A path integral based model for stocks and order dynamics
AA path integral based modelfor stocks and order dynamics
Giovanni Paolinelli Gianni Arioli
Dipartimento di MatematicaPolitecnico di Milanopiazza Leonardo da Vinci 32, 20133 Milano, Italy [email protected], [email protected]
Keywords : Stock prices, econophysics, path integral, gauge theory, fi-nancial markets.
Abstract
We introduce a model for the short-term dynamics of financial as-sets based on an application to finance of quantum gauge theory, de-veloping ideas of Ilinski. We present a numerical algorithm for thecomputation of the probability distribution of prices and compare theresults with
APPLE stocks prices and the
S&P500 index.
In [2] Ilinski introduced a model for the short-term dynamics of stocks andorders based on a gauge interpretation of classical finance. A similar ap-proach has been developed in [10, 11, 12], where a model based on quantumfield theory has been developed. The approach to the financial mathemati-cal problems with the quantum mechanics is called quantum finance. In thelast 20 years, the path integral approach to quantum mechanics introducedby Feynman [15] has turned out to be particularly suitable for financial ap-plications, see also [4, 5, 6, 7, 8, 9]. We further analyse and extend theseideas and develop a model that provides results in good agreement with realmarket data.Ilinski’s model describes the amount of cash and share in the portfolioand all possible trading configurations, and in particular the impact of theorders in the stocks dynamics. Quantum mechanics plays a fundamental rolein providing a robust mathematical background to describe the fundamen-tal ideas of this theory. In particular, the discrete nature of the portfolio,characterized by an integer number of stock and cash is modelled by the1 a r X i v : . [ q -f i n . C P ] M a r oherent-state path-integral. We generalize this model and introduce analgorithm to compute the coherent-state path-integral proposed by Ilinski.The model is tested against data of APPLE stocks and the
S&P500 index with a time step of one minute; the agreement between real data andthe model covers four orders of magnitude. In particular we get a good fitalso of the fat-tails.It is interesting to observe that the fat tails phenomenon can be seenas an effect of the orders. We point out that the shape of the tails has asignificant impact on the risk-free rate. More precisely, in [13] it is shownthat the rate of return of an investment with no risk of financial loss andthe term premium, i.e. the compensation that investors require for bearingthe risk that short-term Treasury yields do not evolve as they expected, aremiscomputed if the fat tails are ignored.We show a direct relationship between the kurtosis of the PDF and thestrength of the perturbation caused by the orders; in fact, the model withoutorders provides results equivalent to the Geometric Brownian Motion.Our model entails five parameters; the same amount of the models in[10] and [2]. We analyse the impact of the parameters on the PDF andprovide a financial interpretation.
