Entropic Dynamics of Exchange Rates and Options
aa r X i v : . [ q -f i n . P R ] A ug Entropic Dynamics of Exchange Rates and Options
Mohammad Abedi ∗ , Daniel Bartolomeo † Department of Physics, University at Albany-SUNY,Albany, NY 12222, USA.
August 20, 2019
Abstract
An Entropic Dynamics of exchange rates is laid down to model the dynamics offoreign exchange rates, FX, and European Options on FX. The main objective is torepresent an alternative framework to model dynamics. Entropic inference is an in-ductive inference framework equipped with proper tools to handle situations whereincomplete information is available. Entropic Dynamics is an application of entropicinference, which is equipped with the entropic notion of time to model dynamics. Thescale invariance is a symmetry of the dynamics of exchange rates, which is manifestedin our formalism. To make the formalism manifestly invariant under this symmetry, wearrive at choosing the logarithm of the exchange rate as the proper variable to model.By taking into account the relevant information about the exchange rates, we derive theGeometric Brownian Motion, GBM, of the exchange rate, which is manifestly invariantunder the scale transformation. Securities should be valued such that there is no ar-bitrage opportunity. To this end, we derive a risk-neutral measure to value EuropeanOptions on FX. The resulting model is the celebrated Garman–Kohlhagen model. ∗ [email protected] † [email protected] eywords: Entropic Inference; Maximum Entropy; Entropic Dynamics; Fokker–Planck equation; Garman–Kohlhagen model; Black–Scholes–Merton partial differentialequation
To understand, describe, and predict phenomena, scientists have come up with thescientific method of reasoning. For a small fraction of the situations where completeinformation about the subject is accessible logic is proposed as a framework to reason.In most situations, where they are faced with not having enough information about thesystem, an extension of the logic is required. To deal with such situations, an inductiveinference framework is designed.
Entropic inference is an inductive inference frameworkdesigned with proper tools to cope with situations where incomplete information is atour disposal [1, 2, 3].The first tool of entropic inference designed to represent a quantitative description ofthe state of partial belief is probability theory. The probability distribution representsthe information we have about the outcome of the system. The second tool is designedto handle the situation when new information about the subject matter is accessible.The relative entropy is designed such that it can incorporate the new information and update the state of partial knowledge, the probability distribution [4, 5]. An importantcriterion in designing the relative entropy is the Principle of Minimal Updating . Thisprinciple ensures that the prior probability distribution is updated only to the extentrequired by the new information. By maximizing the relative entropy, a posteriordistribution is derived, which is the least biased distribution, namely it describes allgiven information, nothing else. It is noteworthy to mention that this notion of entropyis not derived from physics; it is shown that the known notion of entropy in physics is anapplication of the relative entropy [6, 7]. The third tool of inference is the informationgeometry. The space of probability distributions defines an information geometry witha unique metric, which defines the distance between two neighboring distributions [8, 9]. he significance of information geometry in finance will be addressed in future workwhere we will model the dynamics of many equities and address how to invest [10].A long-standing aspiration of scientists has been describing how a quantity of in-terest changes, the dynamics . The conception of time is contrived to further simplifydescribing any dynamics. If entropic inference is any good, as a formalism divorcedfrom science, not derived from it, it ought to come up with a notion of time that leadsto the known notions of time, i.e., the notions of time should emerge from the entropicframework. Entropic Dynamics is an application of the entropic inference framework,which defines entropic time and in turn enables the formalism to model dynamics. Thisentropic time can be tailored to suit modeling different unrelated dynamics where theentropic time pertains to the system of interest rather than having a unique universalnotion of time for all branches of science [11]. Entropic Dynamics has extensively beenapplied to model dynamics in physics [12, 13, 14, 15, 16, 17, 18].Modeling the dynamics of securities, especially stocks, goes back to Bachelier’sthesis [19]. For quite a while, his brilliant work was forgotten until it was rediscoveredby Samuelson [20]. Models were developed in the regime of continuous dynamics