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Smile transformation for price prediction
Options prices are driven by supply and demand in the market while simultaneously being bound by no-arbitrage
restrictions. This makes it difficult to create models for their prediction. Petros Dellaportas and Aleksandar Mijatović use a
simple time series model to propose an option price prediction algorithm that satisfies the no-arbitrage requirement and
yields profitable trading strategies
ntuitively, a set of implied volatilities IV t .Ki ; T /, i D 1; : : : ; n, stochastic alpha beta rho (SABR) model, which we will describe in
allows arbitrage if and only if it is possible to trade (buy and sell) detail later. This not only allows us to circumvent the difficulty arising
the corresponding contracts, at prices given by IV t .Ki ; T /, i D from the no-arbitrage restriction for option prices, but it also expresses
1; : : : ; n, and follow a self-financing trading strategy on the foreign the vanilla option for any strike and a given maturity in terms of the
exchange spot rate (buy and sell the underlying currencies) in such values of three parameters: the current level of the forex spot rate and
a way that at some future time the portfolio of options and the gains the two interest rates in the respective currencies over the period from
from the trading strategy in aggregate have a non-negative value with current time until maturity.
certainty and a strictly positive value with positive probability. In other
Forecasting option prices is impeded by the fact that time series
words, if a market maker quotes option prices that allow arbitrage, a models are not designed to yield arbitrage-free predictions, makcounterparty could enter into a contract with the market maker without ing the problem of setting up profitable trading strategies in options
any risk of loss. It is evident that when the number of option prices intractable. The key contribution of this paper is that it combines
is large (it is in the thousands or even tens of thousands in the case
the notion of the risk-neutral measure with time series analysis for
of market makers), it is very hard to check a priori from the quoted
arbitrage-free forecasting of option prices. We do this in the followoption prices whether they are arbitrage free.
ing way. First, for a given maturity, we encode the time series of the
In practice this is achieved by using the theoretical concept of
historical implied volatilities across strikes in terms of the parameters
the risk-neutral measure, which arises in the fundamental theorem
of a corresponding risk-neutral measure; second, we perform a time
of asset pricing. Simply put, it ensures the implied volatility prices
series analysis on the historical risk-neutral parameters and use them
IV t .Ki ; T /, i D 1; : : : ; n, satisfy a no-arbitrage restriction if they
to predict tomorrow’s option prices.
are obtained as discounted expectations of their respective payoffs
We provide empirical evidence that in the forex derivatives market
under a risk-neutral measure Q. In other words, if they are of the
(where
we have available data), the time series based on the observed
.rd rf /T Q
C
E Œ.ST Ki / , where ST denotes the random
form e
parameter
values exhibits out-of-sample predictive ability for implied
forex rate (for example, US$/U) at future time T , rd (rf ) represents
the dollar (yen) interest rate over the time interval of length T , and volatilities (and hence option prices) through Arma-Garch (autoregressive moving average-generalised autoregressive conditional hetx C D maxf0; xg for any x 2 R.
The defining feature of a risk-neutral measure Q is that the mean eroskedasticity) based statistical models for the risk-neutral parameof the random forex rate ST at time T is equal to the exponential of ters. A major benefit of this approach lies in the fact that in the riskthe interest rate differentials of the two currencies over the time period neutral parameter space, the no-arbitrage constraints are reduced to
from now until expiry T : EQ ŒST D FT , where FT D e.rd rf /T S0 . very simple and easy to enforce conditions on the time series model,
This condition is clearly satisfied by many probability measures Q as making the predicted option prices free of arbitrage. A simple trading
strategy based on our approach using strangles and risk-reversals is
it only fixes the mean of the random variable ST .
Two observations play an important role in how the concept of illustrated at the end of this paper.
the risk-neutral measure is applied in the financial markets. First, the
choice of a risk-neutral measure does not necessarily give the market price for a given derivative contract. All such a choice does is
The problem and a solution
to provide arbitrage-free prices for all traded options with the same
The problem. A call option on the US$/U exchange rate struck
maturity. Note that since the market maker is interested in the consistent pricing of derivatives, this is precisely what they are after. at K and expiring at some future time T is a contract that gives the
Second, the correct price of an option, or equivalently, the correct owner the right, but not the obligation, to purchase $N at expiry T for
choice of the risk-neutral measure, is down to supply and demand the price of UK per dollar. In practice, option traders do not refer to
in the market and crucially depends on the market maker’s view of option prices per se but rather express them through implied volatility,
where the specific contracts should trade. This is why flexible para- which allows one to compare directly option prices across different
metric forms of the risk-neutral measure are useful: they allow the strikes, maturities, expiration times and underlyings in the same units.
market maker to express their view of where the liquid derivatives For this reason we will consider the prediction of the implied volatilshould trade while providing an arbitrage-free pricing mechanism for ities IV.K; T / instead of the option prices C.K; T / (see, for example, Breeden and Litzenberger 1978; Gatheral 2006; Renault 2010).
all derivatives.
