fast fourier transform for discrete asian options
DESCRIPTION
Fast Fourier Transform for Discrete Asian Options. European Finance Management Association Lugano June 2001 Eric Ben-Hamou [email protected]. Outline. Motivations Description of the methods Log-Normal density Non-Log-Normal density Conclusion. Motivations. - PowerPoint PPT PresentationTRANSCRIPT
Fast Fourier Transform for Discrete Asian Options
European Finance Management Association
Lugano June 2001Eric Ben-Hamou
June 2001 Lugano 2001 Conference Slide N°2
Outline
• Motivations• Description of the methods• Log-Normal density• Non-Log-Normal density• Conclusion
June 2001 Lugano 2001 Conference Slide N°3
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.
June 2001 Lugano 2001 Conference Slide N°4
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)
June 2001 Lugano 2001 Conference Slide N°5
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
June 2001 Lugano 2001 Conference Slide N°6
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
June 2001 Lugano 2001 Conference Slide N°7
• 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
June 2001 Lugano 2001 Conference Slide N°8
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..
June 2001 Lugano 2001 Conference Slide N°9
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
June 2001 Lugano 2001 Conference Slide N°10
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
June 2001 Lugano 2001 Conference Slide N°11
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
June 2001 Lugano 2001 Conference Slide N°12
Efficiency of the algorithm
June 2001 Lugano 2001 Conference Slide N°13
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
June 2001 Lugano 2001 Conference Slide N°14
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
June 2001 Lugano 2001 Conference Slide N°15
• 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
June 2001 Lugano 2001 Conference Slide N°16
Student law
• Variance is exactly• density is given by the Gamma function
2nn
21
2
1
2
21
n
nt
nn
n
June 2001 Lugano 2001 Conference Slide N°17
Non log-normal densities
June 2001 Lugano 2001 Conference Slide N°18
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
June 2001 Lugano 2001 Conference Slide N°19
Results
Long maturity before expiry
short maturitybefore expiry
June 2001 Lugano 2001 Conference Slide N°20
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