We present in this section the basic concepts of Ilinski’s theory; the readerinterested in a detailed explanation should refer to [1] and [2].Ilinski’s starting point is the basic idea that it is not possible to earnmoney without risk; if this were the case, then we would have an arbitrageopportunity. Consider an elementary market model where it is possible tobuy a non-risky asset B and a risky asset S with the same initial value. Ifwe assume temporarily that S is not a risky asset, then after a time T thefinal values of B and S must to be equal. Otherwise it would be possibleto perform an arbitrage operation; that is, we could borrow and sell theunder-performing asset, and then use the capital obtained to buy the over-performing one. When we sell the over-performing asset we have enoughmoney to buy back the under-performing asset and also have some moneyleft, the arbitrage revenue. This situation does not occur in the reality sincewe cannot know the value at a future time of S , i.e., we do not know inadvance whether S is the under- or the over-performing asset. This impliesthat a revenue from the previous situation can only be achieved with someamount of risk.In [2] Ilinski shows a strategy involving stocks and cash which generatesa positive revenue independently of the final value of the risky asset S . Suchstrategy is called arbitrage . In classical finance such amount is assumed tobe zero. Ilinski proposes a weaker assumption, which entails a minimization2f the arbitrage.Denote by A g ( { S } ) the gain of the arbitrage described above; in Ilinski’smodel this quantity is treated as the action in a Lagrangian description ofthe dynamics of the system. We assume that the probability associated to A g ( { S } ) is given by P ( { S } ) = N e − βA g ( { S } ) . (1)This assumption represents the main link with the quantum mechanics andwith path integrals in particular. The core idea in the path integral repre-sentation of quantum mechanics consists in the fact that all trajectories areconsidered, but those achieving a lower action are more probable. In Ilinski’sfinancial model the least action ( (cid:126) → numeraire for any assetat any moment of the time. Agents do not start to behave in a different waybecause they are dealing with 100 pence instead of 1 £ or the equivalentamount of money in $. This invariance must be encoded in all quantitiesdescribed in this theory. We assume that the probability of a certain amountof arbitrage A g ( { S } ) does not change if the assets are expressed in a differentcurrency.In physics, the equations that describe the system in a gauge theory areinvariant under the transformation induced by the action of a group, eitherlocal or global. If we want to denote 10$ in £ , i.e, perform a change of gauge,we have to multiply the capital by a positive number which is the conversionratio; the previous number is an element of the gauge group. This is true forall possible currency conversions, therefore the gauge group in this contextis the multiplicative R + .Ilinski uses the previous framework to obtain the conditional probabilityfor the stock price. He considers a discrete time model, where time takesthe values t i = i ∆ for some ∆ >
