F Audo

C Finot

Extinction ratio enhancement by sinusoidal phase modulation and nonlinear spectral focusing

Keywords: Signal processing, spectral compression, nonlinear propagation

We propose a new approach to enhance the extinction ratio of a pulse impaired by a coherent continuous-wave background. Taking advantage of a sinusoidal phase modulation followed by a nonlinear spectral focusing and an optical bandpass filtering, we are able to significantly decrease the level of the unwanted background. The principle of the method is experimentally validated at telecommunication wavelengths.

Introduction

Optical pulses with durations of tens or hundreds of picoseconds are required for a wide range of applications. An efficient cavity-free solution to generate such pulses is to use a continuous wave laser that is externally modulated in the temporal domain by an optical modulator driven by a high-bandwidth electrical waveform generator. However, the extinction ratio of these modulators is finite and is usually limited to around twenty dBs. As a direct consequence, for pulse trains with a low duty cycle, the overall energy remaining in the residual coherent continuous background is not negligible and can be of the same order as the energy contained within the pulse. This may therefore seriously impair the practical use of the pulse train by creating unwanted interferences or by affecting the efficiency of subsequent amplification stages.

In this context, it becomes of high interest to develop all-optical approaches to remove this spurious background. Many solutions have been proposed in the field of optical communications using dedicated components such as saturable absorbers [1,2], or artificial absorbers taking advantage of the nonlinear effects occurring in optical fibers. Indeed, nonlinear optical loop mirrors [3,4], Mamyshev's regenerators [START_REF] Mamyshev | All-optical data regeneration based on self-phase modulation effect[END_REF][START_REF] Provost | [END_REF], spectral-compression based devices [7] or four-wave mixing processes [8] present a nonlinear relationship between the input and output powers, enabling the enhancement by several dB of the extinction ratio.

We propose here a new method based on the use of a sinusoidal phase modulation followed by nonlinear spectral focusing and optical bandpass filtering. After having described the principle of our approach, we experimentally validate its efficiency by demonstrating a reduction by more than 15 dB of the continuous background.

Principle of operation

In order to illustrate our approach, let us consider a chirp-free Gaussian pulse with a temporal profile G(t) lying on a continuous background with a level that is ER = 23 dB lower than the peak power P 0 of the pulse:

  00 ( ) ( ) t P G t ER  
. Our approach can be applied to other pulse waveforms as long as their temporal intensity profile in the central part can be approximated to a good extend by a parabolic fit [9]. We consider here parameters that are typical of the experimental validation that will be described in section 3 and 4. The initial pulses have a 120 ps full-width at half-maximum (FWHM) pulse duration and a repetition rate of 78.13 MHz, leading to a duty-cycle of 0.94%. The spectral intensity profile corresponding to

  0 ( ) ( ) G ER       is plotted on figure 1(b1)
where we can make out in the spectral domain that the Gaussian spectrum linked to the pulsed part is more than 21 dB below the level of the continuous wave. The initial signal is then externally phase modulated by a temporal sinusoidal waveform

(t) =  cos( mod •t+  0 )
with an angular frequency ω mod , an amplitude A and a phase offset  0 . The resulting spectrum is made of replicas of the initial spectrum spaced by ω mod [START_REF] Goodman | Introduction to Fourier Optics[END_REF]:

    mod 0 mod ( ) ( ) n n Jn                  (1)       mod mod mod ( ) ( ) n n J G n ER n                (2)
where J n is the Bessel function of the first kind of the n-th order. The amplitude of the phase modulation is chosen to be  = 2.405 so that the component centered at ω = 0 vanishes, i.e. J 0 (β) = 0. Therefore, as can be seen in panel b2 of figure 1, the central coherent line has disappeared and its energy is now spread into a set of equally spaced spectral lines. Note that other periodic phase modulation schemes may also lead to spectrum splitting into lateral bands [START_REF] Finot | [END_REF], however, they will not be suitable for the second stage of operation.

After the external phase modulation that induces a sinusoidal chirp in the temporal domain, the signal is propagated in a Kerr medium such as a highly nonlinear fiber that induces self-phase modulation (SPM). The resulting chirp is directly linked to the temporal gradient of the initial intensity profile 2 0 () t



and is therefore maximum for the pulsed part 2 (t) G and null for the coherent background. If usually SPM is associated with spectral expansion and the generation of new frequencies in the spectrum [START_REF] Stolen | [END_REF][START_REF] Agrawal | Nonlinear Fiber Optics, Fourth Edition[END_REF], this is not always the case when the input wave is already chirped with a chirp slope opposite to the SPM chirp. In this context, a spectral redistribution of the components occurs, leading to a spectral narrowing of the pulse [START_REF] Stolen | [END_REF][START_REF] Zohrabian | [END_REF]15]. Usually, the input chirp required to observe spectral compression is imparted by an anomalously dispersive element, but it can also be achieved through external temporal phase sinusoidal modulation, provided ϕ 0 = π. Given the temporal duration of the pulse under study, we can consider that the dispersive effects can be neglected [16] and assume a perfectly nonlinear propagation where the SPM impact is limited to an additional temporal phase term γ•

(t) G

•L, with L being the length of the fiber and γ its nonlinear coefficient. Inspired by our previous works [17,18], we have carefully chosen the  mod in order to observe an optimal compensation of the SPM-induced chirp by the sinusoidal phase modulation ( mod = 4.45 GHz). Therefore, for an input peak power of 1.15 W, we can make out in panel (a1) that the chirp is very significantly reduced over most of the pulse extend. This leads to an efficient narrowing, as can be seen in panels (b3) where nearly 95% of the energy of the pulsed signal has gone back around the central wavelength. As the SPM only affects the pulsed part of the signal, the coherent background remains unchanged and no energy from the residual wave is focused back to the center of the spectrum. Therefore, an optical bandpass filter (OBPF) centered at the central wavelength may easily remove the coherent background.

We can clearly observe on the panel b4 that the signature of the coherent wave has nearly completely disappeared. In the temporal domain, the level of the coherent background has been decreased by more than 20 dB (see inset a3). The filtered pulse is extremely close from the Fourier transform limit, indicating that the chirp has been nearly canceled. However, the temporal intensity profile is slightly broadened compared to the initial one. The sharp edge filter under use leads to rapidly decaying oscillations in the temporal profiles as can be noticed when plotted on a logarithmic scale.

Let us note here that self-phase modulation combined with optical bandpass filtering has already been involved in the past for optical regeneration in a device known as Mamyshev regenerator [START_REF] Mamyshev | All-optical data regeneration based on self-phase modulation effect[END_REF][START_REF] Provost | [END_REF]: the offset frequency filtering of the spectrally broadened spectrum can achieve an efficient suppression of the continuous wave. However, contrary to our proposed method, such a scheme imposes a wavelength conversion of the signal and requires higher peak power. Much more recently [7], spectral focusing followed by optical bandpass filtering has also been suggested to increase the signal-to-noise ratio and to reduce the amplitude jitter of a data pulse stream. However, in that work, the pulses obtained after processing have experienced significant changes of their temporal and spectral properties with for example a temporal stretching by a factor above 10.

Experimental setup

The all-fibered experimental setup is sketched in figure 2 and relies exclusively on commercially available components from the telecom industry. The initial continuous wave is delivered by an external cavity laser at 1550 nm and is intensity modulated by a Lithium-Niobate modulator with a bandwidth of 20 GHz. The electrical signal that drives the modulator is the result of the electrical filtering by a 5.4 GHz low-pass filter of an on-off keyed signal from a 10 GHz pattern generator. As a 128-bit sequence with 1 one followed by 127 zeros is chosen, a repetition rate of 78.13 MHz is obtained. A second modulator is driven by a 4.69 GHz electrical clock synchronized with the first electrical signal. It imprints a sinusoidal phase on the optical wave with the appropriate phase offset  0 =  and amplitude  = 2.405. The resulting signal is then amplified by an erbium doped fiber amplifier and sent into a 500 m highly nonlinear fiber with  = 10 W -1 /km -1 . The linear attenuation of the fiber is 0.7 dB/km and the dispersion of the fiber is normal (D = -1 ps/km/nm) in order to prevent from any degradation that may be linked to modulation instability. A narrow bandwidth OBPF with a close to rectangular shape is finally used to remove the unwanted sidebands from the processed signal.

The optical signal is characterized at different stages both in the temporal and spectral domains by a 40 GHz electrical sampling oscilloscope and a high resolution optical spectrum analyzer (100 MHz resolution, APEX 2443B). Let us remark that the order of the external phase-modulation and the SPM stages can be modified as long as dispersion does not play any significant role in the propagation.

