\section{Introduction}

NaI(Tl) has been used as a scintillator for particle detection for many years~\cite{hofstadter_detection_1949}, and its wide-spread availability makes it a convenient low-cost choice for many particle-detection applications.
Undoped NaI has also been known to be an effective scintillator at low temperatures~\cite{van_sciver_fundamental_1958}.
The scintillation properties of these crystals have been characterized, though the difference between crystals can be large~\cite{sibczynski_properties_2011,moszynski_study_2003}, highlighting the importance of characterizing individual crystals in use, or at least batches of crystals.

The DAMA/LIBRA collaboration~\cite{bernabei_final_2013} detect an annual modulation signal~\cite{bernabei_final_2013,bernabei_first_2018} consistent with the presence of a dark matter halo, using an array of radiopure NaI(Tl) crystals.
To this date however, no other experiments have detected a similar signal with other target materials~\cite{kahlhoefer_model-independent_2018}.
The SABRE collaboration~\cite{antonello_sabre_2018,bignell_sabre_2020} aim to verify the DAMA/LIBRA annual modulation observation using the same target material in two distinct locations to remove seasonal effects: the Gran Sasso Lab in Italy and the Stawell Underground Physics Lab in Australia.

Though SABRE plans to use room-temperature NaI(Tl) crystals, others intend to use it a lower temperatures (for instance ASTAROTH~\cite{zani_astaroth_2021} at 87~K).
Moreover, advantages can be gained by introducing background discrimination in the experiment.
It has been shown that using an alkali halide scintillator at cryogenic temperatures, measuring both scintillation light and phonons created by particle interactions, can lead to the detection of a modulation in relatively small exposures~\cite{nadeau_sensitivity_2015}.
This allows the discrimination of electron recoils, usually from electron and $\gamma$ backgrounds, and nuclear recoils which are expected from WIMP interactions, reducing the level of background.
The COSINUS collaboration plans to use undoped NaI at cryogenic temperatures to search for dark matter signals~\cite{angloher_results_2017}.

This paper will present measurements made with SABRE-grown crystals to verify the scintillation properties of these NaI and NaI(Tl) crystals at low temperature.
It is important to ascertain these properties to allow for cryogenic experiments to properly design their light-collection and processing equipment.
It also may be interesting to determine the difference in performance for moderately cold (liquid nitrogen) temperatures to allow for better use for other applications.

\section{Experiment}

The experiment was carried out in a compact optical cryostat based on a Gifford-McMahon cryocooler capable of cooling small $5\times5\times2$~mm$^3$ samples of NaI and NaI(Tl) to any temperature between 3.5~K and 300~K~\cite{di_stefano_counting_2013,clark_particle_2018}.
The samples were taken from the tips of crystals grown by RMD Inc. as part of the SABRE collaboration~\cite{antonello_sabre_2018}.
These crystals were of a standard purity with an unknown crystal orientation and polished to transparency.
The NaI(Tl) sample is doped at roughly 1/1000 atoms Tl/NaI.

\begin{figure}
\centering
\includegraphics[width=0.6\textwidth]{crystal}
\caption{\label{fig:cryst}Photograph of the NaI(Tl) sample mounted inside the cryostat. The crystal is held onto the gold-plated copper finger using silver paint for good thermal contact. A collimated Am-241 source illuminates the large face of the crystal.}
\end{figure}
Each crystal sample  was mounted inside the cryostat and excited by $\alpha$ particles and 60~keV $\gamma$ quanta from an internal collimated $^{241}$Am source mounted to the sample holder (Fig.~\ref{fig:cryst}).
Additionally, an external $^{57}$Co source could be used to test $\gamma$ energies of 122~keV.
The $^{241}$Am $\alpha$ source used in this work was adapted from a common smoke detector where a protective film covers the radioactive material, reducing the energy of the $\alpha$-particles to 4.7~MeV.
The cryostat is housed in an acrylic glovebox to allow for careful control of the humidity.
By flowing pure nitrogen through the glovebox during installation of the crystal and throughout the data-taking, relative humidity within the glovebox was kept below 0.1\%, the sensitivity of our General Tools SAM990DW Psychrometer.

The samples were adhered to a custom-made holder using silver paint, instead of being mechanically held in place, to minimize mechanical stress as the sample cools.
The crystal sample was at an angle of {30\textdegree} with respect to the $^{241}$Am source so that the emitted $\alpha$ particles were incident with one of the smooth, large faces of the sample, but at the same time this face was well exposed to a 28~mm diameter PMT (Hamamatsu R7056). A second PMT was placed opposite of the first one and was exposed to the rear face of the crystal, partially obstructed by the sample holder.
These PMTs have sensitivity to wavelengths down to 185~nm and a maximum quantum efficiency of 25\% at 420~nm.
The PMT that was in view of the unobstructed face of the sample was used as the primary PMT for data acquisition, while the other was used to assist triggering.
Based on the solid angle of the crystal exposed to the primary PMT and transmittance of the windows, we expect a light collection efficiency of roughly $10\%$ for the main PMT~\cite{nadeau_cryogenic_2015}.
Note that no optical filters were used, so all light within the sensitivity of the PMT contributed to the light yield, as is standard in particle detection.

The temperature of the crystal was controlled by a PID such that it was steady to within 0.1~K of the goal temperature during data taking.
Data were taken starting from room temperature, cooling down to the lowest temperature of 3.5~K in a single run to remove the possibility of releasing trapped energy through warming, known as thermoluminescence, which could occur when heating a crystal to a temperature that corresponds to a gap in its energy levels~\cite{sabharwal_thermoluminescence_1985}.
Scintillation pulse shapes were acquired using a modified version of the multiple photon counting coincidence technique~\cite{clark_particle_2018}.
When a coincidence within 30~ns was detected between the two PMTs observing the sample, a 600~$\mu$s acquisition window sampled at $1.25 \times 10^9$~samples/second was opened in which the full digitizer trace was acquired.  The window included a 30~$\mu$s pre-trigger.
This trace was later zero-suppressed to reduce the data volume.
In previous measurements~\cite{clark_particle_2018}, we observed that simply recording information above a threshold caused a bias by not recording some information.
To combat this, we additionally record the 5 samples immediately preceding and following a sample above threshold.
This number was used since a typical single photon pulse was found to be around 10 samples wide.
The threshold for a given pulse was set to 5 standard deviations of the fluctuations of the baseline of that pulse.
An example of the effect of this zero-suppression is shown in Figure~\ref{fig:pulse}.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{rawpulse}
\caption{\label{fig:pulse}An example of a single event in the raw digitizer trace (red line), recording only samples above the threshold (blue circles), and recording those plus the surrounding points (black dots).}
\end{figure}

The signal from the front PMT was passed to two digitizer channels with different ranges to allow for the measurement of both high voltage signals and small single-photon signals, as described previously~\cite{clark_particle_2018}.
This dynamic range was important due to the large energy range of interest between 60~keV gammas and nearly 5~MeV alpha-particles.
The combination of the two channels resulted in a single measurement containing unsaturated, reconstructed pulses with higher detail at low voltages than would be possible with a single channel.
At low temperatures where the light yield is higher (below 30~K for NaI(Tl) and below 150~K for NaI), a 10dB signal attenuator was used to prevent the data acquisition from saturating.
The typical integral of a single photon was measured in both attenuated and non-attenuated configurations, such that the light yields can be directly compared in terms of number of photons.
In addition, full datasets were taken at reference temperatures (50~K for NaI(Tl); 200~K and 150~K for NaI) in both configurations to ensure that the results were consistent.

\section{Data Analysis}

To determine the light yield, the integral of the events was binned and a gaussian fit was performed to find the peak corresponding to each particle excitation.
The mean of each peak was taken to be the average light yield of that excitation.
An example of an alpha excitation peak is shown in Figure~\ref{fig:alpha} and a gamma excitation peak in Figure~\ref{fig:gamma}.
The integral was then converted to a corresponding number of photo-electrons (PE) using the typical single photon integral in those digitizer conditions.
The conversion to number of PE allows us to compare our light yields with and without the signal attenuator directly, using different conversion factors for the two acquisition configurations.

It was noticed during the measurement of NaI that the alpha peak seemed to split into two distinct peaks at temperatures below 60~K.
It was later seen through inspection that the crystal was cracked after the experiment was completed, likely due to differences in thermal contraction between the copper sample holder and the NaI crystal.
We believe that the crack changed the optical properties of the crystal, changing the efficiency of detecting photons from events on either side of the crack.
Due to limited availability of crystal samples, this data will still be studied despite the change in crystal quality, with some added uncertainty after the cracking event.
To compare with the rest of our data, the weighted average of the two peaks was used to determine the light yield at temperatures where a split alpha peak is observed.

For each run, we perform a cut on the first photon time such that the first photon does not arrive before the expected coincidence window of 30~ns.
This removes events that have photons in the pre-trigger, which could indicate that the tail of a previous event is present in our acquisition (pile-up event), which would give an inaccurate measurement of both the light yield and time behaviour.
We then perform a cut on the light yield (integral of the event) to select for a particular particle interaction (4.7~MeV $\alpha$-particle, 60~keV $\gamma$).
This is done using the gaussian fit as shown in Figure~\ref{fig:peaks} where all events within 2 sigma of the mean of the gaussian distribution are chosen to represent each population of events.
If the light yield is low, a minimum integral of two PE is set as the lower bound.
Finally, we perform a cut on the mean arrival time of photons after the first photon.
A large mean arrival time compared to the typical value of the population could indicate another particle interaction occurred during the tail of the event which could interfere with our average pulses.
\begin{figure}
\centering
\begin{subfigure}{0.7\textwidth}
\includegraphics[width=\textwidth]{SABRENaIfit_77att_1gaus.png}
\caption{\label{fig:alpha}4.7~MeV alpha events from Am-241}
\end{subfigure}
\begin{subfigure}{0.7\textwidth}
\includegraphics[width=\textwidth]{SABRENaIfit_60keV_77att.png}
\caption{\label{fig:gamma}60~keV gamma events from Am-241}
\end{subfigure}
\caption{\label{fig:peaks}Histograms of the measured light from events of both excitations from the Am-241 source for NaI at 77~K. In red is the fit to a gaussian curve, to determine the average light yield.  In the case of the gamma spectrum, the lower energy shoulder is attributed to Np X-rays from the source.}
\end{figure}

\section{Results and Discussion}

\subsection{Light yield}

The results of our light yield measurements as a function of temperature are shown in Figure~\ref{fig:LYT}, in addition to results from previous measurements~\cite{sailer_low_2012,nadeau_cryogenic_2015} that have been scaled to be equivalent with these results at room temperature for each type of excitation.
For NaI(Tl) (Fig~\ref{fig:NaITlLY}), the light yield changes by as much as 30\% as the temperature decreases for both $\alpha$ and $\gamma$ excitations.
At room temperature, we observe 1.4~PE/keV  (photoelectrons per keV) for the 60~keV $\gamma$-excitation, which, when accounting for the solid angle of our photomultiplier ($\approx$ 11\%) and the quantum efficiency (24\% at 400~nm), gives a rough estimate of $50 \pm 10$~photons/keV for the absolute light yield.
This is marginally higher than the frequently-quoted value of  38~photons/keV~\cite{holl_measurement_1988}.
At our lowest temperature of 3.5~K, we detect 1.1~PE/keV for $\gamma$ excitation corresponding to a decrease of light yield of 20\%.  This low-temperature value of $40 \pm 8$~photons/keV is slightly above the 29~photons/keV of CaWO4~\cite{mikhailik_performance_2010}.
The light yield of 4.7~MeV $\alpha$ excitations remain relatively stable at  0.65~pe/keV throughout the temperature range.
We observe a drop in light production at 150~K, which coincides with a thermoluminescence peak that has been observed previously for thallium-doped NaI~\cite{sabharwal_thermoluminescence_1985}.
This indicates there could be a shallow trap at the corresponding energy of $kT$ $\approx$ 10~meV.

\begin{figure}
\centering
\begin{subfigure}{0.7\textwidth}
\includegraphics[width=\textwidth]{R83LY_all}
\caption{\label{fig:NaITlLY}NaI(Tl)}
\end{subfigure}
\begin{subfigure}{0.7\textwidth}
\includegraphics[width=\textwidth]{R84LY_phperE}
\caption{\label{fig:NaILY}Undoped NaI}
\end{subfigure}
\caption{\label{fig:LYT}The light yield as a function of temperature for both crystals, along with comparisons to previous measurements~\cite{sailer_low_2012,nadeau_cryogenic_2015}, scaled to our measured value at room temperature. Data with signal attenuation are shown in empty markers, and without attenuation in full markers.  At temperatures where both attenuated and non-attenuated data was taken, results are consistent. Data from Coron (respectively Sailer) are normalized to our room temperature gamma (resp. alpha) value. Data from Coron 2013 are not time-resolved. Errorbars on each point are within the individual markers.}
\end{figure}
In Figure~\ref{fig:NaITlLY}, we have compared the results reported in other studies~\cite{nadeau_cryogenic_2015,sailer_low_2012,coron_study_2013} to our own.
Since these results were reported as relative shifts in light yield, to compare with our results we have scaled each study to be equal to our own measurement at room temperature, for the specific particle interaction that was measured. We see features at approximately the same temperatures as other studies, though not to the same degree.
It should also be noted that Coron et al.~\cite{coron_study_2013} were not measuring time-resolved interactions but a constant illumination with X-rays, in addition to reporting in the warming phase over most of the temperature range, which could add additional complications from thermoluminescence.

