Entropic Dynamics of Stocks and European Options
aa r X i v : . [ q -f i n . P R ] A ug Entropic Dynamics of Stocks and European Options
Mohammad Abedi ∗ , Daniel Bartolomeo † Department of Physics, University at Albany-SUNY,Albany, NY 12222, USA.
August 20, 2019
Abstract
We develop an entropic framework to model the dynamics of stocks and EuropeanOptions. Entropic inference is an inductive inference framework equipped with propertools to handle situations where incomplete information is available. The objective ofthe paper is to lay down an alternative framework for modeling dynamics. An impor-tant information about the dynamics of a stock’s price is scale invariance. By imposingthe scale invariant symmetry, we arrive at choosing the logarithm of the stock’s priceas the proper variable to model. The dynamics of stock log price is derived using twopieces of information, the continuity of motion and the directionality constraint. Theresulting model is the same as the Geometric Brownian Motion, GBM, of the stockprice which is manifestly scale invariant. Furthermore, we come up with the dynamicsof probability density function, which is a Fokker–Planck equation. Next, we extendthe model to value the European Options on a stock. Derivative securities ought to beprices such that there is no arbitrage. To ensure the no-arbitrage pricing, we derive the ∗ [email protected] † [email protected] isk-neutral measure by incorporating the risk-neutral information. Consequently, theBlack–Scholes model and the Black–Scholes-Merton differential equation are derived. Keywords: maximum entropy method, entropic dynamics, geometric Brownian mo-tion, European options, Black–Scholes model, Black–Scholes–Merton equation, put-callparity
In pursuit of understanding and describing phenomena, scholars often encounter situ-ations in which information about the subject of interest is limited. Entropic inferenceis an inductive inference framework equipped with proper tools to handle situationswhere incomplete information is available [1, 2, 3]. Such tools are probability theory,relative entropy, and information geometry. To give a quantitative description of thestate of partial knowledge, we use the probability distribution. Once new informationis available, we can update our state of partial knowledge by maximizing the relativeentropy. It is of crucial importance to notice that this notion of entropy does notoriginate from physics; however, it is the other way around. The celebrated notion ofentropy in physics is indeed an example of the notion of relative entropy in entropicinference [3].The information about the subject matter, the dynamical variable, takes the formof constraint. We use the method of Maximum Entropy, specifically the relative en-tropy, to incorporate the information available to answer questions about the system ofinterest. The Maximum Entropy method is designed to reflect the appealing propertyof Principle of Minimal Updating. The Principle of Minimal Updating ensures thatthe probability distribution is updated to the extent required by the new information.Due to incomplete information, addressing a question renders a probabilistic answer,namely a probability distribution. The relative entropy is designed to update the stateof partial knowledge, namely the probability distribution, whenever a new piece ofinformation is available. The advantage of an entropic framework is the flexibility with hich it can be adapted to deal with a variety of situations: once one realizes howinformation is codified into constraints, it is straightforward to modify the constraintsto create new models. The main challenge is to identify the correct variables and theinformation that happens to be relevant to the problem.Entropic Dynamics, ED, is an example of entropic inference framework in whichthe dynamical theories are modeled. Inference framework does not provide a notion oftime, namely an instant and a duration or clock. These notions need to be appendedto enable inference framework to model dynamics where time plays an important role.In ED, an entropic notion of time [4] is introduced which is suited to the system ofinterest. We are exploiting Mechanics without a Mechanism so as to develop a purelyinferential dynamics completely divorced from physics [5], for example, there is noenergy or momentum conservation. Entropic Dynamics has been extensively used tomodel the dynamics of subatomic particles [5, 6, 7, 8, 9, 10, 11].The practice of modeling the dynamics of stock goes back to Bachelier’s thesis[12] which predates Einstein work on Brownian Motion [13]. Bachelier modeled thedynamics of stock price using the stochastic process and he came up with what is knownas Geometric