0, and then the price of an asset is denotedby S i = S ( t i ) = S ( i ∆), i = 0 , . . . , N . Denote by S the price at time 0 and S the final price at time T = N ∆; Ilinski proves that: P ( T, S | , S ) = N − Y i =1 Z ∞ dS i S i ! exp − σ N − X i =0 R i ! , (2)where R i = S − i e r ∆ S i +1 e − r ∆ + S i e − r ∆ S − i +1 e r ∆ − t i of the double arbitrage and the constants r and r represent the interest rates respectively of the cash and of the risky asset.3he exponential is the probability described in (1), whereas σ is the varianceof the prices of the risky asset. Note that the values R i are gauge invariant.The product of differentials is the path space differential used to sum allover the possible trajectories from S (0) to S ( T ); details about this conceptcan be found in [16]. The measure is the gauge invariant expression dS i /S i .In [2], Ilinski proves that (2) results in a normal PDF. Figure 1 shows theresult of a numerical simulation that corroborates such result.Figure 1: Probability density functions with respect to the final stock price S , in logarithmic scale. Obtained by numerical computation of (2) –red– andwith the analytic formula –black–. Both curves have identical parameters σ = 0 . T = 10, N = 10 and r = r = 0. In [2] Ilinski also introduces a generalization of the previous model consistingin the addition of a perturbation which takes into account the effect ofthe orders in the stocks dynamics. In this paper we show that a modifiedperturbation generates a leptokurtic probability distribution of the returns.Ilinski adds to the action a term which describes the dynamics of theorders, so that the price goes up (down) when somebody buys (sells) thestock. The action is given by − β A g ( S ) = − σ ∆ N − X i =0 (cid:18) log( S i +1 ) − log( S i ) − µ ∆ − N i λ (cid:19) ; (3)4he derivation can be found in [2].The terms N i represent the net amount of the orders at time t i , its valueis positive if the net amount is a buy order, negative otherwise, while λ represents the share liquidity.The initial allocation of the portfolio is described by the pair ( n , m )which represents the amounts of cash and share at the initial time; the finalallocation is denoted by ( n, m ). Because of the gauge invariance, we canexpress the share and money values in the same unit, so that a unit of cashcan be traded for a share.We assume the closed environment hypothesis, which implies a constantamount of lots at all times; denoting by M the total number of traded lots(both money and shares) we have n + m = n + m = M . In this contextthe initial configuration of the system is the capital allocation ( n , m ); ateach time step i , the system evolves to the capital allocation ( n i , m i ) up totime T , where it achieves the final allocation ( n, m ). In order to considerthe effect of the orders on the price dynamics, it is necessary to compute allthe possible paths in the capital allocation space and add the effect of eachpath.This computation is achieved with the Coherent State Path Integral(CSPI), see [2, 16, 17]. The numbers ( n i , m i ) are integers, and in the contextof the CSPI they are described as n i = ¯ ψ ,i ψ ,i ,m i = ¯ ψ ,i ψ ,i , where { ψ j,k } are complex numbers, corresponding to the creation/annihilationoperators.We denote an order N i via the ψ j,k operators. Buying k stocks we lose k units of cash, and vice versa. We have: N i = δ i [ ¯ ψ ,i ψ ,i − ¯ ψ ,i ψ ,i ] . Where the symbol δ i stands for the forward different quotient, i.e., δ i h ( i ) = h ( i + 1) − h ( i )∆ , with ∆ equal to the minimal