In contrast, undoped NaI (Fig~\ref{fig:NaILY}) shows a large increase from  0.04~PE/keV at room temperature for $\gamma$ excitation, reaching a maximum of 1.15~PE/keV at 70~K, increasing by roughly a factor of 30.
The light yield then decreases, but remains high at 0.85~PE/keV until the lowest temperature, roughly $30 \pm 6$ photons/keV in absolute light yield.
This is a much higher relative increase than has been seen previously, with previous measurements increasing by at most a factor of 10~\cite{nadeau_cryogenic_2015}.

It should be noted that the decrease in observed light yield coincides with the temperature at which the crystal is suspected to have cracked.
This may indicate that thermal stresses within the crystal, which were released when the crystal cracked, had an effect on the light yield. Another factor may be changes in optical efficiency.

\subsection{Quenching factor}

From the previous measurements, we have also determined the $\alpha$/$\gamma$ ratio of both NaI and NaI(Tl) as a function of temperature.
The results are shown in Figure~\ref{fig:ag}.
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{both_ag_np}
\caption{\label{fig:ag}Measured $\alpha$/$\gamma$ ratio for both NaI and NaI(Tl) as a function of temperature.  Errorbars on each point are within the individual markers.}
\end{figure}
Since these results compare $\alpha$ and $\gamma$ excitations of different energies, we have employed a theory of inorganic scintillator non-proportionality to correct for this, transforming the 60~keV $\gamma$ light yield to that of a 4.7~MeV $\gamma$, as has been described in previous publications~\cite{clark_particle_2018}.

NaI(Tl) shows a slight increase with decreasing temperature, with a value of 0.7 at 3.5~K.
A peak value of 0.7 is also seen at the same location in the dip in light yield, corresponding again with the likely location of a shallow trap at the corresponding energy~\cite{sabharwal_thermoluminescence_1985}.

Undoped NaI also shows an evolution over temperature, but in this case drops from 0.7 at room temperature to 0.35 at 160~K, returning to 0.7 at 40~K once again.
This result differs from our previous measurements~\cite{nadeau_cryogenic_2015}, in  which the relative humidity was only kept below 20\% during sample handling, as opposed to the more stringent upper limit of 0.1\% in the present work.  This indicates that light yield of surface events, caused by alphas, depends on the state of the sample surface.  An alternative explanation is that the discrepancy is caused by differences in trace contamination between the different lots of samples.

\subsection{Time Structure}

We have also investigated the pulse shapes induced by alpha and gamma interactions at different temperatures (Fig.~\ref{fig:AllPulses_cum}).  The time constants tend to increase as the temperature decreases, as observed previously in various materials (eg~\cite{verdier2011scintillation}) including NaI~\cite{birks_theory_1964}.
A more precise determination of the time constants is complicated by the presence of afterpulsing in the PMs, which has been independently confirmed by exposing the PMs to brief light pulses from an LED (Fig.~\ref{fig:aftpcomp}).  At the standard operating voltage of the PMs, a main narrow structure was observed $\sim 300$~ns after the excitation, as was a broad secondary feature $\sim 1.5$~$\mu$s after the excitation.  This corresponds with the features in the scintillation pulse, as can be seen in Fig.~\ref{fig:aftpcomp}.
The integral of the afterpulses represents 3\% of the signal from the LED, and thus has a small effect on the light yield measurement.
Attempts to deconvolve the afterpulsing from the overall pulse were hindered by the fact that the afterpulsing appears to have increased in the 2 years between the main experiment and the tests.
\begin{figure}
\centering
\includegraphics[width=1\textwidth]{NaITlGamma_cumulative} \\
\includegraphics[width=1\textwidth]{NaITlAlpha_cumulative} \\
\includegraphics[width=1\textwidth]{NaIGamma_cumulative} \\
\includegraphics[width=1\textwidth]{NaIAlpha_cumulative}
\caption{\label{fig:AllPulses_cum}The average pulses from NaI(Tl) (top two rows) and NaI (bottom two rows) (Run 83) under $\gamma$ excitation (odd rows)  and $\alpha$ excitation (even rows), at 3 different temperatures. Pretrigger is 30~$\mu$s long. Trend is for pulses to become longer as temperature decreases. }
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.8\textwidth]{AftPulseCompare.png}
\caption{\label{fig:aftpcomp}The LED measurement of afterpulsing overlaid with data from $\alpha$ excitation in NaI at 60~K. The largest afterpulse corresponds with the pulse shape at low time, and a later bump may correspond with the later bump in the pulse shape.  Pulses have been shifted to correspond to a pretrigger of 30~$\mu$s.}
\end{figure}

To present a quantitative measure of the changing time structure as a function of temperature, Figure~\ref{fig:allMAT} shows the mean arrival time of photons after the first detected photon, for $\alpha$ and $\gamma$ events for each crystal.
For each event of a given type of particle (alpha or gamma), we calculate the mean arrival time of photons after the first one; the standard deviation of these values for a given population are used as error bars.
Unusually, the slowing trend in the data as the temperature is cooled is not monotonic.  All the data seem to show particularly long pulses at $\sim 40$~K and $\sim 175$~K.

We considered if the attenuator, used at low temperatures, could have played a role in this, since it might have made it harder to detect straggling single photons that appear at the low temperature.  Checks were therefore carried out with and without attenuator at 50~K for NaI(Tl) and at 150~K and 200~K for NaI; results were found to be consistent, making the attenuator an unlikely explanation.
We also note that the attenuator would have had little effect on the light yield, because for both samples at all temperatures, the bulk of the photons are not resolved individually.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{MATplot_NaITl.png}
\includegraphics[width=0.45\textwidth]{MATplot_NaI.png}
\caption{\label{fig:allMAT}The evolution of mean photon arrival time as a function of temperature for the NaI(Tl) sample (left) and NaI sample (right).  Filled (respectively empty) markers are data taken without (with) attenuator. For NaI(Tl), cross check at 50~K showed consistent results with and without attenuator.  For NaI, cross checks, at 150~K and 200~K, were also consistent. }
\end{figure}

\section{Conclusion}

In this work, we have shown that the light yield of the NaI and NaI(Tl) crystals grown by the SABRE collaboration perform very well in terms of scintillation yield at low temperatures.
NaI(Tl) maintains its high light yield at all measured temperatures, while undoped NaI sees a large increase in light yield as temperature decreases, up to a factor of 30 times higher than at room temperature.
The evolution of the light yield of NaI(Tl) as a function of temperature is quite irregular above $\sim 50$~K, which is consistent with previous experiments.
For gamma excitations at 4~K, compared to CaWO4, the light yield of NaI(Tl) is slightly greater while that of NaI is comparable.
Because of our instrumentation and relatively high light collection, we are able to measure both $\alpha$ and $\gamma$ excitations under the same conditions.
We have reported here the $\alpha$/$\gamma$ ratio as a function of temperature, showing that it changes over the temperature range, with a maximum value of 0.7.
The response to alphas may very well depend on the state of the surface of the crystal, which itself depends on how much humidity the samples have been exposed to.
Therefore, care was taken to handle these samples under dry nitrogen to maintain a good surface quality.
In addition, all measurements of a given sample were made during a single cooling cycle to avoid contributions from thermoluminescence.
Another factor that may be relevant is the presence of the impurities in a given sample.
These various factors could explain the differences between this result and previous measurements in light yield and $\alpha$/$\gamma$ ratio.

Time-resolved measurements of the scintillation light show presence of  light out to 600~$\mu$s after the initial excitation. At cryogenic temperatures, undoped NaI shows an increased proportion of light at long times, where NaI(Tl) shows a larger increase at short times, even if the absolute light yield at long times increases.  The mean arrival times of photons from both $\alpha$ and $\gamma$ excitations do not increase monotonically, showing a complicated evolution as the temperature decreases.

Significant efforts were made to shelter the samples from humidity and to handle them delicately to reduce thermal stress during cooling. The sensitivity of NaI, both pure and doped, to humidity and the mechanical fragility of the crystals underscore the challenges of using these materials in cryogenic applications, though the potential for background discrimination may make it worth the effort.

\acknowledgments
Funding in Canada has been provided by NSERC through SAPIN grants, by CFI-LOF and ORF-SIF.
Queen's summer student Miaofen Zheng contributed to the data taking.  Queen's Visiting Research Student Ilian Moundib contributed to the afterpulsing analysis.
F.~Froborg has received funding from the European Unions Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 703650.  F.~Froborg's participation in the experiment was made possible by a grant from the Queen's Principal's Development Fund.
We thank Radiation Monitoring Devices (RMD) and the SABRE Collaboration for providing the test samples.

\bibliographystyle{JHEP}


\section{Introduction}
\label{sec:Intro}
Muons observed in underground laboratories are produced mainly in
decays of pions and kaons generated by the interaction of primary
cosmic rays with the upper atmosphere.  Since muons loose energy
crossing the rock overburden, only high energy muons can be detected,
with an energy threshold depending on depth,
usually expressed in metre water equivalent (m.w.e.).
It is known that the flux of atmospheric muons detected deep underground shows
seasonal time variations correlated with the temperature of the stratosphere,
where the primary cosmic rays interact~\cite{bib:season}.
This effect has been reported by other experiments at the underground Gran Sasso Laboratory (LNGS)
(MACRO~\cite{bib:macro1, bib:macro1bis, bib:macro2},
LVD~\cite{bib:lvd}, Borexino~\cite{bib:borex, bib:borex2} and GERDA~\cite{bib:gerda})
and elsewhere (AMANDA~\cite{bib:amanda}, IceCube~\cite{bib:icecube}, MINOS~\cite{bib:minosf, bib:minosn},
Double Chooz~\cite{bib:db} and Daya Bay~\cite{bib:dayabay}).
An increase in temperature of the stratosphere, indeed,
causes a decrease of
the air density, thus reducing the chance of $\pi, K$ mesons to interact, and
resulting in a larger fraction decaying into muons.
So, the atmospheric muon rate changes during the year
increasing in summer and decreasing in winter.
The variation can be modeled as a sinusoidal function, which is only a first
order approximation since the average temperature is not precisely constant
over the years and short term effects occur, leading to local maxima and
minima, like the ``sudden stratospheric warming'' events~\cite{bib:ssw}.

The OPERA experiment~\cite{bib:detector} discovered
$\nu_\mu$ into $\nu_\tau$ oscillations in appearance mode
using the CERN Neutrino to Gran Sasso (CNGS) beam~\cite{bib:prl2015, bib:prl2018}.
The experiment was
located
in the Hall C of
LNGS laboratory, at 3800 m.w.e. depth.
To identify $\nu_\tau$ Charged Current interactions, the Emulsion Cloud Chamber
technique was used, with 1 mm lead sheets alternated with
nuclear emulsion
films, for a total target mass of about 1.2 kt.
Lead and emulsions were organised into units called ``bricks''.
The detector was divided into two identical Super-Modules (SM), each made of a target section,
composed by 31 brick walls interleaved with target tracker layers to locate bricks with neutrino interactions,
and of a muon
spectrometer to optimise the muon identification probability and to
measure momentum and charge.
In each SM, the Target Tracker (TT)~\cite{bib:TTref}
consisted of 31 pairs of orthogonal planes made of
2.6~cm wide
scintillator strips,
read-out by means of
WLS fibers.
The fibers from the 6.7~m long strips were collected into four
groups of 64 and coupled to 64-channel Hamamatsu H7546
photomultipliers.
The muon
spectrometers were made by an iron-core dipole magnet with
drift tubes used as precision trackers and 22 layers of resistive
plate chambers (RPC)~\cite{bib:RPC} inside the magnetised iron.
The 2D read-out was performed by
means of 2.6~cm pitch and 8~m long vertical strips, which measured the
coordinate in the bending plane, and 3.5~cm pitch and 8.7~m long
horizontal strips, measuring the orthogonal coordinate.

The analysis presented here is based on TT and RPC data recorded
during about five years from January 2008 to March 2013.
In the TT,
the trigger condition required either hits in the horizontal and
vertical views of at least two
planes or
at least 10
hits in a single plane
with a signal greater than
30 photo-electrons for 2008 and 2009 runs;
from 2010 up to 2013
the latter requirements were lowered to
4 hits and 10
photo-electrons,
respectively.
Data from the RPCs of each
spectrometer were acquired
in presence of at least 3 planes fired in a
time window of 200~ns.
Events were recorded in presence of at least 5
TT and/or RPC hits in each view, horizontal and vertical, within a time window of 500~ns.