Brownian Motion. Bachelier’s work was rediscovered by Samuelson [14].The models were developed with the assumption of no jump [14, 15]. The GeometricBrownian Motion model was used by Black and Scholes to value Options [16, 17]. Thedynamics of stocks and pricing of Options were further developed by Merton to includejumps [18]. Numerous extensions and applications were proposed such as introducingstochastic volatility [19, 20, 21, 22, 23, 24, 25, 26, 27, 28]. Our model differs from thestochastic process approach since we do not make an assumption about the process,but we derive the dynamics of price i.e., GBM. Entropic modeling complements thestochastic process modeling. An advantage of Entropic Dynamics is that it can unifythe models developed in different branches of science.In this work, we model the dynamics of stock price. The stock price is changing andthere are other factors affecting the price to change. Due to a lack of information, wecannot keep track of the exact factors affecting the price; instead, based on the infor- ation available to us, we can proceed to predict the distribution of the price providingchange happens. We do not take into account that jumps can happen; we develop ourmodel in the regime of continuous evolution. The ED of the stock model yields theGeometric Brownian Motion dynamics. This is the new contribution where we derive the assumption of stock price undergoing a Geometric Brownian Motion. Furthermore,we address the question of how the probability density distribution changes over timewhich will be a Fokker–Planck equation.The main objective this work is to lay down an alternative framework to modeldynamics. While the main focus of other articles on the subject are to assume differentstochastic models for the dynamics of stock price, we derive the stochastic dynamicmodel from our formalism. This is where we contribute to the literature on the subjectby introduction a formalism to derive dynamics. A simple comparison can be madewith the Newton’s theory of motion where the dynamics are assumed, the Newton’slaws, and the Lagrangian formalism where the dynamics can be derived.Next, we proceed to apply the entropic stock model to price European Options. Todo so, we construct the risk-neutral probability distribution, applying which to valueOptions ensures the no-arbitrage pricing. Deriving the risk-neutral distributions is donewithin the Entropic Dynamics formalism, i.e., we use the risk-neutral information toderive the appropriate measure. We will arrive at the Black–Scholes, BS, model [16, 17]by assuming that the volatility and risk free rate are constant over time. The celebratedBlack–Scholes-Merton, BSM, differential equation [18] is derived by taking the timederivative of the expected payoff at maturity. We want to model the dynamics of stock’s price. Given the current price of the stock,how will the price change? In this work, we do not need to address how we price orvalue the stock, namely how market determines the price, but instead, assuming thatis valued, we would like to come up with a model to describe how the price would hange. Specifying the subject matter, the underlying dynamical variable to model, is of crucialimportance. Prior information about the dynamical variable provides guidance tospecify the proper dynamical variable and the prior probability density.The dynamics should manifest scale invariance symmetry. As we will see, thestochastic process we derive will be manifestly scale invariant. However, to have ourframework be manifestly scale invariant, we try to construct the probability densitiesthat are invariant under scale transformation. This choice of having a formalism to bemanifestly invariant will lead to choosing a logarithm of price as the proper dynamicalvariable to model.Where does this symmetry come from? How do we know the symmetry before wederive the model? The symmetry is due to the interest of investors in investing in thestocks with higher returns. For any investor, it is not important what the price is, but,if they invest in a stock, how much return they will receive from that investment. Astock with a low price could have a higher return compared with a stock with a higherprice. This would make the stock with a higher return more favorable. The demandto purchase and invest in the stock with a higher return will increase, which, in turn,will lead to a rise in the price of the stock. This rise of the stock price will decrease thestock return. Supply and demand forces not only determine the market value of thestocks but most importantly they will equilibrate the return of the stocks such that allstocks, which have the same volatility, have the same return.The scale invariant objects are