time frame.The dynamics of the variables ¯ ψ, ψ is described by a Hamiltonian H ( t )( ¯ ψ, ψ ) = H ( i ) jk ¯ ψ j,i +1 ψ k,i , which links ψ ( j,k ) ,i to S ( i ∆) = S i . The following expression is derived in [2]: H ( i ) jk = γS ˜ βi e − ˜ βr ∆ e − ˜ βµt γS − ˜ βi e − ˜ βr ∆ e ˜ βµt , (4)5 = (1 − tc ) / ∆ . Here tc is the relative cost of the transaction; ∆ is the time step of themodel in the Hamiltonian dynamics and ˜ β denotes the amplitude of theprice variations.In our simulations we assume µ = r = r = tc = 0 and ∆ = ∆. Giventhe Hamiltonian, we compute the propagator: h ¯ ψ ,N , ..., ¯ ψ Z,N | | U ( T = N ∆ , | | ψ , , ..., ψ Z, i = N − Y j =1 Z Y k =1 Z dψ k,j d ¯ ψ k,j iπ × exp (cid:20) − N − X j =1 Z X k =1 ψ k,j ¯ ψ k,j + N − X j =1 Z X k =1 ψ k,j ¯ ψ k,j +1 (cid:21) × (5)exp (cid:20) ∆ Z X j,k =1 H ( N − jk ¯ ψ i,N ψ k,N − + . . . + ∆ Z X j,k =1 H (0) jk ¯ ψ i, ψ k, (cid:21) . The quantity inside the first square bracket corresponds to Z X k =1 ¯ ψ k, ψ k, + Z X k =1 N − X j =0 ( ¯ ψ k,j +1 − ¯ ψ k,j ) ψ k,j . (6)Substituting (4) and (6) in (5) we obtain: h ¯ ψ N | ˆ U ( T = N ∆ , | ψ i = e ¯ ψ , ψ , + ¯ ψ , ψ , Z Y k =1 , N − Y i =1 dψ k,j d ¯ ψ k,j iπ × exp (cid:20) N − X i =0 ( ¯ ψ ,i +1 ψ ,i − ¯ ψ ,i ψ ,i + ¯ ψ ,i +1 ψ ,i − ¯ ψ ,i ψ ,i + S ˜ βi ¯ ψ ,i +1 ψ ,i + S − ˜ βi ¯ ψ ,i +1 ψ ,i ) (cid:21) . (7)We introduce the hydrodynamical variables¯ ψ k = p M ρ k e iφ k ψ k = p M ρ k e − iφ k , (8)where ρ ∈ [0 ,
1] and φ ∈ [0 , π ]. Note that, because of the close environmentassumption, M ( ρ + ρ ) = M and then ρ = 1 − ρ .We write (7) in the hydrodynamical variables. Starting from¯ ψ ,i +1 ψ ,i − ¯ ψ ,i ψ ,i = [ √ ρ ,i +1 e i ( φ ,i +1 − φ ,i ) − √ ρ ,i ] √ ρ ,i , (9)if we assume e i ( φ ,i +1 − φ ,i ) ’ i ( φ ,i +1 − φ ,i )then (9) becomes:¯ ψ ,i +1 ψ ,i − ¯ ψ ,i ψ ,i = [ √ ρ ,i +1 − √ ρ ,i + i √ ρ ,i +1 ( φ ,i +1 − φ ,i )] √ ρ ,i ;6ecalling the definition of the forward difference quotient, we obtain:¯ ψ ,i +1 ψ ,i − ¯ ψ ,i ψ ,i = ∆ √ ρ ,i ( δ i √ ρ ,i ) + i ∆ √ ρ ,i +1 ρ ,i ( δ i φ ,i );recalling that √ ρ ,i ( δ i √ ρ ,i ) = 12 δ i ρ ,i , we get ¯ ψ ,i +1 ψ ,i − ¯ ψ ,i ψ ,i + ¯ ψ ,i +1 ψ ,i − ¯ ψ ,i ψ ,i == i (cid:20) ( φ ,i +1 − φ ,i ) √ ρ i +1 ρ i + ( φ ,i +1 − φ ,i ) q (1 − ρ i +1 )(1 − ρ i ) (cid:21) , (10)where the conservation law ρ + ρ = 1 has been used.We also have S ˜ βi ¯ ψ ,i +1 ψ ,i + S − ˜ βi ¯ ψ ,i +1 ψ ,i == S ˜ βi q ρ i +1 (1 − ρ i ) e i ( φ ,i +1 − φ ,i ) + S − ˜ βi q (1 − ρ i +1 ) ρ i e i ( φ ,i +1 − φ ,i ) . (11)Using the equation (10) and (11) in (7) we obtain the formula for the prop-agator: h ¯ ψ N | ˆ U ( T = N ∆ , | ψ i = e M Y k =1 , N − Y i =1 Z dρ k,i ρ k,i π Z π dφ k,i × exp (cid:20) M N − X i =0 (cid:18) i { ( φ ,i +1 − φ ,i ) √ ρ ,i +1 ρ ,i + ( φ ,i +1 − φ ,i ) √ ρ ,i +1 ρ ,i } ++ S ˜ βi √ ρ ,i +1 ρ ,i e i ( φ ,i +1 − φ ,i ) + S − ˜ βi √ ρ ,i +1 ρ ,i e i ( φ ,i +1 − φ ,i ) (cid:19)(cid:21) . The previous propagator depends on all possible paths in the portfolio space;in order to obtain the conditional probability we need to link it with the pricedynamics using (3). We can write N i as: N i = δ i [ ¯ ψ ,i ψ ,i − ¯ ψ ,i ψ ,i ] = δ i [ ρ − ρ ] = 2 δ i ρ i , (12)so that equation (3) becomes − β A g ( S ) = N − X i =0 − σ ∆ (cid:18) log( S i +1 ) − log( S i ) − α [( ρ i +1 − ρ i )] (cid:19) with α = M/λ.