whereno jump happens [19, 20, 21]. The resulting Geometric Brownian Motion dynamicswas used by Black and Scholes and later by Merton to value Options [22, 23, 24]. Themodels were extended in many directions such as including stochastic volatility [25,26, 27, 28, 29, 30, 31, 32, 33, 34]. Pricing European Options on the exchange rate wasdeveloped by Garman and Kohlhagen [35]. In later works, the Garman–Kohlhagenmodel was extended in various directions [36, 37, 38, 39].We present an alternative framework to model dynamics. In our formalism, the dy-namical models are derived by maximizing the relative entropy. The relative entropy is designed as a tool of inference to update the state of partial belief when new informationis available. We do not resort to a principle such as an action principle or Hamiltonianmechanism, which is useful in a particular branch of science, to derive dynamics, nor dowe assume the dynamics like in the stochastic mechanics formalism. The significanceof our formalism is that we derive those ad hoc principles from our formalism. This is n important breakthrough in that we unify the scientific theories and show that theyare derived from a more fundamental approach. The main challenges of our formalism,as an alternative framework to model dynamics, are figuring out the proper variable,the microstate, to model and the relevant information about the system. Once therelevant information about the system of interest is found, then it is straightforwardto manipulate that information and modify the models. In addition, we show that ourmodel derives the stochastic process, which is an advantage of our formalism over thestochastic mechanics where the stochastic process is assumed. Deriving the stochasticprocess has the advantage of being explicit in what assumptions need to be made toyield such a process.In this work, we wish to model the dynamics of an exchange rate. Scale invarianceis the symmetry of the dynamics of the exchange rate, which should be incorporated inour model. Basically, investors are in favor of investing in securities with higher returnthan the securities with lower return given that they have the same volatility [41].It does not matter what the numeric value of the security is, but instead, the returnof such a security plays the crucial role. In order to have our formalism be manifestly scale invariant, we wish to formulate our formalism such that the probability densitiesare scalar functions. This choice will lead to choosing the logarithm of the exchange rateas the proper variable to model. Therefore, we want to model how the log exchange ratewould change given the current log exchange rate, to be specific, what the transitionprobability density P (ln u ′ | ln u ) is, where u = S f S d represents the current exchangerate of foreign currency to domestic currency and u ′ is the next exchange rate. It isimportant to notice that any extension of our model, such as including jumps, shouldbe such that the scale symmetry is not broken, otherwise there will be an arbitrageopportunity.In Entropic Dynamics, the information about the subject matter takes the form ofa constraint equation on the probability density function. Two pieces of informationrelevant to the dynamics of the log exchange rate are the continuity of the dynamicsand the directionality. In this work, we do not take into account that jumps can appen. Including jumps in our model is left for future work. The method of MaximumEntropy, maximizing the relative entropy, is used to assign and update the probabilitydensity. By maximizing the relative entropy subject to the constraints, we arriveat the transition probability distribution. The transition probability density will bea Gaussian distribution in the log exchange rate, which amounts to the GeometricBrownian Motion of the exchange rate. Apart from the dynamics of the exchangerate, the dynamics of the probability density is crucial, especially for the purposeof forecasting. We will show that the probability density will evolve according to aFokker–Planck equation.In Section 3, we apply our entropic model of the exchange rate to value EuropeanOptions on the exchange rate from the perspective of a domestic investor. Derivativesecurities should be valued such that there is no arbitrage opportunity. To establisha no-arbitrage valuation, we derive the risk-neutral probability density. Risk-neutralinformation is used and imposed on the entropic exchange rate model to derive a risk-neutral measure. The European Options premium is computed by taking the expectedvalues of the Options at maturity and discounting it with a risk free rate. The resultingmodel is the Garman–Kohlhagen model, which is the counterpart of the Black–Scholesmodel of European Options on stocks. Then, the call-put parity is derived, whichfurther certifies the no-arbitrage valuation. Using the same