As it turns out, in the forex option markets there is a standard para- Assume we have chosen strikes Ki , i D 1; : : : ; n, and would like
metric form for the choice of the risk-neutral measure, based on the to model the evolution of the corresponding implied volatility vector
Co
py
rig
ht
In
cis
ive
M
ed
ia
I
62
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Cutting edge: Option pricing
1 US$/U
×10–3
Prediction of one day ahead: mean absolute error between predicted and true implied volatilities
3.8
3.6
3.4
3.2
3.0
2.8
0.900S
0.925S
0.950S
0.975S
S
1.025S
1.050S
1.075S
1.100S
ia
Strike prices
ed
×10–3
Prediction of two days ahead: mean absolute error between predicted and true implied volatilities
M
6.0
5.5
ive
5.0
4.5
In
cis
4.0
3.5
0.900S
0.925S
0.950S
0.975S
S
1.025S
1.050S
1.075S
1.100S
×10–3
Prediction of three days ahead: mean absolute error between predicted and true implied volatilities
rig
7.5
ht
Strike prices
py
7.0
6.5
Co
6.0
5.5
5.0
4.5
0.900S
0.925S
0.950S
0.975S
S
Strike prices
1.025S
1.050S
1.075S
1.100S
Note: red crosses, our predictions; blue dots, random walk predictions
Y t with coordinates Y t .i/ WD IV t .Ki ; T /, i D 1; : : : ; n, over time
indexed by t 2 N. IV t .Ki ; T / here describes the midday implied
volatilities on day t for the strike Ki and it is assumed that we are
given IV t .Ki ; T / over the time period t D 1; : : : ; t0 . The problem is
finding a time series model for Y t such that its coordinates are guaranteed to be arbitrage-free for all t > t0 and its predictions support a
profitable trading strategy.
A solution. We propose a way to predict the implied volatility
vector Y tC1 that will be shown to outperform random walk predictions
consistently1 in a large out-of-sample exercise (see figure 1 for the
numerical results).
We first apply a non-linear transformation to the implied volatilities
data up to and including time t . This one-to-one transformation maps
1 The
random walk (or white noise) model is a standard benchmark in the
related literature for forecasting equities and it is typically very hard to
beat (see, for example, Welch and Goyal 2008). It simply forecasts the
option prices at time t C 1 from the observed price at the time t and it is
arbitrage-free by construction.
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Cutting edge: Option pricing
the implied volatilities into the parameters of a risk-neutral measure
Q. The core of our approach rests on the fact that every element in the
space of Q-parameters yields an arbitrage-free set of implied volatilities across all strikes. This allows us to perform the statistical analysis in the space of Q-parameters. We use the daily recorded parameters that characterise the choice of the risk-neutral measure given
by the quoted implied volatilities on that day to forecast, via a time
series model, tomorrow’s Q-parameter values. The predicted implied
volatilities YO tC1 are then computed via a non-linear pricing function
that guarantees the no-arbitrage restrictions are satisfied. A schematic
description of our approach is given in the following diagram:
Prediction: YO tC1
IV data: 1Wt
O
Inverse pricing formula
BS.FT ; K; T; / WD erd T ŒFT N.dC / KN.d /
d˙ D
Arbitrage-free option prices in forex
py
rig
ht
As mentioned before, in the forex option markets there is a standard
parametric form for choosing the risk-neutral measure. It expresses
options in terms of the current level of the forex spot rate and the
two interest rates in the respective currencies over the period from
current time until maturity. There are two reasons why this particular
parametrisation of option prices, based on the SABR formula, has been
adopted as the market standard:
Co
(i) the parsimonious description of all option prices with a given
maturity is very robust and easy for traders to use; and
(ii) each of the three parameter values (the overall level of option
prices and the skewness and kurtosis of the risk-neutral law of
the forex spot rate ST ) is closely related to the three fundamental
features of the risk-neutral distribution (which determine option
prices).