We consider now the propagator above together with the stock price action(3), and we obtain the new propagator which takes into account both thetrajectories in the spaces of portfolio allocation and the stock prices.7 ¯ ψ N | ˆ U ( T = N ∆ , | ψ i = e M Y k =1 , N − Y i =1 Z dρ i ρ i π Z π dφ k,i Z ∞−∞ d log( S i ) × exp[ − β A g ( S ) + ˜ S ] . with: − β A g ( S ) = N − X i =0 − σ ∆ (cid:18) log( S i +1 ) − log( S i ) − α [( ρ i +1 − ρ i )] (cid:19) ˜ S = M N − X i =0 (cid:18) i (cid:20) ( φ ,i +1 − φ ,i ) √ ρ i +1 ρ i +( φ ,i +1 − φ ,i ) q (1 − ρ i +1 )(1 − ρ i ) (cid:21) ++ S ˜ βi q ρ i +1 (1 − ρ i ) e i ( φ ,i +1 − φ ,i ) + S − ˜ βi q (1 − ρ i +1 ) ρ i e i ( φ ,i +1 − φ ,i ) (cid:19) . We note that the coherent states at initial and final times are not integratedin the previous formula.By the quantum mechanics formalism, the conditional probability isgiven by P ( S ( T ) , ( n, m ) | S (0) , ( n , m )) == Z Y i,k πi d ¯ ψ i,k dψ i,k e − ¯ ψ i,k ψ i,k h n, m | | ψ ,k i h ¯ ψ N | ˆ U ( T = N ∆ , | ψ i h ¯ ψ ,k | | n , m i , where: h n, m | = h | ψ n ,N ψ m ,N n ! m ! | n , m i = ¯ ψ n , ¯ ψ m , | i With some computations explained in [2] , we obtain the formula P ( S ( T ) , ( n, m ) | S (0) , ( n , m )) = 1 n ! m ! S ( T ) − ˜ β ( n − m )2 S (0) ˜ β ( n − m (13) × Z d ¯ ψdψ h ¯ ψ N | ˆ U ( T = N ∆ , | ψ i ¯ ψ n , ¯ ψ m , ψ n ,N ψ m ,N e − M , where: Z d ¯ ψdψ = Y k =1 , Y i =0 ,N Z πi d ¯ ψ k,i dψ k,i . The previous integral is expressed in the coherent state variables, whichmeans that in order to compute it, we need to transform the whole expressionin the hydrodynamical variables; Note that, as in quantum mechanics, theresult of the integral is a complex number, while the probability is its module pg.s. 136, 166 and 277-281 S ( T ) with a final portfolio allocation ( n, m ), giventhe initial price S (0) and the portfolio allocation ( n , m ). Due to gauge-invariance, we can choose S (0) = 1. This is the choice we make in allsimulations. The numerical simulation requires the computation of the approximate valueof an integral in high dimension. We provide here a brief description of thestrategy for the numerical integration. We first note that the integral (13)involves four variables for each time step: ρ i , φ ,i , φ ,i and S i . The first three variables describe the orders dynamics; we denote the spaceof these variables the hydrodynamical space. The last variable is the stockprice.The algorithm first selects a particular configuration in the hydrodynam-ical space, which corresponds to a particular trading pattern, then it samplesthe associated stock price variables with the Metropolis-Hastings algorithm,which is a Markov chain Monte Carlo method, using the potentialexp (cid:20) − N − X i =0 σ ∆ (cid:18) log( S i +1 ) − log( S i ) − α [( ρ i +1 − ρ i )] (cid:19) (cid:21) . The sampled values S i are then used to compute the integral (13). Detailsabout the Metropolis-Hastings algorithm can be found in [18]. We recallthat the following assumption have been introduced:• µ = r = r = 0 ,• tc=0• ˜ β =2.5,• M=2 n =2 m =2 n =2 m =100.The first assumption is not restrictive, and makes the results more transpar-ent. The second one consists in neglecting transaction costs, but we pointout that these could be easily introduced in the model. We choose ˜ β = 2 . β does not change the qualita-tive behaviour of the model. Indeed, the Hamiltonian is invariant under thetransformation S i → S ˜ β − i , σ → σ ˜ β − / β .The final assumption is chosen as a compromise between the compu-tational complexity and the accuracy; simulations performed with highervalues of M show similar results. The dynamics of the model is a perturbedGeometric Brownian Motion, the perturbation being proportional to theparameter α . The parameters of the Brownian Motion are the same of thesimulation discussed above. The results of simulations are shown in loga-rithmic scale in Figures 2 and 3. It turns out that this perturbation onlyFigure 2: Normal distribution (red) and PDF of the perturbed model with α = 0 .