the probabilities and the relative entropy and con-straints. The probability densities need not be invariant under scaling transformation;however, to have a formalism that is manifestly invariant under scale transformation,we require the probability densities to be invariant under scale transformation. Thescale transformation of the price is the following: S = l S, (1)where l is a positive constant scaling factor. Choosing the price of the stock as thedynamical variable would lead to a probability density distribution that is not invariantunder scaling transformation. To have a manifestly invariant formalism, we would liketo find a representation such that the probability densities are invariant under thescaling transformation. Mathematically, we are looking for a function of the price f ( S )such that the probability densities are scalars: P (cid:16) f ( ˜ S ) (cid:17) = P (cid:16) f ( S ) (cid:17) . (2)Since probabilities are required to be scalar, then we have the following relation: df ( ˜ S ) = df ( S ) . (3)This equation would lead to the following relation for f ( S ): f ( ˜ S ) = f ( S ) + C, (4)where C is independent of price. Using Equation (1), we get f ( lS ) = f ( S ) + C ( l ) . (5)This relation would constrain the constant function C ( l ). Using the same equation fora different scaling factor, f ( l ′ lS ) = f ( lS ) + C ( l ′ ) = f ( S ) + C ( l ) + C ( l ′ ) = f ( S ) + C ( l ′ l ) , (6)which results in C ( l ) + C ( l ′ ) = C ( l ′ l ) . (7) he solution to Equation (7) is the following: C ( l ) = ln l. (8)Plugging Equation (8) in Equation (5) yields f ( lS ) = f ( S ) + ln l. (9)The unique solution to the above functional equation is the logarithm function, f ( S ) = ln S. (10)We can simply check that the scaling transformation of the price would shift the f ( S )by a constant that satisfies the desired property of Equation (3). All in all, we con-clude that choosing the microstate as the logarithm of the price would be beneficiali.e., the probability density distribution would be a scalar function under the scalingtransformation of the price. Our subject matter is the logarithm of the price. Therefore, the question we wish toaddress is given the current logarithm of the price, ln S , how will the log price change?Given incomplete information about the subject, we can give a probabilistic answer P (ln S ′ | ln S ) that is through the method of Maximum Entropy, ME. Assigning thedistribution P (ln S ′ | ln S ) requires utilizing the relative entropy, S [ P, Q ] = − Z d ln S ′ P (ln S ′ | ln S ) ln P (ln S ′ | ln S ) Q (ln S ′ | ln S ) , (11)where Q (ln S ′ | ln S ) is the prior probability distribution, and specifying it is the subjectof the next section. The relative entropy is designed as a tool of inference to updatethe state of partial belief whenever new information is accessible [29, 30]. It is note-worthy that the notion of entropy does not belong to physics. In the development oftheoretical physics, it became manifest that entropy as a tool of inference can be used o model statistical physics where the thermodynamic entropy is shown to be derivedfrom inference [31, 32, 33]. Maximizing the entropy, updating the distribution, withrespect to no information but normalization constraint would yield a posterior distri-bution that is exactly the same as the prior. This is exactly what we expect; given nonew information, we would have no reason to update our belief, the distribution. The prior distribution presents the information at hand before any new information isaccessed. The prior distribution can be found from maximizing the relative entropy, S [ Q, µ ] = − Z d ln S ′ Q (ln S ′ | ln S ) ln Q (ln S ′ | ln S )Ω(ln S ′ | ln S ) , (12)where Ω(ln S ′ | ln S ) reflects our belief when no information is available. In this case ofextreme uncertainty where the stock’s price can undergo any change irrespective of thecurrent price, we have to assign uniform distribution. However, a priori, we know thatthe log price would make a small change in vicinity of the current log price, namelythe motion of log price is continuous. Such information is harnessed as the followingconstraint: (cid:10) (∆ ln S ) (cid:11) Q = *(cid:18) ln S ′ S (cid:19) + Q = k. (13)Maximizing the relative entropy Equation (12) subject to the prior information andnormalization yields Q (ln S ′ | ln S ) = 1 η exp " − α (cid:18) ln S ′ S (cid:19) , (14)where α is a Lagrange multiplier that is large and will be specified later on and η = R d ln S ′ exp (cid:20) − α (cid:16) ln S ′ S (cid:17) (cid:21) . Notice that, for large α , the prior Equation (14) is a verysharp Gaussian distribution anchored at the current log price ln S . Next, we considerthe interpretation of the