A Macroscopic Portfolio Model: From Rational Agents to Bounded Rationality
AA Macroscopic Portfolio Model:From Rational Agents to Bounded Rationality
Torsten Trimborn ∗ October 29, 2018
Abstract
We introduce a microscopic model of interacting financial agents, where each agent ischaracterized by two portfolios; money invested in bonds and money invested in stocks.Furthermore, each agent is faced with an optimization problem in order to determine theoptimal asset allocation. The stock price evolution is driven by the aggregated investmentdecision of all agents. In fact, we are faced with a differential game since all agents aim toinvest optimal. Mathematically such a problem is ill posed and we introduce the conceptof Nash equilibrium solutions to ensure the existence of a solution. Especially, we denotean agent who solves this Nash equilibrium exactly a rational agent. As next step we usemodel predictive control to approximate the control problem. This enables us to derivea precise mathematical characterization of the degree of rationality of a financial agent.This is a novel concept in portfolio optimization and can be regarded as a general ap-proach. In a second step we consider the case of a fully myopic agent, where we can solvethe optimal investment decision of investors analytically. We select the running cost tobe the expected missed revenue of an agent and we assume quadratic transaction costs.More precisely the expected revenues are determined by a combination of a fundamen-talist or chartist strategy. Then we derive the mean field limit of the microscopic modelin order to obtain a macroscopic portfolio model. The novelty in comparison to existentmacroeconomic models in literature is that our model is derived from microeconomic dy-namics. The resulting portfolio model is a three dimensional ODE system which enablesus to derive analytical results. The conducted simulations reveal that the model sharesmany dynamical properties with existing models in literature. Thus, our model is ableto replicate the most prominent features of financial markets, namely booms and crashes.In the case of random fundamental prices the model is even able to reproduce fat tails inlogarithmic stock price return data. Mathematically, the model can be regarded as themoment model of the recently introduced mesoscopic kinetic portfolio model [46].
Keywords: portfolio optimization, model predictive control, stock market, boundedrationality, crashes, booms ∗ IGPM, RWTH Aachen, Templergraben 55, 52056 Aachen, Germany, [email protected] a r X i v : . [ q -f i n . P M ] O c t Introduction
For many years the Efficient Market Hypothesis (EMH) by Eugene Fama [16] has been thedominant paradigm for modeling asset pricing models. Many famous theoretical models infinance such as Merton’s optimal portfolio model [36] or the Black and Scholes model [24]presume the correctness of the EMH. In the past years there has been a shift from rationaland representative financial agents to bounded rational and heterogeneous agents [22]. Theformer notion of agents is in agreement with the EMH, whereas the latter one contradicts theEMH and has to be understood in the sense of Simon [42]. The drift away from the EMHhas been supported by several empirical studies [34, 28] and the financial crashes of the pastdecade [27, 2].Bounded rational agents are widely used in econophysical asset pricing models, particularlyin agent-based-computational financial market models. These models usually consider a largenumber of interacting heterogeneous financial agents. These large complex systems are studiedby means of Monte Carlo simulations. Major contributions are for example the Levy-Levy-Solomon [29], Cont-Bochaud [13] and the Lux-Marchesi [33] model. The benefit of thesemodels are first explanations for the existence of stylized facts, such as fat tails in asset re-turns or volatility clustering. They have even led to alternative market hypothesis, such asthe adaptive market hypothesis by Lo [30] or the interacting agent hypotheses by Lux [33]as alternative to the EMH. The disadvantage of these agent-based asset pricing models arethe impracticability to apply analytical methods. Furthermore, it has been shown in severalstudies that many agent-based models exhibit finite size effects [45, 15].Another popular approach are simple low dimensional dynamic asset pricing models whichusually consider two types of financial agents. In most cases one agent follows a chartist strat-egy and the other a fundamental strategy. With a chartist strategy we mean an investor whobases his decision on technical trading rules, whereas a fundamentalist originates his invest-ments from deviations of fundamentals to the stock price. These models are often formulatedas two dimensional difference equations or as ordinary differential equations (ODE). They allhave in common that the stock price