461 (green). We present only one simulation in order to show theeffect of the perturbation.Figure 3: In this case all three simulations, α = 0 .
266 (yellow), α = 0 . α = 0 .
461 (green) are shown together to highlight the relationshipbetween the perturbation intensity and α .causes an increase of the variance σ . This can be observed in Figure 4, wherethe PDF of the model without orders, with α = 0 .
266 and σ = 0 . . σ (blue linewith squares) are shown.The numerical computation of the mean, variance and kurtosis of thesimulated probability density function shows that the previous values are10igure 4: The red line is the normal distribution, whereas the blue linewith the squares is the result of the simulation of the previous model with α = 0 .
266 .equal to those corresponding to the Geometric Brownian Motion with σ =0 . . ∗ . Ilinski suggests a second kind of perturbation v = − α [( ρ i +1 − ρ i )] k − α ( ρ i +1 − ρ i ) ∆ log( S i ) n /M − / , and he computes the probability distribution of S ( T ) with the saddle pointmethod and other approximations in the case k = 1. The probability distri-bution displayed in [2, p. 148, Fig. 6.15] is very accurate in the central part,but it behaves badly in the deep tails. If we analyse the probability densityfunction derived by Ilinski’s model, we note that its wings exhibit a linearrelationship between log( P ( S ( T ))) and log( S ( T )).This behaviour is not in good agreement with the stocks dynamics. Theoverlap between the computed and observed probability density functionsis not very accurate in the wings region. In particular we can see thatthe relation between log( P ( S ( T ))) and log( S ( T )) showed in the PDF is ofpolynomial type; i.e, log( P ( S ( T ))) = α log( S ( T )) Γ We propose a different perturbation: Ilinski’s action − β A g ( S ) = N − X i =0 − σ ∆ (cid:18) log( S i +1 ) − log( S i ) − α [( ρ i +1 − ρ i )] (cid:19) , ρ i +1 − ρ i ). We introduce the action − β ˜ A g ( S ) = N − X i =0 − σ ∆ (cid:18) log( S i +1 ) − log( S i ) − J X k =1 α k ( ρ i +1 − ρ i ) | ρ i +1 − ρ i | Γ k − (cid:19) , where J ≥ k ≥ J small to avoid to overfit the data; it turns out that thisperturbation with J = 2 provides results in good agreement with all thereal data that we analysed. Still, at first we present a result with J = 1 andΓ = 3 i.e. J X k =1 α k ( ρ i +1 − ρ i ) | ρ i +1 − ρ i | Γ k − = 2 α ( δρ ) . The result of the numerical simulation is displayed in Figure 5, which showsthe leptokurtic behaviourFigure 5: The red line is the normal distribution, whereas the blue line isthe generalized model with α = 0 . S&P500 index and
APPLE stocks from 01/05/2017 to 26/07/2017. The samplingfrequency is τ = 60 s ; each dataset consists of about 25000 prices. In orderto obtain the probability density function associated to the index and thestock, we follow the method introduced in reference [14], i.e. we consider aset of historical data as instances of a stochastic variable. We first compute X i by X i = log( P ( t i ) /P ( t i − )) t i − t i − = τ, then we build a histogram of the values X i with N bins. The histogramsshown in Figure 6 , , C in each his-togram bin, divided by the bin width ∆ S/N . The result is then normalizedin order to approximate a probability density function.The error bars for the real data are estimated as σ bins ∆ S/N ; where ∆ S isthe width of the histogram x -bars. In order to compare the simulations with12he results obtained in [2, 3, 10], we plot the probability density functionsin logarithmic scale.We plot the results of some simulations performed with the intent