The TT systems were operative during most of the time in
the considered years, while the RPCs had a lower run-time, being
operative only during CNGS runs
and switched off during the CNGS
winter shutdowns.
More details about the electronic
detectors used for the cosmic ray muon analysis can be found in
\cite{bib:detector}.

\section{Cosmic ray muon flux measurement and its modulation}
\label{sec:Cray}
Cosmic ray induced events in the OPERA detector were selected, through
their absolute time,
outside of the CNGS spill window.
Once the event was tagged as ``off-beam'' it was classified as cosmic and
processed in a dedicated way.
The
standard reconstruction package ``OpRec''~\cite{oprec}
was complemented with a set of algorithms developed for the different
cosmic and beam event topologies.
The reconstruction was effective at identifying single and multiple muon tracks
(muon bundles).
For this analysis, a total of about 4 million single muon events have been
selected requiring a single track reconstructed in both views.

Different atmospheric muon rates have been measured in periods with and without RPC acquisition.
The scale factor between the two rates has been evaluated directly from data over the full data taking period,
extracting the two constant terms
$I^0_{TT+RPC} = (3359 \pm 5)~\mu$/day and $I^0_{TT-only} = (1960 \pm 5)~\mu$/day from a maximum likelihood fit on the two data sets.

In Fig.~\ref{fig:murate} (top panel) the flux of atmospheric single muons measured
from 1 January 2008 (day 1 in the plot) to
March 2013 is shown.
After data quality cuts,
our data set is composed of 1274 live days, out of which 919 days with TT+RPC.
The longer downtime period corresponds to the first 5 months of 2009, when the acquisition was
stopped due to a DAQ upgrade, at first, and then to the earthquake in L'Aquila.
Other shorter downtime periods are present in winter due to maintenance operations.
Data with TT-only acquisition have been rescaled to the TT+RPC average rate.
The flux has been fitted to
\begin{equation}
I_{\mu}(t) = I^0_\mu + I^1_\mu \cos \frac{2 \pi} {T} (t-\phi)
\label{murate}
\end{equation}
\begin{figure}[]
\centerline{\includegraphics[width=\linewidth]{plot_murate.pdf}}
\centerline{\includegraphics[width=\linewidth]{plot_Teff7.pdf}}
\caption[]{Single muon rate measured by the OPERA detector (top) and
effective atmospheric temperature (bottom) from January 2008 to March 2013.
Fit results are superimposed to the data sets.
Symbols in the legends are defined in the text and in Eqs.~\ref{murate} and~\ref{trate}.
}
\label{fig:murate}
\end{figure}
The presence of a sinusoidal
component with a period $T=(359 \pm 2)$ days is observed, with an
amplitude amounting to $\alpha= I_\mu^1/I_\mu^0 = (1.55 \pm 0.08)$\% of the average flux,
and a phase $\phi=(197 \pm 5)$~days, with a $\chi^2/\textrm{dof} = 1.46$.
In accordance with other LNGS experiments and
with the fit result on the temperature reported in Sec.~\ref{sec:Temperature},
we take as the best estimate of the phase the result obtained fixing the period to one year of 365 days.
The maximum is then observed
at
day
$\phi = (186 \pm 3)$, corresponding to July 5,
with a $\chi^2$/dof value of 1.47. The result is also shown in Fig.~~\ref{fig:murate} (top panel).

The Lomb-Scargle
periodogram \cite{bib:LS1,bib:LS2} is a common tool to analyze
unevenly spaced data
to detect a periodic variation
independently of the modulation phase.
For the analysis presented here, we have
exploited the generalised Lomb-Scargle
periodogram, proposed in \cite{bib:LSgen}, which takes into account the
non-zero average value of the event rate.
The periodogram obtained for the single muon event rate is shown in Fig.~\ref{fig:LS} (left panel).
To assess the significance of the periodogram peaks, 10$^5$ toy
experiments with a constant rate of 3359~$\mu$/day in the
detector have been simulated, and the corresponding periodograms
reconstructed.  In Fig.~\ref{fig:LS} the 99\% significance level,
defined as the value for which 99\% of the toy experiments result in a
lower spectral power, is also shown.

The most significant peak is around one year ($P_{max}$ at $T \sim 365$ days), but
other less significant peaks are also present, as a consequence of the fact
that Eq.~\ref{murate} is
an approximation.
A simulated experiment has been performed extracting
daily rates according to the result of our fit and comparing the periodograms obtained
with and without data in the days of detector downtime.
It shows that the amplitude of the peaks around 200 days
increases with the detector downtime.

\begin{figure}
\includegraphics[width=0.5\linewidth]{plot_GLS_mu.pdf}
\includegraphics[width=0.5\linewidth]{plot_GLS_T6.pdf}
\caption[]{Generalised Lomb-Scargle periodograms for the measured muon
  rate (left) and the effective atmospheric temperature (right).
The 99\% significance level is also drawn as a reference in both periodograms. }
\label{fig:LS}
\end{figure}

We checked possible systematic effects on the phase $\phi$ coming from the data taking stability along the years.
Possible variations in DAQ efficiency are likely to coincide with calendar years by the fact that RPC were turned on and off on a yearly basis and DAQ maintenance and interventions were done during the CNGS winter shutdowns.
By applying scale factors on a yearly basis,
a constant average rate is achieved by definition.
The normalisation, different at the few per mille level, was applied
to TT-only and TT+RPC data, which were then rescaled one to the other.
The rate has been fitted to Eq.~\ref{murate}
fixing the period to 365 days, the modulation maximum results on day~$\phi = (177 \pm 3)$ with a $\chi^2/\textrm{dof} = 1.64$.
Comparing these results with those obtained with a constant scale factor,
we evaluate a systematic error on the phase as the semi-difference between
the two $\phi$ values
extracted
with period fixed to one year,
i.e. $\delta \phi_{\textrm{sys}} = 5$~days.
Our best estimate of the muon rate maximum, obtained at fixed period $T = 365$~days, is found on day
$\phi = 186 \pm 3_{\textrm{stat}} \pm 5_{\textrm{sys}}$, i.e. July 5.

\section{Atmospheric temperature modulation}
\label{sec:Temperature}
To measure the atmospheric temperature modulation, we have used data
from the European Center for Medium-range Weather Forecasts
(ECMWF)~\cite{bib:ECMWF}.
The center provides temperature values at different altitudes above given
locations, obtained by means of interpolations based on measurements
of various kinds around
the planet and on a global atmospheric model.
The coordinates are those used in \cite{bib:borex}:
13.5333$^\circ$~E, 42.4275$^\circ$~N.
The atmospheric temperature is provided at 37 discrete pressure levels
ranging from 1 to 1000 hPa four times in each day (0.00~h, 6.00~h, 12.00~h
and 18.00~h).
Averaging these temperature values using weights accounting for the production
of pions and kaons at different altitudes (see Appendix~\ref{sec:Appendix}),
the effective atmospheric temperature $T_{eff}$ has been calculated four times
a day.
The four measurements are then averaged and the variance used as an estimate
of the uncertainty on the mean value.

The weights used in this analysis have been computed as described in~\cite{bib:Grashorn} and in
previous experimental papers~(\cite{bib:macro1}-\cite{bib:dayabay}). They depend on the inclusive meson
production in the forward region, on the attenuation lengths of
cosmic ray primaries,
pions and kaons, as well as on the average value
$\langle E_{thr} \cos \theta \rangle$, where $E_{thr}$ is the minimum energy required
for a muon to reach the considered underground site and $\theta$ is
the angle between the muon and the vertical direction.
From a full Monte Carlo simulation taking into account the rock map above Hall C of LNGS~\cite{chargeratio}, it results that
$E_{thr} = 1.4$~TeV and $\langle E_{thr} \cos \theta \rangle = (1.1 \pm 0.2)$~TeV.
A previous analysis~\cite{bib:borex} quoted a higher value, $E_{thr} = 1.8$~TeV,
extracted from numerical methods
assuming a flat overburden~\cite{bib:Grashorn}.
All other parameters in the weight functions are site independent.
More details about the effective atmospheric temperature calculation are
given in Appendix~\ref{sec:Appendix}.

The $T_{eff}$ values are shown in Fig.~\ref{fig:murate}~(bottom panel) in the
time period of the
data taking, from January 2008 to March 2013.
The temperature has been fitted to a sinusoidal function similar to Eq.~\ref{murate}:
\begin{equation}
T_{eff}(t) = T^0_{eff} + T^1_{eff} \cos \frac{2 \pi} {T} (t-\phi)
\label{trate}
\end{equation}
The fit results are also shown in Fig.~\ref{fig:murate}.
The average effective temperature is 220.9~K, and a modulation is
observed with an amplitude $\alpha = T_{eff}^1/T_{eff}^0 = (1.726 \pm 0.002)\%$ of $T^0_{eff}$.
The period $T = (364.9 \pm 0.1)$~days and phase $\phi=(184.6 \pm 0.1)$~days are similar to those observed for the
single muon rate.
A more refined study about the time correlation between temperature and
muon rate is
presented
in the next Section.
In Fig.~\ref{fig:LS} (right panel) the generalised Lomb-Scargle periodogram is displayed also for
$T_{eff}$. As for the muon rate, the most significant peak is around 365 days
and other less significant peaks are present.

\section{Cosmic ray flux and effective atmospheric temperature correlation}
\label{sec:correlation}
The possible presence of a time shift $\tau$ between the modulated components of
the cosmic ray muon rate
and of the effective atmospheric temperature has been
investigated using the cross correlation function defined as:

\begin{equation}
R(\tau)~ =~ \int_0^{\Delta t} \frac{I_\mu (t)-I^0_\mu}{I^1_\mu}~\frac{T_{eff}(t-\tau)-T^0_{eff}}{T^1_{eff}}~\frac{dt}{\Delta t} ~\simeq~ \frac{1}{N_{d}}~\Sigma_i~\frac{I_\mu (t_i)-I^0_\mu}{I^1_\mu}~\frac{T_{eff}(t_i-\tau)-T^0_{eff}}{T^1_{eff}}
\label{eqtimecor}
\end{equation}
$I^0_\mu$ and $T^0_{eff}$ are the average values as obtained in the fits of
Sec.~\ref{sec:Cray}, for the single muon cosmic events, and
of Sec.~\ref{sec:Temperature}, for the effective atmospheric temperature, while
$I^1_\mu$ and $T^1_{eff}$ are the corresponding amplitudes of the
modulated components.
The sum runs over the $N_{d}$ days with both measurements.

The correlation function is shown in Fig.~\ref{fig:correphase},
together with the 99\% C.L., in dashed red, evaluated by producing with Monte Carlo
techniques 10$^5$ toy experiments, each one consisting of two time
series, one for the temperature and the other for the muon rate.  The
99\% C.L. value is defined as that for which 99\% of the toy
experiments have a correlation value for $\tau=0$ day lower than it.
Both temperatures and rates have been generated according to the results
of the fits reported in the previous sections, fixing the time period
to 365 days and the phase to 185 days both for the simulated cosmic ray flux
and effective atmospheric temperature.
In the same figure, in dotted blue, the cross correlation function is reported, as
a reference, for one of the toy experiments.
In real data
a peak with maximum at $\tau = 0$ day
is observed
above the expected
contribution of the modulated components.
This peak is due to correlated short term deviations (few
days scale) from the fitted functions in the atmospheric muon rate and the effective temperature.
Both the sinusoidal components and the short term variations of
the two
time series are synchronous.

\begin{figure}[]
\centerline{\includegraphics[width=0.7\linewidth]{correphase6.pdf}}
\caption[]{Cross correlation function (continuous black) between the measured daily muon rate
  and the effective atmospheric temperature. In dotted blue the result of a toy
  Monte Carlo simulation described in the text
  is reported, where the muon rate and the effective
  temperature have been extracted according to the fit results, but with equal
  time period and phase. In dashed red the 99\% significance level is also
  shown (see text for the definition).}
\label{fig:correphase}
\end{figure}

In Fig.~\ref{fig:alphaT} the percentage deviation of the single muon
flux, $\Delta I_\mu/I^0_\mu=(I_\mu-I^0_\mu)/I^0_\mu$, is shown as a function of
the relative effective temperature variation,
$\Delta T_{eff}/T^0_{eff}=(T_{eff}-T^0_{eff})/T^0_{eff}$.
According to the model described in Appendix \ref{sec:Appendix}, a
proportionality relation is expected:
\begin{equation}
\frac{\Delta I_\mu (t)}{I_\mu^0} = \alpha_T ~\frac{\Delta T_{eff} (t)}{T_{eff}^0}
\label{eqalphat}
\end{equation}
The effective temperature coefficient $\alpha_T$ depends
on the energy threshold for atmospheric muons to reach the detector
and on the ratio of kaon/pion production in cosmic rays interactions with
the atmosphere.
The linear fit performed on the plot of Fig.~\ref{fig:alphaT}
gives $\alpha_T=0.95\pm0.04$,
consistent with expectations and with results
from other LNGS experiments (\cite{bib:macro1}-\cite{bib:gerda}).
In the fit, a constant term, $a_0$, has been added
to verify the
consistency of our data with Eq.~\ref{eqalphat}.
Its measured value  $a_0 = -0.08 \pm 0.05$ is consistent with zero, as expected.
The correlation coefficient is $R=0.50$.
It is worth noticing that $R$ is proportional to the cross correlation function
evaluated for $\tau=0$, and can be obtained from Eq.~\ref{eqtimecor}
replacing $I^1_\mu$ and $T^1_{eff}$, the modulation amplitudes, by the
respective standard deviations.