equation is driven by the excess demand of financialagents. In comparison to agent-based models, these models can be regarded as macroscopic,since they consider aggregated quantities. In literature there are numerous models of thatkind, for example by Beja and Goldman [3], Day and Huang [14], Lux [31], Brock and Hommes[6, 7], Chiarella [10] and Franke and Westerhoff [17]. These asset pricing models feature arich body of complex phenomena, such as limit cycles, chaotic behavior and bifurcations.Economically, these models study for example the impact of behavioral and psychologicalfactors such as risk tolerance on the price behavior. Furthermore, they discover the origins ofstylized facts. More precisely they study the source of booms and crashes and excess volatility.The financial agents are always modeled as bounded rational agents in the sense of Simon [42]and possess behavioral factors. The importance of psychological influences in agent modelinghas been emphasized by several authors [41, 25, 32, 12]. One may note that the precise formof agent demand is not established from microscopic dynamics and the connection to rationalagents is unclear. Albeit the form of the agent demand in the models [7, 11] is derived by amean-variance wealth maximization the expected stock return over bond return is modeledmacroscopically. In addition, these models neglect the impact of wealth evolution on the2gent demand. One example of a bounded rational agent is a myopic agent, who basis hisaction only on currently available informations [8]. To our knowledge, a precise mathematicalnotion of rationality in the context of portfolio optimization is missing. In addition, there isa lack of explanation concerning the interrelations between the action of a rational agent andfor example a fully myopic agent.For these reasons we introduce a rather general mathematical framework in order to quantifythe level of rationality of each agent in the context of portfolio optimization. More precisely,we formulate a model of rational agents on the microscopic level. Thus, each agent is facedwith an optimal control problem in order to optimize their portfolio and determine the optimalinvestment decision. In fact, each agents’ portfolio dynamics is divided in the time evolutionof the two asset classes bonds and stocks. We employ the notion of Nash equilibrium solutions[5] to ensure that the optimization problem is well posed. In addition, we define a rationalagent to be an agent who solves the differential game exactly, respectively obtains the Nashequilibrium solution. In a next step we apply model predictive control [9] to approximatethe optimization problem. This enables us to give a precise mathematical definition of thelevel of rationality of an investor. Especially, thanks to this methodology we obtain a naturalconnections between rational and bounded rational agents. Up to this step the approach isfully generic since each agent is equipped with rather general wealth dynamics. The stockprice equation is driven by the aggregated excess demand of agents in agreement with theBeja-Goldman [3] or Day-Huang [14] model. In a second step we consider the situation ofa fully myopic agent which enables us to compute the optimal control explicitly. We modelthe cost function to measure the expected lost revenue of the investor. The return estimateis a convex combination of a pure chartist or pure fundamental trading strategy. The weightbetween both strategies is determined by an instantaneous comparison of the chartist andfundamental return estimate and is closely connected to the strategy change in the Lux-Marchesi model [33]. We obtain a large dynamical system in the spirit of known agent-basedfinancial market models. The disadvantage is that such models are far to complex to studythem by analytical methods. For that reason we derive the mean field limit [44, 18] of oursystem. More precisely we derive the time evolution of the average money invested in stocksand the average money invested in bonds. Hence, we obtain a macroeconomic portfolio modelof three ODEs. This time continuous model shares many similarities with macroeconomicmodels in literature e.g. with the model by Lux [31]. The novelty of our approach is that theresulting macroeconomic model is supported by precise microeconomic dynamics. In generalthe advantage of a time continuous model is the possibility to use many analytical tools inorder to quantify the dynamic behavior of the model. A second advantage is that the modelcan be studied on arbitrary time scales since we consider dimensionless quantities.The outline of this paper is as follows: First we consider microscopic agent dynamics. Infact, we first introduce a microscopic model of rational agents. As second step, we approxi-mate the complex optimization problem and give a connection between rational and boundedrational agents. Then finally, we derive the macroscopic portfolio model in the case of fullymyopic agents. In section 4 we study the qualitative behavior of our model. Thus, we discusspossible steady states and study the dynamics caused by a pure fundamental or pure chartiststrategy. In the following section, we present numerous simulation results and analyze theimpact of several model parameters. Finally, we give a conclusion and short discussion of ourresults. 3