Lagrange multiplier α . .2.2 Volatility and Entropic Clock Inference is not equipped with the notion of time; we need to add this notion to enablethe entropic inference framework to model dynamics. The same as entropy, time is alsoconsidered as a quantity belonging to physics. Time in physics, the same as entropy,is an example of entropic time [4]. Here, we introduce an entropic notion of clock, aduration that is tailored to stock price dynamics. We introduce the notion of entropicclock as follows: α = 1 σ ∆ t , (15)where σ is the volatility of the stock log price. If the volatile happened to be constant,then this notion of clock resembles Newtonian time [6]. Entropic clock could be alsodefined in such a way to be the same as relativistic time [8, 9, 10]. Upon the systemof interest, a relevant entropic time could be introduced, which would simplify thedynamics. In entropic inference framework, the information relevant to dynamical variable isintroduced in the form of constraint. Notice that, if we take into account wronginformation, we will get an incorrect model. The only piece of information pertainingto change of price is: (cid:28) ln S ′ S (cid:29) P = k ′ , (16)where k ′ will be determined shortly and P stands for the posterior transition density.Next, by Taylor expanding the log function to the second order, we getln S ′ S ≈ ∆ SS − (cid:18) ∆ SS (cid:19) (17)and then take the expectation with respect to the transition distribution, (cid:28) ln S ′ S (cid:29) P ≈ (cid:28) ∆ SS (cid:29) P − *(cid:18) ∆ SS (cid:19) + P , (18) here the first term of the expansion is specified by a drift, (cid:28) ∆ SS (cid:29) P = µ ∆ t. (19)In our model, we do not need to know what the value of the drift is; however, to applythe model, the drift is a piece of information that ought to be found. Each stock hasits own drift that reflects the performance of the company. Another separate entropicmodel can be developed to take into account the fundamental ratios and relevantinformation of the company to derive the drift term.To specify the second term of the expansion in Equation (18), we maximize the entropysubject to normalization and the constraint Equation (16). This will yield the transitionprobability: P (ln S ′ | ln S ) = 1 ξ exp " − α (cid:18) ln S ′ S (cid:19) + β ln S ′ S , (20)where β is a Lagrange multiplier corresponding to the constraint Equation (16) andthe normalization factor is ξ = R d ln S ′ exp (cid:20) − α (cid:16) ln S ′ S (cid:17) + β ln S ′ S (cid:21) . We can rewritethe distribution Equation (20) as a Gaussian distribution in log S ′ , P (ln S ′ | ln S ) = 1 Z ( α, β, ln S ) exp " − α (cid:18) ln S ′ S − βα (cid:19) , (21)where the new normalization factor is Z = R d ln S ′ exp (cid:20) − α (cid:16) ln S ′ S − βα (cid:17) (cid:21) . We attainthe Weiner process for the logarithm of price,ln S ′ S = (cid:28) ln S ′ S (cid:29) + ∆ W, (22) (cid:28) ln S ′ S (cid:29) = β σ ∆ t h ∆ W i = 0 , (cid:10) (∆ W ) (cid:11) = 1 α = σ ∆ t. (23)Now, to find the second term in Equation (18), D(cid:0) ∆ SS (cid:1) E P , we square the Taylorexpansion Equation (17) and then take the expectation with respect to transitiondistribution, (cid:18) ln S ′ S (cid:19) + P = *(cid:18) ∆ SS (cid:19) + P = σ ∆ t. (24)All in all, we specify the constraint Equation (16) and the Lagrange multiplier β , (cid:28) ln S ′ S (cid:29) ≈ (cid:28) ∆ SS (cid:29) − *(cid:18) ∆ SS (cid:19) + = µ ∆ t − σ ∆ t = k ′ (25)and βα = β σ ∆ t = k ′ = µ ∆ t − σ ∆ t , thus we have β = µσ − . To summarize, wecan rewrite the transition probability, P (ln S ′ | ln S ) = 1 Z exp " − σ ∆ t (cid:18) ln S ′ − (cid:0) ln S + µ ∆ t − σ ∆ t (cid:1)(cid:19) . (26)This is the normal distribution for the log price that leads to a Wiener process for thelog price, ln S ′ S = (cid:28) ln S ′ S (cid:29) P + ∆ W, (27)where (cid:28) ln S ′ S (cid:29) P = µ ∆ t − σ ∆ t h ∆ W i P = 0 (28)with the following fluctuation, (cid:10) (∆ W ) (cid:11) P = σ ∆ t. (29)It is noteworthy that we can see why, in Taylor expansion, we kept the second orderin Equations (17),(24). The higher order terms are proportional to higher order of∆ t . Thus, in the regime of continuous motion, we only kept terms that converge tothe left-hand side in probability. To rewrite the transition probability in terms of theprice of the stock, we just transform from log price back to price using the followingtransformation: d ln S ′ P (ln S ′ | ln S ) = dS ′ S ′ P (ln S ′ | ln S ) = dS ′ P ( S ′ | S ) . (30) hen, we get the probability of price, P ( S ′ | S ) = 1 Z S ′ exp − σ ∆ t " ln S ′ − (cid:16) ln S + µ ∆ t − σ ∆ t (cid:17) . (31)This is the lognormal distribution for the S ′ . Notice that this is the transition dis-tribution for a very short time