Experimental results

Temporal and spectral properties of the pulse at different stages of the setup are summarized in figure 3. The initial pulse obtained after optimization of the working parameters of the intensity modulator (especially the bias voltage in order to reduce as much as possible the background) has a FWHM pulse duration of 120 ps, leading to a duty cycle of 0.94%. Let us note that the noise performance of our temporal detection does not allow us to determine the exact level of the background. Given the large continuous component visible in the initial spectrum (panel b1), the energy contained in the coherent background is far from being negligible. From the spectral measurements, we estimate than 35% of the energy is contained within the central coherent component, leading to an extinction ratio of 23 dB, which is fully consistent with the technical datasheet of the commercial intensity modulator. -4) is for the input spectrum. The circles used in panel 4a refers to the intensity profile that can be inferred from the optical spectrum in the case of a pulse in the Fourier transform limit.

The power driving the phase modulator is adjusted in order to be as close as possible from the point where the central continuous harmonics vanishes. For this amplitude of phase modulation, the energy is transferred to higher-order sidebands as shown in panel b2. The nonlinear propagation within the hnlf enables the spectral focusing of the energy associated to the pulse. The injected optical power in the hnlf is optimized to observe the optimum spectral narrowing of the pulsed part. We observe in panel b3 that experimentally the spectral compression stage does not affect the lateral coherent sidebands created by the phase modulation. The temporal intensity profiles are not affected neither by the phase modulation nor by the spectral compression.

The final stage is to take advantage of a narrow bandwidth filter with sharp edges to isolate the central part of the spectrum and to eliminate the sidebands associated to the continuous background. The resulting spectral profile is given in panel b4 that highlights that the signatures of the continuous background have been dramatically reduced. From the output spectrum, we estimate that the energy in the coherent background now represents only 0.5% of the total energy of the signal, leading to an enhancement of 20 dB of the extinction ratio. Let us however note that as our setup involves an amplification stage in order to reach the adequate peak power, some amplified spontaneous emission may slightly degrade the intrapulse OSNR. The use of an optical filter with sharp edges limits this unwanted ASE. The temporal intensity profile does not show any deleterious degradation linked to the filtering process. On the contrary, a small ripple visible in the trailing edge of the initial pulse has been removed in the process. We can finally compare the recorded intensity profile with the one expected from the optical spectrum when assuming a Fourier transform profile: the temporal durations and shapes of the temporal profiles are rather similar, indicating that the final pulse obtained experimental is reasonably close from the Fourier-transform limit.

Conclusion

We have presented a new, simple approach to enhance the extinction ratio of a pulse train affected by a coherent residual background. Combining a sinusoidal phase modulation with nonlinear spectral focusing and spectral filtering, we have theoretically and experimentally demonstrated the possibility to enhance by more than 10 dB the extinction ratio of the signal. With phase modulators sustaining operation up to 40 GHz and low-dispersion highly nonlinear fibers, our approach can be adapted to pulse durations as short as 15 picoseconds. The scheme can also be combined during the nonlinear propagation with distributed amplification [18]. Given the low level of SPM that is required, the process is suitable for onchip operation [19][20][21]. We have experimentally demonstrated the improvement of signal degraded by the finite modulation depth of a Lithium-Niobate Modulator. Our approach could also work with others cavity-free pulse generation schemes method such as the one based on the parametric amplification of a frequency shifted continuous wave [22]. Given the space-time duality [23], we may also extend our approach to the cleaning of a residual coherent background of a spatial beam.

Figure 1 -

 1 Figure 1 -Evolution of the temporal and spectral properties of the signal (subplots a and b, respectively). The initial temporal and spectral intensity profiles are provided in panels a2 (blue line) and b1. Panel b2 shows the spectrum obtained after phase modulation whereas the blue curve of panel a1 stands for the chirp induced by phase modulation in the temporal domain. The chirp induced by SPM of the Gaussian pulse is plotted with red curve in panel a1 and the resulting chirp of the signal after nonlinear propagation is plotted in yellow curve. Plot b3 shows the resulting spectral intensity profile (blue line) after spectral focusing as well as the filter under use (red curve). Panel b4 represents the spectral profile after filtering. The corresponding intensity profile is plotted in yellow in panel a2 and is compared with the profile assuming a Fourier-limited pulse (circle). Panel a3 shows the input and filtered intensity profiles on a logarithmic scale (red and blue curves respectively).

Figure 2 -

 2 Figure 2 -Experimental setup. IM: Intensity Modulator ; PM: Phase Modulator ; EDFA: Erbium Doped Fiber Amplifier ; HNLF: Highly Nonlinear Fiber ; OSA: Optical Spectrum Analyzer ; ESO: Electrical Sampling Oscilloscope ; OBPF: Optical Bandpass Filter.

Figure 3 -

 3 Figure 3 -Temporal and spectral intensity profiles of the pulse train (panels a and b respectively). Spectral intensity profiles obtained directly after intensity modulation, after phase modulation, after nonlinear propagation and after spectral filtering are plotted with solid lines in panels b1, b2, b3 and b4, respectively. The dotted line in panels b(2-4) is for the input spectrum. The circles used in panel 4a refers to the intensity profile that can be inferred from the optical spectrum in the case of a pulse in the Fourier transform limit.

Acknowledgements

The article has benefited from the PICASSO experimental platform of the University of Burgundy. We thank Sonia Boscolo for unvaluable discussions on nonlinear spectral focusing. The authors would like to acknowledge support by the Région Bourgogne (Pari Photcom) and the Agence Nationale de la Recherche (Labex Action ANR-11-LABX-01-01). This work was supported by the European Union within the framework of the operational Program FEDER-FSE Bourgogne 2014-2020 and by the Institut Universitaire de France.


Kyritsis Konstantinos

Sotiropoulos Ioannis

Gogos Christos

Efstratios Kypriotelis

NOTE ON THE EFFECT OF THE 11 YEARS GLOBAL CLIMATE CYCLE ON THE PRICES OF THE CAPITAL MARKETS

Keywords: Stock Exchanges, volatility index, 11-years sunspot cycle, spectral analysis. Jel: C32, C4, C53, E32, E37, E41

) till 2011, a next major maximum of the prices during 2010-2011 and a fall of the prices for the period 2011-2017. We discuss sequences of causal explanations based on the balance of demand and supply and comment on the general value of such an omniscience wisdom that involves at least five different sciences.

Introduction

In this short note we shall try to describe the effect of the 11 years sunspot and meteorological global climate cycle, on the growth and decay of the prices in the stock Exchanges. We shall also try to give a sketchy sequence of causalities based on the balance of demand and supply that may help to understand this effect. We are not of course the first to have discovered this effect. In the contrary it is an old discovery, there is extensive (although not much known) literature, and the last 20 years, it has been considered to be more and more important. Nevertheless the first of the authors re-discovered this effect during 2000-2001 while performing simulation and statistical analysis of time series of stock exchanges. The more we studied it, the more we overcome our hesitations and doubts, and the more we are scientifically convinced for its validity and reality. We must mention that the study of cycles in time series, even when they are weather data time series, is a difficult, and tricky endeavor, and only the last decades, due to the advance of computers and massive data analysis, it has started to become clear how to detect and analyze them. The idea of business cycles is of course old (e.g. see Henry, Ludwell, Moore in references) but it was never handled in statically, sufficient robust way, by the economists. The same, surprisingly, was also the situation with the idea of weather cycles, in academic and practical applications, research. (see Burroughs Williams James in references). We shall also discuss the value of such a knowledge. If we are in search for a common systematic behavior of the capital markets, other than the assumed constant long run trend (drift), then an eternal of astronomic source cycle, of nature seems the only first good solution. If we are in search for a systematic reliable behavior that is not incidental to specific historic contingencies of the evolution of economies , and thus non-repeatable, or it is not a short-term shockwave of non-repeatable historic situations, the resorting to the effect of eternal "pace" originating in nature (e.g. meteorology) seems to us to be the first best solution. Of course the seasonal cycles (winter-summer ) is a traditional concept of how nature may have an effect on the prices. But searching for other not so obvious cycles like the 11-years cycle, that can be also very strong and significant, seems to be an intriguing matter. In addition forecasting or risk management of investments, that is based on such eternal cycles, is entirely more reliable and scientifically clear, compared to forecasting based on news from newspapers, that either are already late , or of unknown eventual effect on the prices of the capital markets. In involving sources of causalities in macroeconomics, that go beyond classical macroeconomics, to the areas of ecology and meteorology, widens our concept of "world" in economic causalities. It creates a significant "omniscience wisdom" that may be of benefit to all, and also makes us more conscious of the overall functions of life and society. This eventually increases the awareness of ourselves and the determination we have to our fate, and the success of our goals. Each time we take advantage and go parallel to the flow of time and power of evolution, in larger time intervals, we increase our ability of success in our goals, and we also increase the awareness of our consciousness. The psychologist reassure us that increasing awareness in our activities unavoidably increases our self-esteem and success as individuals and collectively. Including nature in economics is probably an important ingredient of the philosophy contained in the newly developing trend of "green economics" that claims to transcend classical economics. The effects are also invertible. If we know the details of the sequence of how nature influences the changes in the prices of business, then we are also more aware of how business in their turn have an effect on nature. As an important development in this direction is the issuing and trading of weather financial derivatives , in the Stock Exchanges, that are based on meteorological variables, like average temperature, humidity etc of various town in USA, and Europe. The main indented value of such financial derivatives is the hedging in the prices of commodities of energy consumption (like electricity, oil, natural gas etc) to eliminate waves of prices based on seasonality, but also to hedge for the systematic increase of the average temperature of the planet, and for the extreme weather phenomena. Finally there is, of course, the restricted financial benefit in investments for individuals, enterprises, or domestic national economies, when forecasting based on existing eternal cycles of nature. Budgeting becomes more successful. When the individuals are investing in capital markets, they are unavoidably based on faith, and even the metaphysical element is involved. Such a knowledge helps them to overcome fears, avoid bad decisions, based on public panic, control more their emotions , keep an optimistic metaphysical context, with positive hopes, helps them to avoid the worn-out of their soul, while at the same time it contributes in taking more successful financial risk management decisions.