Economic Microfoundations
We consider N financial agents equipped with their personal monetary wealth w i ≥
0. Wedenote all microscopic quantities with small letters. The non-negativity condition means thatno debts are allowed. This wealth is divided in the wealth invested in the asset class stocks x i ( t ) > y i ( t ) >
0. We neglect all other asset classes andassume that w i ( t ) = x i ( t ) + y i ( t ) holds. The time evolution of the risk-free asset is describedby a fixed non-negative interest rate r ≥ S ( t ) + D ( t ) S ( t ) , where S ( t ) is the stock price at time t and D ( t ) ≥ u i . Thus, we have thedynamics ˙ x i ( t ) = ˙ S ( t ) + D ( t ) S ( t ) x i ( t ) + u i ( t )˙ y i ( t ) = r y i ( t ) − u i ( t ) . Notice, that u i determines the investment decision of agents and implicitly specifies the assetallocation between both portfolios. We still need to describe the time evolution of the stockprice S . We define the aggregated excess demand ED N of all financial agents as the averageof all investment decisions of the agents. ED N ( t ) := 1 N N (cid:88) i =1 u i ( t ) . The aggregated demand ED N is the average of agents’ excess demand u i . The agents’ excessdemand u i was defined as the investment decision of agents and can be interpreted as thedemand minus the supply of each agent. Hence, the excess demand is positive if the investorsbuy more stocks than they sell. For further details regarding the aggregated excess demandwe refer to [35, 43, 13, 48, 45]. Thus, the macroscopic stock price evolution is driven bythe excess demand and is given by ˙ S ( t ) = κ ED N ( t ) S ( t ) . (1)where the constant κ > Microscopic portfolio optimization
As in classical economic theory, u i will be a solutionof a risk or cost minimization. The precise model of the objective function Ψ i ( t ) is left openat this point. We want to emphasize that Ψ i may depend on the stock price S and the wealthof the agent’s portfolios. The agent tries to minimize the running costs (cid:90) T (cid:16) µ u i ( t ) + Ψ i ( t ) (cid:17) dt.
4e consider a finite time interval [0 , T ] and have added a penalty term that punishes transac-tions. The penalty term is necessary to convexify the problem but is also reasonable, becauseit describes transaction costs. The transaction costs are modeled to be quadratic which is afrequently used assumption in portfolio optimization [4, 38, 39, 19]. Furthermore, we assumethat there are no final costs at final time T present.Hence, in summary, the microscopic model is given by˙ x i ( t ) = ˙ S ( t ) + D ( t ) S ( t ) x i ( t ) + u ∗ i ( t )˙ y i ( t ) = r y i ( t ) − u ∗ i ( t )˙ S ( t ) = κ ED N ( t ) S ( t ) (2) u ∗ i := argmax u i :[0 ,T ] → R (cid:90) T (cid:16) µ u i ( t ) + Ψ i ( t ) (cid:17) dt. The microscopic model is an optimal control problem. The dynamics are strongly coupled bythe stock price in a non-linear fashion. In fact, it is impossible that all agents minimize theirindividual cost function since all agents play a game against each other. We are faced witha non-cooperative differential game. We choose the concept of Nash equilibria which will beexplained in detail in the next section.
As we have defined the microscopic portfolio model we want to specify different solution of (2)with respect to their economic interpretation. We define the Nash equilibrium of a differentialgame.
Definition 1.
A vector of control functions t (cid:55)→ ( u ∗ , ..., u ∗ N ) T is a Nash equilibrium for adifferential game argmin u i : [0 ,T ] → R q i ( x ( T )) − T (cid:90) L i ( t, x , u ) dt, where x = ( x , ..., x N ) T ∈ R dN , x i = ( x i , ..., x di ) T ∈ R d , ≤ i ≤ N, d ≥ , u =( u , ..., u N ) T ∈ R N and with the state dynamics ˙ x = f ( t, x , u ) , x (0) = x ∈ R dN , (3) with f = ( f , ..., f N ) T , L = ( L , ..., L N ) T , q = ( q , ..., q N ) T are defined on: f i : [0 , T ] × R dN × R N → R d ,L i : [0 , T ] × R dN × R N → R ,q i : R dN → R . n the class of open-loop strategies if the following holds.The control u ∗ i provides a solution to the optimal control problem for player i: argmin u i :[0 ,T ] → R q i ( x ( T )) − T (cid:90) L i ( t, x , u i , u ∗− i ) dt, with the dynamics (3) . Here, L i denotes the running cost and q i the terminal cost. For details regarding differential games and the idea of Nash equilibrium solutions werefer to [5]. This mathematical equilibrium concept enables us to give a precise definition ofrationality in economics.