toreproduce real data. Here we did not plot the simulation errors in order tokeep the pictures as clear as possible; such errors have been estimated andthey are within 1%.The picture show in blue the empirical distributions of S&P500 index;the black curves represent the results of the simulations with the generalizedmodel.The red curves represent the normal distribution with the same σ and T ;Figure 6: S&P500: σ = 0 . T = 10, α = 0 . N = 60 and∆ log( S ) = 0 . ;yet in the tails, the model underestimates the value of the probability densityfunction. This behaviour concerns the simulations of both the indices andthe Apple stock. Figure 7: APPL σ = 0 . T = 10, α = 0 . N = 80 and ∆ log( S ) = 0 . This can be explained by the absence of large jumps in the simulation, log( S (0)) = 0. | δρ | ’
1. For the
APPLE share and
S&P500 index two different kinds of perturbations areused: J X k =1 α k ( ρ i +1 − ρ i ) | ρ i +1 − ρ i | Γ k − = 2 α ( δρ ) + 2 α ( δρ ) APPL , J X k =1 α k ( ρ i +1 − ρ i ) | ρ i +1 − ρ i | Γ k − = 2 α ( δρ ) + 2 α ( δρ ) S&P500.The numerical simulations associated with the previous perturbations givethe following results: Figure 8: APPL σ = 0 . T = 10, α = 0 . α = 0 . N = 80 and ∆ log( S ) = 0 . The measure that we used to quantify the agreement between real dataand the simulations is the overlap amplitude between the numerical and realFigure 9: S&P500 σ = 0 . T = 10 , α = 0 . α = 0 . N = 60 and∆ log( S ) = 0 . . . τ = 1 min. Better results havebeen achieved in [10] by Dupoyet and Fiebig using a quantum lattice modelwhich reproduces the probability density function of NSDAQ index with anagreement about four orders of magnitude with the same τ . Albeit [10] is aconsiderable improvement over [3], it suffers of the same problem of Ilinski’s,that is, it underestimates the probabilities of large market corrections.Our model fits the APPLE stock and S&P index probability densityfunctions with an agreement of almost four order of magnitude, and in par-ticular, when compared with the other models mentioned above, it providesa good fit or the distributions in the deep tails region. We provide a quantitative relationship between the statistical propertiesof the simulations and the parameters values. Two tables are presentedwith different values of α and α ; in the first line we write the parametersassociated to a GBM with the same σ and T of the generalized model. Fromthe tables it is possible to infer a direct relationship between the strengthof the perturbation and the value of α k . The greatest differences betweenthe two models can be seen in the last lines of the table, where α ’
0, i.e.,where the perturbation generated by α is stronger. The estimate of theparameters is obtained by the classical formulak-th moment( X ) = Z ( x − µ ) k f ( x ) dx, where f stands for the probability density function of the variable X and µ is its mean.The error in the variance, kurtosis and the other moments is estimatedcomputing 5 times the probability density function and then evaluating foreach result the parameters. The values written are the mean and standarddeviation computed over the previous results.15 , Variance Kurtosis 6th-Moment 8th-MomentGBM 3.93 e-8 3.00 9.19 e-22 2.53 e-28 α = 2 . α = 1 . ± ± ± ± α = 1 . α = 1 . ± ± ± ± α = 6 . α = 1 . ± ± ± ± α = 3 .
25 e-4 α = 1 . ± ± ± ± α = 1 .
65 e-4 α = 1 . ± ± ± ± , Variance Kurtosis 6th-Moment 8th-MomentGBM 3.93 e-8 3.00 9.19 e-22 2.53 e-28 α = 2 . α = 0 1.9595 e-07 ± ± ± ± α = 1 . α = 0 8.5264 e-08 ± ± ± ± α = 6 . α = 0 5.0286 e-08 ± ± ± ± α = 3 .
25 e-4 α = 0 4.4560 e-08 ± ± ± ± α = 1 .
