the discrete fourier transform as a tool in mathematical finance
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The Discrete Fourier Transform as aTool in Mathematical Finance:Computation of Index Returns
Jean-François Emmenegger, Anna Serbinenko
Department of Quantitative Economics, University of Fribourg,SwitzerlandJean-Francois.Emmenegger@unifr.ch, Ganna.Serbinenko@unifr.ch
Swiss Statistics Meeting, 14.-16. November 2007, Lucerne
J.-F. Emmenegger Mathematical Finance
International Project
Analysis of Economic and Environmental Time Series1 Dr. J.-F. Emmenegger, Department of Quantitative
Economics, University of Fribourg, Switzerland2 Dr. Tamara Bardadym, V. M. Glushkov Institute of
Cybernectis of the NASU, Kyiv, Ukraine3 Prof. Elena Pervukhina, Sevastopol National Technical
University, Ukraine4 Ass. Prof. Ivanka Stamova, Bourgas Free University,
Bulgaria5 Ass. Prof. Gani T. Stamov, Technical University of
Sliven, Bulgaria6 Anna Serbinenko, Department of Quantitative
Economics, University of Fribourg, Switzerland
J.-F. Emmenegger Mathematical Finance
Introduction
Gaussian Hypothesis : the distribution of returns isGaussian , Bachelier (1900)
Paretian Hypothesis : the distribution of returns isα-stable, Lévy (1925), Mandelbrot (1963)
Models of volatile time series : GARCH, Bollerslev(1987)
GARCH-Models of 19 Eastern European Index Time
α-stable and normal distribution models for the returns
Risk evaluation
J.-F. Emmenegger Mathematical Finance
The efficient market Hypothesis (EMH) (1)
What determines the price of a share of stock ?
Answer : demand and supply
A : The efficient market hypothesis states that it is notpossible to consistently outperform the market byusing any information that the market already knows,except through luck.
B : The EMH says : It is impossible to beat the market !
Also, work by Alfred Cowles in the 30s and 40s showedthat professional investors were in general unable toout perform the market.
The efficient market hypothesis (EMH) was firstexpressed by Louis Bachelier (1900), then bySamuelson and Fama (1965)
J.-F. Emmenegger Mathematical Finance
EMH -> Random walk -> Brownian Motion
The economists concluded that the EMH should implyrandom walk for the stock prices
This is the thesis of Bachelier (1900)
This means : predictability is impossible
Only new information changes prices
Random walk implies : Brownian Motion -> NormaldistributionEmpirical evidence that support normal distribution ofthe returns ! :
1 Kendall (1953) : weekly price changes of Chicago wheatand British common stock are approximately normaldistributed
2 Bachelier’s process leads to Brownian motion, seeDonsker (1951) : Cent. Limit Th. , re-scaled random walk
J.-F. Emmenegger Mathematical Finance
Counter observations to Normal Distribution
Efficient Market Theory <-> Random walk theory ofstock pricesEmpirical evidence against the Gaussian Hypothesisfrom empirical economists :
1 "The empirical distributions of price changes areusually too peaked to be relative to samples fromGaussian populations, see W. C. Michell (1915)
2 "The central bells remind one of the Gaussian ogive.But there are typically so many outliers that ogivesfitted to the mean square of price changes are muchlower and flatter than the distribution and the datathemselves, see Mandelbrot (1963)
3 "Asset prices fluctuate considerably more than onewould expect from the volatility of their fundamentaldeterminants", see Shiller (1981)
J.-F. Emmenegger Mathematical Finance
The stable Paretian Hypothesis
Mandelbrot and Fama (1963) : Departure fromnormality of the empirical distributions of pricechanges are constantly observedNew radical approach : The Stable Paretian hypothesis
1 the variances of empirical distributions behave as ifthey were infinite
2 the empirical distributions conform best tonon-Gaussian members of a limiting distribution
3 the heavy tail character and the asymmetry of theempirical distributions of price changes has beenrepeatedly observed
4 Mandelbrot (1963), Rachev (2000), Olszewski (2005)suggest applying the α-stable distributions
5 Isssued from the Generalized Central Limit Theorem
J.-F. Emmenegger Mathematical Finance
The α-stable distribution (1)
An α-stable distribution is denoted by X ∼ Sα(σ, β, µ) andits characteristic function admits the representation(Stoyanov, Racheva-Iotova, 2003 and 2003a) :
ΦX (t) = EeitX ={
exp{−σα|t|α(1− iβ t|t| tan(πα2 )) + iµt}, α 6= 1
exp{−σ|t|(1 + iβ 2π
t|t| ln(|t|)) + iµt}, α = 1,
(1)where 0 < α ≤ 2 is the index of stability, −1 ≤ β ≤ 1 is theskewness parameter, σ > 0 is a scale parameter and µ ∈ Ris a location parameter.
