The implementation of the stochastic interacting traders model is a modication of the model (19), built by M. Jablonska in her paper [21] and introduced in Section 1.3.1.
The global interaction component, f2(k,Xt) of equation (19), has been replaced by a global kernel HN(XNk(t)) depending on the Heaviside function, dened as (77). This kernel gives information about the distribution of the total population with respect to the k−th individual.
The local interaction function, f3(k,Xt), has been replaced with an interaction kernel dened as a symmetric functionKN rescaled byN via a symmetric probability density K1 and depending on the interaction scaleβ.
The reason for applying these changes arises from the need to study in depth the existence of a link between the stochastic dierential equation and the Burgers equation as mentioned in [21]. Also the wishes to improve the predictions for commodity markets drove us to conduct of this analysis.
3 THE DYNAMICS OF TRADERS: A NEW PROPOSAL 89 The goodness of the t of the stochastic interacting traders model for Silver and New Zealand electricity market, described in Section 3.2.2, can be compared with the pre-vious model obtained using the system of stochastic dierential equations of the type (19), as described in Section 1.3.1. But the analysis cannot be conducted only compar-ing the statistical features of the two models, and this diculty is due to their dierent implementations.
The individual based model developed by M.Jablonska is implemented day by day, which means that at each day t the prediction for the next 91 days is computed. We can say that this model is adaptive because at each day it receives new information and computes again the prediction adding these news. While the interacting traders model has been tested on a time windows of 30 days, i.e. at day t is computed the prediction for the next 30 days, which means that the information are transferred to the model only one time (day) per month.
In the Jablonska model the parameters are already optimized which means that the results cannot be improved by changing parameter values and that the performance are at maximum level. Instead in the model proposed in this thesis the optimization of the parameters is still ongoing.
A comparison is done through the following gures:
3 THE DYNAMICS OF TRADERS: A NEW PROPOSAL 90 Silver market
Figure 36: Silver data simulations at dierent times Hb made with the Jablonska individual based model.
Figure 37: Root mean square error.
3 THE DYNAMICS OF TRADERS: A NEW PROPOSAL 91
Figure 38: Silver data simulated at each 30 days with the stochastic interacting traders model, γN = 15, N = 100, β= 0.05and σ = 1
(a) Root mean square error for Gaussian
ker-nel. (b) Root mean square error for kernel K.
Figure 39: Root mean square error for Silver data with the two dierent kernels.
3 THE DYNAMICS OF TRADERS: A NEW PROPOSAL 92 New Zealand electricity market
Figure 40: New Zealand electricity data simulations at dierent times Hb made with the individual besed model.
Figure 41: Root mean square error.
3 THE DYNAMICS OF TRADERS: A NEW PROPOSAL 93
Figure 42: New Zealand data simulated at each 30 days with the stochastic interacting traders model,γN = 2, N = 100,β = 0.1and σ = 120
(a) Root mean square error for Gaussian
ker-nel. (b) Root mean square error for kernel K.
Figure 43: Root mean square error for New Zealand electricity data with the two dierent kernels.
Despite the diculty to compare the two models, Figures 38, 39, for Silver, and Fig-ures 42, 43, for New Zealand electricity, show the eective existence of improvements achieved by using the new proposed model with respect to Figures 36, 37 and Figures
4 DISCUSSION AND FUTURE PERSPECTIVES 94 40, 41; the new model is also capable of producing high price spikes.
4 Discussion and future perspectives
In this work we studied the mean reverting process of Ornstein-Uhlenbeck type, the Burgers equation and its connection with a stochastic dierential equation. In particu-lar, we focused on a specic interaction kernel reached through the Heaviside function, which is linked to the Burgers equation.
The aim of this study was to build a stochastic model for commodity markets and show its convergence to the Burgers equation when the number of market participants increases.
Our analysis and our numerical experiments show the goodness of the model to forecast the two tested markets, and its improvements are conrmed by the ability of the model to reproduce big spikes when their eects occur in a suciently long period of time.