65 e-4 α = 0 3.9401 e-08 ± ± ± ± α k values are proportional to σ . In thefirst generalized model simulation α/σ ’ − / − = 10 , which is equalto the real data cases α/σ ’ − / − = 10 . This is clear if we observethe whole action˜ A g ( S ) ’ (cid:18) δ log( S ) − α ( δρ ) Γ − α ( δρ ) Γ (cid:19) . In the Metropolis-Hastings algorithm, in order to obtain a mixing ratio of25%, the S fluctuations are proportional to σ , while δρ is always distributedin [ − , δ log( S ) ’ σ, − α ( δρ ) Γ − α ( δρ ) Γ ’ ± α ± α . In order to perturb in a proper way the price variation we need σ ’ ± α ± α , therefore a change in the order of magnitude of σ must correspond to asimilar change in the parameters α and α . The terms α k ( δρ ) Γ k introduced above allow us to obtain a good agreementbetween the simulated and the real PDF. In this section we provide a finan-cial interpretation of such quantities. Given the perturbation α ( δρ ) Γ , we observe that 17 Γ affects the jumps size,• α affects the probability of the jumps.We consider Γ first. Figure 10 shows the result of a single perturbed model α ( δρ ) Γ with different values of Γ and fixed α .Figure 10: α ( δρ ) Γ Γ=5 -blue-, 7 -green-, 9 -yellow-, 11 -purple- and 13 -black-.The red curve represents the normal distribution with the same σT . The Geometric Brownian Motion yields a mesokurtic probability densityfunction and all the trajectories simulated with this model have a contin-uous path. However, it is possible to obtain big fluctuations between theinitial price S (0) and the final price S ( T ) by setting a large variance σ .Increasing σ does not affect the trajectory continuity, since the model re-mains a GBM. The price fluctuations are directly linked with the variance σ ; but the continuity is not affected by the value of this parameter. Thesefacts suggest an interpretation of σ as the frequency of the orders with smallspread ∆ S = | S ( i ) − S ( i + 1) | . A larger value of σ corresponds to a largernumber of orders per unit time, yielding a larger price fluctuation. Since weare considering small ∆ S variations, continuity is preserved.This model is too simple for a real market description; in particular somemassive price corrections may happen in a unit time frame. This events arecalled jumps.In the real financial context, massive price corrections appear when thereis an external change in the macroeconomic scenario; when this happens,the original price may be greatly underestimated or overestimated. In theprevious situation the orders given immediately after the macroeconomicchange will have a large spread ∆ S .Within this framework, large price corrections are more likely; whichimplies that the tails of the PDF are fatter. Given a model which allowsonly x ∆ S jumps with ∆ S = | S ( T ) − S (0) | , the probability of a price change y ∆ S where y << x , it is equal to the case without jumps. Instead, if y ≥ x the probability will be much higher with respect the jumps-less case.18enoting as ˜ S the price where the wings start to exhibit their presence,we can observe that it is proportional to Γ. In fact we note that the overlapbetween the normal distribution and the black line, with Γ = 13, is longerwith respect to the overlap of the blue line, with Γ = 5; moreover all theprice variations y∆ S ≥ | ˜ S − S (0) | are more likely to happen with respectto the Geometric Brownian Motion model. This is in agreement with ourinterpretation on Γ.We also recall that the perturbation with Γ = 1 is equal to a GeometricBrownian Motion with increased σ ; suggesting again that Γ k is related withjumps size present in the model.To consider the effect of α , we show the results of a simulation with theperturbation α ( δρ ) and different values of α in Figure 11.Figure 11: α ( δρ ) α = 0.00181 -green-, 0.00158 -yellow-, 0.00140 -blue-, 0.00126 -purple- and 0.00114-black-. The previous simulation, along with the previous tables, shows that thekurtosis and higher even moments of the distribution are directly linkedwith the value of α .The mass under the tails quantify the presence of massive price variation,occurring in presence of jumps; which means that the probability associatedto this variations is directly related with the jumps probability. This showsthe relation between jumps and the shape of the tails. References [1] K. Ilinski,
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