\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{alphat6.pdf}
\caption[]{Correlation between the muon rate and the effective
  temperature relative variations. Symbols in the legend are defined in the text. }
\label{fig:alphaT}
\end{figure}

Two sources of systematic errors have been investigated: the energy
threshold on cosmic ray muons collected by the
detector, affecting
the calculation of the effective atmospheric temperatures, and the
muon rate data-set rescaling.
Using $\langle E_{thr} \cos \theta \rangle \sim 1.8$~TeV, as done by other LNGS experiments,
the weights
have been re-evaluated
resulting
in an effective temperature which is, on average,
0.2~K higher, with no appreciable
effect on $\alpha_T$ measurement and also on the other analyses reported here.

The evaluation of $\alpha_T$ has been performed also
applying scaling factors, different year by year, to the muon flux
measurements,
independently
with and without the RPC system
as done in Sec.~\ref{sec:Cray}; possible effects
due to small changes in detector settings and data acquisition
would be taken into account.
The obtained effective temperature coefficient
$\alpha_T = (0.93 \pm 0.04)$
is compatible with the previously
quoted value within statistical uncertainties.
The systematic error on $\alpha_T$ has been therefore neglected with
respect to the statistical one.

In Fig.~\ref{fig:alphaTvsE} our result is shown together with
the values measured by other experiments as a function of the energy threshold $\langle E_{thr} \cos \theta \rangle$.
The model accounting for pion and kaon contributions to the muon flux is represented by the continuous black line, where the kaon/pion ratio
$r_{K/\pi} = Z_{NK}/Z_{N\pi} = 0.144$ is fixed to the value inferred by the muon charge ratio analysis~\cite{bib:chargeratio2014}.

\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{alphaT_vs_energy7.pdf}
\caption[]{ Experimental values of the correlation coefficient $\alpha_T$ as a function of $\langle E_{thr} \cos \theta \rangle$. The result of this analysis is displayed in black, measurements
reported by other experiments in grey (AMANDA~\cite{bib:amanda}, IceCube~\cite{bib:icecube}, MINOS~\cite{bib:minosf, bib:minosn}, Double Chooz~\cite{bib:db} and Daya Bay~\cite{bib:dayabay}).
The seven measurements performed by five LNGS
experiments (MACRO~\cite{bib:macro1bis}, Borexino~\cite{bib:borex, bib:borex2}, GERDA~\cite{bib:gerda}, LVD~\cite{bib:lvd} and OPERA) are shown in the insert plot.
The LNGS energy threshold - assumed to be $(1.1 \pm 0.2)$~TeV, as evaluated in this paper - is artificially displaced for each data point
for the sake of visualisation.
The continuous black line represents the model
accounting for pions and kaons, with $r_{K/\pi} = 0.144$~\cite{bib:chargeratio2014}. }
\label{fig:alphaTvsE}
\end{figure}

\section{Conclusions}
In this paper we report on studies about the seasonal variation of
the flux of single muons generated
in cosmic rays interactions in the high atmosphere, as measured by the OPERA electronic detectors located in the LNGS underground laboratory.
The observed dependence is
approximated by a
constant
flux
with the presence of a
$(1.55 \pm 0.08) \%$ modulated component
with one year period
and maximum
at
$(186 \pm 3_{\textrm{stat}} \pm 5_{\textrm{sys}})$~day, corresponding to July 5.

An effective atmospheric temperature has been defined as the average of the
temperatures of the air column above the experimental site, weighted by the
production probability of high energy muons, detectable by our underground
detector.
The effective atmospheric temperature
shows a time variation similar to cosmic ray rate changes.

A cross correlation function based study has demonstrated the presence of
short term deviations from a pure yearly modulated model, both in the
cosmic ray flux and in the effective temperature.
These short term deviations and the modulated components appear to be
simultaneous in the two time series.

The effective atmospheric temperature and muon rate variations are positively
correlated ($R=~0.50$). The effective temperature
coefficient is measured to be $\alpha_T = (0.95 \pm 0.04)$, consistent
with the model of $\pi$ and $K$ production in primary cosmic ray
interactions and with previous experimental results. Both the phase and the effective temperature coefficient were measured by several LNGS experiments over different time periods in different decades, and the results agree very well within the quoted uncertainties.