Parameters optimization is still challenging for the proposed model, because of lack of information about the individual traders' oers.
We also gured out that the model fails to follow the trend of the real price, due to its inability to have an inclination. One option to avoid this eect is to use the gradient or replace the Wiener process with dierent noise and prove that under this assumption the convergence to the Burgers equation still holds. An alternative way to solve this problem can be to create a mean reversion with respect to the trend, or adding an extra component that enriches the model with this information.
It follows that the stochastic interacting traders model is a good starting point for future improvements.
After these failures and adjustments have been completed, the model can be applied in dierent ways.
The stochastic interacting traders model can be used alone to test its ability in simulat-ing dierent commodity markets, or it can be added to the two or three-factor models discussed in the introduction, so that the traders' component becomes a factor that, together with stock price, convenience yields and interest rate has an impact on price formation.
A FUNDAMENTALS OF STOCHASTIC PROCESSES 95 Starting from the existing traders model, two dierent systems of stochastic dierential equations, for buyer and for seller, could be developed and the nal price can be computed, as in the reality, as the point at which demand meets supply.
After testing the capabilities of the stochastic interacting traders model, it might be interesting to shift to a multidimensional case, in which dierent kinds of markets and their derivatives are considered to interact, and each of them are assumed to be estimated as the solution of the described system of N stochastic dierential equations.
These and many other applications represent the future of the stochastic interacting traders model.
A Fundamentals of Stochastic Processes
A.1 Stochastic processes and their properties
Here we introduce basic properties of the stochastic processes and of the Wiener Process in particular the reader may refer to [43] for more details.
Denition A.1. Let (Ω,F,P) be a probability space, T an index set, and (E,B) a measurable space. An(E,B)−valued stochastic process on(Ω,F,P)is a family(Xt)t∈T
of random variables Xt: (Ω,F)→(E,B) for t∈T.
(Ω,F,P)is the underlying probability space of the(Xt)t∈T,(E,B)is the state space, also called, phase space. Fixing t ∈ T, the random variable Xt is the state of the process at time t. For all ω ∈ Ω, the mapping X(·, ω) : t ∈ T → Xt(ω) ∈ E is called the trajectory or path of the process corresponding to ω.
Denition A.2. Let (Ω,F,P) be a probability space. A ltration {F}t≥0 on(Ω,F,P) is a family of sub sigma−algebras of a sigma−algebra F, with the properties that if s≤t, then Fs⊂Ft. Ft=σ(X(s),0≤s≤t), t ∈R+ is called the generated or natural ltration of the processXt.
A probability space(Ω,F,P)endowed a ltration (Ft)t≥0 is called a ltered probability space.
Denition A.3. A stochastic process (Xt)t≥0 on(Ω,F,P)is adapted to the ltration (Ft) if, for each t ≥0, Xt is Ft−measurable.
A FUNDAMENTALS OF STOCHASTIC PROCESSES 96 Denition A.4. A ltered complete probability space (Ω,F,P,(Ft)t∈R+) is said to satisfy the usual hypotheses if
1. F0 contains all P−null sets of F.
2. Ft=T
s>tFs, for all t∈R+, i.e, the ltration(Ft)t∈R+ is righ−continuous.
Denition A.5. Let (Xt)r∈R+ be a stochastic process on a probability space, value in (E,B) and adapted to the increasing family (F)t∈R+ of a σ−algebra of subsets of F. (Xt)t∈R+ is a Markov process with respect to (F)t∈R+ if the following conditions is satised:
∀B ∈B,∀(s, t)∈R+×R+, s < t: P(Xt∈B|Fs) = P(Xt∈B|Xs) a.s.
That mean, the future is determined only by the present and not by the past.
Denition A.6. A real−vaued stochastic process(Xt)t∈R+ is continuous in probability if
∀t∈R+ and > 0, P(|Xs−Xt| ≥)→0 as s →t.