The 11 years sunspot and the 11-years global climate meteorological cycle

In this paragraph we present same basic and useful to have in mind (while reading this paper) facts of the 11 years cycle. We have to enter the jurisdictions of three sciences : astronomy, meteorology, and ecology.

Figure 1

From Astronomy: There much material in the internet and some of the interesting sites and panels are mentioned in the references. The north and south magnetic poles of the sun, interchange every 11 years. After 22 years the magnetic north pole has returned back to its initial position. This phenomenon is parallel with a cycle in the number of appearing sunspots on the surface of the sun. The sun spots are observable as darker spots on the surface, but essentially they are a known effect of the "meteorology" of the solar plasma, as most probably are anticyclones of the solar plasma, in other words at this spot the plasma is rotating fast (like a vast tornado) generating strong magnetic field, and with a inward to the center of the sun direction. This blocks the radiation from inner layers to the surface, giving the appearance of the darker spot. In the chart in this paragraph we see the cyclic behavior of the number of appearing sunspots when plotted in time. We must mention that although an astronomic cycle the period of 11 years, is statistically only constant with variations that maybe longer than one year! We see in the figures 1, and 2 that the last peak of the sunspots activity was during 2000, and the next peak is estimated during 2011 or starting 2012. The predictions mentions a delay of the next peak towards the end of 2011 or during 2012. In addition it is predicted that the intensity of this peak shall be much higher by a factor 30%-50% .

Figure 2

From meteorology: A very good book on weather cycles is that of Burroughs (see references) Satellite measurements since 1978 confirm that the total output of the solar energy varies in cyclic way every 11-years in exact proportionality with the number of appearing sunspots. Also measurements of the earth's magnetic field give that the 11-years sun's cycle produces an 20-22-years earthly magnetic field cycle (Hale cycle). Thus through radiation, and through the solar wind that reaches the earth, (and which is responsible for the polar aurora phenomenon) the meteorological phenomena are influenced by the 11-years sun-cycle, to produce the 11-years cycle in the global climate. This cycle includes cyclic behavior of successive maxima-minima of earthly surface temperature, level of rainfalls, drought severities , electromagnetic storms, frequency of appearing tornados etc. A seemingly related famous meteorological cycle of the global climate, is the QBO (quasi-biennial oscillation) that seems to have period of 2.75 years (11/2=5.5, 5.5/2=2.75) . The QBO appears as a cycle of the speed of the stratospheric wind. In this short paper we presented only one major cycle that of the 11 years of the global climate. But we have found that there are at least 5 more cycles with their harmonics, that range from the 11 years to 5 minutes. The presentation and analysis of them cannot be the content of the present little paper. We hope to enlarge on them in other papers or in a book dedicated to them. All of the above cycles have also been discovered in meteorology. Meteorology has also discovered and studied longer cycles that range from 300 years to thousands of years that cannot be related with the modern form of capital markets.

From Ecology: Finally the sequence of causalities reaches the growth of life on earth. A most impressive example are the tree-rings. If we cut vertically a tree, we observe in its main body, rings that are the growth of the thickness of the body of the tree every year. These are the tree-rings. It is impressive that every 11-years the tree-rings are thicker. This means a more intense growth of the trees (dendroclimatology) This seems to be due to increased rainfalls, increased radiation, surface temperature extremes etc. Similar cycles appear in varves (layer lines) of clay, corals , ice cores, etc. The same 11-years cyclic behavior is observed with the water level of the lakes. It is obvious that the growth of grains in agriculture is more intense and follows the same 11-years cyclic pattern.

It has been also measured an impressive coincidence of the 11-years cycle with the number of lynx furs purchased by Hudson's Bay Company in the McKenzie river region from 1821 to 1910. This is seen in the figure 3, This reflects the periodic fluctuations of the species that traditionally is described by the Voltera's Prey-predator equations that do create oscillations. Nevertheless the real measurements prove that the coefficients of the Voltera equations are such that the actual period coincides with the 11-years cycle of the global climate. We see the chart below. (Data from the Global Population Dynamics database, University College, London, England) Figure 3.

How are the cycles detected in the time series?

In searching for cycles in time series it is of key importance to know technical ways to detect them. As with statistics, so with time series too, it is very easy to be misled when analyzing with standard ways the data. For example it is very easy to miss obvious cycles, or to think you have found cycles when in fact it is only a peculiarity of a particular piece of the sample. The present state of the art in academic research is such that on the same sample data many different and often contradicting models may fit quite well. Thus, the successful research, requires not only standard fit of a model, but at first exhaustive familiarization with the data . We have to observe them many hours and make many simple calculations with a calculator, in relation with the actual situation from which they emerge, before we start analyzing them in a more sophisticated way with many different techniques, in many different areas of the sample etc. After the wide success of the theory of the random walk and the efficiency of the markets, most of the research for cycles or cycles-based systematic correlations, became the least publicly enforceable category of ideas. We must mention also that different sciences are oriented to different techniques for cycles detection in time series. For example electric engineering signal experts use FIR and IRR filters, and spectral analysis, of which the classical economist are not very familiar. Experts in seismology, use much different techniques. Physicists may, still use other techniques, like operator theory of quantum oscillators, that engineers or economists, or meteorologist may know very little. We must mention also that the Box-Jenkins parsimony of low p, q order e.g. ARMA(p,q) time series, would entirely make invisible long run cycles, that involve higher order, longer term memory. To detect cycles in time series we may assume either stationary or non-stationary time series. But if the time series is assumed non-stationary, then the classical spectral analysis cannot be claimed to apply, as almost all of its theorems are proved for stationary time series. It is an assumption after all significant for the multitude of paths, rather than a single path. The fact that most time series models are fitted on a single path, explains why so many different models, fit to the same data. After all a stochastic process is defined on a sample of paths. Any oneelement sample would be vague enough to support many different assumptions. A more reliable method is to produce, in a standard way many overlapping paths, of the same time interval (from a single long path or even better e.g. for indexes from all components securities of the index) , and then let the data decide clearly if it is stationary or not process. In other words it is best to use the non-parametric data-generated process or time series. An other phenomenon that, any researcher that has studied extensively, time series, will come to it, and seems to escape the dominating mentality of academic researchers of time series is the next: A spectral analysis based on the finest bins of the data, would not give the same cyclic behaviors, with analysis based on many different time scales, with many different larger bins. In other words the element of time scale and time bins, almost defines a different world of phenomena in the data, not derivable by a flat spectral analysis on the finest (smallest) time steps (bins). Similarly the relative size of the sample and the suspected major period of cyclic behavior, is very important factor in extracting the necessary information. We find that the methods that meteorologist use to detect cycles in time series data, are the most natural, simple, and the ways that they fail to detect existing cycles have been quite well studied. We mention them briefly:

a The equations is so chosen that the least squares fit of it is very easily done, e.g. in Microsoft excel with the function Linest, and the coefficient a is estimated as slope of a least squares line. Such an equation, if the constant a, turns out to be negative, has complex, characteristic root, and thus models an harmonic cycle. If a is non-negative no cycle exists. If a cyclic frequency is found then it is the best frequency, of period T and afterwards we fit a classical Fourier series trigonometric term, on this period (defined by a) to find the best fit amplitude and phase.