\section{Introduction}
The link between the classical Black-Scholes equation and quantum mechanics and the application of quantum formalism to Mathematical Finance has been investigated by several authors. For example: \cite{AccBoukas}-\cite{Choustova}, \cite{Haven}-\cite{Hidalgo}, \cite{Maslov}, and \cite{McCloud_1}-\cite{Segal}. In particular, the approach of modelling derivative prices using self-adjoint operators on a Hilbert space was suggested by Segal \& Segal in \cite{Segal}. In this paper the authors noted that, in the real world, the market operates with imperfect information and that different observables, such as underlying price and option delta, are usually not simultaneously observable. This fact makes the non-commutative extension of the Black-Scholes framework a natural step. The authors point out that this approach addresses some of the limitations of the classical Black-Scholes model, such as the underestimation of the probability of extreme events - so called ``fat tails". In this sense, non-commutative Quantum models present an alternative means to capture complex market dynamics, without the addition of new stochastic variables.\newline
In \cite{AccBoukas}, Accardi \& Boukas derive a general form for the Quantum Black-Scholes equation based on the Hudson-Parthasarathy calculus (cf \cite{Hudson-Parthasarathy}) and show that a commutative unitary time development operator acting on the market state, leads to a classical Black-Scholes system. Further they give the quantum stochastic differential equation governing the time development operator, and demonstrate how unitary transformations can lead to non-commutativity. An example of a non-commutative Quantum Black-Scholes partial differential equation is derived, although the authors work in an abstract setting and do not discuss specific unitary transformations and Hilbert space representations of financial markets.\newline
Therefore, one objective of this work is to use the Accardi-Boukas framework to look at how different unitary transformations can be used to transform the classical Black-Scholes equation, and to understand how quantum effects become apparent. We then go on to explore an example application of the approach in the modelling of bid-offer spread dynamics. The final objective of the current work is to identify suitable Monte-Carlo methods, which can be used for the simulation of solutions.\newline
In section 2, we give an overview of the Accardi-Boukas derivation of the general form for the Quantum Black-Scholes equation, from \cite{AccBoukas}. With the objective of looking at ``near classical'' Black-Scholes worlds, we then derive specific forms for the resulting partial differential equations that result from small translations, and rotations. This in turn involves the extension of the Accardi-Boukas equation to systems with more than one underlying variable. We go on to discuss how this approach can be applied to the modelling of bid-offer dynamics.\newline
In section 3, we show how this can be linked to the nonlocal diffusion processes discussed by Luczka, H{\"a}nggi and Gadomski in \cite{Luczka}. Here the impact of the diffusion differential operator is spread out through the convolution with a ``blurring" function. The Kramers-Moyal expansion of the nonlocal Fokker-Planck equations allows us to derive the moments of the blurring function for the ``near classical'' quantum system.\newline
This approach allows a natural route to the visualisation of the quantum effects on the system using McKean SDEs (cf \cite{McKean}). The Monte-Carlo methods, developed by Guyon, and Henry-Labord\`ere in \cite{Guyon}, can then be adapted to the simulation of solutions. This is discussed in section 4, where we present numerical results and show how, by introducing small transformations to the system, the stochastic process now reacts to a market downturn by returning higher volatility. This effect is observed even where there is a single static Black-Scholes type volatility.
\section{Quantum Black-Scholes equation}
In this section we follow the notation given, by Accardi \& Boukas, in \cite{AccBoukas}. The current market is represented by a vector in a Hilbert space: $\mathbb{H}$, which contains all relevant information about the state of the market at an instant in time. The tradeable price for a security is represented by an self-adjoint operator on $\mathbb{H}$: $X$, and the the spectrum of $X$ represents possible prices.\newline
\break
Let $L^2[\mathbb{R}^+;\mathbb{H}]$ represent functions from the positive real axis (time) to the Hilbert space $\mathbb{H}$. Then the random behaviour of tradeable securities can be modelled using the tensor product of $\mathbb{H}$ with the bosonic Fock space: $\mathbb{H}\otimes\Gamma (L^2[\mathbb{R}^+;\mathbb{H}])$. We term this the ``market space". The operator that returns the current price becomes $X\otimes \mathbb{I}$, where $\mathbb{I}$ represents the identity operator. The time development of $X\otimes \mathbb{I}$ into the future is modelled by:\newline
\break
$j_t(X)=U_t^*X\otimes\mathbb{I}U_t$\newline
\break
$\mathbb{H}$ carries the initial state of the market and $U_t$ acts by introducing random fluctuations that fill up the empty states in: $\Gamma (L^2[\mathbb{R}^+;\mathbb{H}])$. The functional form for $U_t$ is derived by Hudson \& Parthasarathy in \cite{Hudson-Parthasarathy}, and is given by:\newline
\break
$dU_t=-\Bigg(\bigg(iH+\frac{1}{2}L^*L\bigg)dt+L^*SdA_t-LdA^{\dagger}_t+\bigg(1-S\bigg)d\Lambda_t\Bigg)U_t$\newline
\break
$dA^{\dagger}_t, dA_t, d\Lambda_t$ represent the standard creation, annihilation, and Poisson operators of quantum stochastic calculus. $H, S$ and $L$ also operate on the market space, with $S$ unitary, and $H$ self-adjoint. The multiplication rules of the Hudson-Parthasarathy calculus are given below (cf \cite{Hudson-Parthasarathy}):\newline
\break
\begin{tabular}{p{1cm}|p{1cm}p{1cm}p{1cm}p{1cm}}
-&$dA^{\dagger}_t$&$d\Lambda_t$&$dA_t$&$dt$\\
\hline
$dA^{\dagger}_t$&0&0&0&0\\
$d\Lambda_t$&$dA^{\dagger}_t$&$d\Lambda_t$&0&0\\
$dA_t$&$dt$&$dA_t$&0&0\\
$dt$&0&0&0&0
\end{tabular}\newline
\break
The first thing to note is that, for $S\neq 1$, there is a non-zero Poisson term and the time development operator is non-commutative.\newline
\break
The next thing to note is that, where $S=1$, the Poisson term disappears. The model can be written using the Ito calculus in place of the more general Hudson-Parthasarathy framework. The Wiener process $dW_t$ can be modelled using: $dA_t+dA^{\dagger}_t$.\newline
\break
Let $V_T=j_T(X-K)^+$, represent the option price process as at final expiry $T$, and $K$ the operator given by multiplying by the strike. Further, for $V_t=j_t(X-K)^+$ the following expansion is assumed:\newline
\break
$V_t=F(t,x)=\sum_{n,k} a_{n,k}(t-t_0)^n(x-x_0)^k$\newline
\break
The Hudson-Parthasarathy multiplication rules can be applied to this expansion to give a quantum stochastic differential equation for $V_t$, that corresponds to the usual Ito expansion used in the derivation of the classical Black-Scholes. By assuming one can construct a hedge portfolio by holding the underlying and a risk free numeraire asset, Accardi \& Boukas are able to derive the general form the Quantum Black Scholes equation using the assumption that any portfolio must be self financing. Proposition 1, from \cite{AccBoukas} gives the full Quantum Black-Scholes equation:\newline
\break
\begin{equation}\label{QBS}
a_{1,0}(t,j_t(X))+a_{0,1}(t,j_t(X))j_t(\theta)+\sum_{k=2}^{\infty} a_{0,k}(t,j_t(X))j_t(\alpha\lambda^{k-2}\alpha^{\dagger})=a_tj_t(\theta)+V_tr-a_tj_t(X)r
\end{equation}
Here, $a_t$ represents the holding in the underlying asset and is given by the boundary conditions:\newline
\break
$\sum_{k=1}^{\infty} a_{0,k}(t,j_t(X))j_t(\lambda^{k-1}\alpha^{\dagger})=a_tj_t(\alpha^{\dagger})$\newline
$\sum_{k=1}^{\infty} a_{0,k}(t,j_t(X))j_t(\alpha\lambda^{k-1})=a_tj_t(\alpha)$\newline
$\sum_{k=1}^{\infty} a_{0,k}(t,j_t(X))j_t(\lambda^{k})=a_tj_t(\lambda)$\newline
\break
Further, $\theta, \alpha$ and $\lambda$ are given by:\newline
\break
$\alpha=[L^*,X]S$, $\lambda=S^*XS-X$, $\theta=i[H,X]-\frac{1}{2}\{L^*LX+XL^*L+2L^*XL\}$. In this case the boundary conditions arise because when the Poisson term: $d\Lambda$ is non-zero, unlike Ito calculus where expansion terms with order above 2 can be ignored, higher order terms still contain non-vanishing contribution.
\subsection{Translation}
The natural Hilbert space for an equity price (say the FTSE price) is: $\mathbb{H}=L^2[\mathbb{R}]$. In this case, the only unitary transactions we can use are the translations:\newline
\break
$T_\epsilon:f(x)\rightarrow f(x-\varepsilon)$\newline
\break
Here we have, for a translation invariant Lebesgue measure $\mu$:\newline
\break
$\langle T_\varepsilon f|T_\varepsilon g\rangle = \int_{\mathbb{R}} \compconj{f(x-\varepsilon)}g(x-\varepsilon)d\mu=\int_{\mathbb{R}} \compconj{f(x)}g(x)d\mu=\langle f|g\rangle$\newline
\break
So S is unitary in this case. Therefore, translating by $\varepsilon$ we get:\newline
\break
$\lambda=T_{-\varepsilon}XT_\varepsilon f(x) - Xf(x)=T_{-\varepsilon}xf(x-\varepsilon )-xf(x)=(x+\varepsilon )f(x)-xf(x)=\varepsilon f(x)$\newline
\break
So we have $\lambda=\varepsilon$, and it is clear the example given in \cite{AccBoukas} relates to a translation by $\varepsilon=1$. Following the key steps from \cite{AccBoukas} Proposition 3, and inserting this back into equation \ref{QBS}, we get the following Quantum Black-Scholes partial differential equation for this system:\begin{lemma}
Let $u(t,x)$ represent the price at time t, of a derivative contract in the system described above under small translation $\varepsilon$, and with interest rate $r$. Then the quantum Black-Scholes equation becomes:
\begin{equation} \label{eq:1}
\frac{\partial u(t,x)}{\partial t}=rx\frac{\partial u(t,x)}{\partial x}-u(t,x)r+\sum_{k=2}^{\infty}\frac{\varepsilon^{k-2}}{k!}\frac{\partial^k u(t,x)}{\partial x^k}g(x)
\end{equation}
\end{lemma}
\begin{proof}
The proof follows the same steps Accardi \& Boukas outline in \cite{AccBoukas} proposition 3, with small modifications.
\end{proof}
For $\varepsilon=0$, the last term drops out, and the equation reverts to the classical Black-Scholes. We investigate the impact of non-zero $\varepsilon$ in section 4.
\subsection{Rotation}
For the one dimensional market space: $L^2[\mathbb{R}]$, the Lebesgue invariant translations, are the only unitary transformations available. However, the true current state of the financial market contains a much richer variety of information than just a single price, and by increasing the dimensionality of the Market space accordingly we introduce a wider variety of unitary transformations, that can introduce non-commutativity. For example, let $x$ represent the FTSE mid-price, and $\epsilon$ half of the bid-offer spread so that $(x+\epsilon)$ represents the best offer-price and $(x-\epsilon)$ the best bid-price. Now the market is represented by the Hilbert space: $\mathbb{H}=L^2[\mathbb{R}^2]$, and we can apply rotations, in addition to translations.\newline
\break
We make the simplifying assumption that market participants can trade the mid-price: $x$ (for example during the end of day auction process) and that the market has sufficient liquidity to enable participants to alternatively act as market makers (receiving bid-offer spread) or as hedgers (crossing bid-offer spread) and therefore trade the bid-offer spread: $\epsilon$. Therefore we make the following assumption:
\begin{assumption}\label{ass_1}
For any derivative payout $V(x_T,\epsilon_T)$, we can construct a hedged portfolio, and can proceed with the derivation of the Quantum Black Scholes equation following the basic methodology from \cite{AccBoukas}.
\end{assumption}
We now have separate creation, annihilation and Poisson operators, for $x$ and $\epsilon$; $dA_x$, $dA_{\epsilon}$ etc. These can be combined using the multiplication table (\cite{Hudson-Parthasarathy}, Theorem 4.5), by making the assumption that the bid-offer is uncorrelated with the equity price. This corresponds to assumption