J.-F. Emmenegger Mathematical Finance
The α-stable distribution (2)
FIG.: Pdf of α-stable distributions : www.wikipedia.org, 25.7.2006
ΦX (t) = E(eitX ) =∫ ∞−∞
eitx fX (x)dx. (2)
fX (x) = 12π
∫ ∞−∞
ΦX (t)e−itxdt. (3)
J.-F. Emmenegger Mathematical Finance
Fourier Transforms and the characteristic function
The characteristic function ΦX (u) of a random variable X
ΦX (u) = E(eiuX ) =∫ ∞−∞
eiuxdFX (x) =∫ ∞−∞
eiux fX (x)dx. (4)
fX (x) = 12π
∫ ∞−∞
e−iuxΦX (u)du. (5)
(4), (5) are Fourier Transforms, Bracewell (1986)
ΦX (2πw) = IFT (fX ) =∫ ∞−∞
e2πiwx fX (x)dx. (6)
fX (x) = FT (ΦX ) =∫ ∞−∞
e−2πiuxΦX (u)du. (7)
J.-F. Emmenegger Mathematical Finance
Example : Uniform Distribution U0.5
-7.5 -5 -2.5 2.5 5 7.5u
-0.4
-0.2
0.2
0.4
0.6
0.8
1�X�u�
-7.5 -5 -2.5 2.5 5 7.5w
-0.4
-0.2
0.2
0.4
0.6
0.8
1F�w��IFT�R0.5�
-2 -1 1 2 x
0.2
0.4
0.6
0.8
1
1.2R0.5�x�
-2 -1 1 2 x
0.2
0.4
0.6
0.8
1
1.2R0.5�x�
Ra(x) ={
12a ; |x| ≤ a0 ; |x| > a (8)
F(w) = IFT (Ra) =∫ ∞−∞
e2πiwxRa(x)dx = sin(2πaw)2πaw := sinc(2aw).
(9)
ΦX (u) = E(eiuX ) =∫ ∞−∞
eiuxRa(x)dx = sin(au)au = sinc(au
π).
(10)J.-F. Emmenegger Mathematical Finance
The Discrete Fourier Transform
N ∈ N,x = 0,∆x,..., (N − 1)∆x,yi = f (i∆x), i = 0, ..., (N − 1),
Yk =N−1∑j=0
yje−2πiN jk ; k = 0, ...N − 1, (11)
yj = 1N
N−1∑k=0
Yke2πiN jk ; j = 0, ...N − 1. (12)
J.-F. Emmenegger Mathematical Finance
The sampling theorem
The sampling theorem states that it is possible torecover the intervening values with full accuracy fromsampled values.
It is used the other way round ! The condition is thatthe function f is band-limited ; that is, its FourierTransform fX (x) = FT (ΦX ) (7) is only nonzero over thefinite range x ≤ |xc|.Clearly, the interval ∆t between samples yj = Φ(j∆t) iscrucial. It determines the upper bound of the frequencyinterval, the so-called Nyquist frequency XN = 1
2∆t thatmust be greater or equal than the cut-off frequency xc,XN ≥ xc.
