoidal rule very accurate for f analytic in a stripz ∈ C : |=(z)| < d∣∣∣∣∣

∫R

f (x)dx −∞∑

m=−∞f (kh)h

∣∣∣∣∣ ≤ Ce−2πd/h

1− e−2πd/h

We use trapezoidal rule to compute Fourier inverse integral




Backward induction in state spaceBackward induction in Fourier spaceDiscrete approximation

Bermudan style vanilla options

Bermudan put: payoff G (S) = (K −S)+, discrete monitoring

T = t0, t1, · · · , tN = 0,∆, 2∆, · · · ,N∆ = T

Exponential Levy model: Xt a Levy process in (Ω,F ,F,P),start from ln(S0/K ), equivalent martingale measure P is given

St = KeXt

Optimal stopping

V 0(S0) = supτ

E0[e−rτG (Sτ )]





Backward induction in state space

Variable change x = ln(S/K ),

g(x) = G (Kex) = K (1− ex)+, f 0(x) = V 0(Kex)

Backward induction

f N(x) = g(x)

f j(x) = max(g(x), e−r∆Ej∆,x [f

j+1(X(j+1)∆)]), 0 ≤ j < N

V 0(S0) = f 0(ln(S0/K ))





Direct implementation

Knowing the transition density p∆(·)

Ej∆,x [fj+1(X(j+1)∆)] =

∫R

f j+1(y)p∆(y − x)dy

When discretized, the convolution becomes a Toeplitz matrixvector 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 modeland Merton’s model

For general Levy models: COS method Fang & Oosterlee 08





Dampening for integrability

For α > 0 (for puts), define

f jα(x) = eαx f j(x), gα(x) = eαxg(x)

Backward induction

f Nα (x) = gα(x)

f jα(x) = max

(gα(x), eαx−r∆Ej∆,x [e

−αX(j+1)∆f j+1α (X(j+1)∆)]

),

V 0(S0) = e−α ln(S0/K)f 0α (ln(S0/K ))





Esscher transform

Define new measure Pα via

dPα

dP|Ft =

e−αXt

φt(iα)

Esscher transformed Levy process is still a Levy process

φαt (ξ) =φt(ξ + iα)

φt(iα)

Applying property Eαs [Yt ] = Es [YtZt/Zs ]

Ej∆,x [e−αX(j+1)∆f j+1

α (X(j+1)∆)] = e−αx−∆ψ(iα)Eαj∆,x [f j+1α (X(j+1)∆)]





Dampened backward induction

Dampened backward induction

f Nα (x) = gα(x)

For 0 ≤ j < N

f jα(x) = max

(gα(x), e−(r+ψ(iα))∆Eαj∆,x [f j+1

α (X(j+1)∆)])

= gα(x)1(−∞,x∗j )(x)

+e−(r+ψ(iα))∆Eαj∆,x[f j+1α (X(j+1)∆)

]1[x∗j ,∞)(x)

where x∗j is the early exercise boundary at time j∆





Backward induction in Fourier space

By convolution theorem, Fourier transform of

Eαj∆,x[f j+1α (X(j+1)∆)

]=

∫R

f j+1α (y)pα∆(y − x)dy

is a product f j+1α (ξ)φα∆(−ξ)

Backward induction in Fourier space

f Nα (ξ) = gα(ξ)

f jα(ξ) = F(gα1(−∞,x∗j ))(ξ) + e−(r+ψ(iα))∆

(1

2f j+1α (ξ)φα∆(−ξ)

+i

2e iξx∗j H

(e−iηx∗j f j+1

α (η)φα∆(−η))

(ξ)

)Liming Feng Pricing Bermudan options in Levy models




Early exercise boundary

Early exercise boundary solves

gα(x) = e−(r+ψ(iα))∆Eαj∆,x [f j+1α (X(j+1)∆)]

Using Fourier inverse representation

gα(x) =1

2πe−(r+ψ(iα))∆

∫R

e−iξx f j+1α (ξ)φα∆(ξ)dξ

x∗N = K . To solve for x∗j , use Newton-Raphson, with startingpoint x∗j+1





Algorithm summarized

Start with Fourier transform of dampened payoff f Nα = gα

At time j∆, with f j+1α , compute early exercise boundary x∗j

using Newton Raphson

Compute f jα from f j+1

α and x∗j (Hilbert transform)