Denition A.7. The stochastic process(Ω,F,P,(Xt)t∈R+)with state space(E,B), is called a process with independent increments if, for alln ∈Nand for all (t1, . . . , tn)∈ Rn+, where t1 < . . . < tn, the random variables Xt1, Xt2 −Xt1, . . . , Xtn −Xtn−1 are independent.
The increments are said to be stationary if, for any t > s and h > 0, the distribution of Xs+h−Xt+h is the same as the distribution of Xs−Xt.
The stochastic process related to the movement of the particles, entirely chaotic, is the Wiener process.
Denition A.8. The real−valued process (Wt)t∈R+ is a Wiener process if it satises the following conditions:
1. W0 = 0 almost surely.
2. (Wt)t∈R+ is a process with stationary independent increments.
3. Wt−Ws is normally distributed, N(0, t−s), (0≤s < t). Proposition A.1. If (Wt)t∈R+ is a Wiener process, then
A FUNDAMENTALS OF STOCHASTIC PROCESSES 97
1. E[Wt] = 0 for all t∈R+
2. K(s, t) = Cov[Wt, Ws] = mins, t, s, t ∈R+
A particular and common used stochastic process is a Gaussian Process.
Denition A.9. A real−valued stochastic process(Ω,F,P,(Xt)t∈Rn+)is called a Gaus-sian process if, for all n ∈Nand for all (t1, . . . , tn)∈Rn+, the n−dimensional random vector X = (Xt1, . . . , Xtn)0 has a multivariate Gaussian distribution, with probability density
ft1,...,tn(x) = 1 (2π)n2√
detkexp{−1
2(x−m)0K−1(x−m)}, (92) where
mi =E[Xti]∈Ri= 1, . . . , n, Kij =Cov[Xti, Xtj]∈Ri, j = 1, . . . , n.
The covariance matrix K = (σij) is taken as positive−denite, i.e.,for all a ∈ Rn : Pn
i,j=1aiKijaj >0.
Proposition A.2. The Wiener process is a Gaussian process.
Theorem A.1. If (Wt)t∈R+ is a real−valued Wiener process, then it has continuous trajectories almost surely.
Proposition A.3. Let (Xt)t∈R+ be a real−valued continuous process starting at 0 at time 0. If the process is Gaussian process satisfying
1. E[Xt] = 0 for all t∈R+
2. K(s, t) = Cov[Xt, Xs] = mins, t s, t,∈R+
then it is a Wiener process.
Theorem A.2. Almost every trajectory of the Wiener process (Wt)t∈R+ is nowhere dierentiable.
Proposition A.4. Let (Wt)t∈R+ be a Wiener process; then the following properties hold:
A FUNDAMENTALS OF STOCHASTIC PROCESSES 98 1. (Symmetry) The process(−Wt)t∈R+ is a Wiener process.
2. (Time Scaling) The time−scaled process ( ¯Wt)t∈R+ dened by ( ¯Wt)t∈R+ =tW1
t, t >0, W¯0 = 0 is also a Wiener process.
3. (Space Scaling) For any c >0, the space−scaled process ( ¯Wt)t∈R+ dened by ( ¯Wt)t∈R+ =cW1
c2, t >0, W¯0 = 0 is also a Wiener process.
Proposition A.5. (Strong law of large numbers) Let (Wt)t∈R+ be a Wiener process. Then
Wt
t →0, as t→ ∞, a.s.
Denition A.10. (multi−dimensional Wiener process)
The real−valued process(W1(t), . . . , Wn(t))0t≥0 is said to be ann−dimensional Wiener process (or Brownian motion) if
1. For all i∈ {1, . . . , n}, (Wi(t))t≥0 is a Wiener process 2. The processes (Wi(t))t≥0, i={1, . . . , n} are independent,
thus the σ−algebras σ(Wi(t), t≥0)i= 1, . . . , nare independent.