interval. Assuming that the drift µ and volatility areuniform; then, for a finite time interval T , we will get a lognormal distribution with∆ t replaced by the finite time T . However, if we relax the assumption of uniformity,we will no longer end up with a lognormal distribution. The resultant distribution willbe the solution of a Fokker–Planck equation that is the subject of the next section. To complete the notion of entropic time, we need to define entropic instant. An entropicinstant is defined as p (ln S ′ ) t ′ = Z d ln S P (ln S ′ | ln S ) p (ln S ) t . (32)The distribution p (ln S ) t represents all available information at an instant t and thenext instant is defined as p (ln S ′ ) t ′ . For simplicity, we write p (ln S, t ) instead of p (ln S ) t .This parameter t has a nice property of being ordered and having an arrow [4]. Theintegral Equation (32) can be written in a differential form: ∂ t p (ln S, t ) = − ∂∂ ln S ( µ − σ ) p (ln S, t ) ! + 12 ∂ ∂ (ln S ) (cid:18) σ p (ln S, t ) (cid:19) . (33)This is a Fokker–Planck equation that governs the dynamics of density distribution. In this section, we use the entropic stock model to derive the risk-neutral probabilitydensity. Using the risk-neutral measure for valuation amounts to a no-arbitrage pricing. e simply value the options at maturity by its expected payoff using the risk-neutralmeasure and then the premium is calculated as the discounted expected payoff. Next,we derive the Black–Scholes-Merton partial differential equation governing the timeevolution of the options premium following the same argument. Derivative securities should be priced such that there is no arbitrage opportunity. Tohave a no-arbitrage pricing, we derive the risk-neutral probability density. Risk-neutralmeasure is derived by imposing the risk-neutrality constraint [34]: µ = r f , (34)where r f is the risk free rate. To derive the risk-neutral measure, we follow the sameprocedure as we did to derive the distribution for the underlying security, but, instead,on the drift in Equation (19), we impose the constraint Equation (34). The resultantrisk neutral probability density is as follows: P (ln S ′ | ln S ) = 1 Z exp " − σ ∆ t (cid:18) ln S ′ − (cid:0) ln S + r f ∆ t − σ ∆ t (cid:1)(cid:19) . (35)To value the European Call option, we start with computing the expected payoffat maturity using the risk neutral probability assuming that the the risk free rate andthe volatility are uniform, constant in time and independent of price. The expectedpayoff, denoted by V c , at maturity is given by the difference between the expected saleprice and the expected purchase price, V c = h Sale i LN,T − h
Purchase i LN,T , (36)where we have h Sale i T = Z ∞ K dS P ( S, T | S ) S, (37) h Purchase i T = Z ∞ K dS P ( S, T | S ) K, here K is the strike price and S is the current stock price. We integrate from thestrike price since, if the price is less than the strike price, we will not exercise the calloption. Then, the expected payoff can be written as V c = Z ∞ K dS P ( S, T | S ) ( S − K ) . (38)The Premium for call option is just the discounted value of the payoff. The secondpiece of information about the risk-neutral valuation is to discount the future valueswith the risk free rate [34]: C = e − r f T V c . (39)Since we know the current price S and we have already assumed the risk free rate andthe volatility to be uniform, then the probability distribution at maturity is given by P ( S T | S ) = Z d ˜ S P ( S T | ˜ S ) P ( ˜ S | S ) = Z d ˜ S, P ( S T | ˜ S ) δ ( ˜ S − S ) , (40) ∼ LN (ln S + r f T − σ T , σ √ T ) . This is still a lognormal distribution of price for a finite time interval T . Notice that thenice lognormal distribution is achieved since the interest rate and volatility are time andprice independent. Relaxing these assumptions would lead to a different distributionthat is a solution of the Fokker–Planck equation. Calculating the expected sale priceyields h Sale i LN,T = S exp[ rT ] N ( d ) , (41)where d = ln S + r f T + σ T − ln Kσ √ T and N ( d ) is the standard normal cumulative distribu-tion function N ( d ) = 1 √ π Z d −∞ dx e − x . (42) n addition, the expected purchase price can be computed, h Purchase i LN,T = KN ( d ) , (43)where d + σ √ T = d . Then, the premium of call option is C = S N ( d ) − e − r f T K N ( d ) . (44)This is the Black–Scholes model for a European call option. We can follow the samelogic to value a European put option. The expected payoff at the maturity for a putoption is V p = Z K dS P ( S, T | S ) ( S − K ) . (45)Notice that we integrate from zero to the strike price K because, if the price is greaterthan the strike price, we will not exercise the put option. Discounting this expectedpayoff