With f 1α , option value at time 0

f 0α (x) = max

(gα(x),

1

2πe−(r+ψ(iα))∆

∫R

e−iξx f 1α (ξ)φα∆(ξ)dξ

)





Discrete approximation

Need to repeatedly evaluate a Fourier inverse integral andHϕ(ξ)

Trapezoidal rule for Fourier inverse integral, truncate infiniteseries with truncation level M, computational cost O(M)

Replace Hϕ by discrete Hilbert transform, truncate resultinginfinite series

Hϕ(ξ) ⇐M∑

m=−M

ϕ(mh)1− cos[π(ξ −mh)/h]

π(ξ −mh)/h





Error estimate

Discretization error ∼ O(exp(−πd/h)

With φt(ξ) ∼ exp(−ct|ξ|ν), truncation error is essentially

O(exp(−∆c(Mh)ν))

Select h = h(M) according to

h(M) =

(πd

∆c

) 11+ν

M− ν1+ν

Total error: O(exp(−CMν

1+ν ))





Toeplitz matrix vector multiplication

Evaluate

Hψ(ξ) ⇐M∑

m=−M

ψ(mh)1− cos[π(ξ −mh)/h]

π(ξ −mh)/h

for ξ = −Mh, · · · ,Mh

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))





NIG model

Pricing Bermudan put option in NIG

NIG process: time changed Brownian motion, with an inverseGaussian process as the stochastic time

Characteristic exponent

ψ(ξ) = −iµξ+ δNIG (√α2

NIG − (βNIG + iξ)2−√α2

NIG − β2NIG )

φt(ξ) has exponential tails with ν = 1, error estimate in M:

O(e−C√

M)





Bermudan put in the NIG model

Figure: T = 1,N = 252,S0 = 100,K = 110, r = 0.1, q = 0, αNIG =25, βNIG = −5, δNIG = 0.5, implemented in Matlab using a Laptop withCPU 2GHz, RAM 1G, last node takes 2.9s




Barrier optionsLookback options

Bermudan barrier 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∆Ej∆,x [fj+1(X(j+1)∆)])

= g(x)1(l ,∞)(x)1(−∞,x∗j )(x)

+e−r∆Ej∆,x [fj+1(X(j+1)∆)]1(l ,∞)(x)1(x∗j ,∞,)(x)





CGMY model

Pricing Bermudan down-and-out put in CGMY

Levy density

π(x) =CeGx

|x |1+Y1x<0 +

Ce−Mx

|x |1+Y1x>0


ψ(ξ) = −iµξ−CΓ(−Y )((M − iξ)Y −MY +(G + iξ)Y −GY )

Characteristic function has exponential tails with ν = Y , errorestimate in M: O(exp(−CMY /(1+Y )))





Bermudan DOP in the CGMY model

Figure: T = 1,N = 252,S0 = 100,K = 110, r = 0.1, q = 0,C = 3,G =22,M = 28,Y = 0.8, implemented in Matlab using a Laptop with CPU2GHz, RAM 1G, last node takes 2.0s





Bermudan floating strike lookbacks

Standard backward induction involves two state variables:asset price, maximum asset price

Can be reduced to one state variable, maximum assetprice/asset price

f j(y) = max(ey − 1, e−q∆E∗j∆,y [f j+1(eY(j+1)∆)]

Double exponential fast Gauss transform method (Yamamoto05 ), limited to BSM and Merton’s models





Kou’s model

Pricing Bermudan floating strike lookback put

Log asset price follows

Xt = µt+σBt+Nt∑

n=1

Zn, Zn ∼ pη1e−η1x1x>0+(1−p)η2e

η2x1x<0


ψ(ξ) = −iµξ +1

2σ2ξ2 + λ

(1− pη1

η1 − iξ− (1− p)η2

η2 + iξ

)Characteristic function has exponential tails with ν = 2, errorestimate O(exp(−CM2/3))





Bermudan floating strike lookback put in Kou’s model

Figure: T = 1,N = 252,S0 = 100, r = 0.1, q = 0, σ = 0.25, λ = 3, p =0.3, η1 = 17, η2 = 12, implemented in Matlab using a Laptop with CPU2GHz, RAM 1G, last node takes 1.1s





Summary

Hilbert transform method for pricing Bermudan style optionsin Levy process models

Very accurate with exponentially decay errors

Fast with computational cost O(NM log(M))

European vanilla, barrier, lookback, defaultable bonds,Bermudan vanilla, barrier, floating strike lookback, montecarlo simulation etc.


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