procedure, we derive thedynamics of European Options, which is the Black–Scholes–Merton partial differentialequation. We would like to model how the exchange rate changes. However, as will unfold inthe following, the proper variable to model turns out be the logarithm of the exchangerate. Prior information about the subject and the scale invariance symmetry lead us tochoose the logarithm of the exchange rate; therefore, we want to model the dynamicsof the logarithm of the exchange rate. cale invariance is an important symmetry, which ought to be manifest in thedynamics of the exchange rate, namely in the stochastic process that will be derived.Where does this symmetry come from? Investors do no care much about the value ofthe exchange rate, but if they invest in one, they would like to have a high return fromthat investment. Therefore, from the investment perspective, the exchange rate witha higher return would be more favorable than the one with a lower return assumingthe two assets have the same volatility. If we have two assets, with the same amountof risk associated with them, they are expected to have the same return, otherwisean arbitrage opportunity will emerge. Investors will take advantage of that arbitrageopportunity and equilibrate the market such that the two assets will have the samereturn. To be more specific, the demand for the asset with higher return will increase,which leads to an increase in the value of that asset. This increase in the value of theasset will lead to a lower return for that asset to the extent that both assets will havethe same return. Investors in the market, through the supply and demand forces, willuse the arbitrage opportunities to equilibrate the market. This is the essence of thescale invariance.A simple way of manifesting the scale invariance symmetry is to choose the rightfunction of the exchange rate such that the probability measures are invariant, scalarfunctions, under the scale transformation. Let us denote the exchange rate as u = S f S d ,where S f and S d represent the foreign currency and domestic currency, respectively.The scale transformation is given by, ˜ u = l u (1)where l is a positive constant called the scaling factor. We are looking for a function ofthe exchange rate f ( u ) such that the probability density P (cid:0) f ( u ) (cid:1) is invariant underthe scale transformation Equation (1), i.e., P (cid:0) f (˜ u ) (cid:1) = P (cid:0) f ( u ) (cid:1) (2)This leads to a constraint equation for f ( u ), (˜ u ) = f ( u ) + C (3)Using the scale transformation Equation (1) twice with two scaling factors l and l ′ ,we get a constraint on C , C ( l ) + C ( l ′ ) = C ( l l ′ ) (4)The unique solution to Equation (4) is C ( l ) = ln l , which in turn leads to a uniquesolution to Equation (3), f ( u ) = ln u (5)Therefore, to have a manifestly scale invariant formalism, we need to choose thelogarithm of the exchange rate as our subject matter to model. Notice that once thedynamics of the logarithm of the exchange rate is laid down, we can do any trans-formation, i.e., any change of variable, to find the dynamics of other functions of theexchange rate. The proper variable to model is the logarithm of the exchange rate denoted by ln u =ln S f S d . To come up with the dynamics, we address the question: How will the logexchange rate change given the current log exchange rate? In the entropic inferenceframework, we address such a question by assigning a transition probability distribu-tion, P (ln u ′ | ln u ). We use the method of maximum entropy to assign the transitiondistribution, S [ P, Q ] = − Z d ln u ′ P (ln u ′ | ln u ) ln P (ln u ′ | ln u ) Q (ln u ′ | ln u ) , (6)where Q (ln u ′ | ln u ) is called the prior, which captures prior information when we aremaking the inference. Next, we specify the prior distribution. .2 The Prior The prior distribution can be specified or assigned by using the method of maximumentropy and imposing the prior information. To assign the prior distribution, we max-imize the relative entropy S [ Q, q ] subject to the prior information, S [ Q, q ] = − Z d ln u ′ Q (ln u ′ | ln u ) ln Q (ln u ′ | ln u ) q (ln u ′ | ln u ) , (7)where q (ln u ′ | ln u ) is a uniform prior, which represents the situation of extreme igno-rance. Notice that q (ln u ′ | ln u ) is a prior for Q (ln u ′ | ln u ), which itself is a prior for P (ln u ′ | ln u ). Any non-uniform distribution amounts to prioritizing some outcomesover others, which is in contrast with having no information. The prior informationwe have about the subject is that the log exchange rate will have a small change,which amounts to saying that the dynamics is continuous. This continuity of dynamicsinformation is represented by the following constraint, (cid:10) (∆ ln u ) (cid:11) Q = *(cid:18) ln u ′ u (cid:19) + Q = k , (8)where k is small and will be determined in next part. Maximizing