\ref{ass_2}:
\begin{assumption}\label{ass_2}
$dA_xd\Lambda_{\epsilon}=dA_{\epsilon}d\Lambda_x=d\Lambda_xd\Lambda_{\epsilon}=d\Lambda_{\epsilon}d\Lambda_x=dA_xdA^{\dagger}_{\epsilon}=dA_{\epsilon}dA^{\dagger}_x=d\Lambda_xdA^{\dagger}_{\epsilon}=d\Lambda_{\epsilon}dA^{\dagger}_x=0$.
\end{assumption}
Lastly, we make the assumption that we can expand the derivative payout as before:
\begin{assumption}\label{ass_3}
$V_t=F(t,x,\epsilon)=\sum_{n,k,l} a_{n,l,k}(t-t_0)^n(x-x_0)^k(\epsilon-\epsilon_0)^l$
\end{assumption}
We can now derive the relevant Quantum Black-Scholes equation:
\begin{proposition}
Let $\mathbb{H}=L^2[\mathbb{R}^2]$, and let $X\otimes 1$ and $\epsilon\otimes 1$ operate on the market space: $\mathbb{H}\otimes\Gamma (L^2[\mathbb{R}^+;\mathbb{H}])$, to return the mid-price, and bid-offer spread for a tradeable security respectively. Further, let the notation from \cite{AccBoukas}, and the above assumptions apply.\newline
\break
Then the Quantum Black-Scholes equation in this case is given by:\newline
\begin{equation}\label{QBS2}
\begin{split}
a_{1,0,0}(t,j_t(X),j_t(\epsilon))+a_{0,1,0}(t,j_t(X),j_t(\epsilon))j_t(\theta_x)+a_{0,0,1}(t,j_t(X),j_t(\epsilon))j_t(\theta_{\epsilon})\\
+\sum_{k=2}^{\infty} a_{0,k,0}(t,j_t(X),j_t(\epsilon))j_t(\alpha_x\lambda_x^{k-2}\alpha_x^{\dagger})+\sum_{l=2}^{\infty} a_{0,0,l}(t,j_t(X),j_t(\epsilon))j_t(\alpha_{\epsilon}\lambda_{\epsilon}^{l-2}\alpha_{\epsilon}^{\dagger})\\
=a_{x,t}j_t(\theta_x)+a_{\epsilon,t}j_t(\theta_{\epsilon})+V_tr-a_{x,t}j_t(X)r-a_{\epsilon,t}j_t(\epsilon)r
\end{split}
\end{equation}
Where for $j_t(X)$:
\begin{equation}\label{bc_x}
\begin{split}
\sum_{k=1}^{\infty} a_{0,k,0}(t,j_t(X),j_t(\epsilon))j_t(\lambda_x^{k-1}\alpha_x^{\dagger})=a_{x,t}j_t(\alpha_x^{\dagger})\\
\sum_{k=1}^{\infty} a_{0,k,0}(t,j_t(X),j_t(\epsilon))j_t(\alpha_x\lambda_x^{k-1})=a_{x,t}j_t(\alpha_x)\\
\sum_{k=1}^{\infty} a_{0,k,0}(t,j_t(X),j_t(\epsilon))j_t(\lambda_x^{k})=a_{x,t}j_t(\lambda_x)\\
\end{split}
\end{equation}
and for $j_t(\epsilon)$:\newline
\begin{equation}\label{bc_e}
\begin{split}
\sum_{l=1}^{\infty} a_{0,0,l}(t,j_t(X),j_t(\epsilon))j_t(\lambda_{\epsilon}^{l-1}\alpha_{\epsilon}^{\dagger})=a_{\epsilon,t}j_t(\alpha_{\epsilon}^{\dagger})\\
\sum_{l=1}^{\infty} a_{0,0,l}(t,j_t(X),j_t(\epsilon))j_t(\alpha_{\epsilon}\lambda_{\epsilon}^{l-1})=a_{\epsilon,t}j_t(\alpha_{\epsilon})\\
\sum_{l=1}^{\infty} a_{0,0,l}(t,j_t(X),j_t(\epsilon))j_t(\lambda_{\epsilon}^{l})=a_{\epsilon,t}j_t(\lambda_{\epsilon})
\end{split}
\end{equation}
\end{proposition}
\begin{proof}
First, the equations for time-development operators for $X\otimes 1$, and $\epsilon\otimes 1$ become:\newline
\break
$dU_{x,t}=-\Bigg(\bigg(iH+\frac{1}{2}L_x^*L_x\bigg)dt+L_x^*SdA_x-L_xdA_x^{\dagger}+\bigg(1-S\bigg)d\Lambda_x$\newline
$dU_{\epsilon,t}=-\Bigg(\bigg(iH+\frac{1}{2}L_{\epsilon}^*L_{\epsilon}\bigg)dt+L_{\epsilon}^*SdA_{\epsilon}-L_{\epsilon}dA_{\epsilon}^{\dagger}+\bigg(1-S\bigg)d\Lambda_{\epsilon}$\newline
\break
Then, applying the Hudson-Parthasarathy multiplication rules to the expansion given in assumption \ref{ass_3} gives:
\begin{equation}\label{expansion}
\begin{split}
dV_t=\bigg(a_{1,0,0}(t,j_t(x),j_t(\epsilon))+a_{0,1,0}(t,j_t(x),j_t(\epsilon))j_t(\theta_x)+a_{0,0,1}(t,j_t(x),j_t(\epsilon))j_t(\theta_{\epsilon})\\
+\sum_{k=2}^{\infty} a_{0,k,0}(t,j_t(X),j_t(\epsilon))j_t(\alpha_x\lambda_x^{k-2}\alpha_x^{\dagger})+\sum_{l=2}^{\infty} a_{0,0,l}(t,j_t(X),j_t(\epsilon))j_t(\alpha_{\epsilon}\lambda_{\epsilon}^{l-2}\alpha_{\epsilon}^{\dagger})\bigg)dt\\
+\bigg(a_{0,1,0}(t,j_t(X),j_t(\epsilon))j_t(\alpha_x)+\sum_{k=2}^{\infty} a_{0,k,0}(t,j_t(X),j_t(\epsilon))j_t(\alpha_x\lambda_x^{k-1})\bigg)dA_x\\
+\bigg(a_{0,0,1}(t,j_t(X),j_t(\epsilon))j_t(\alpha_{\epsilon})+\sum_{l=2}^{\infty} a_{0,0,l}(t,j_t(X),j_t(\epsilon))j_t(\alpha_{\epsilon}\lambda_{\epsilon}^{k-1})\bigg)dA_{\epsilon}\\
+\bigg(a_{0,1,0}(t,j_t(X),j_t(\epsilon))j_t(\alpha_x^{\dagger})+\sum_{k=2}^{\infty} a_{0,k,0}(t,j_t(X),j_t(\epsilon))j_t(\lambda_x^{k-1}\alpha_x^{\dagger})\bigg)dA_x^{\dagger}\\
+\bigg(a_{0,0,1}(t,j_t(X),j_t(\epsilon))j_t(\alpha_{\epsilon}^{\dagger})+\sum_{l=2}^{\infty} a_{0,0,l}(t,j_t(X),j_t(\epsilon))j_t(\lambda_{\epsilon}^{k-1}\alpha_{\epsilon}^{\dagger})\bigg)dA_{\epsilon}^{\dagger}
\end{split}
\end{equation}
Where $\theta_x, \theta_{\epsilon}$ are given by:\newline
\break
$\theta_x=i[H,X]-\frac{1}{2}\bigg(L_x^*L_xX+XL_x^*L_x-2L_x^*XL_x\bigg)$\newline
$\theta_{\epsilon}=i[H,\epsilon]-\frac{1}{2}\bigg(L_{\epsilon}^*L_{\epsilon}\epsilon+\epsilon L_{\epsilon}^*L_{\epsilon}-2L_{\epsilon}^*\epsilon L_{\epsilon}\bigg)$\newline
\break
$\alpha_x, \alpha_{\epsilon}$ are given by:\newline
\break
$\alpha_x=[L_x^*,X]S$\newline
$\alpha_{\epsilon}=[L_{\epsilon}^*,\epsilon]S$\newline
\break
and finally $\lambda_x, \lambda_{\epsilon}$ are given by:\newline
\break
$\lambda_x=S^*XS-X$\newline
$\lambda_{\epsilon}=S^*\epsilon S-\epsilon$\newline
\break
By assumption \ref{ass_1} we can form a hedge portfolio which we now use:\newline
\break
$V_t=a_{x,t}j_t(X)+a_{\epsilon,t}j_t(\epsilon)+b_t\beta$, for risk free numeraire asset $\beta$.\newline
\break
$dV_t=a_{x,t}dj_t(X)+a_{\epsilon,t}dj_t(\epsilon)+b_t\beta rdt$\newline
\break
Applying the unitary time development operators for $\epsilon$ and $x$ we have:\newline
\break
\begin{equation}\label{hedges}
\begin{split}
dV_t=a_{x,t}\big(j_t(\alpha_x^{\dagger})dA_x^{\dagger}+j_t(\lambda_x)d\Lambda_x+j_t(\alpha_x)dA_x\big)\\
+a_{\epsilon,t}\big(j_t(\alpha_{\epsilon}^{\dagger})dA_{\epsilon}^{\dagger}+j_t(\lambda_{\epsilon})d\Lambda_{\epsilon}+j_t(\alpha_{\epsilon})dA_{\epsilon}\big)\\
+\big(j_t(\theta_x)+(V_t-a_{x,t}j_t(X)-a_{\epsilon,t}j_t(\epsilon))r\big)dt
\end{split}
\end{equation}
\break
Equating the risky terms between equations (\ref{expansion}), and (\ref{hedges}) leads to the boundary conditions, (\ref{bc_x}) and (\ref{bc_e}) on $a_{x,t}$ and $a_{\epsilon,t}$. Similarly, equating the $dt$ terms, leads to the Quantum Black-Scholes equation for this system: equation (\ref{QBS2}).
\end{proof}
Now, let $f\big(x,\epsilon\big)$ represent a vector in $\mathbb{H}$, and apply a rotation matrix:\newline
$S=\begin{bmatrix} cos(\phi) && -sin(\phi) \\ sin(\phi) && cos(\phi) \end{bmatrix}$\newline
\break
We have:\newline
\break
$Sf\big(x,\epsilon\big)=f\big(cos(\phi)x-sin(\phi)\epsilon,cos(\phi)\epsilon+sin(\phi)x\big)$\newline
\break
$XSf=xf\big(cos(\phi)x-sin(\phi)\epsilon,cos(\phi)\epsilon+sin(\phi)x\big)$\newline
\break
$S^*XSf=\big(cos(\phi)x+sin(\phi)\epsilon\big)f(x,\epsilon)$\newline
\break
So, we end up with:\newline
\break
$\lambda_x=\bigg(\big(cos(\phi)-1\big)x+sin(\phi)\epsilon\bigg)$, $\lambda_{\epsilon}=\bigg(\big(cos(\phi)-1\big)\epsilon-sin(\phi)x\bigg)$.\newline
\break
Finally, inserting this back into equation (\ref{QBS2}), we get the Black-Scholes equation for the system (following notation from \cite{AccBoukas}):
\begin{proposition}
Let $u(t,x,\epsilon)$ represent the price at time t, of a derivative contract in the system described above under rotation $\phi$, and with interest rate $r$. Then the quantum Black-Scholes equation becomes:
\begin{equation}\label{eq:2}
\begin{split}
\frac{\partial u(t,x,\epsilon)}{\partial t}=rx\frac{\partial u(t,x,\epsilon)}{\partial x}+r\epsilon\frac{\partial u(t,x,\epsilon)}{\partial \epsilon}-u(t,x,\epsilon)r\\
+\sum_{k=2}^{\infty}\frac{((cos(\phi)-1)x+sin(\phi)\epsilon)^{k-2}}{k!}\frac{\partial^k u(t,x,\epsilon)}{\partial x^k}g_1(x,\epsilon)\\
+\sum_{l=2}^{\infty}\frac{((cos(\phi)-1)\epsilon-sin(\phi)x)^{l-2}}{l!}\frac{\partial^l u(t,x,\epsilon)}{\partial \epsilon^l}g_2(x,\epsilon)
\end{split}
\end{equation}
\end{proposition}
\begin{proof}
We assume that the operators $L_x, L_x^*,L_{\epsilon}, L_{\epsilon}^*$ involve multiplication by a polynomial in $x, \epsilon$, and therefore commute with $\lambda_x, \lambda_{\epsilon}$. Therefore, from the boundary conditions we have:\newline
\break
$\sum_{k=1}^{\infty} a_{0,k,0}(t,j_t(X),j_t(\epsilon))j_t(\lambda_x^{k-1})=a_{x,t}$\newline
\break
$\sum_{l=1}^{\infty} a_{0,0,l}(t,j_t(X),j_t(\epsilon))j_t(\lambda_{\epsilon}^{l-1})=a_{\epsilon,t}$\newline
\break
Inserting this into \ref{QBS2} gives:
\begin{equation}
\begin{split}
a_{1,0,0}(t,j_t(X),j_t(\epsilon))+a_{0,1,0}(t,j_t(X),j_t(\epsilon))j_t(X)r+a_{0,0,1}(t,j_t(X),j_t(\epsilon))j_t(\epsilon)r\\
+\sum_{k=2}^{\infty} a_{0,k,0}(t,j_t(X),j_t(\epsilon))j_t(\lambda_x^{k-2}(\alpha_x\alpha_x^*-\lambda_x(\theta_x-xr)))\\
+\sum_{l=2}^{\infty} a_{0,0,l}(t,j_t(X),j_t(\epsilon))j_t(\lambda_{\epsilon}^{l-2}(\alpha_{\epsilon}\alpha_{\epsilon}^*-\lambda_{\epsilon}(\theta_{\epsilon}-\epsilon r)))\\
=V_tr
\end{split}
\end{equation}
Now writing $g_1(x,\epsilon)=j_t(\alpha_x\alpha_x^*-\lambda_x(\theta_x-xr))$, $g_2(x,\epsilon)=j_t(\alpha_{\epsilon}\alpha_{\epsilon}^*-\lambda_{\epsilon}(\theta_{\epsilon}-\epsilon r))$, and $a_{0,k,0}(t,j_t(X),j_t(\epsilon))=\frac{1}{k!}\frac{\partial^k u}{\partial x^k}, a_{0,0,l}(t,j_t(X),j_t(\epsilon))=\frac{1}{l!}\frac{\partial^l u}{\partial {\epsilon}^l}$, we have the result given.
\end{proof}
For small rotations, we have $cos(\phi)=1-\frac{\varepsilon^2}{2}+o(\varepsilon^2)$, and $sin(\phi)=\varepsilon + o(\varepsilon^2)$. Inserting this into equation (\ref{eq:2}), we have a new partial differential equation, where the coefficient of the $k$th partial derivative, for $k\geq 3$, with respect to $x, \epsilon$, is correct to $o(\varepsilon^{2(k-2)})$. This form for small rotations is more amenable to the methods we apply in section 3.
\begin{equation}\label{eq:3}
\begin{split}
\frac{\partial u(t,x,\epsilon)}{\partial t}=rx\frac{\partial u(t,x,\epsilon)}{\partial x}+r\epsilon\frac{\partial u(t,x,\epsilon)}{\partial \epsilon}-u(t,x,\epsilon)r\\
+\sum_{k=2}^{\infty}\frac{(\varepsilon\epsilon-(\varepsilon^2/2)x)^{k-2}}{k!}\frac{\partial^k u(t,x,\epsilon)}{\partial x^k}g_1(x,\epsilon)\\
+\sum_{l=2}^{\infty}\frac{(-\varepsilon x-(\varepsilon^2/2)\epsilon)^{l-2}}{l!}\frac{\partial^l u(t,x,\epsilon)}{\partial \epsilon^l}g_2(x,\epsilon)
\end{split}
\end{equation}
As is the case for equation (\ref{eq:1}), this reduces to the classical Black-Scholes for 2 uncorrelated random variables (in this case price: $x$, and bid-offer spread: $\epsilon$) when $\varepsilon=0$.\newline
\break
For the classical case, the addition of the bid-offer spread is in some ways unnecessary when using the model for derivative pricing. For derivative contracts depending on the close price, one can usually hedge daily at the closing price during the end of day auction process. For many trading desks this may be sufficient in practice, and terms involving the bid-offer spread will drop out of the model. In the quantum case, examination of equations (\ref{eq:2}) and (\ref{eq:3}) shows that we expect interference between the bid-offer spread dynamics and the price dynamics. For small rotations, these equations are singular PDEs, and we expect the behaviour in most regions to approximate classical behaviour. However, when the higher derivative terms are larger, quantum interference may be significant. We discuss this more in sections 3 and 4.
\section{Nonlocal Diffusions}
In this section, we derive the Fokker-Planck equations associated to the Quantum Black-Scholes equations: (\ref{eq:1}), and (\ref{eq:3}). We show how these can be written in integral form, by using the Kramers-Moyal expansion (see for example \cite{Frank}). This enables us to link the Quantum Black-Scholes models of the previous section to nonlocal diffusions (see for example the paper by Luczka, H{\"a}nggi and Gadomski: \cite{Luczka}). We assume zero interest rates in this section to help clarify the notation without changing the key dynamics. The integral form for the Fokker-Planck equations is given by:
\begin{equation}\label{FokkerPlanck1}
\begin{split}
\frac{\partial p(t,x,\epsilon)}{\partial t}=\frac{1}{2}\frac{\partial^2}{\partial x^2}\bigg(\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\big(H(y_x,y_{\epsilon}|x,\epsilon)g_1(x,\epsilon)p(x-y_x,\epsilon-y_{\epsilon},t)\big)dy_xdy_{\epsilon}\bigg)\\