When analyzing time series for cyclic behavior it would give more vivid results if we "detrend" the original time series. In this note we adopt a rather classical method: We apply the analysis not on the original series, but on the volatility time series of the original (volatility=standard deviation of the continuous time, one-step percentage rate of change). Even if the original time series was an "almost" (non-stationary) random walk, the detrended in this way would be an (stationary) "almost" white noise.

In this short note we shall use only the classical spectral analysis technique, to prove the existence of the 11-years global climate cycle in the prices of the Stock Exchanges.

The 11 years global climate cycle in the stock Exchange markets.

To prove the effect of the 11-years global climate cycle on the stock exchange markets we select the most highly traded index of S&P500 of the CME (Chicago USA). We present a chart of its data from the period 1985-2006 in figure 4 . We put a vertical line in this figure at the year of most recent solar sunspots peak. In the figure 5 we present the options implied volatility index VIX of options on S&P500, of the CBOE exchange. We put a vertical line in this figure at the year of most recent solar sunspots peak. We also plot a least squares fit of a 5-order polynomial that shows its slow periodic motion. This index is a kind of "detrending" or "pre-whitening" of the index S&P500. We prefer nevertheless to calculate directly a "de-trending" or "pre-whitening" of the index S&P500 by taking its volatility. In particular we take its 10 days volatility adjusted for year scale. In other words we take the x n =Squarer-root(252)*Stanrad-deviation-for-10days-of-Log(P n /P n-1 ) Where P n is the prices of the initial S&P500 index . The volatility is thus the standard deviation of the continuous time daily rate of change. We present this time series in the figure 6. We put a vertical line in this figure at the year of most recent solar sunspots peak. Then we apply spectral analysis for this volatility time series, that the periodogram by frequency is shown in the figure 7. The spectral analysis is calculated by the software SPSS. We put a vertical line at the frequency of the 11-years. We see also clearly the frequencies of harmonics of this cycle especially of the half 5.5 years cycle.

Sequences of causalities , demand-supply and cosmic habitforce.

In this paragraph we shall discuss the causalities behind the 11-years cyclic behavior of the capital markets. We have shown this behavior only for large stock Exchanges, like that of the CME, CBOE, ECBOT (futures on the index S&P500) at Chicago, in USA. But we have discovered the same cycle in many other large exchanges, including the vast Interbank Market of the cross-currencies exchanges rates , that is operated by the banks. The techniques and the volume of the results could not be appropriate for this short note.

Let us analyze, shortly, how the seasonality in the prices e.g. of grains like Soya, are explained by the balance of demand and supply in the classical seasonal cycle. The Soya is not collected all the weeks of the year, therefore the supply of this grain is concentrated in a single season. The supply follows a cyclic annual pattern in time as time series. Assuming an average constant demand during the year, the balance of demand-supply (on higher supply on the same demand the prices decrease) would result in a time-series of prices that is annually cyclic in time. We shall not complicate the reader with various types of equations of the balance of demand-supply (there is an extensive literature on it), but we shall be constant that almost all of the systems of equations of demand and supply would give the above mentioned result. The previous is a well known and accepted fact. It is easy to extend it for an 11-years cyclic pattern of prices, if we know that the supply of Soya follows in addition, an 11-years cyclic pattern. This is plausible from what has been mentioned in the paragraph 2 on the 11-years cycle in ecology. Similar arguments can be produced for the energy commodities like electricity, crude oil, heating oil, gasoline, natural gas etc. This time the supply may be considered in the average constant, while the demand by cyclic in time with period 11-years. In fact all the volume of the activities follows this cyclic patterns of 11-years. In the paragraph 3 we proved it for the 10-days volatility of CME index S&P500 . It seems that it holds that the time series of volumes of transactions is very strongly directly correlated to the volatility time series of the transaction prices (see e.g. Chen, Gong-Meng, Michael Firth, Oliver M. Rui, 2001 in the references) Therefore we may safely conclude that the 11-years cycles exists also in the volume of transactions of the stock Exchanges. As the volume of transactions is directly related to the cash, and exchanges of currencies , we deduce that such a cycle is expected to be found in the currencies commodities too of the Interbank Market. In fact this 11-years cycle is not confined to the prices of the capital markets. It exists in all business activities, including real estate. It is only that in capital markets the data are easier available, and more sensitive in time.

Still we believe that the various paths of demand-supply balance in various industries, and through e.g. input-output equations among them, is not the whole system of the causalities that could explain this phenomenon. Maybe it is only one side of the coin. Probably there are deeper lines of causalities that go parallel to he demand-supply causalities. Andrew Carnegies the well know major owner of the steel industry in the beginning of the 20 th century in USA, mentioned the term cosmic habitforce to discuss how the various human activities in business tune between them like precession motion, have their own momentum conservation and eventually are compiled to the human subconscious. The size of this paper would not permit us to enlarge on it.

The value of the knowledge of the effect of nature's cycles on the growth of the wealth of nations

It is difficult to underestimate the value of such a prognostication, as the above, that was sketchy only presented, in the present short paper. We prefer to talk about prognostication which is a rather philosophical and social concept that mere stochastic process forecasting. The reason is that, as it seems, in modern sciences what is missing is not techniques and analysis, but a strong merging with good values and goals. It is usually said that prognosticating is of two kinds objective (Ïf I do nothing such and such will happen) and subjective (if I do so and so, such and such will happen). At first we shall mention some of the values in such a prognostication that refers not in the forecasting itself but rather on the nature of involved concepts and methods. As a scientific method the value has the next basic characteristics: 1) It is not narrow statistically phenomenological (mere technical analysis or mere econometrics) 2) It is multi-science 3) It is synergistically wise 4) It is based on inside the planet and solar system celestial physics predictable action and flow of energy. 5) It is based on the functions of life on earth 6) It is intriguing for self-knowledge and knowledge of interrelations of phenomena in different cosmic scales (astronomical, ecological, social etc) 7) It is still mainly unexplored in its full details from the scientific point of view 8) It is essentially optimistic as far as scientist's ability to handle the random, to forecast and realize that "pure randomness" and "chaos" may be in the scientist minds rather than in the hidden reality of the observed phenomena. 9) It gives to scientists, who love their quantitative work, the opportunity to prove that knowledge may bring directly good luck in wealth creation, when there is diligence for good risk management and the character in practice is strong, following not only the laws of the intellect, but also the laws of the emotions. In this way they may avoid the classical accusation of "paralysis by analysis" from the world of business to the world of academics 10) It gives hopes, for avoiding , public panic, economic disasters, suicides of individuals etc

During 20 th century lived Buckminster Fuller, (see references) a rare and fine and global thinker that had omniscience wisdom, besides being a mellontologist (futurist). He was awarded by 42 honourable titles of various universities during the 20 th century. We would like to quote him as he expresses once more the value of prognostication which is neither bad news neither good news but it can become good news to those that would respond appropriately and bad news to those that would ignore it. "…I am firmly convinced that I can see clearly a number of coming events, and I am therefore vitally, eager that people should not be hurt by the coming of these events, particularly when I can see ways in which it would be possible not only for them to avoid hurt, but even to prosper by and enjoy what now seems to me to be inevitable." According to him wealth starts from physical energy, which is so abundant as coming from outside the planet, that humanity can consider to consists from a few billions of "billionaires" in this energy and power. To quote him again: "…Wealth is, then, the already organized human capability and know-how to employ the fixed inanimate, planetary assets and omni-cosmically operative and only celestially emanating, natural energy income, in such a manner as to predictably cope with so many forward days of so many human lives by providing for their (1) protection, (2) comfort, and (3) nurturing, and for (4) the accommodation of the ongoing development by humans of their as-yet-untapped store of intellectual and aesthetic faculties, while (5) continually eliminate restraints and (6) increasing the range and depth of their information-accumulating experience."

In the next, lets run through the scales from the individual to a government and a group of nations to state some of the benefits of the prognostication of the 11 years cycle.

1) The individuals in their investments and wealth creation can use such a prognostication so as to know what sequence of 5-6 years shall have almost year after year gains and which 5-6 years shall have year after year losses. Some of the successful millionaires have revealed in their books, one of their "secret rules of success in the capital markets". The rule is: "Buy whenever you have the funds and keep the assets at least for 10 years. Then during the next 15 years, find a year that the capital gains are satisfactory and sell". Obviously such a rule holds good because of the 11-years cycle. Furthermore this rule does not require that you know the phase of the 11-years cycle, in other words which year is the peak and which year the minimum. If they would also know the concept of the 11-years cycle and its source in nature, they could improve the rule, and reduce its duration from 25 years to just at most 11.