the relative entropyEquation (7) subject to normalization and the continuity constraint Equation (8), weget the prior distribution, Q (ln u ′ | ln u ) = 1 η exp " − α (cid:18) ln u ′ u (cid:19) (9)where α is large, which will be specified next, and the normalization factor is η = R ∞−∞ d ln u ′ exp (cid:20) − α (cid:16) ln u ′ u (cid:17) (cid:21) . We need to construct time in our formalism to be able to model the dynamics. Whywould we need to construct entropic time? Why can we not use the time that has beenused in physics or everyday life? We are putting forth an entropic framework, which s divorced from physics or finance. If our formalism is any good, we ought to be ableto model dynamics without resorting to the other theories or formalisms.The notion of time we introduce here will eventually be the same as the usualconception of time we use to model any stochastic process. Here, we introduce anotion of time that is a convenient tool to keep track of change [11]. We introduce thenotion of the entropic clock ∆ t as following, α (ln u ) = 1 σ (ln u ) ∆ t (10)where σ is the volatility of the log exchange rate. Notice that in Equation (10), wedefine α in terms of two variables, which later on are specified as the time durationand volatility. The value for k in constraint Equation (8) can be computed as k = α = σ ∆ t . If volatility were independent of exchange rate, then this notion of time wouldresemble Newtonian time, otherwise it is similar to a relativistic time. To complete thenotion of entropic time, we need to define an entropic instant . An entropic instant isdefined as, p (ln u ′ ) t ′ = Z d ln u P (ln u ′ | ln u ) p (ln u ) t , (11)If the distribution p (ln u ) t were to represent information at one instant, then thenext instance is defined as p (ln u ′ ) t ′ , where t ′ = t + ∆ t . Notice that an instant isdefined by a single state, where in our case, the state is represented by a probabilitydistribution. In addition, Equation (11) specifies ∆ t as the time interval; if t ′ is thenext instant to t , then the time interval is t ′ − t = ∆ t . For simplicity, we write p (ln u, t ),instead of p (ln u ) t . This parameter t has a nice property of being ordered and havingan arrow. The stochastic process that the prior distribution represents is a Brownian Motion ofthe log exchange rate with no drift. The new piece of information we have is that thereis a drift in the log price. This information is captured in the following constraint, ln u ′ u (cid:29) P = k ′ ( u ) → , (12)where k ′ ( u ) will be determined shortly. We can Taylor expand the log function,ln u ′ u ≈ ∆ uu − (cid:18) ∆ uu (cid:19) , (13)and then take the expectation with respect to the posterior distribution, (cid:28) ln u ′ u (cid:29) P ≈ (cid:28) ∆ uu (cid:29) P − *(cid:18) ∆ uu (cid:19) + P , (14)where the first term of the expansion defines a drift, (cid:28) ∆ uu (cid:29) = ( µ d − µ f ) ∆ t , (15)where µ d − µ f is the difference of domestic and foreign drifts. At this point, we do notneed to specify these drifts; however, another model needs to be developed to specifythe drifts. To specify the second term of the expansion, we maximize the entropyEquation (6) subject to normalization and the directionality constraint Equation (15).This will yield the transition probability, P (ln u ′ | ln u ) = 1 ξ exp " − α (cid:18) ln u ′ u (cid:19) + β ( u ) ln u ′ u (16)where β is a Lagrange multiplier corresponding to the directionality constraint and α isgiven in Equation (10). The normalization factor is ξ = R ∞−∞ d ln u ′ exp (cid:20) − α (cid:16) ln u ′ u (cid:17) + β ( u ) ln u ′ u (cid:21) .The transition probability distribution can be rewritten in a Gaussian form, P (ln u ′ | ln u ) = 1 Z ( α, β , ln u ) exp " − α (cid:18) ln u ′ u − βα (cid:19) (17)where the new normalization factor is Z = R ∞−∞ d ln u ′ exp (cid:20) − α (cid:16) ln u ′ u − βα (cid:17) (cid:21) . Thistransition probability density leads to a Wiener process of the logarithm of the exchangerate, n u ′ u = (cid:28) ln u ′ u (cid:29) P + ∆ W (18)where the drift and the Brownian Motion are, (cid:28) ln u ′ u (cid:29) P = β σ ∆ t , h ∆ W i P = 0 , D (∆ W ) E P = 1 α = σ ∆ t (19)Next, we need to specify the second term in Equation (14). We take the square ofEquation (13), and taking the expectation, calculation is skipped; we get, *(cid:18) ln u ′ u (cid:19) + P = *(cid:18) ∆ uu (cid:19) + P = σ ∆ t (20)Therefore, our directionality constraint is found to be, (cid:28) ln u ′ u (cid:29) P ≈ (cid:28) ∆ uu (cid:29) P − *(cid:18) ∆ uu (cid:19) + P = ( µ d − µ f ) ∆ t − σ ∆ t = k ′ (21)The Lagrange multiplier β can now be specified, β = ( µ d − µ f ) σ −