It should be stressed here that, in general, the analytical SABR
formula (see (2) below) yields arbitrage-free option prices except in the
extreme wings (that is, for the strikes that are many standard deviations
away from the at-the-money value). However, the points made above,
and the fact that for such extreme strikes the notion of the marketdefined price is questionable due to the lack of liquidity, suggest it is
as reasonable to apply the SABR parametrisation of the risk-neutral
measure for the purposes of our problem as it is to use it for the pricing,
hedging and risk management of the portfolios of options, which is a
very common day-to-day practice in the forex options markets.
2 This
ia
and N./ is the standard normal cumulative distribution function.
Implied volatility is well defined since the function:
7! BS.FT ; K; T; /
is strictly increasing for positive (the vega of a call option
p
.@BS=@ /.FT ; K; T; / D erd T FT N 0 .dC / T is clearly strictly
positive) and the market price C.K; T / lies in the image of the BlackScholes formula if and only if the call option price C.K; T / satisfies a
no-arbitrage restriction. It is clear from the definition, albeit suppressed
in the notation, that the implied volatility IV.K; T / also depends on the
current level of the forex spot rate S0 and the interest rate differential
between the two currencies.
The implied volatility IV.K; T / is nothing more than a convenient
number. The convenience, from the perspective of market participants,
lies in the fact that IV.K; T / contains information about the price of
a call option that is independent of the forex rate and can therefore be
easily compared across currency pairs. Furthermore, implied volatility
is being used in the forex option markets to specify parametrically the
call option prices for all strikes and a given maturity.
It is well-known (see, for example, Breeden and Litzenberger 1978)
that the information contained in knowing the prices of the implied
volatilities for all strikes and a given maturity T is equivalent to specifying the risk-neutral law of the forex rate ST . Therefore, the SABR
formula for the implied volatility IV.K; T / given in the next section together with the Black-Scholes formula in (1) specify an easy
parametrisation of the risk-neutral law of the spot rate ST .
The parametric form of the risk-neutral measure in the forex
markets. The version of the SABR formula for the implied volatility
that was derived in Hagan et al (2002), which will be used in this paper,
takes the form:
In
cis
We note here that the arbitrage-free evolution of option prices under
a risk-neutral measure has been studied extensively (see, for example, Carmona and Nadtochiy (2009), Kallsen and Krühner (2010),
Schweizer and Wissel (2008) and the references therein). In this paper
we model the evolution of the implied volatility smile under the realworld measure and are solely concerned with the arbitrage-free constraint in a static sense; that is, at each moment of the running time,2
in order to obtain viable predictions for option prices.
ed
/ Predicted Q-parameters
M
Stochastic model
log.FT =K/ ˙ 2 T =2
p
T
ive
(1)
where:
Pricing formula
Q-parameters
is completely analogous to the way pricing theory is applied in the
derivatives markets.
64
A further argument in favour of using the SABR formula, besides
the lack of liquidity in the wings, is that the price of options in the
extreme wings is more uncertain than in the ‘near’ wings due to the
increase in the bid-offer spread for such derivatives.
Implied volatility in the forex markets. The value BS.FT ; K;
T; / of the European call option with strike K, the price of the underlying FT and expiry T in a Black-Scholes model with constant volatility
> 0 is given by the Black-Scholes formula:
IV.K; T /
1 ˛=2
2 32 2
˛2
1
C
C
z
424 .FT K/1=2 4 .FT K/1=4
24
D˛
1
F
F
1
x.z/
T
T
.FT K/1=4 1C
log2
C
log4
424
K
161920
K
(2)
1CT
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Cutting edge: Option pricing
FT
.FT K/1=4 log
;
˛
K
p
1 2z C z 2 C z x.z/ D log
1
zD
ht
In
cis
ive
The market data in this formula consists of the current forex spot rate
S0 and the interest rate differential for the maturity T , which features
in FT D e.rd rf /T S0 . The parameters allowing the market participant
to express their view on the price of the call option struck at K with
maturity T are given by the instantaneous current level of volatility ˛,
the volatility of the volatility , and the instantaneous correlation .
