The Discrete Fourier Transform as a Tool in Mathematical Finance

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,[email protected], [email protected]

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


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


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)


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


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)


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


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.



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)


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)


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)


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) !



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.



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.


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


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



FIG.: Eastern European Market Indices



FIG.: Index Returns of Eastern European Market Indices


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.


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)


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)


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.


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


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.


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


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The Discrete Fourier Transform as a Tool in Mathematical Finance