12 (22)Summarizing our findings, we get the transition probability density to a be lognor-mal distribution, P (ln u ′ | ln u ) = 1 Z exp " − σ ∆ t (cid:18) ln u ′ u − (cid:16) µ d − µ f − σ (cid:17) ∆ t (cid:19) (23)with the stochastic process for the log exchange rate as a Brownian Motion with adrift, ln u ′ u = (cid:28) ln u ′ u (cid:29) P + ∆ W (24) (cid:28) ln u ′ u (cid:29) P = (cid:18) µ d − µ f − σ (cid:19) ∆ t , h ∆ W i P = 0 , D (∆ W ) E P = σ ∆ t (25) his is the Brownian Motion for the log exchange rate, which amounts to a Geo-metric Brownian Motion of the exchange rate. In hindsight, it becomes obvious why wetook the expansion in the Taylor expansion only to the second order, simply becausethe higher orders of expansion are proportional to a higher order of ∆ t , which in theregime of continuous motion can be neglected.It is noteworthy to mention that we can observe explicitly that the transitiondensity Equation (23) is invariant under scaling transformation, i.e., P (ln u ′ | ln u ) = P (ln ˜ u ′ | ln ˜ u ). Under scaling transformation, the log exchange ratio gets shifted, ln ˜ u =ln lu = ln u + ln l , which in turn leads to a shift in the mean of the transition density.Both ln u ′ and ln u are shifted, and they cancel out, leaving the transition probabilitydensity invariant. Further, we can address how the probability density would evolve over time. Equation(11) can be written in a differential equation form, ∂ t p (ln u, t ) = − ∂∂ ln u (cid:18)(cid:16) µ d − µ f − σ (cid:17) p (ln u, t ) (cid:19) + 12 ∂ ∂ (ln u ) (cid:16) σ p (ln u, t ) (cid:17) (26)This is the Fokker–Planck equation for the distribution p (ln u, t ). If the drifts andthe volatility happen to be constant over time and independent of the exchange rate,namely uniform, then for a finite time interval, the transition probability density will bea lognormal distribution of the exchange rate with ∆ t = T . Notice that the dynamicsof the probability density is invariant under the scaling transformation. European Options are valued in a risk-neutral universe, which is equivalent to a no-arbitrage pricing. In this section, we construct the risk-neural probability distributionand use it to value the European Options on an exchange rate. The model developed s a counterpart of the Black–Scholes model and is known as the German-Kohlhagenmodel. Next, we derive the Black–Scholes–Merton partial differential equation for thedynamics of European Options. A risk-neutral universe has two main constraints: the expected drift is the same as therisk-free rate, and the rate with which we discount should be the risk-free rate [42]. Todrive the risk-neutral measure, we impose the first risk-neutral constraint on Equation(15), (cid:28) ∆ uu (cid:29) P = ( r d − r f ) ∆ t , (27)where r d and r f are the domestic and foreign risk-free rates. Notice that the deriva-tion of the risk-neutral probability density distribution is the same as the lognormaldistribution we derived for the transition probability Equation (23) with the exceptionthat instead of the drift of the exchange rate, we have the risk-free rates. Further, weassume that the risk-free rate and the volatility are uniform, and we get the risk-neutralmeasure, P (ln u ′ | ln u ) = 1 Z exp " − σ ∆ t (cid:18) ln u ′ u − (cid:16) r d − r f − σ (cid:17) ∆ t (cid:19) (28)Notice that with the assumption of the uniformity of the risk-free rate and thevolatility, this is the risk-neutral transition probability for any finite time interval ∆ t .If we relax these assumptions, we need to solve the Fokker–Planck equation to solvefor the risk-neutral distribution at any time in the future.Now, we can proceed to value European Options with the risk-neutral measure. Wesimply compute the expected payoff of the Options at maturity and discount it usinga risk-free rate to get the premium. The expected payoff of a call Option, denoted by V c , for a domestic investor at maturity is given by the difference between the expectedsale value and the expected purchase value, c = h Sale i LN,T − h
Purchase i LN,T (29)where we have, h Sale i T = Z ∞ K du P ( u, T | u ) u (30) h Purchase i T = Z ∞ K du P ( u, T | u ) K where K is the strike exchange rate and u is the current exchange rate. We integratefrom the strike rate since, if the rate is less than the strike rate, we will not exercisethe call Option. Then, the payoff can be written as, V c = Z ∞ K du P ( u, T | u ) ( u − K ) (31)The Premium for the call Option is just the discounted value of the payoff, wediscount the expected payoff with the domestic risk-free interest rate, C = e − r d T V c = e − r d T h Sale i LN,T − e − r d T h Purchase i LN,T (32)Since we know the current exchange rate u , then the risk-neutral probability dis-tribution at maturity is given by, P ( u T | u ) = Z d ˜ u P ( u T | ˜ u ) P (˜ u | u ) = Z d ˜ u P ( u T | ˜ u ) δ (˜ u − u ) (33) ∼ LN (ln u + ( r d − r f ) T − σ T , σ √ T )where LN stands for lognormal distribution. Next, we compute the expected sale valueat maturity, and we get, h Sale i LN,T = u exp[( r d − r f ) T ] N ( d ) (34)where d = ln