All is okay : fX is considered as band-limited, when ∆tis chosen small enough (continuity) !
J.-F. Emmenegger Mathematical Finance
The α-stable distribution (1)
X ∼ Sα(σ, β, µ)α=1.5, β=1, σ=1, µ=0
-1 -0.5 0.5 1x
0.5
1
1.5
2Re�fX�x��
-1 -0.5 0.5 1x
0.5
1
1.5
2Im�fX�x��
-4 -2 2 4w
-0.2
0.2
0.4
0.6
0.8
1Re��X�w��
-4 -2 2 4w
-0.4
-0.2
0.2
0.4
Im��X�w��
ΦX (t) = exp{−σα|t|α(1− iβ t|t| tan(πα
2)) + iµt}, (13)
fX (x) = 12π
∫ ∞−∞
exp{−σα|t|α(1− iβ t|t| tan(πα
2)) + iµt}e−itxdt.
(14)J.-F. Emmenegger Mathematical Finance
The α-stable distribution (2)
X ∼ Sα(σ, β, µ)α=1.5, β=-1, σ=4, µ=0
-3 -2 -1 1 2 3x
0.1
0.2
0.3
0.4
0.5Re�fX�x��
-3 -2 -1 1 2 3x
0.1
0.2
0.3
0.4
0.5Im�fX�x��
-1 -0.5 0.5 1w
-0.2
0.2
0.4
0.6
0.8
1Re��X�w��
-1 -0.5 0.5 1w
-0.4
-0.2
0.2
0.4
Im��X�w��
ϕX (t) = exp{−σα|t|α(1− iβ t|t| tan(πα
2)) + iµt}, (15)
fX (x) = 12π
∫ ∞−∞
exp{−σα|t|α(1− iβ t|t| tan(πα
2)) + iµt}e−itxdt.
(16)J.-F. Emmenegger Mathematical Finance
Analysis of Eastern European Indices (1)
Idea of the study OutlineEastern Europeans Indices
ARMA-GARCH modeling
Gaussian or α-stable distribution models for thereturns
Risk measures
Comparison of risk measures
J.-F. Emmenegger Mathematical Finance
Analysis of eastern European Indices (2)
Data presentation, analysis and modelingSeries Frequen- Period Description(1) cy (2) (3) (4)RSI (Russia) daily 01.09.1995-06.06.2006 Russian StExISSEI (Slovenia) daily 05.01.1993-01.06.2006 Slovene StExIBET (Romania) daily 01.01.2003-28.04.2006 Bucharest StExIOMXT (Estonia) daily 03.01.2000-01.06.2006 Tallin StExICrobex (Croatia) daily 18.03.1997-01.06.2006 Croatia StExIPX50 (Czech Rep.) monthly 01.2001-05.2006 Prague StExIMBI (Macedonia) daily 08.01.2002-01.06.2006 Maced. StExISofix (Bulgaria) daily 23.10.2000-31.05.2006 Sofia StExIOMXV (Lithuania) daily 03.01.2000-01.06.2006 Vilnius StExIOMXR (Latvia) daily 03.01.2000-01.06.2006 Riga StExIRURUSD (Russia) daily 01.09.1995-28.12.2005 ExR RURUSDSLTUSD (Slovenia) daily 02.09.2003-14.11.2006 ExR SLTUSDSKKUSD (Slovakia) daily 14.11.1996-14.11.2006 ExR SKKUSDROLUSD (Romania) daily 14.11.1996-14.11.2006 ExR ROLUSDUAHUSD (Ukraine) monthly 01.2001-06.2006 ExR UAHUSDCZKUSD (Czech R.) daily 08.08.1996-14.11.2006 ExR CZKUSDBLGUSD (Bulgaria) daily 14.11.1996-17.08.2006 ExR BLGUSDHUFUSD (Hungary) daily 16.11.1995-14.11.2006 ExR HUFUSDPLNUSD (Poland) daily 16.11.1995-14.11.2006 ExR PLNUSD
Table1 :
Nineteen series taken from Eastern European markets
J.-F. Emmenegger Mathematical Finance
Analysis of Eastern European Indices (3)
FIG.: Eastern European Market Indices
J.-F. Emmenegger Mathematical Finance
Analysis of Eastern European Indices (4)
FIG.: Index Returns of Eastern European Market Indices
J.-F. Emmenegger Mathematical Finance
GARCH Modeling the residuals(5)
Bollerslev (1986) Let ψt−1 design the information set knownat the moment t. The GARCH(p, q) model is
εt |ψt−1 ∼ N (0, ht) (17)
ht = α0 +p∑
i=1βiht−i +
q∑i=1
αje2t−i (18)
where p ≥ 0, q > 0 are parameters, α0 > 0, αi ≥ 0, i = 1, ..., q,βi ≥ 0, i = 1, ..., p are coefficients, and ht is the conditionalvariance, variable over time.