will yield the put premium, P = e − r f T K N ( − d ) − S N ( − d ) . (46)We can simply check that the call and put premium satisfy the so-called call-put parityrelation, C − P = e − r f T ( F − K ) , (47)where F is the forward price, S = e − r f T F . Therefore, we can conclude that pricingthe European options based on expected payoff is a no-arbitrage valuation. To derive the Black–Scholes-Merton differential equation, we start with the expectedpayoff equation, V (ln S, K, t ) = Z d ln S T P (ln S T , T | ln S, t ) ( S T − K ) . (48) otice that we left the bounds of the integral such that we can use it both for call andput options. Next, we take time derivative, ∂ t V = Z d ln S T ( S T − K ) , ∂ t P (ln S T , T | ln S, t ) , (49)where the time derivative of the transition probability is given by a Backward–Kolmogorovequation, ∂ t P (ln S T , T | ln S, t ) = − (cid:18) r f − σ (cid:19) ∂P (ln S T , T | ln S, t ) ∂ ln S − σ ∂ P (ln S T , T | ln S, t ) ∂ (ln S ) . (50)Substituting Equation (50) into Equation (49), we get ∂ t V = Z d ln S T ( S T − K ) (cid:20) − (cid:18) r f − σ (cid:19) ∂P∂ ln S − σ ∂ P∂ (ln S ) (cid:21) (51)= − (cid:18) r f − σ (cid:19) ∂∂ ln S Z d ln S T ( S T − K ) P (ln S T , T | ln S, t ) − σ ∂ ∂ (ln S ) Z d ln S T ( S T − K ) P (ln S T , T | ln S, t )= − (cid:18) r f − σ (cid:19) ∂V∂ ln S − σ ∂ V∂ (ln S ) . We can rewrite this equation as ∂ t V + r f S ∂V∂S + σ S ∂ V∂S = 0 . (52)The partial differential equation for the option premium is derived just by substituting E = e − r f ( T − t ) V into the above equation ∂ t E + r f S ∂E∂S + 12 σ S , ∂ E∂S − r f E = 0 , (53)where E stands for both call and put options. This is the celebrated Black–Scholes–Merton equation for European options. To solve the BSM equation for put or calloptions, we need to apply the right boundary conditions. We laid down an entropic framework to model the dynamics of stocks and Europeanoptions[35]. In our formalism, the dynamical model is derived by maximizing the elative entropy subject to the information relevant to system of interest. An importantcontribution of our work is introducing this alternative framework, which leads toderiving a stochastic process. The Geometric Brownian Motion model of stock priceis derived by taking into account two pieces of information and imposing the scalingsymmetry. It is noteworthy to mention that other literature on the subject starts byassuming an ad hoc stochastic process to model the dynamics, whereas we derive thedynamics.Next, we extended our entropic stock model to value European options on stocks.Derivative securities ought to be priced such that there is no arbitrage opportunity.To this end, we incorporated the no arbitrage, or the risk-neutral, information in ourformalism and the risk-neutral probability density was derived. To value the optionspremium, we discounted the expected payoff the options at maturity. The resultingmodel is the same as the Black–Scholes model and a differential equation was derivedto value options at any time, which is the Black–Scholes–Merton differential equation.Our formalism makes the assumptions made to derive the GBM clear. Incorporatingnew information or relaxing the uniformity of the drift or volatility not only lead toan extension of the dynamics of stock price, but also to a new model of pricing thederivatives. Another relevant piece of information about the dynamics of a stock priceis that jumps happen, which will be addressed in future work.What if we have a different security like a Foreign Exchange? In another work, weshowed that it is straightforward to extend our formalism to model the dynamics ofa FX. In addition, we derive the dynamics of FX value and the corresponding Black–Scholes model for European Options, known as the Garman–Kohlhagen model, onforeign exchange [36]. It is remarkable that our framework can be easily adapted tomodel different securities.An interesting extension is modeling the dynamics of a set of stocks. An importantquestion about such system is how to construct an optimized portfolio. Markowitz’sportfolio theory of mean-variance has addressed such a question. We would like to fur-ther develop our model to address such a question. Since our model is based on taking nto account information relevant to the problem, we will also be able to incorporateother information, or prior beliefs, available to private investors into account. Thiswould lead to a modified portfolio theory [37]. Acknowledgments:
We would like to thank Ariel Caticha, Lewis Segal, Amos Golan,and the Information Group of the University at Albany for many insightful discussionson Finance, Economics and Entropic Dynamics.
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