u +( r d − r f ) T − σ T − ln Kσ √ T and N ( d ) is the standard normal cumulative dis-tribution function, ( d ) = 1 √ π Z d −∞ dx e − x (35)The expected purchase value can be computed, h Purchase i LN,T = KN ( d ) (36)where d + σ √ T = d . Then, the premium of the call Option is: C = u e − r f T N ( d ) − e − r d T K N ( d ) (37)This is the celebrated Garman–Kohlhagen model for the call Option. To value aEuropean put Option, we follow the same procedure as the call Option. The expectedpayoff at the maturity for the put Option is, V p = Z K ∞ du P ( u, T | u ) ( u − K ) (38)Notice that we integrate to the strike rate K because if the rate is greater than thestrike rate, we will not exercise the put Option. Discounting this expected payoff willyield the put premium, P = e − r d T V p = e − r d T K N ( − d ) − u e − r f T N ( − d ) (39)Which is the Garman–Kohlhagen model for the European put Option. We cansimply check that the call and put premium satisfy the so-called call-put parity relation, C − P = e − r f T u − e − r d T K (40)The call-put parity relation ensures that this was a no-arbitrage valuation of Euro-pean Options. We can drive the differential equation for the European Options, which is known asthe Black–Scholes–Merton differential equation. To derive the differential equation, we tart with the expected payoff equation, V (ln u, K, t ) = Z d ln u T P (ln u T , T | ln u, t ) ( u T − K ) (41)The boundaries of the integral are left out to get the differential equation for bothcall and put Options. Next, we take the time derivative of both sides, ∂ t V = Z d ln u T ( u T − K ) ∂ t P (ln u T , T | ln u, t ) (42)where the time derivative of the transition probability is given by the backward Kol-mogorov equation; the derivation is skipped, ∂ t P (ln u T , T | ln u, t ) = − (cid:18) ( r d − r f ) − σ (cid:19) ∂P (ln u T , T | ln u, t ) ∂ ln u − σ ∂ P (ln u T , T | ln u, t ) ∂ (ln u ) (43)Substituting Equation (43) into Equation (42), we get, ∂ t V = Z d ln u T ( u T − K ) (cid:20) − (cid:18) ( r d − r f ) − σ (cid:19) ∂P∂ ln u − σ ∂ P∂ (ln u ) (cid:21) (44)= − (cid:18) ( r d − r f ) − σ (cid:19) ∂∂ ln u Z d ln u T ( u T − K ) P (ln u T , T | ln u, t ) − σ ∂ ∂ (ln u ) Z d ln u T ( u T − K ) P (ln u T , T | ln u, t )= − (cid:18) ( r d − r f ) − σ (cid:19) ∂V∂ ln u − σ ∂ V∂ (ln u ) We can rewrite this equation as, ∂ t V ( u, t ) + ( r d − r f ) u ∂V∂u + σ u ∂ V∂u = 0 (45)The partial differential equation for the European Option premium is derived justby substituting E = e r d ( t − T ) V into the above equation, ∂ t E + ( r d − r f ) u ∂E∂u + 12 σ u ∂ E∂u − r d E = 0 (46)where by applying the boundary conditions for the call/put Option, we get the solutionfor the call/put Option. Summary and Discussion
We put forth an entropic framework to model the dynamics of exchange rates[40]. Thisalternative framework is complementary to the stochastic process modeling becausewe derived the stochastic dynamics from maximizing a relative entropy. To derivethe transition distribution, we needed to take into account the relevant information.The scale symmetry of the dynamics is a significant piece of information, which ledus to choose the logarithm of the exchange rate as the proper variable to model. Thecontinuity of the motion and the directionality were the other pieces of information wehad about the exchange rates, which were formulated as a constraint equation. Theresulting model was a Geometric Brownian Motion for the exchange rate. Further,a dynamics for the probability density distribution was found to be a Fokker–Planckequation.Next, we applied the entropic exchange rate model to value European Optionson FX . We derived the risk-neutral probability density by imposing the risk-neutralconstraints. Using the risk-neutral measure, we valued the European Options, whichwas the same as the known Garman–Kohlhagen model. The dynamics of the EuropeanOptions was found to be the Black–Scholes–Merton partial differential equation.An extension to our model could be allowing the dynamics to have jumps. Weimposed the continuity of the motion to derive the Geometric Brownian Motion of theexchange rate. By taking into account the information about the jump process, we canextend our model, which further will lead to a modified Options value.We have extended our framework to model the dynamics of stocks and valuingEuropean Options on stocks [41]. It was shown that under similar constraints, wecould yield a Geometric Brownian Motion of stocks. Then, a no-arbitrage pricing ofEuropean Options on the stock was provided, which gave rise to the Black–Scholesmodel, and the dynamics of the Option premium was found to be the Black–Scholes–Merton differential equation. Another extension could be modeling the dynamics ofmany stocks [10]. cknowledgments: 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, and Entropic Dynamics.