2) The brokers are also much interested in such a prognostication, as the volumes of their transactions follow also this cycle. We mentioned that the volatility of the index is directly correlated with the volumes of the transactions. Therefore the brokerage companies can budget reliably for their revenues and thus for the size of their human capital.

3) As the enterprises assets contain investments in the capital markets, or real estate, such a forecasting, is valuable to anticipate their revenues. 4) Insurance companies, may very well adjust the hazard rates of many insurable events. The hazard rates (not only of disasters originating from extreme weather phenomena, but also of car accidents, and human diseases) wave following the 11years cycle. Thus insurance premiums can be in other sequence of years, cheaper, and in other sequence of years, more expensive, in all cases better handled by both insurance companies and insured customers. 5) Also for governments budget such a prognostication is valuable. The government can know when to expect capital gains or losses and also when to expect public satisfaction or discontent (usually their personal gains or losses in the capital markets are related faulty by the public to bad or good financial policies by the government) 6) Finally whole unions of nations can benefit by such a prognostication in the same way that a domestic government can. Industries production and operations, and not only capital markets are influenced by the 11-years cycle. Budgeting that takes in consideration this cycle is obviously a more realistic and successful budget.

If more investors are aware of this cycle, in their investment decisions, will have as effect by the supply-demand balance to smooth-out and reduce the severity of the wave. This is exactly what the Buy-and-Holders would wish for. But to realize and make best use of such a prognostication in society and unions of nations, we must adopt the philosophical and practical attitude of win-win in the deals and interrelations. The creation of new wealth by new technology and procedures is unavoidably a wealth of all society not only of parts of it. This excludes the concept and belief of "zero-sum game" in the capital markets. Only the win-win attitude and principle is appropriate and true in the global economy. This win-win attitude goes parallel with a mentality of abundance of resources and technology, rather than a scarcity mentality of resources and technology in business and investments.

Figure 4 Figure 5 0

 45 Figure 4Figure5

  . Sherman's statistic: It is simple statistical test based on a chart, that decides of randomly spaced, in time, events, are uniformly random, or are clustered in concentrations, or have a cyclic periodicity. b. Spectral Analysis: (see Lambert H. Koopmans in references) A classical tool based on Fourier transformation, and available in many software packages. Its limitations are quite well known. We may add here new relevant techniques like the maximum entropy spectral analysis (mesa), that tries make analysis based on assumptions for the available data , that are maximally non-committal with regard to the unavailable data. Then the singular spectrum analysis that is detecting best anharmonic oscillations to the basic frequency (resorting to classical techniques of orthogonal functions and principal components). c. ARMA, SARIMA, FIR, IRR filters etc: The classical fit techniques of linear recursive models. If the characteristic roots are complex and not real, and of norm one, we detect cyclic effects. Economists usually avoid complex roots. d. Wavelet analysis: Originally used to study the way that the spectrum changes, when we move to successive intervals in the data. (see Parcival D.B. et al in the references) It reproduces the time series as superposition not of constant repeating cycles but as superposition of "shock waves" which, at a frequency start with low amplitude go to a maximum and then diminish again. Much like the sound of "vowels" in speaking. It is best for time series with many abrupt peaks, and the orthogonal base of "wavelets" can be after all, any conceivable pattern. e. Best frequency finder: This is a technique devised by us, that we have used extensively in our research. It is a technique that does not seem to be known by the majority of the researchers, or at least used in their research. . It would deserve a separate paper to describe it. Its goal is not to find all frequencies with their amplitudes, and phases, but to find the one and most strong frequency if it exist at all. It consists in fitting with ordinary least squares method, a linear recursive equation of the type Δ(x n -x n-1 )=a x n . (See Goldberg Samuel in references).


Dr Costas Kyritsis

THE IMPACT OF THE CONVERGENCE OF THE GREEK ECONOMY TO EMI IN THE STOCKMARKET: BAYES, NESTED ESTIMATION OF THE STOCK TRENDS

Introduction

The time when an economy enters the first world economy is a very interesting time. It is even more interesting if the group of nations where it enters, in this case European Union, becomes gradually, with respect to some parameters, the strongest economy in the world. Although the Greek economy is by far not perfect or advanced, there is the firm decision to handle its indices, as much as possible, so as to qualify according to the standards of European Monetary Integration. These standards are set, for the Greek economy, mainly in the next profile: a) The Inflation rate less than 1.5% b) The deficit of the Government less than 0.9% of the Gross National Product c) The national debt less than 100% of the Gross National Product d) Growth rate of the Gross National Product at least 4.5%.

Macroeconomics factors influencing the prices in the Athens Stockmarket

There is no doubt that the previous standards of EMI make a profile of a mature economy and also no doubt that a young state like the Greek (less than 2 hundred years old) has major difficulties in qualifying in the profile of EMI, before 2001 .It is worth trying nevertheless, even only for the benefit of eliminating the continuous currency devaluation of the national wealth through the exchange rates. Experience has showed that the basic magnitudes of Macroeconomics that have significant impact on the changes of prices of stocks in the Stockmarket are: a) The average rate of deposit in the banks, or the rate of change of the time-value of money.

b) The exchange rates c) Mass-media information about other economies and changes of prices in other Stockmarkets. The procedures with which the previous factors influence the changes of prices in Stockmarket is always through the aggregate demand and supply for each stock: 1) Surplus of demand to purchase stocks in the computer waiting lines creates growth of the price of the stock (Bull-market) 2) Surplus of supply to sell stocks in the computers waiting lines creates falling of the price of the stock (Bear-market) The exact equations of how stochastic demand and supply results in to the random variables of price and volume and their changes, is not an issue to cover in the present paper. It is not of intractable difficulty to formulate though.

We shall state, nevertheless, the basic equations of competition of demand and supply for each stock. The equations of two populations in competition have been a topic of systematic study. It may not be surprising that such equations have been studied and solved not in the science of Economics but in Ecology. They are a standard topic in an area initiated by Volterra and his equations for populations. Let us denote by x (t n ) and y(t n ) the average value, at time t n of the random variable of the volume of orders of the demand to buy and of the volume of orders of the supply to sell a stock. The next equations describe the interplay of demand and supply:

(

1) x(t n+1 )= x (t n )(a-b x (t n )-c y(t n )) (2) y(t n+1 )= y (t n )(e-f x (t n )-g y(t n ))
The symbols a, b, c, d, e, f. g are constants defining the competition. Such equations, formulated in continuous time and deterministic mode are the well known equations of competition in Ecology (see e.g. [Maynard S.J] p 59 formula 36).We notice that they are non-linear equations. They have been solved numerically, studied and applied in many situations of populations in competition .The populations involved here are of the investors who want to buy and those who want to sell .The equations describe the effect in demand and supply of the automatic negotiation algorithm in the computers waiting lines . These equations if formulated in continuous time they do not involve oscillations. But when formulated in discrete time and as stochastic processes or time series, they do involve (non-linear) oscillations which is the common experience for anyone that has spent some time in front of a monitor of a Stockmarket company. If we make use of the prey-predator or host-parasitoid , Volterra equations that different from the equations (1), (2) only at a plus sign instead of a minus sign at he coefficient f in (2), then we get larger scale oscillation. During 1997 there was a major impact on the price growth in the Athens Stockmarket of the size, at year base, close to 50% .It is supposed that it was created by the fall of the deposit rates of the banks (factor a) mentioned in this paragraph ). During 1998 there was an even larger impact on the price growth of a size close to 70%. It is supposed that it was created mainly by the currency devaluation in the exchange rates decided by the government in March 1998. As the latter case was the most dramatic, we shall try to analyze it with a new statistical method.

