Fast Fourier Transform for Discrete Asian Options

Fast Fourier Transform for Discrete Asian Options

European Finance Management Association

Lugano June 2001Eric Ben-Hamou

[email protected]

June 2001 Lugano 2001 Conference Slide N°2

Outline

• Motivations• Description of the methods• Log-Normal density• Non-Log-Normal density• Conclusion


Motivations• When pricing a derivative, one should keep

in mind:• Within a model, the quality of the approximation.• model risk, reflected by the uncertainty on the

model parameters• objectivity of the model

• So need to examine the risk of a certain derivative, here Asian option.


Asian option’s characteristics• academic case is geometric Brownian motion

continuous time average: no easy-closed forms (Geman Yor 93 Madan Yor 96)

• discrete averaging (see Vorst 92 Turnbull and Wakeman 91 Levy 92 Jacques 96 Bouazi et al. 98 Milevsky Posner 97 Zhang 98 Andreasen 98)

• empirical literature suggests fat-tailed distribution (Mandelbrot 63 Fama 65) and smile literature (Kon 84 Jorion 88 Bates 96 Dupire 94 Derman et al 94 see Dumas et al. 95) stochastic volatility (Hull and White 87 Heston 93)


Asian option risk• Main risks:

• jump of delta risk, or reset risk at each fixing (Vorst 92 Turnbull and Wakeman 91 Levy 92…)

• distribution risk underlying the jump of delta at each fixing dates…

• other issues like modeling of discrete dividends (Benhamou Duguet 2000)

• Good method:• tackles discret averaging• non lognormal densities


Aim:

• Assume that returns are iid with a well known density (known numerically)

• Target of the method: get the density of any type of sum of underlying at different dates..

with

tttt A

nSSS

n ...

21

itittt

i

RRtt eSS ,11,0

0

...

ititR ,1


• Standard hypotheses:… complete markets and no arbitrage.

• If the density of is known then simple numerical integration.

• Density of a sum of independent variables is the convolution of the individual densities

FFT O(Nlog2(N) ) typically N=2p so that p2p (like binomial tree)

KAeEP TrTQ

TA


New insights

• Old method first offered by Caverhill Clewlow 92 but inefficient and only for lognormal densities

• One way of improving it is to systematically re-center the convolution at each outcome

• Second, we examine the impact of non-lognormal law and see important changes in the delta..


Factorization

• Hodges (90)

• Recursive scheme

ntntntnttttt RRRRtt eeeeSnA ,11,22,11,0

01...111

ntntRentntR

ettR

e

tt eSnA

,11log1,2...1log1,0

0

1log


Algorithm 1 Result

1

0

1

1

1

log

,

,

1logB

t

Btti

ttn

eSnA

eRB

RBi

ii

nn

Algorithm 2 Result

11

0

11

1

1

1

11

log

,

,

,,

1log

1log

ABt

BAttii

Atti

ttnttnn

eSnA

eRAB

eA

ARAB

ii

ii

i

ii

nnnn


leads to

This imposes at each step to interpolate the densities obtained at the previous step.

interpolation

dyJdpdq

XfY

yfyfY 11

dyAepe

eydpi

i

i

i

iXiA Ayi

AyXAy

Ay

Ae

11log

1 11log 1


Efficiency of the algorithm


Grid specification (lognormal case)• Centered grid with• 212 =4096 points per grid• Simpson numerical integration for the final

procedure• interpolation done by standard cubic spline• Standard FFT algorithm as described in

Press et al.• Proxy for the mean efficient for volatility

lower than 30%

dtn9


Non log-normal case• Smile can be seen as a proof of non log-

normal densities with fatter tails. (excess kurtosis and skewness)

• Lognormal case already requires numerical methods

• So instead use of Student distributions, Pareto, generalized Pareto, power-laws distributions.. Here took Student density


• Student additional advantage to converge to normal distribution… which gives the geometric Brownian motion

• Assumptions:

follows a Student density of dfnormal case

12

1

2

, 21

ii

iitt

tt

ttrRii

122

12

1

Example of Student law

1


Student law

• Variance is exactly• density is given by the Gamma function

2nn

21

2

1

2

21

n

nt

nn

n


Non log-normal densities


Delta hedging

• Strong impact on the delta whereas small impact on the price.

• Price effect offsets by the overpricing of the lognormal approximation

• This justifies the use of lognormal approximation as a way of incorporating fat tails… but wrong for the delta


Results

Long maturity before expiry

short maturitybefore expiry


Conclusion• Offered an efficient method for the pricing of

discrete Asian options with non lognormal laws.• Shows that fat tails impacts much more for the

Greeks than the price• future work:

• adaptation to floating strike option• use of other fat tailed distribution• raises the issue of deriving an efficient methodology for

deriving densities from Market prices

Fast Fourier Transform for Discrete Asian Options

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