References [1] Caticha, A. Entropic Inference.
AIP Conf. Proc. , , 20;doi:10.1063/1.3573619.[2] Golan, A. Foundations of Info-Metrics ; Oxford University Press: Oxford, UK,2018; doi:10.1093/ajae/aay085.[3] Caticha, A.
Entropic Inference and the Foundations of Physics ;University at Albany: S˜ao Paulo, Brazil, 2012; Available online: (accessed on1/1/2019).[4] Shore, J.; Johnson, R. Axiomatic derivation of the principle of maximum entropyand the principle of minimum cross-entropy.
IEEE Trans. Inf. Theory , , ,26–37; doi:10.1109/TIT.1980.1056144.[5] Vanslette, K. Entropic Updating of Probabilities and Density Matrices. Entropy , , 664; doi:10.3390/e19120664.[6] Jaynes, E.T. Information theory and statistical mechan-ics. Phys. Rev. , , 620; doi:10.1103/PhysRev.106.620. https://link.aps.org/doi/10.1103/PhysRev.106.620 [7] Jaynes, E.T. Gibbs vs. Boltzmann Entropies. Am. J. Phys. , , 391;doi:10.1119/1.1971557.[8] Amari, S. Differential Geometrical Methods in Statistics ; Springer-Verlag: NewYork, USA, 1985.[9] Caticha, A. Geometry from Information Geometry. Available online: https://arxiv.org/pdf/1512.09076.pdf (accessed on 1/10/2019).
10] Abedi, M.; Bartolomeo, D. Entropic Dynamics Many Stocks and Investment,Forthcoming.[11] Caticha, A. Entropic time.
AIP Conf. Proc. , 200–207.[12] Caticha, A. Entropic dynamics: Mechanics without mechanism. arXiv ,arXiv:1704.02663. Available online: https://arxiv.org/abs/1704.02663 (ac-cessed on 1/1/2019).[13] Nawaz, S.; Abedi, M.; Caticha, A. Entropic dynamics on curved spaces.
AIPConf. Proc. , , 030004; doi:10.1063/1.4959053.[14] Bartolomeo, D.; Caticha, A. Trading drift and fluctuations in entropic dynamics:Quantum dynamics as an emergent universality class. J. Phys. Conf. Ser. , , 012009; doi:10.1088/1742-6596/701/1/012009.[15] Ipek, S.; Abedi, M.; Caticha, A. A covariant approach to entropic dynamics. AIPConf. Proc. , , 090002; doi:10.1063/1.4985371.[16] Ipek, S.; Abedi, M.; Caticha, A. Entropic dynamics: Reconstructing quantumfield theory in curved space-time. arXiv , arXiv:1803.07493; Available on-line: https://arxiv.org/abs/1803.07493 (accessed on 1/1/2019).[17] Ipek, S.; Abedi, M.; Caticha, A. Entropic dynamics of quantum scalarfields in curved spacetime. arXiv , arXiv:1803.07493v1; Available online: https://arxiv.org/abs/1803.07493v1 (accessed on 1/1/2019).[18] Pessoa, P.; Caticha, A. Exact renormalization groups as a form of entropic dy-namics. Entropy , , 25; doi:10.3390/e20010025.[19] Bachelier, L. Th´eorie de la sp´eculation. Annales Scientifiquesde l ´Ecole Normale Sup´erieure , , 21–86; Available online: http://archive.numdam.org/article/ASENS_1900_3_17__21_0.pdf (ac-cessed on 1/10/2019). (In French)[20] Samuelson, P. Proof that properly anticipated prices fluctuate randomly. Ind.Manag. Rev. , , 13; doi:10.1142/9789814566926 0002.