Bayes fractal-like nested estimation of time series

As it is known there is a topic in statistics called Bayes estimators. (See e.g. [MOOD A.-GRAYBAILL A.F.-BOES D.C.] pp 339-351). The main idea is that when we have a parameter in a distribution that we must estimate, we may assume as a metalevel that it is already a random variable with an a priori given distribution . For example if we are estimating a Gaussian (normal) random variable N(m,s) we may assume that we have a double variation and a second stochastic level and that the parameters m, s are already Gaussian (normal) random variables with means m m , m s and variances S m S s . It is not that we want to make the computations more complicated but that we need to fit a more flexible model to the real situation. For doubly stochastic time series see [TONG H.] pp 117-118. We shall describe a general method to refine autoregressive time series models, such that at each refinement, it appears higher order variability and higher Bayes order as discussed above. For the sake of clarity we shall apply it to the Black-Scholes lognormal model of the prices of stocks .The model is known in stochastic processes and stochastic differential equations as the geometric Brownian motion . (see [OKSENDAL B.] pp 59-61 ,198-199 and 223-225 Although much popularity is related to this model, it cannot describe but the «buyand-hold» situation in the Stockmarket . We may try to vary this model with the idea of Bayes so as to include reversal patterns and price motion with or without resistance. We supplement the idea of Bayes by corresponding to each new stochastic or Bayes level a finer grid of the argument .In this way different models appear to different scale regimes, but still something is repeated thus we follow also the basic idea of self-similarity introduced graphically by Mandelbrot with fractals and multi-fractals . Mandelbrot has applied his idea of self-similar fractals to the Stockmarkets, arguing that much of the oscillating effects of stock prices are not observed in the Black-Scholes model. There are many new results of qualitative dynamics of dynamic systems under the term «chaos». The ideas are not irrelevant but in order to apply them in a professional way to Stockmarkets we require them in stochastic differential equations or time series (see [TONG H.]) The idea of nested patterns of «tides» (trend of a year or more) ,«waves» (in seasonal horizon) and «ripples» (day or intra-day oscillations ) goes back to the theory of Dow and Elliot in the Technical Analysis of stocks (see [MURPHY J.J.] pp 24-35 ,371-414). [MURPHY J.J.] . It is also obvious the relevancy of the Elliot wave theory with Spectral Analysis and fast Fourier transformation in time series. The way to enhance the «buy-and-hold» model of Black-Scholes is as follows: 1) We define a nested system of grids in the time argument .For example starting with an horizon of a year we partition it to smaller seasonal horizons (e.g. 60 Stockmarket days). We may continue in this way to monthly, weekly and finally daily horizons . 2) For the first one year horizon we perform an ordinary estimation of the Black-Scholes model .It gives the buy-and-hold trend.

3) In the seasonal horizon we increase the Bayes stochastic order. For each season in the one year horizon we estimate a second Bayes order model. The four seasonal models are pasted automatically to a more flexible overall model than the Black-Scholes 4) We continue to increase the Bayes order by one for each finer horizon, of a month, a week or a day and we estimate a new model for each smaller horizon. The resulting time series fits pretty well to the real life surprises of the Stockmarket . The method resembles the splines in numerical analysis only that it is not performed on polynomials and the models are not deterministic but stochastic. A good question is how we increase the Bayes order. A simple method is to consider the constant coefficients of the initial model as varying linearly relative to time. This introduces for estimation new constant parameters .At each finer grid we assume the previous constant parameters as varying linearly and we estimate the new constant parameters.

In the next paragraph we shall perform the method at two only horizons of one year and a seasonal of 60 Stockmarket days .

An example: The impact of the currency devaluation in the spring of 1998.

As we mentioned in the previous paragraph the Black-Scholes model of the prices of stocks is the geometric Brownian motion in other words defined in continuous time by the stochastic differential equation:

(3) dx=rxdt+σxdz. Where x is the price of the stock and z is a Brownian motion.

In this example we implement the discrete time, non-homogeneous time-series version defined by the equation (4) x n+1 =(r+s e n )x n We make use of a close relative to it, which is the next time series in explicit form:

(5) x n =exp(rn+se n ) Where e n is a normal error or innovation. We do not insist on any stationarity assumption. We make the assumption that the «noise» or innovation term is additive in the exponent instead of multiplicative and of constant variance, that is, an homoskedasticity assumption that makes the variance of the residual, in the exponent, constant in time. This simplifies the estimation of the parameters of the time series The application of the original model of constant coefficients for an one year horizon is straightforward and is very well known. We proceed with the nested Bayes estimation that we described in the previous paragraph .We assume for the four seasonal (3-months) horizons of one year that the model has variable coefficients and that the coefficients vary linearly with respect to time. This introduces new constant coefficients a, b in (5) :

(r n = an+b) The exponent becomes now quadratic with respect to time. (6) x n =exp((an+b)n+se n ) More generally we estimate the equation (7) x n =exp((an+b)n+c+ se n ) We notice that the equation is almost the normal curve except of a linear term or sign reversal.

To estimate it we take the logarithm of the prices and apply polynomial regression. The exponent is in general an at most quadratic polynomial .If the coefficient of the quadratic term is negative, we have an instance of an almost Gaussian (normal) curve, which is interpreted as follows: 1) Increase of the prices with an asymptotic upper resistance, which becomes a reversal pattern (first part of the curve) 2) Decrease of the prices with an obvious asymptotic lower resistance at zero, thus practically without resistance (second part of the curve) If the coefficient of the quadratic term of the exponent is positive then the probable cases are: 3) Increase of the prices very fast (faster than the simple exponential growth) without upper resistance (second part of the curve) 4) Decrease of the prices with lower asymptotic resistance that becomes a reversal pattern (first part of the curve) Thus the qualitative dynamics of the stock at each time are described by the above four dynamic states

The results of the least squares estimation of this linear model with time variable coeficients are given below. The estimated model between the dates 10/03/1998 (n=1) and 05/06/1998 (n=60),that is 60 Stockmarket days is (8) x n =exp(((-0,00025)n+0,023223)n+7,317873+ e n ) The maximum of the normal curve occurs in the day n=47 that is in 19/05/98. In this date the model gives a clear selling signal .Of course we cannot trade with the general index .But it would give one if we had applied it for a particular stock . The author scored code in visual basic in Excell in order to analyse the buying and selling signals during the year.The results were quite positive for forecasting .For further analysis of optimal trading se bibliography below from BREIMAN L.1961 to GENCAY R. 1998. The variance of the residual and the goodness of fit are given below: (9) S= 8409,733584 (10) R= 92,62713729 The reader should be warned nevertheless, that a high goodness of fit of a forecasting model, for a particular short time interval, as the above, is not adequate for a repetitive, trading based on it and for a long time (years). For a model to be used for repetitive trading and for a long time (years), it should be tested that for the goodness of fit at repetitive forecasting does remains high for long times intervals, that must me at least 2 to 5 years, but even better 20-25 years.

In figure 2 we have an superimposed form the general index and the estimated "normal" curve for the seasonal horizon of 60 days . In table 1 they are given the numerical data of the chart .As soon as we have estimated the model by continuing it in a resaonable forward horizon we have an effective forecasting .The forecasting is corrected at best every day so that the buing or selling signals are with minimum time delay . We have used data of closing daily prices and not intra-day data . The Bayes nested estimation can be extended for shorter horizons and the exponent becames a polynomial of order higher than the quadratic .

  Figure 2

  or [KARLIN S-TAYLOR H.M.] pp 267-269 ,357 ,363,385 and [MALLARIS A.G.-BROCK W.A.] pp 220-223. It is a linear SDE of constant coefficients and multiplicative «noise» or innovation.

Table 1

 1

	Date General	Normal	Date	General
	Index	Smoothing		Index


Dr Konstantinos Kyritsis

Dr N Antoniadis

OPTION PRICING BASED ON THE CONCEPT OF INSURANCE: MARKET MODELS-FREE METHODS THAT GIVE AS SPECIAL CASE THE BLACK-SCHOLES OPTION PRICING

In this paper, alternative methods to the Black-Scholes method of option pricing are given, yielding the latter as special case. The alternative methods are similar to the methods of insurance policies pricing in actuarial mathematics. The choice of the model that represents the changes of the price of the underlying exchange market is left open. Numerical examples are given and the proposed method is compared to the traditional Black-Scholes method. The resulting advantages are discussed.

1.