A general version of the SABR formula contains a further parameter,
ˇ, the value of which is taken to be 12 in (2). In practice, ˇ D 1 is also
used, marginally changing the analytical expression in (2). However,
this makes no material difference for pricing purposes as a change
in ˇ can be compensated by a change in the value of since both
parameters mainly affect the skew of the smile.
The formula in (2) yields a natural parametrisation of the risk-neutral
law of ST . It was developed in Hagan et al (2002) based on an assumption that the forex spot rate process .S t / t>0 evolves under a riskneutral measure as a stochastic volatility process: ˛ denotes the value
of the volatility process in this model at time zero (that is, the current
value); is the correlation between the two Brownian motions driving
the spot and volatility processes; and is the volatility of the stochastic
volatility process (see Hagan et al (2002) for more details).
A simple and yet important fact is that for each value of the state
vector .˛; ; /, where ˛ and are positive and is between 1 and
1, the function:
Our data set consists of the time series for the parameter values
.˛; ; / implied by the US$/U liquid one-month options, the forex
spot rate level and the interest rate differentials at London midday
over the period September 29, 2006 to December 16, 2011. The data
set has been provided by the RBS forex options desk, one of the largest
market makers in forex options. The option quotes are taken at London midday to ensure all the option prices are temporally consistent. In
other words, if the option prices used to obtain the parameters .˛; ; /
are not recorded simultaneously, there is no guarantee that the parameter triplet will reflect the market view on the risk-neutral law of the
forex spot rate.
Since London midday is the only time when the trading desk records
all the liquid option prices simultaneously, the parameters .˛; ; /
obtained in this way describe all the option prices with expiry T without the usual difficulties with options data that arise from illiquidity
in the market. Furthermore, this also gives us a very clear interpretation of the predicted parameters: our prediction yields a risk-neutral
law of ST , which can be converted to a vector of option prices and
compared (out-of-sample) with the option prices that are observed at
future London midday fixing times.
Time series modelling. We have obtained the data set .˛ t ; t ; t /
for t D 1; : : : ; 1360 for the US$/U exchange rate. Our modelling
construction is executed as follows: we use the first n0 D 1000 values
to fit a model and the next n1 D 360 values to test its predictive ability.
We first transform our data using:
ia
ed
M
where:
rig
K 7! BS.FT ; K; T; IV.K; T //
Co
py
given by the formulas (1) and (2), represents an arbitrage-free collection of option prices for any positive strike K. The parameters .˛; ; /
are used to control the risk-neutral distribution in the following way: a
change in the parameter ˛ has the effect of changing the overall level
of option prices, while the parameters and control the kurtosis and
skewness of the risk-neutral distribution of ST . This natural interpretation of the parameters has made (2) a standard parametrisation of a
risk-neutral law for the spot in the forex option markets.
Note that our approach does not depend on the specific parametric
form of a risk-neutral law. More generally, in a different market the
method outlined here can be applied to the parameters of the riskneutral law given by the standard of that particular market. For example, in the equity option markets one can use Heston’s parametrisation
of the risk-neutral law (Heston 1993).
Arbitrage-free modelling of co-terminal options in forex
Description of the data set and forex option market conventions.
In forex the liquid options of a given market-defined maturity T (T D
1 month, for example) have a rolling expiry with respect to the current
time t. In other words, at time t a liquid call option expiring in one
month would cover the time interval Œt; t C T . The next day, at time
t C 1, a liquid one-month option will be a different derivative contract
covering the time interval Œt C 1; t C 1 C T .
A t WD log.˛ t /
N t WD log. t /
and
R t WD log
t C 1
1 t
so that all transformed parameters lie on the real line. Model building has been guided through an Arma-Garch time-series methodology, which includes autocorrelation and partial autocorrelation plots,
the Akaike information criterion (AIC), the Bayesian information
criterion (BIC) and parameter significance tests (see, for example,
Brockwell 2005). AIC and BIC are widely used Bayesian criteria for
choosing a model, designed to balance the number of free parameters with the value of the likelihood attained by the model given the
observations.
In particular, in all series there was overwhelming evidence that the
hypothesis of unit root cannot be rejected (with augmented DickeyFuller and Phillips-Peron tests), so the difference operator was applied
to all series and the unit root hypothesis was rejected in the resulting
series. All (differenced) series exhibited strong Garch effects (that is,
the time series exhibit autoregressive conditional heteroskedasticity)
that were evident both by inspecting the autocorrelation plots of their
squares and by Engle’s Arch test.