21] Fama, E.F. The behavior of stock-market prices.
J. Bus. , , 34–105; Avail-able online: (accessed on 1/10/2019).[22] Black, F.; Scholes, M. The valuation of option contracts and a test of marketefficiency. J. Financ. , , 399–417; doi:10.2307/2978484.[23] Black, F.; Scholes, M. The pricing of options and corporate liabilities. J. Polit.Econ. , , 637–654; doi:10.1086/260062.[24] Merton, R.C. Theory of rational option pricing. Bell J. Econ. Manag. Sci. , , 141–183; doi:10.2307/3003143.[25] Cox, J.C.; Ross, S.A. The valuation of options for alternative stochastic processes. J. Financ. Econ. , , 145–166; doi:10.1016/0304-405X(76)90023-4.[26] Eisenberg, L.K; Jarrow, R.A. Option pricing with random volatilities in completemarkets. Rev. Quant. Financ. Account . , , 5–17; doi:10.1007/BF01082661.[27] Heston, S.L. A closed-form solution for options with stochas-tic volatility with applications to bond and currency op-tions. Rev. Financ. Stud. , , 327–343; Available online: https://EconPapers.repec.org/RePEc:oup:rfinst:v:6:y:1993:i:2:p:327-43 (accessed on 1/10/2019).[28] Dupire, B. Pricing with a smile. Risk , , 18–20.[29] Hull, J.; White, A. The pricing of options on assets with stochastic volatilities. J. Financ. , , 281–300; doi:10.1111/j.1540-6261.1987.tb02568.x.[30] Wiggins, J.B. Option Values under Stochastic Volatilities. J. Financ. Econ. , , 129–145; doi:10.1016/0304-405X(87)90009-2.[31] Ritchken, P.; Trevor, R. Pricing options under generalized GARCH and stochas-tic volatilities. Int. J. Theor. Appl. Financ. , , 377–402; doi:10.1111/0022-1082.00109.[32] Christoffersen, P.; Jeston, S.; Jacobs, K. Option valuation with conditional skew-ness. J. Econom. , , 253–284; doi:10.1016/j.jeconom.2005.01.010.
33] Makate, N.; Sattayanthan, P. Stochastic volatility jump-diffusion model for op-tion pricing.
J. Math. Financ. , , 90–97; doi:10.4236/jmf.2011.13012.[34] Makate, N.; Sattayanthan, P. European option pricing for a stochastic volatilitylevy model. J. Math. Financ. , , 98–108; doi:10.4236/jmf.2011.13013.[35] Garman, M.; Kohlhagen, S. Foreign currency options values. J. Intern. MoneyFinanc. , , 231–237; doi:10.1016/S0261-5606(83)80001-1.[36] Evans, M.D.D. FX Trading and Exchange Rate Dynamics. NBERWorking Paper No. 8116, Issued in February 2001. Available online: (accessed on 1/1/2019).[37] Engel, C.; West, K.D. Exchange rates and fundamentals. J. Polit. Econ. , , 485–517; doi:10.1086/429137.[38] Ahn, C.M.; Cho, D.C.; Park, K. The Pricing of Foreign Currency Op-tions under Jump-Diffusion Processes. J. Futur. Mark. , , 669–695;doi:10.1002/fut.20261.[39] Engel, C. Exchange rates and interest parity. NBER Work-ing Paper No. 19336, Issued in August 2013. Available online: (accessed on 1/1/2019).[40] Abedi, M.; Bartolomeo, D. Entropic Dynamics of Exchange Rates and Options. Entropy , , 586. https://doi.org/10.3390/e21060586 [41] Abedi, M.; Bartolomeo, D. Entropic Dynamics of Stocks and European Options. Entropy , , 765. https://doi.org/10.3390/e21080765 [42] Hull, J.C.; Basu, S. Options, Futures, and Other Derivatives ; Pearson Education,Inc: India, 2018.; Pearson Education,Inc: India, 2018.