Introduction

It is often said that derivatives are used either for speculation or hedging. In particular, it seems that from the derivatives, options were originally designed as insurance contracts for the case of loss or damage in positions on other assets, indexes or securities but especially of futures. We say especially for futures, because positions in futures can easily lead to bankruptcy, due to the existence of margin and leverage, while positions in securities, literally, do not. By buying contracts of put options we can insure long positions in futures while by buying contracts of call options we can insure short positions on futures. This paper requires a familiarity with derivatives, either as practical experience or education on them. For the definitions of put and call options, the reader may refer to Hull (2000) (chapter 1, pp 1-5) or [START_REF] Wilmot | Derivatives[END_REF]. We do require this familiarity so as not to have to define in this paper terms like margin, leverage, market maker etc. Alternative models to the Black-Scholes model have been published by [START_REF] Cox | The Valuation of Options for Alternative Stochastic Processes[END_REF], by [START_REF] Duan J-C | The GARCH Option pricing model[END_REF], by [START_REF] Hull | The Pricing of Options on Assets with Stochastic Volatilities[END_REF] etc., all of which are based on different ideas, than the present papers idea. The authors, introduce in this paper, a new model for option fair pricing, which is based on principles of actuarial mathematics and insurance, and which includes the Black-Scholes as a special case. The main advantage of the present suggested model, is that the final option's fair price, does depend and includes, the trend of the underlying asset. In the Black-Scholes model, the option's fair price is independent from the trend of the underlying asset, and the only way for the market-makers to include it in the computation of a fair price, is to use different volatilities of the underlying asset (volatility smiles etc), for different strikeprices. This nevertheless is a contradiction of the definition of volatility for the underlying and a fairly complicated and irrational technique. In the present approach, the market maker, can make use, one only volatility, for the underlying asset, plus a trend of it, that can be positive or negative, and accounts for his expected, or forecasted trend of the market for the underlying asset till the expiration. The practice of the market makers as far as their forecasting of the trend of the underlying, through adjustments of the parameter of volatility, of the Black-Scholes formulae, is not based on any standard statistical forecasting technique, but rather on arbitrary intuitive choices. In the present approach the market maker could make use of a statistical measurement of the trend of the underlying, as he can do for the volatility too. But obviously, the widespread tendency is to put an arbitrary expected trend, at each time, according to their speculations. From this point of view the present suggested approach for the option fair price, does not include necessarily, a better forecasting for the underlying, as usually, there is not a standard forecasting technique for this. Of course in the approach suggested in this paper, we could use more appropriate forecasting techniques for the underlying asset, than those used by the Black-Scholes model, or we could use exactly the same forecasting. The present option fair pricing does suggests nevertheless a better system of choices for the market maker or the investor, based on two parameters rather than one, the trend and the volatility of the underlying. When the trend of the underlying is put equal to the risk-free rate, then the present formulae for the option's fair price coincide with those of Black-Scholes model.

The present suggested option fair pricing, based on principles of insurance, is not included of course in any treatise or publication (as far as we know) of actuarial mathematics, as it is standard that in the applications of actuarial mathematics is not included the Financial Derivatives, and options, that is a topic of finance rather than insurance.

2.

Properties of the Black-Scholes Options Pricing Model

The model that is used by the market makers of most markets for options fair pricing is the Black-Scholes model. An outline of the characteristics of this model is given below.

Before we proceed we have to outline the model for the movements of underlying assets that is assumed in the Black-Scholes Model as well.

0.

Underlying assets follow a geometric Brownian motion. The geometric Brownian motion is like a random geometric progression. of, say a random interest compound. The exact definition is the following:

1.

If S is the spot price of an asset, it is assumed that S follows an (Ito) stochastic process (see [START_REF] Oksendal | Stochastic Differential equations[END_REF])

Sdz Sdt dS     (1)
where dz is a Wiener process or Brownian motion. In other words it has the next two properties a)

The increment or change z  during time

t  is t   
, where ε is a standardized normal random variable b)

The increments z  for two different intervals of time t  are independent, as random variables.

Equivalently, we could postulate that the spot price of the asset follows a geometric Brownian or Wiener process, in other words, the logarithm S ln of the spot price follows a Wiener or Brownian process, generally with drift  [see also Karlin and Taylor (1975), p. 357].  is called the drift of the process and  its coefficient of volatility.

The drift parameter μ and the coefficient of volatility σ in discrete time and a sufficient fine grid or resolution of time step δt, can be estimated by the formulae:

   N i i R t N 1 1   (2), where i i i i S S S R   1
(3) and

     N i R R i t N 1 2 ) ( ) 1 ( 1   (4) where t R   (5)
Obviously if the time unit is the pixel δt , then μ and σ are the average and standard deviation of the rates of return per time step which is also the time unit.

For the option fair pricing model of Black-Scholes, the following axioms are assumed:

1) The stock or underlying price follows a geometric Wiener process with volatility σ and drift μ .

2) The short selling of the underlying asset is permitted

3) There are no transaction costs or taxes, and the invested value of the underlying can be of any continuous size (the values of contracts are not necessarily integers) 4) There are no dividends during the life cycle of the option.

5) There are no riskless arbitrage opportunities 6) Trading of the underlying is continuous 7) The risk-free rate is constant during the life cycle of the option.

As a result of the above axioms, it can be proved that an option price f has to satisfy the next Partial Differential equation known also as Black-Scholes-Merton equation.

rf S f S S f rS t f          2 2 2 2 ) 2 / 1 (  (6)
The solution of this equation with boundary final condition

) 0 , max( ) , ( X T f S S T T  
when t=T, which is the payoff for a standard European call option at expiry, and at time t=0 is:

) 2 ( ) 1 ( 0 d N Xe d N S c rT    (7)
where

T T r X S d   ) 2 / ( ) / ln( 2 2 0    (8) T T r X S d   ) 2 / ( ) / ln( 1 2 0    (9)
and N(x) is the distribution function of a normal random variable.

Various modifications to this model have been presented. For example, [START_REF] Huang | Warants Pricing: Stochastic Volatility vs. Black-Scholes[END_REF] propose the use of a model having a stochastic volatility parameter. [START_REF] Lehar | GARCH vs. Stochastic Volatility: Option Pricing and Risk Management[END_REF] propose the GARCH model and compare it to the stochastic volatility model. A common point of reference for these works is the use of a volatility parameter, on which certain assumptions are imposed.

Introduction to the insurance model of Option Pricing

In the B-S option fair pricing model , the drift of the underlying is different in general from the risk-free continuous time rate, but after all the axioms of the model is does not enter eventually in the final formula of fair price.

An alternative approach would be to abandon the erroneous axioms of the B-S model and to allow the drift of the underlying asset to enter the formula. The fact that options are actually insurance contracts of loss in long or short positions, gives an obvious way to derive their fair premium price, in a way exactly the same with insurance contracts in actuarial mathematics. The principle of pricing insurance contracts in actuarial mathematics is based on the following simple equation:

The average value of the outflow from the insurance company to the customer due to the occurrence of the events being insured, discounted in present value should be equal to the sum of inflows in the company from the customer , discounted in present value.

This principle of course requires a model to estimate the probabilities of the insured events, and thus derive the average value of the outflows from the company to the customer. In the case of options this model is the model of the price changes of the underlying asset.

We may restate the previous general principle for the case of options with which we derive option fair prices:

THE GENERAL INSURANCE PRINCIPLE OF OPTION FAIR PRICING.

The options fair price P at a present time moment t is the discount with the risk-free rate ρ at this moment t, of the average paid value at expiration T, given that the value of the price random variable, S Τ, of the underlying asset at expiration is calculated by an assumed model M and by the spot price S t at the present time t.

We notice here that the above concept of fair pricing corresponds to buying the contract of the option at time t and exercising it at expiration, without intermediate trading. This way, no assumptions about zero transaction costs or continuous delta-hedging trading or infinite continuous divisibility of invested size, are necessary.

As we shall see in the next paragraph, the Black-Scholes option fair pricing falls too under this general principle, if we assume that the model M is a geometric Brownian motion with a drift equal to the continuous-time, risk-free rate. Although in the assumptions of the B-S model the geometric Brownian motion of the underlying asset does not necessarily have a drift equal to the risk-free rate, the final fair premium when included in the above principle corresponds to a choice of geometric Brownian motion with drift equal to the risk-free rate.

Black-Scholes model: a special case of the insurance option pricing model

To prove that the Black-Scholes option fair price formula is a special case of the above general insurance principle of option fair pricing, the following lemma is required (Hull (2000), p.268).

Lemma: If V is lognormal distributed with average value m and the standard deviation of V ln is s , then

    ) ( ) ( 0 , max 2 1 d XN d mN X V E    (10)
where

s s X m d 2 ) ln( 2 1   s s X m d 2 ) ln( 2 2   (11) ] [A E
denotes the average value of the random variable A and ) (x N is the distribution function of a normal variable x .

Using this result we may derive the general form of the option fair price , and the exact formula in the case we assume (as is also assumed in the B-S model) that the underlying follows a geometric Brownian motion (lognormal distribution).