J.-F. Emmenegger Mathematical Finance
Best fitted models (6)
Series Levels Residuals Distribution of residuals(1) (2) (3) (4)SSEI AR(1) GARCH(2,1) alpha-stableBET εt - alpha-stableOMXT AR(1) - alpha-stableCrobex AR(2) GARCH(1,1) alpha-stablePX50 εt - normalMBI AR(1) GARCH(1,1) alpha-stableSofix ARMA(1,1) GARCH(2,1) alpha-stableOMXV AR(2) GARCH(1,1) alpha-stableOMXR ARMA(1,1) GARCH(1,1) alpha-stableRURUSD ARMA(15,5) GARCH(1,1) alpha-stableUAHUSD εt - alpha-stable
Validated models for levels and residuals - the distributionsof the residuals
(1− ϕ1(σ1)
L − ϕ2(σ2)
L2)rt = (1 + θ1(δ1)
L)εt (19)
εt |ψt−1 ∼ N (0, ht),ht = 0.28
(0.01)ht−1 + 0.75
(0.01)e2
t−1(20)
J.-F. Emmenegger Mathematical Finance
Estimation of the parameters (7)The quantile method of McCulloch (1986) is applied
Series Estimation α β σ µ(1) for residuals (3) (4) (5) (6)RUR/USD (Russia) GARCH 1.02 0.10 0.0006 0.0020SSEI (Slovenia) GARCH 1.28 0 0.0046 0.0003BET (Roumania) AR(0) 1.52 0.11 0.0070 0.0004OMXT (Estonia) ARMA 1.12 0.11 0.0029 0.0015UAH/USD (Ukraine) AR(0) 0.69 -0.21 0.0002 0.0012Crobex (Croatia) GARCH 1.33 0.07 0.0071 0.0008MBI (Macedonia) GARCH 0.60 0.89 0.0020 0.0049Sofix (Bulgaria) GARCH 1.20 0.12 0.0053 0.0003OMXV (Lithuania) GARCH 1.12 0.12 0.0023 0.0012OMXR (Latvia) GARCH 1.01 0.19 0.0027 0.0229Table 13 : Estimation of the parameters of alpha-stable distributions.
This method, calculates stepwise the parameters using the5-th, 25-th, 50-th, 75-th and the 95-th quantiles of the seriesof the residuals. Suggestion : A "goodness of fit" throughthe pseudo-metric, see Bauer (1968) :
I (fX ,e(x), fX ,th(x)) = 12
∫ ∞−∞|fX ,e(x)− fX ,th(x)|dx (21)
J.-F. Emmenegger Mathematical Finance
Comparison of distributions (8)
Series In Iα Preferred Series In Iα Preferreddistr-n distr-n
RURUSD 0.663665 0.04960945 alpha SLTUSD 0.21114 0.092536 alphaSSEI 0.2562825 0.04564535 alpha SKKUSD 0.250971 0.0966085 alphaBET 0.3846425 0.0606755 alpha ROLUSD 0.38835 0.3042545 alpha
OMXT 0.284278 0.126986 alpha CZKUSD 0.2415515 0.079608 alphaUAHUSD 0.53294 0.575065 normal BLGUSD 0.4513335 0.083314 alphaCrobex 0.239545 0.0555565 alpha HUFUSD 0.2482185 0.1170735 alpha
MBI 0.287078 0.67889 normal PLNUSD 0.253183 0.1013285 alphaSofix 0.329729 0.114359 alpha
OMXV 0.266521 0.0812595 alphaOMXR 0.387376 0.631795 normal
RSI 0.1637865 0.069851 alpha
Result : alpha distribution is more appropriate than anormal distribution in 83% of cases. The averageunexplained part of the residuals’ behavior is 32% fornormal distribution versus 19% for alpha-stabledistributions.