After removing the Garch effects (in which Student-t errors seemed
to perform much better than ‘normal’ errors when BIC and AIC values were compared and residuals inspected), the autocorrelation and
partial autocorrelation plots of the residual series indicated that forms
of Arma-Garch models might be appropriate. Our subsequent modelling choice procedure consisted of fitting a series of such models,
risk.net
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Cutting edge: Option pricing
j D1
˛
t WD N t N t1
0.7807
(0.0413)
0.2269
(0.0441)
3.8903
(0.5788)
0.1096
(0.0428)
0.0317
(0.0145)
0.9397
(0.0311)
6.1987
(1.3818)
0.2753
(0.0341)
0.0445
(0.0249)
0.9262
(0.0444)
2.9694
(0.4177)
An important observation is that the predicted risk-neutral measure in our framework depends not only on the SABR parameters,
which our methodology predicts, but also on the future level of the
forex forward rate. Since the latter is notoriously difficult to predict,
in our approach we use the random walk prediction for it. Figure 1
clearly shows that our forecasts outperform the random walk by a
larger amount for options that are struck further out of the money. The
shape of the payoff of such options dictates that their price depends
to a greater extent on the tail and less on the mean of the predicted
risk-neutral distribution.
Furthermore, it is interesting to note that, as clearly shown by figure 1, historical option price data contains sufficient information to
improve the prediction of future option prices compared with a random walk forecast, despite the fact that future option prices depend on
the future forex forward rate, which is very hard to forecast. Perhaps
this offers an avenue for the improved prediction of the forex rate itself.
A simple trading strategy for strangles and risk reversals.
Based on the results of the previous section, we perform an illustrative trading exercise to demonstrate the potential of our methodology. We choose to buy or sell strangles (P .K ; T / C C.KC ; T /) and
risk-reversals (C.KC ; T / P .K ; T /) at strikes K D 0:9S and
KC D 1:1S, where S is the current level of the spot. We emphasise
here that this choice of K˙ is arbitrary and that structures consisting
of any finite combinations of options with maturity T may be traded
since we have arbitrage-free forecasts for the entire implied volatility
smile at T . A realistic strategy requires a trading rule in which trades
are executed only when the return forecast of a strangle/risk-reversal
exceeds a given value of ı > 0. Put differently, we go long a strangle
if the current price (P .K ; T /; C.KC ; T /) and the predicted price
(PQ .K ; T /; CQ .KC ; T /) satisfy:
In
cis
and
t WD R t R t1
py
rig
ht
for t D 2; : : : ; n0 , and where the error terms t˛ , t , t follow a
Garch.1; 1/ model with Student-t errors.
The resulting estimated parameters, based on the initial time series
of 1,000 data points, are given in table A.
The out-of-sample prediction exercise was performed by fitting the
model of table A as each new data point t D 1; 001; 1; 002; : : : ; 1; 360
arrives and then predicting the triplet of the parameters through the
fitted model for one, two and three days ahead. These predicted triplets
were then transformed back to predicted implied volatilities via (2).
The three ingredients that were unknown in (2), namely the future spot
rate S0 and the two interest rates rd , rf (yielding the future forward
rate FT ), have been predicted with a simple random walk model and
were therefore taken to be equal to current values.
Strikes K were selected to be 40 equally spaced values between
90% and 110% of the current spot rate. For every future day and
strike we have calculated the predicted implied volatility and recorded
the absolute error against the implied volatility derived using the true
values of the parameters ˛, , and the realised values of the spot rate
S0 and the interest rates rd , rf . Figure 1 illustrates that the mean error
varies between 28 basis points (bp) for predictions one day ahead and
may even reach 65bp for predictions three days ahead.
In figure 1 we have depicted (with blue dots) the corresponding
errors produced by the naive random walk predictor. Our methodology
clearly outperforms the random walk forecasts for all strikes, with the
improvement reaching as high as 4bp for out-of-the-money options.
Beyond the three-day time horizon this improvement vanishes, indicating that our Arma-Garch model does not have further predictive
ability.