+\frac{1}{2}\frac{\partial^2}{\partial {\epsilon}^2}\bigg(\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\big(H(y_x,y_{\epsilon}|x,\epsilon)g_2(x,\epsilon)p(x-y_x,\epsilon-y_{\epsilon},t)\big)dy_xdy_{\epsilon}\bigg)
\end{split}
\end{equation}
The function $H(y_x,y_{\epsilon}|x,\epsilon)$ has the effect of "blurring" the impact of the diffusion operator. In the case that $H(y_x,y_{\epsilon}|x,\epsilon)$ is a Dirac delta function, the diffusion operator is localised as usual, and the associated Fokker-Planck equation reduces to the standard Kolmogorov forward equation associated with the classical Black-Scholes.\newline
\break
We start with the following general form for equations (\ref{eq:1}) and (\ref{eq:3}):
\begin{equation}\label{eq:4}
\frac{\partial u(t,x,\epsilon)}{\partial t}=g_1(x,\epsilon)\sum_{k=2}^{\infty}\frac{f_1(x,\epsilon,\varepsilon)^{k-2}}{k!}\frac{\partial^k u(t,x,\epsilon)}{\partial x^k}+g_2(x,\epsilon)\sum_{l=2}^{\infty}\frac{f_2(x,\epsilon,\varepsilon)^{l-2}}{l!}\frac{\partial^l u(t,x,\epsilon)}{\partial \epsilon^l}
\end{equation}
\begin{proposition}
The Fokker-Planck equation associated to equation (\ref{eq:4}), with $r=0$ is given by:
\begin{equation}\label{FokkerPlanck2}
\begin{split}
\frac{\partial p(t,x,\epsilon)}{\partial t}=\sum_{k=2}^{\infty}\frac{(-1)^k}{k!}\frac{\partial^k \big(g_1(x,\epsilon)f_1(x,\epsilon,\varepsilon)^{k-2}p(t,x,\epsilon)\big)}{\partial x^k}\\
+\sum_{l=2}^{\infty}\frac{(-1)^l}{l!}\frac{\partial^l \big(g_2(x,\epsilon)f_2(x,\epsilon,\varepsilon)^{l-2}p(t,x,\epsilon)\big)}{\partial \epsilon^l}
\end{split}
\end{equation}
\end{proposition}
\begin{proof}
For a derivative payout $h(x,\epsilon)$, with zero interest rates, we have the following price in risk neutral measure $Q$:\newline
\break
$u(x_t,\epsilon_t, t)=E^{Q}\big[h(x_T,\epsilon_T)\big]=\int_{\mathbb{R}^2} h(y_x,y_{\epsilon})p(y_x,y_{\epsilon}|x,\epsilon,t)dy_xdy_{\epsilon}$\newline
\break
Where $p(y_x,y_{\epsilon}|x,\epsilon,t)$ represents the risk neutral probability density for the variables observed at time $T$, conditional on the values at time $t$. $h(x,\epsilon)$ represents a derivative payout at $T$. We then write the right hand integral as:\newline
\break
$\int_{\mathbb{R}^2} g(y_x,y_{\epsilon})p(y_x,y_{\epsilon}|x,\epsilon,t)dy_xdy_{\epsilon}=\int_0^t\int_{\mathbb{R}^2} Lh(y_x,y_{\epsilon})p(y_x,y_{\epsilon}|x,\epsilon,s)dy_xdy_{\epsilon}ds$\newline
\break
Where $L$ represents the operator:\newline
\break
$Lh(x,\epsilon)=\bigg(g_1(x,\epsilon)\sum_{k=2}^{\infty}\frac{f_1(x,\epsilon,\varepsilon)^{k-2}}{k!}\frac{\partial^k}{\partial x^k}+g_2(x,\epsilon)\sum_{l=2}^{\infty}\frac{f_2(x,\epsilon,\varepsilon)^{l-2}}{l!}\frac{\partial^l}{\partial \epsilon^l}\bigg)h(x,\epsilon)$\newline
\break
The Fokker-Planck equation, is given by the adjoint operator $L^*$. Therefore, since:\newline
\break
$\int_0^t\int_{\mathbb{R}^2} Lh(y_x,y_{\epsilon})p(y_x,y_{\epsilon}|x,\epsilon,s)dy_xdy_{\epsilon}ds=\int_0^t\int_{\mathbb{R}^2} h(y_x,y_{\epsilon})L^*p(y_x,y_{\epsilon}|x,\epsilon,s)dy_xdy_{\epsilon}ds$\newline
\break
If we truncate equation (\ref{eq:4}) at a certain order for the derivative: $N$, the result follows by integrating by parts $N$ times. Proceeding with higher and higher $N$, we can match the derivative terms of any arbitary order, and the result follows.
\end{proof}
The objective now, is to write equation (\ref{FokkerPlanck2}) in the form of (\ref{FokkerPlanck1}). To do this we can follow a Moment Matching algorithm. We use the following expansion:\newline
\break
$g(x,\epsilon)p(x-y_x,\epsilon-y_{\epsilon},t)=\sum_{i,j=0}^{\infty} \frac{(-1)^{(i+j)}}{(i+j)!}y_x^iy_{\epsilon}^j\frac{d^{i+j}(g(x,\epsilon)p(x,\epsilon))}{dx^id{\epsilon}^j}$\newline
\break
Inserting this into equation (\ref{FokkerPlanck1}) gives:
\begin{equation}\label{eq:5}
\begin{split}
\frac{\partial p(t,x,\epsilon)}{\partial t}=\frac{1}{2}\frac{\partial^2}{\partial x^2}\bigg(\sum_{i,j=0}^{\infty}\frac{(-1)^{(i+j)}}{(i+j)!}\frac{\partial^{i+j}(g_1(x,\epsilon)p(x,\epsilon))}{\partial x^i\partial\epsilon^j}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}H(y_x,y_{\epsilon}|x,\epsilon)y_x^iy_{\epsilon}^jdy_xdy_{\epsilon}\bigg)\\
+\frac{1}{2}\frac{\partial^2}{\partial {\epsilon}^2}\bigg(\sum_{i,j=0}^{\infty}\frac{(-1)^{(i+j)}}{(i+j)!}\frac{\partial^{i+j}(g_2(x,\epsilon)p(x,\epsilon))}{\partial x^i\partial\epsilon^j}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}H(y_x,y_{\epsilon}|x,\epsilon)y_x^iy_{\epsilon}^jdy_xdy_{\epsilon}\bigg)
\end{split}
\end{equation}
Now by equating the coefficients of the derivatives with respect to $x$ and $\epsilon$, between equations (\ref{eq:5}) and (\ref{FokkerPlanck2}) one can calculate the moments of the ``blurring" function $H(y_x,y_{\epsilon}|x,\epsilon)$. For the translation case, $g_2(x,\epsilon)=0$, and the probability density is a function of $x$ only.
\subsection{Moment Matching: Translation Case}
In the translation case, of section 2.1, since the coefficients of each differential term in equation (\ref{eq:1}) is a constant multiplied by $g(x)$, the moments of the ``blurring'' function $H(y)$ will not depend of $x$. Equation (\ref{eq:5}) becomes:
\begin{equation}\label{eq:6}
\frac{\partial p(t,x)}{\partial t}=\frac{1}{2}\bigg(\sum_{j=0}^{\infty}\frac{(-1)^{(j)}}{j!}\frac{d^{(j+2)}(g(x)p(x))}{dx^{(j+2)}}\int_{-\infty}^{\infty}H(y)y^jdy\bigg)
\end{equation}
Similarly, the Fokker-Planck associated with equation (\ref{eq:1}), with $r=0$, is given by:
\begin{equation}\label{eq:7}
\frac{\partial p(t,x)}{\partial t}=\sum_{k=2}^{\infty}\frac{(-1)^k\varepsilon^{k-2}}{k!}\frac{\partial^k (g(x)p(t,x))}{\partial x^k}
\end{equation}
Now the moments of the ``blurring'' function can be matched by equating directly equations (\ref{eq:6}) and (\ref{eq:7}):
\begin{proposition}\label{trans_mom}
Let $H_i$ represent the $i^{th}$ moment of $H(y)$, for the Fokker-Planck equation (\ref{FokkerPlanck2}), relating to the translation case described in section 2.1. Then, $H_i$ is given by:\newline
\break
$H_i=\frac{2(-\varepsilon)^i}{(i+1)(i+2)}$
\end{proposition}
\begin{proof}
$H_i$ follows (for $i\geq 0$) by equating the coefficients for: $\frac{\partial^{(i+2)}}{\partial x^{(i+2)}}$, between equations (\ref{eq:6}) and (\ref{eq:7}).
\end{proof}
We find that, in this case, $H(y)$ is a normalised function that tends to a Dirac function as $\varepsilon$ tends to zero, and for $\varepsilon=0$ we end up with classical 2nd order Fokker-Planck equation. This is discussed further in section 4.
\subsection{Moment Matching: Rotation Case}
In the rotation case of section 2.2, the coefficients of each differential term in equation (\ref{FokkerPlanck2}) are functions of $x$ and $\epsilon$. Therefore, we require the moments for the ``blurring'' function also to be functions of $x$, and $\epsilon$: $H(y_x,y_{\epsilon}|e,\epsilon)$. Once we have calculated the coefficients for the differential terms, we can use these to form an inhomogeneous 2nd order differential equation for the moments of $H(y_x,y_{\epsilon}|e,\epsilon)$.\newline
\break
In this case, from equation (\ref{FokkerPlanck2}) we have: $f_1(x,\epsilon)=\varepsilon\epsilon-(\varepsilon^2/2)x$, and $f_2(x,\epsilon)=-\varepsilon x-(\varepsilon^2/2)\epsilon$. Therefore, the Fokker-Planck equation associated with equation (\ref{eq:3}), with $r=0$, is given by:
\begin{equation}\label{eq:8}
\begin{split}
\frac{\partial p(t,x,\epsilon)}{\partial t}=\sum_{k=2}^{\infty}\frac{1}{k!}\frac{\partial^k \bigg(\big((\varepsilon^2/2)x-\varepsilon\epsilon\big)^{k-2}g_1(x,\epsilon)p(t,x,\epsilon)\bigg)}{\partial x^k}\\
+\sum_{l=2}^{\infty}\frac{1}{l!}\frac{\partial^l \bigg(\big(\varepsilon x+(\varepsilon^2/2)\epsilon\big)^{l-2}g_2(x,\epsilon)p(t,x,\epsilon)\bigg)}{\partial \epsilon^l}
\end{split}
\end{equation}
The moments of the ``blurring'' function will now follow by equating coefficients for the differential terms between equations (\ref{eq:5}), and (\ref{eq:8}).
\begin{proposition}\label{rot_mom}
Where the moments of the ``blurring'' function: $H(y_x,y_{\epsilon}|x,\epsilon)$ are given by:\newline
\break
$a^i_{x}=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}H(y_x,y_{\epsilon}|x,\epsilon)y_x^idy_xdy_{\epsilon}$\newline
$a^j_{\epsilon}=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}H(y_x,y_{\epsilon}|x,\epsilon)y_{\epsilon}^jdy_xdy_{\epsilon}$\newline
\break
and $a^0, a^1_x, a^1_{\epsilon}$ are assumed to be:\newline
\break
$a^0=1, a^1_x, a^1_{\epsilon}=0$\newline
\break
Then for the higher moments we have, for $n\geq 2$:
\begin{equation}
\frac{(-1)^na^{n-2}_x+2n\frac{\partial a^{n-1}_x}{\partial x}+n(n-1)\frac{\partial^2 a^{n}_x}{\partial x^2}}{n!}=\frac{((\varepsilon^2/2)x-\varepsilon\epsilon)^{n-2}}{n(n-1)(1-(\varepsilon^2/2))^{(n-1)}}
\end{equation}
\begin{equation}
\frac{(-1)^na^{n-2}_{\epsilon}+2n\frac{\partial a^{n-1}_{\epsilon}}{\partial \epsilon}+n(n-1)\frac{\partial^2 a^{n}_{\epsilon}}{\partial {\epsilon}^2}}{n!}=\frac{((\varepsilon^2/2)\epsilon+\varepsilon x)^{n-2}}{n(n-1)(1-(\varepsilon^2/2))^{(n-1)}}
\end{equation}
\end{proposition}
\begin{proof}
We first calculate the coefficients for $\frac{\partial^n (g_1(x,\epsilon)p(x,\epsilon))}{\partial x^n}$ from equation (\ref{eq:8}).\newline
\break
The 2nd order coefficient is given by:\newline
\break
$\sum_{i\geq 2} \frac{(i-2)!(\varepsilon^2/2)^{i-2}\binom{i}{2}}{i!}=\frac{1}{2}\sum_{i\geq 0} (\varepsilon^2/2)^i=\frac{1}{2(1-(\varepsilon^2/2))}$\newline
\break
Similarly, the 3rd order coefficient is given by:\newline
\break
$\sum_{i\geq 3} \frac{(i-2)!(\varepsilon^2/2)^{(i-2)}\binom{i}{3}((\varepsilon^2/2)x-\varepsilon\epsilon)}{i!}=\frac{((\varepsilon^2/2)x-\varepsilon\epsilon)}{3!}\sum_{i\geq 0} (i+1)(\varepsilon^2/2)^i=\frac{((\varepsilon^2/2)x-\varepsilon\epsilon)}{3!(1-(\varepsilon^2/2))^2}$\newline
\break
In general, the nth order coefficient is given by:\newline
\break
$\sum_{i\geq n} \frac{(i-2)!(\varepsilon^2/2)^{(i-2)}\binom{i}{n}((\varepsilon^2/2)x-\varepsilon\epsilon)^{n-2}}{i!(n-2)!}$\newline
\break
$=\frac{((\varepsilon^2/2)x-\varepsilon\epsilon)^{(n-2)}}{n!}\sum_{i\geq 0} (i+1)(i+2)...(i+n-2)(\varepsilon^2/2)^i$\newline
\break
The final summation can be calculated by differentiating $(n-2)$ times, the infinite sum $1/(1-v)$, where $v=(\varepsilon^2/2)$.\newline
\break
Therefore, the coefficient for $n\geq 2$ is given by:
\begin{equation}\label{x_coeff}
\frac{((\varepsilon^2/2)x-\varepsilon\epsilon)^{n-2}}{n(n-1)(1-(\varepsilon^2/2))^{(n-1)}}\frac{\partial^n (g_1(x,\epsilon)p(x,\epsilon))}{\partial x^n}
\end{equation}
Following similar logic for $\epsilon$ we have the coefficient:
\begin{equation}\label{epsilon_coeff}
\frac{((\varepsilon^2/2)\epsilon+\varepsilon x)^{n-2}}{n(n-1)(1-(\varepsilon^2/2))^{(n-1)}}\frac{\partial^n (g_2(x,\epsilon)p(x,\epsilon))}{\partial \epsilon^n}
\end{equation}
These coefficients can now be used to calculate a 2nd order inhomogeneous differential equation for the moments of $H(y_x,y_{\epsilon}|x,\epsilon)$. We start by expanding the $\partial^2/\partial x^2$, and $\partial^2/\partial\epsilon^2$ in equation (\ref{eq:5}).\newline
\break
Since, we assume from section 2.2, that $x,\epsilon$ are uncorrelated, equation (\ref{eq:5}) can be written:
\begin{equation}\label{eq:9}
\begin{split}
\frac{\partial p(t,x,\epsilon)}{\partial t}=
\frac{1}{2}\sum_{i=0}^{\infty}\frac{(-1)^{(i)}}{i!}\Bigg(\frac{\partial^{i}(g_1(x,\epsilon)p(x,\epsilon))}{\partial x^i}\frac{\partial^2 a^i_{x}}{\partial x^2}+\frac{\partial^{i+2}(g_1(x,\epsilon)p(x,\epsilon))}{\partial x^{i+2}}a^i_{x}\\+2\frac{\partial^{i+1}(g_1(x,\epsilon)p(x,\epsilon))}{\partial x^{i+1}}\frac{\partial a^i_{x}}{\partial x}\Bigg)\\
+\frac{1}{2}\sum_{j=0}^{\infty}\frac{(-1)^{(j)}}{j!}\Bigg(\frac{\partial^{j}(g_2(x,\epsilon)p(x,\epsilon))}{\partial \epsilon^j}\frac{\partial^2 a^j_{\epsilon}}{\partial \epsilon^2}+\frac{\partial^{j+2}(g_1(x,\epsilon)p(x,\epsilon))}{\partial \epsilon^j}a^j_{\epsilon}\\+2\frac{\partial^{j+1}(g_1(x,\epsilon)p(x,\epsilon))}{\partial \epsilon^{j+1}}\frac{\partial a^j_{\epsilon}}{\partial \epsilon}\Bigg)\\