J.-F. Emmenegger Mathematical Finance
Data presentation, analysis and modeling (9)
Figure : Estimated probability density functions of the modeled series, comparative graph.
Discrete Fourier Transforms, with estimated parameters α,β, µ, σ
J.-F. Emmenegger Mathematical Finance
Risk management and comparison (10)
The Conditional VaR looks at how severe the average(catastrophic) loss is, if VaR exceeded :
CVaRa100%(r) = E(l|l > VaR(1−a)100%(r)),where r is the return given over time horizon, andl = −r is the loss.The Sharpe ratio (Sharpe, 1966) is a measure ofrisk-adjusted performance of an investment asset, or atrading strategy. It is defined as :
S = E [R − Rf ]σ
, (22)
[3.] The Rachev ratio with parameters α and β isdefined as :
ρ(r) =ETLα100%(rf−r)
CVaRβ100%(r − rf ):= R − ratio(α, β)
.J.-F. Emmenegger Mathematical Finance
Risk Evaluation (11)
Series STARR (0.5) R (0.2, 0.8) Rsk greaternormal alpha normal alpha
BET -1.35607 -1.5852 -1.71493 -2.3734 normalBLG -1.59737 -1.1388 -2.48277 -0.911243 alpha
Crobex -1.46603 -1.46116 -2.05698 -1.8476 alphaCZK -1.7758 -1.7583 -3.22248 -2.72273 alphaHUF -1.83941 -1.65608 -3.36041 -2.80512 alphaMBI -0.237259 -1.52654 0.643838 -2.09988 normal
OMXR -1.61217 -1.42038 -2.49544 -1.71646 alphaOMXT -1.40284 -1.59113 -1.84915 -2.17722 normalOMXV -1.76725 -1.52436 -3.07403 -1.89109 alpha
PLN -1.80811 -1.61664 -3.32686 -2.5856 alphaPX50 -14.275 -1.54502 -5.9959 -2.2492 alphaROL -0.964778 -0.974218 -0.73317 -0.539879RSI -1.39841 -1.3251 -1.84442 -1.41552 alphaRUR -0.690653 -1.61416 -0.155695 -2.45248 normalSKK -1.73052 -1.61158 -2.90426 -2.51167 alphaSLT -1.81885 -1.53308 -3.15329 -2.07819 alpha
Sofix -1.46758 -1.5005 -2.04689 -1.95431SSEI -0.730546 -1.71868 -0.233861 -2.6967 normalUAH -1.33943 -0.924447 -3.15991 -0.648556 alpha
Result : risk under valuated because of the normaldistribution assumption in 63% on STARR ratio and in 74%on Rachev-ratio.
J.-F. Emmenegger Mathematical Finance
Conclusions
The Efficient Market Hypothesis (EMH) leads to theGaussian Hypothesis ! Why ?
The Paretian Hypothesis is better appropriated toempirical results in financial economics, leading to theflexible α-stable distributions
With Fourier Transform α-stable distributions aretreatableFor Eastern European Markets empirics show
1 the α-distribution is more appropriate than the normaldistribution in 83% of cases
2 STARR and R-ratios are appropriate as risk measuresunder α-distribution assumption
3 The assumption of normal distribution under evaluatesthe risk in 63-74% of the treated cases
J.-F. Emmenegger Mathematical Finance
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