Co
0.1801
(0.0444)
0.2279
(0.0372)
ive
˛ t WD A t A t1
0.0002
(0.0001)
0.9104
(0.0160)
0.9789
(0.0053)
0.0002
(0.0001)
0.1844
(0.1082)
0.0042
(0.1105)
0.0000
(0.0000)
0.0013
(0.0022)
0.0520
(0.0273)
0.0004
(0.0003)
ia
˛
where:
66
˛
ed
iD1
A. Parameter estimates for the US$/U series (standard errors in
brackets) for the time series models. The four parameters correspond
to the Garch(1,1) model estimates: the constant, the Arch parameter, the
Garch parameter and the degrees of freedom of the t -density,
respectively. The columns correspond to the lags in the model, which are
different for each parameter
M
and finally the best models have been chosen with respect to the best
AIC/BIC values attained.
In some Arma-Garch models the problem of near root cancellation, which results in misleading inferences (see, for example, Ansley
and Newbold 1980), was observed, so we chose to remove the moving average component. Put differently, the AR and MA polynomials
have common roots, making the parameters of the Arma model nonidentifiable; by setting the MA parameter to zero the model becomes
identifiable. Parameter estimates have been obtained through Matlab
software. The best model, chosen as described above, is:
9
p˛
q˛
X
X
>
>
˛
>
i˛ ˛ ti C
j˛ tj
C t˛ >
˛ t D ˛ C
>
>
>
>
iD1
j D1
>
>
>
>
p
q
=
X
X
t D C
i ti C
j tj C t
(3)
>
>
iD1
j D1
>
>
>
>
>
p
q
>
X
X
>
>
t D C
i ti C
j tj C t >
>
;
min
Q
C .KC ; T / C.KC ; T / PQ .K ; T / P .K ; T /
;
>ı
C.KC ; T /
P .K ; T /
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Cutting edge: Option pricing
and the average daily return of the strategy. Notwithstanding that the
P&L reported in figure 2 is gross of transaction costs, we believe there
is potential to adopt the arbitrage-free option price forecasting methodology described in this paper, both for risk management purposes and
as a profitable quantitative trading strategy. It is worth emphasising in
this regard that both the trading strategy and the forecasting model we
have used were chosen for their simplicity rather than their ability to
capture the full potential of the approach.
2 Performance of an out-of-sample trading strategy
2
4
1
2
0
0
1
2
3
4
5
Trading threshold δ
In
cis
Note: dashed red line, average daily return; solid blue line, average daily
standardised return (that is, the average daily return divided by the
standard deviation of the return). It should be noted that for ı 105 ,
the strategy trades on 82% of days (that is, 281 out of 342), and for
ı 5 104 , it trades on 3.2% of days (that is, 11 out of 342). We
could, of course, increase the trading frequency for any fixed threshold
ı by exploiting all possible structures of vanilla options (expiring at T ) to
obtain a stronger signal. This is possible because we have at our
disposal all arbitrage-free option prices for maturity T
M
0
ed
–2
6
×10–4
1
Conclusion
We use an Arma-Garch time series model, combined with the SABR
formula for the implied volatility, to predict option prices in US$/U
option markets. First, the implied volatility data was put through
a non-linear transformation by mapping it on to the SABR riskneutral parameters. In this parameter space, the no-arbitrage constraint
becomes easy to enforce, making it possible to predict the future riskneutral parameters using an Arma-Garch model. A non-linear pricing
function converted the parameters to predicted implied volatilities.
Our predictions were not only internally consistent (that is, free of
arbitrage) but also outperformed out of sample – for the data used
in our empirical study, the predictions based on today’s prices – a
natural arbitrage-free predictor that is very hard to outperform. We also
provide a simple example of how this method can be used to formulate
a profitable trading strategy using strangles and risk reversals. R
ia
6
ive
Daily standardised return
3
Daily return
×10–3
8
4
Co
py
rig
ht
and act analogously in the case of a risk-reversal. The realised profit
and loss (P&L) of the trade is the next day’s price of the structure
minus P .K ; T / C C.KC ; T / (if we were long a strangle on the
previous day).
Figure 2 depicts the out-of-sample performance of this strategy for
the data used in the previous section. The results indicate a consistent
positive return, expressed by both the average daily standardised return
Petros Dellaportas is a professor in the department of statistics
at Athens University of Economics and Business. Aleksandar
Mijatovic is a reader in probability in the department of mathematics at Imperial College in London. Email: [email protected],
[email protected]
This work is funded in part by the European Union (European Social Fund – ESF) and Greek national funds through
the Operational Program ‘Education and Lifelong Learning’ of
the National Strategic Reference Framework (NSRF), project
ARISTEIA-LIKEJUMPS
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