\end{split}
\end{equation}
The coefficients for $\frac{\partial^n (g_1(x,\epsilon)p(x,\epsilon))}{\partial x^n}$ from equation (\ref{eq:9}) are now given by: $\frac{\partial^2 a^0}{\partial x^2}(g_1(x,\epsilon)p(x,\epsilon))$ for $n=0$, $(\frac{\partial^2 a^1_x}{\partial x^2}+2\frac{\partial a^0}{\partial x})\frac{\partial (g_1(x,\epsilon)p(x,\epsilon))}{\partial x}$ for $n=1$, and:
\begin{equation}\label{eq:10}
\frac{(-1)^na^{n-2}_x+2n\frac{\partial a^{n-1}_x}{\partial x}+n(n-1)\frac{\partial^2 a^{n}_x}{\partial x^2}}{n!}\frac{\partial^n (g_1(x,\epsilon)p(x,\epsilon))}{\partial x^n}
\end{equation}
\ for $n\geq 2$. Similarly, for $\epsilon$ we have:
\begin{equation}\label{eq:11}
\frac{(-1)^na^{n-2}_{\epsilon}+2n\frac{\partial a^{n-1}_{\epsilon}}{\partial \epsilon}+n(n-1)\frac{\partial^2 a^{n}_{\epsilon}}{\partial {\epsilon}^2}}{n!}\frac{\partial^n (g_2(x,\epsilon)p(x,\epsilon))}{\partial \epsilon^n}
\end{equation}
We now make the assumption that $H$ is a normalised probability distribution with expectation zero for $x$ and $\epsilon$. Ie, $\frac{\partial a_0}{\partial x}=0$, $a^1_x=0$, and $a^1_{\epsilon}=0$. These assumptions ensure the coefficients with $n=0,1$ equate to zero on both sides of equation (\ref{eq:8}). The proposition follows by equating equations (\ref{x_coeff})/(\ref{eq:10}) and (\ref{epsilon_coeff})/(\ref{eq:11}).
\end{proof}
\section{Monte-Carlo Methods \& Numerical Simulations}
In this section, we give a brief overview of McKean stochastic differential equations, before introducing how the particle method, discussed in the book by Guyon \& Henry-Labord\`ere: \cite{Guyon}, can be used in their simulation. We then go on to present numerical results from the bid-offer model discussed above, placing particular emphasis on understanding how quantum effects become apparent through small transformations applied to a classical Black-Scholes system.
\subsection{McKean Stochastic Differential Equations}
McKean nonlinear stochastic differential equations were introduced in \cite{McKean}, and refer to SDEs, where the drift \& volatility coefficients depend on the underlying probability law for the stochastic process. Following notation from \cite{Guyon} we have:\newline
\break
$dX_t=b(t,X_t,\mathbb{P}_t)dt+\sigma(t,X_t,\mathbb{P}_t)dW_t$\newline
\break
These are then related to the nonlinear Fokker Planck equation:
\begin{equation}
\frac{\partial p}{\partial t}=\frac{1}{2}\sum_{i,j}\frac{\partial^2 (\sigma_i(t,x,\mathbb{P}_t)\sigma_j(t,x,\mathbb{P}_t)p(t,x))}{\partial x_i\partial x_j}-\sum_{i}\frac{\partial (b^i(t,x,\mathbb{P}_t))}{\partial x_i}
\end{equation}
In this case, we can write equation (\ref{FokkerPlanck1}) in this form. We have for $r=0$, $b^1(t,x,\epsilon,\mathbb{P}_t)=b^2(t,x,\epsilon,\mathbb{P}_t)=0$ and $\sigma_1(t,x,\epsilon,\mathbb{P}_t)=\sqrt{g_1(x,\epsilon)\mathbb{E}^p\bigg[\frac{H(x-y_x,\epsilon-y_{\epsilon}|x,\epsilon)}{p(x,\epsilon,t)}\bigg]}$, $\sigma_2(t,x,\epsilon,\mathbb{P}_t)=\sqrt{g_2(x,\epsilon)\mathbb{E}^p\bigg[\frac{H(x-y_x,\epsilon-y_{\epsilon}|x,\epsilon)}{p(x,\epsilon,t)}\bigg]}$.\newline
\break
Therefore, we can simulate the solution to equation (\ref{FokkerPlanck1}) by first calculating the function $H(x-y_x,\epsilon-y_{\epsilon})$ using a moment matching algorithm, and then simulating the following McKean SDE, with uncorrelated Wiener processes $dW^1, dW^2$:
\begin{equation}\label{McKean}
\begin{split}
dx=\sqrt{\frac{g_1(x,\epsilon)}{p(x,\epsilon,t)}\mathbb{E}^{p(y)}\big[H(x-y_x,\epsilon-y_{\epsilon}|x,\epsilon)\big]}dW^1\\
d\epsilon=\sqrt{\frac{g_2(x,\epsilon)}{p(x,\epsilon,t)}\mathbb{E}^{p(y)}\big[H(x-y_{\epsilon},\epsilon-y_{\epsilon}|x,\epsilon)\big]}dW^2\\
\end{split}
\end{equation}
The simulation of the above SDE relies on the {\it particle method} outlined in Guyon \& Henry-Labord\`ere's book {\it Nonlinear Option Pricing} chapters 10, 11 (cf: \cite {Guyon}).\newline
\break
Each path $(x^i,\epsilon^i)$ now interacts with the other paths: $(x^j,\epsilon^j), j\neq i$ during the simulation process, and the convergence of the method relies on the so called {\it propagation of the chaos} property. This states:
\begin{definition}
For all functions $\phi(x,\epsilon,t)\in C_0(\mathbb{R}^2)$:
\begin{equation}
\frac{1}{N}\sum_{j=1}^{N} \phi(x^j,\epsilon^j)\xrightarrow{N\rightarrow\infty} \int_{\mathbb{R}^2} \phi(x,\epsilon,t)p(x,\epsilon,t)dxd\epsilon
\end{equation}
\end{definition}
In our case, the SDE (\ref{McKean}), is a McKean-Vlasov process, and we have from Guyon, Henry-Labord\`ere (cf: \cite{Guyon} Theorem 10.3), and originally Sznitman (cf: \cite{Sznitman}), that the propagation of the chaos property holds.
\subsection{Particle Method}
The first step is to discretize the SDE: (\ref{McKean}), as follows:
\begin{equation}\label{discrete_SDE}
\begin{split}
dx^i=\bigg(\sum_{j=1}^{N} H(x^j-x^i,\epsilon^j-\epsilon^i)\frac{P(x^j,\epsilon^j)}{P(x^i,\epsilon^i)}g_1(x^i,\epsilon^i)\bigg)^{0.5}dW^{1,i}\\
d\epsilon^i=\bigg(\sum_{j=1}^{N} H(x^j-x^i,\epsilon^j-\epsilon^i)\frac{P(x^j,\epsilon^j)}{P(x^i,\epsilon^i)}g_2(x^i,\epsilon^i)\bigg)^{0.5}dW^{2,i}
\end{split}
\end{equation}
Where $P(x^j,\epsilon^j)$ represents a suitably discretized probability function. The algorithm then proceeds as follows:
\begin{enumerate}
\item Solve for the moments of the ``blurring'' function $H(x-y_x,\epsilon-y_{\epsilon}|x,\epsilon)$ using propositions \ref{trans_mom}, and \ref{rot_mom}.
\item Choose a parameterised distribution to approximate $H(x-y_x,\epsilon-y_{\epsilon}|x,\epsilon)$, and fit the parameters using the calculated moments. For example, approximate $H(x-y_x,\epsilon-y_{\epsilon}|x,\epsilon)$ as a univariate/bivariate normal distribution.
\item Simulate the 1st time step, $t_1$, using the value of $H(0,0|x_0,\epsilon_0)$, for starting positions $x_0,\epsilon_0$.
\item After each simulation, allocate the simulated paths into discrete probability buckets: $P(x^j,\epsilon^j)$, for paths $j=1$ to $N$.
\item Proceed from the $t_{k-1}$ to $t_k$ timestep, using (\ref{discrete_SDE}), the value of $H(x-y_x,\epsilon-y_{\epsilon}|x,\epsilon)$, and the discrete buckets at $t_{k-1}$.
\item Iterate steps 4 \& 5 until the final maturity: $t_F$.
\end{enumerate}
\subsection{Modelling the Market Fear Factor}
We can see from (\ref{discrete_SDE}), that small translations, will lead to a variance scaling factor:\newline
\break
$\sum_{j=1}^{N} H(x^j-x^i,\epsilon^j-\epsilon^i)\frac{P(x^j,\epsilon^j)}{P(x^i,\epsilon^i)}$\newline
\break
This will have the impact of reducing the volatility of those paths which lie in the middle of the ``bell curve'', owing to the negative curvature of the probability law at these points - probability mass is spread by the ``blurring'' function to lower probability points.\newline
\break
Similarly, at the extremes of the probability density curve where the curvature is positive, probability mass is spread to areas with net higher probability. In essence the market memory of a recent extreme event, will lead to a higher market volatility at the next time step.\newline
\break
This effect differs from the negative skew observed in local volatility models (for example the work by Dupire: cf \cite{Dupire}), and from stochastic volatility models (for example Heston: \cite{Heston}), in the sense that the increase in volatility is linked to recent random moves in the tail of the probability distribution, rather than to the level of the stochastic volatility or a static function of the price, and time.\newline
\break
To highlight the difference, in the process given by equation (\ref{discrete_SDE}), one could allow for periodic rebalancing of the process. For example, one could replace the unconditional probability, with the probability conditional on the previous step. In this way, the level of the volatility would depend purely on a ``memory'' of recent price history, rather than on the absolute level of the market price, or an additional random variable. The market responds to large moves with a heightened fear factor. The study of modelling such processes with rebalancing, will involve advanced techniques for calculating the conditional probabilities, and we defer detailed study to a future work.\newline
\break
\subsection{Numerical Results}
In this section, we simulate the one-factor process described in section 2.1, and 3.1. In this case, we approximate $H(y)$ using a normal distribution using the moments from proposition 3.2: $N(\frac{\varepsilon}{3},\frac{\varepsilon^2}{18})$.\newline
\break
The non-zero 1st moment, will lead to an upside/downside bias to the ``market fear factor'' effect. Essentially, by introducing a translation in the negative $x$ direction, one introduces downside `fear' into the model.\newline
\break
Figures 1 \& 2 below, illustrate the results from a 2 step Monte-Carlo process, with $g(x)=0.01x^2$, starting value: $x_0=1$, 100K Monte-Carlo paths, and 500 discrete probability buckets. The scatter plot shows the magnitude of the proportional return on the 1st time-step on the horizontal axis, and the second time-step on the vertical axis:\newline
\break
\includegraphics[scale=0.5]{Fig1.jpg}\newline
Figure 1: $\varepsilon=0$, horizontal axis represents the proportional return for the first time-step, vertical axis represents the second second time-step.\newline
\break
\includegraphics[scale=0.5]{Fig2.jpg}\newline
Figure 2: The results for $\varepsilon=0.02$, horizontal axis represents the proportional return for the first time-step, vertical axis represents the second time-step.\newline
\break
Figure 1 shows the results for $\varepsilon=0$. This is a classical Black-Scholes system, and there is no correlation between the magnitude \& direction of the 1st and 2nd time-steps.\newline
\break
Figure 2 shows the proportional returns for $\varepsilon=0.02$ (in blue), overlaid on top of the $\varepsilon=0$ results (in orange). The volatility of the second step is reduced on those paths where the first time-step has been small. There is a slight increased second step volatility for those paths with large positive first steps, and significant second step volatility for those paths with a large negative first step. In effect, the drop in market prices has introduced ``fear'' into these paths.\newline
\break
The final chart shows the probability distributions for the natural logarithm of the simulated value after 50 one day time-steps. The non-zero translation results in a natural skewness in the distribution.\newline
\break
\includegraphics[scale=0.5]{Fig3.jpg}\newline
Figure 3: Distribution for the natural log of the final price after 50 one day time-steps. 100K Monte-Carlo paths, and 500 discrete probability buckets.
\section{Conclusions}
In this paper, we demonstrate how unitary transformations can be used to model novel quantum effects in the Quantum Black-Scholes system of Accardi \& Boukas (cf \cite{AccBoukas}).\newline
We show how these quantum stochastic processes can also be modelled using nonlocal diffusions, and simulated using the particle method outlined by Guyon \& Henry-Labord\`ere in \cite{Guyon}.\newline
By introducing a bid-offer spread parameter, and extending the Accardi-Boukas framework to 2 variables, we show how rotations, in addition to translations, can be applied. Thus, a richer representation of the information contained in the current market leads to a wider variety of unitary transformations that can be used.\newline
In section 4, using a Monte-Carlo simulation, we illustrate how introducing a translation to the one dimensional model leads to a skewed distribution, whereby recent market down moves leads to increased volatility going forward. In effect, the market retains memory of recent significant moves.\newline
In \cite{Dupire}, Dupire shows how to calibrate a local volatility to the current vanilla option smile. This enables a Monte-Carlo simulation that is fully consistent with current market option prices. Carrying out the same analysis, using the new Quantum Fokker-Planck equations, is another important next step to consider as a future development of the current work.