If we interpret the average value m as the average value of the price of one unit of the underlying (one item of the security, if the underlying is a stock) at expiration, and the standard deviation s as the standard deviation of the price of the underlying at expiration, then the average value of payoff of one contract of a call option at expiration of exercise price X (and assuming that one contract insures one item of the underlying) is

    0 , max X V E P   (12)
The general principle of pricing requires of course to have this average payoff value discounted at present values, if the pricing is not at expiration. So if the current time is t , and expiration is time T , while the risk-free (continuous time) rate by which we discount is  , then the present fair price of the option is,

    0 , max ) ( X V E e P t T     
(13) This is the general formula of the option fair price, which is model-free in the sense that any model may be assumed for the changes of the prices of the underlying.

If we assume that the price of the underlying at expiration is also lognormally distributed with average value m and standard deviation s , then we may apply the above lemma to transform the general formula into:

)) ( ) ( ( 2 1 ) ( d XN d mN e P t T      (14) where s s X m d 2 ) ln( 2 1   s s X m d 2 ) ln( 2 2   (15)
This is again a general formula of the option fair price, which is also model-free in the sense that, we may assume any model for the changes of the prices of the underlying, provided it is lognormally distributed at expiration, with average value m and standard deviation s .

Let us now choose a particular model for the changes of the prices of the underlying, which is the same as the one assumed by the Black-Scholes model, namely a geometric Brownian motion of drift r and volatility s . Then the average value of the price of the underlying at expiration T is

) )( 5 . 0 ( 2 t T s r e S m    ( 16
)
where t is the present time and S is the present value of the underlying. This yields the following formula for the fair option's premium:

)) 2 ( ) 1 ( ( ) )( 5 . 0 ( ) ( 2 d XN d N e Se P t T s r t T        (17) where s s X Se d t T s r / )) 2 / ( ) / (ln( 1 2 ) )( 5 . 0 ( 2     (18) s s X Se d t T s r / )) 2 / ( ) / (ln( 2 2 ) )( 5 . 0 ( 2     (19)
The reader should notice that both the drift r and the risk-free rate  enter the formula; in general, these two figures are different.

If we now assume that the drift of the underlying r is equal to the risk-free rate  , then the above formula reduces to the familiar formula of option fair price (for call options) of the Black-Scholes Model (Wilmot (1999), p 97):

) 2 ( ) 1 ( ) ( d N Xe d N P t T      (20) where t T s s X S d     ) 2 / ( ) / ln( 2 1  (21) t T s s X S d     ) 2 / ( ) / ln( 2 2  ( 22
)
Thus we have proved:

The insurance model of option fair pricing has as special case the Black-Scholes model, when in the insurance model we assume that the drift of the underlying is equal to the risk-free rate.

Numerical comparisons

In this section, some numerical comparisons between the Black-Scholes model and the insurance model of option fair pricing are presented. In particular we notice that the underlying (chart 1) has negative descending trend . And the options fair price as computed by the present insurance model (lower series 2 of chart 2) is indeed of lower price as it should, compared to that the option's fair price computed by the Black-Scholes model (Middle series 1 of chart 2). According to the assumptions of the models both option prices are fair, but usually the market-makers when expecting a descending trend of the underlying, as it is here, the alter the volatility of the underlying (which here is put 40% as it is measured), to a lower value, thus achieving the effect, of a lower fair price for the option. With the suggested model, it is not necessary to put a lower volatility, but rather a negative trend for the underlying. So the advantage is rather not of a better forecasting (as in both models exactly the same forecasting for the underlying can be used), but a better way to vary the option fair price according to the speculations of the market maker, or investor, in a logically consistent way, and given that it is the standard practice of the market makers to do so, but only in a artificial and logically inconvenient way through the parameter of the volatility. Of course in the insurance option fair pricing we could make use of more appropriate techniques of forecasting of the underlying, than this in the Black-Scholes model, or we could use exactly the same forecasting.

We take real data of the underlying asset, that in this case is the index FTSE/ase20 (high capitalization) of the Athens Stock Exchange Market for the time interval from January 3 rd , 2002 to January 22 nd , 2002. Using this data, we estimate the daily option fair price by Black-Scholes model (for a call option of a fixed exercise price) and, parallel to it, the option fair price by the insurance model, with a drift for FTSE estimated on an initial previous time interval. These series of data are presented in Table 1. The calculations below have been carried out using a software application written by the first of the authors, who used Visual Basic for Applications to develop an Option Simulator in the environment of Microsoft Office Excel. The values for the FTSE-20 is plotted as a time series in Chart 1, while the premiums for the three models are plotted in Chart 2.

Date

Dates

Price

Series1 Series2

Series3

Chart 2: Theoretical prices for Call options, using different models

The reader may notice the sensitivity of the fair price of the insurance model on the trend of the stock market. If the trend is ascending, the insurance model yields a higher price for the call option, thus accounting for the fact that the option is likely to be exercised yielding a higher profit to the investor. If the trend is descending, then the fair price is lower, since it is likely that the stock market will go lower, thus making the call option unattractive to the investor.

In the table above we took at random a time interval of 14 days of the index FTSE-20 (High Capitalization) of the Athens Stock Exchange market for the time interval 3-01 to 21-01 of 2002 (14 Days). We made the assumption that a call option expires at the end of this time interval and we estimated the corresponding fair prices of such an option. A volatility of 40% for the underlying and a risk-free rate at 4% are assumed. The middle line in chart 2 (data in column 2 of the table) is the fair price calculated by the Black-Scholes Model. The lower line (column 3 of the table) is the fair price calculated by the insurance model with assumed drift for the underlying equal to -4% (downwards trend). The upper line is the fair price by the insurance model with assumed drift for the underlying equal to 8% (upward trend). Although the differences for this range of timeto-expire are small, the additional parameter of drift of the underlying allows for flexibility in defining the fair price, as a function of the markets trend. In fact, the drift can be automatically estimated for a given past time interval (e.g. of the same length as at the estimation of the historic volatility) by least squares best fit. There are standard formulae for the determination of the drift at the geometric Brownian Motion (see formula 2). It is important to notice that, in trying to adjust the options price to the trend of the market, the market makers traditionally vary the assumed volatility, which results in changing in an undesired way both the call and the put options prices; this is clearly undesirable and leads to adopting the use of different volatilities for different exercise prices. This is totally unnatural in respect to the definition of the volatility in the Black-Scholes model as referring to the underlying asset, and thus being independent from the exercise price of the options. The proposed model has been used for investment purposes for a year (2001), in the fair option pricing in the Athens Derivatives exchange, by the first of the authors.

Conclusion

We saw how we can formulate an alternative option fair pricing model based on the basic equation of pricing of premiums in insurance contracts with the following characteristics 1) It is left open which is the model of price movements of the underlying, while of course we can as well assume a geometric Brownian motion, or even an other non-Markovian model like ARMA models of time series that may include models of oscillations of the underlying asset. (see e.g. [DUAN J-C (1995)])

2) As the option fair pricing is model free, according to the model or even for a fixed model, the fair option price is sensitive to whether the spot market is descending, ascending or neither. For example, if the model for the underlying is the geometric Brownian motion, it is sensitive to its drift parameter. This permits easier variation of the option fair price by easily measurable parameters of the spot market, and therefore more flexible option fair pricing.

3) The above approach gives an alternative formulation and derivation of the Black-Scholes option fair pricing formula that completely avoids the ironic, ambiguous and controversial Black-Scholes assumptions about risk less arbitrage opportunities, continuous delta hedging trading, zero transaction costs and infinite continuous divisibility of invested size. To derive exactly the Black-Scholes option fair pricing formula we only require the assumption that in the average the underlying has a drift, equal to the risk-free rate. To account for other drifts, and with the same natural assumptions we should resort to the insurance option fair pricing.

Ftse-20 Call Option Premium by Black- Scholes Call Option Premium drift -4% by Insurance Model Call Option Premium drift 8% by Insurance Model

	04-01-02 1447,47	72,44	69,74	73,73
	07-01-02 1441,46	66,29	63,89	67,45
	08-01-02 1423,27	52,17	50,22	53,11
	09-01-02 1407,41	40,56	39,02	41,31
	10-01-02 1406,60	37,89	36,53	38,55
	11-01-02 1397,43	30,65	29,56	31,17
	14-01-02 1365,50	14,81	14,22	15,09
	15-01-02 1364,56	12,23	11,78	12,45
	16-01-02 1355,68	7,53	7,26	7,66
	17-01-02 1393,09	17,26	16,84	17,46
	18-01-02 1390,56	12,31	12,06	12,43
	21-01-02 1382,26	4,89	4,81	4,93
	22-01-02 1392,14
	03-01-02 1441,71	70,22	67,40	71,57

Table 1 :

 1 Values for the FTSE/ase20 (high capitalization) index of the Athens Stock Exchange Market from 3/1/02 to 22/1/02 and estimates for the daily option fair price resulting from the different models.