Definition 2.
We denote any agent who computes the exact Nash equilibrium solution of thesystem (2) a rational agent . This definition fits to the economic theory of rationality [16, 34], so each agent is awareof the correct dynamic and acts fully optimal in the context of Nash equilibria. We want topoint out that in case of many agents, we have a large system of optimization problems (2).Such a system is very expensive to solve. Thus, not only from the perspective of behavioralfinance but also from a pure computational aspect such a rational setting seems to be veryunrealistic. Hence, we want to give a precise mathematical definition of so called boundedrational agents in the sense of Simon [42].
Definition 3.
We denote any agent who computes a numerical approximation of the micro-scopic system (2) a bounded rational agent . We approximate the objective functional in (2) by linear model-predictive control (MPC)[37, 9]. This methods approximates a finite horizon optimal control problem in two ways.First, one predicts the dynamics over a predict horizon T P ≤ T and secondly the control is onlyselected on a control horizon T C ≤ T P . Thus for an arbitrary initial time ¯ t the optimization(2) is performed on [¯ t, ¯ t + T P ]. Then the computed admissible control ¯ u : [¯ t, ¯ t + T P ] → R can be applied on [¯ t, ¯ t + T C ] and the state dynamics evolve accordingly on [¯ t, ¯ t + T C ]. Thenthe whole procedure is repeated with updated initial conditions at ¯ t new := ¯ t + T C , shiftedprediction interval [¯ t new , ¯ t new + T P ] and control interval [¯ t new , ¯ t new + T C ]. The algorithmterminates if the time T is included in the prediction interval. A schematic illustration ofone step of the linear MPC method is depicted in Figure 2.1. For sure, we can only expectto obtain a sub-optimal strategy compared to the original mode (2). This procedure can beconsidered as a repeated open-loop control in a feedback fashion. The computation on theprediction horizon is open-loop. Then one evolves the system on the control horizon andthus obtains a feedback by the system, which is used as initial condition for the next open-loop optimization. In order to apply MPC on our model we assume that T C = T P holds.Furthermore, we discretize our dynamics on time intervals of length ∆ t > T = M ∆ t and we ensure that ∆ t ≤ T C holds. Then the prediction and control horizon can be definedas T C = T P = p ∆ t, < p ≤ M .In fact, we assume that the optimal control u ∗ i is well approximated by piecewise constantcontrols of length ∆ t on [¯ t, ¯ t + M ∆ t ]. u ∗ i ( t ) ≈ M − (cid:88) k =0 u ki [ t k ,t k +1 ) ( t ) , t k = ¯ t + k ∆ t, k = 0 , ..., M. futurepast T C T P x ( t ) u ∗ ( t )¯ t + ∆ t ¯ t Figure 1: Schematic illustration of the MPC methodology.We choose the penalty parameter µ in the running costs to be proportional to the timeinterval so that µ = ν ∆ t for some ν . This can be motivated by checking the units of thevariables in the cost functional ( K is a rate, thus measured in 1 / time, Ψ is wealth / time, u wealth / time). We see that the penalty parameter µ must be a time unit. Furthermore,we insert the right-hand side of the stock price equation into the stock return. Thus, thesemi-discretized constrained optimization problem on [¯ t, ¯ t + p ∆ t ] reads (cid:90) ¯ t + p ∆ t ¯ t (cid:18) ν ∆ t u i ( t ) + Ψ i ( t ) (cid:19) dt → min˙ x i ( t ) = κ ED N ( t ) x i ( t ) + D ( t ) S ( t ) x i ( t ) + u i ( t ) , x i (¯ t ) = ¯ x i , ˙ y i ( t ) = ry i ( t ) − u i ( t ) , y i (¯ t ) = ¯ y i , ˙ S ( t ) = κ ED N ( t ) S ( t ) , S (¯ t ) = ¯ S. Thus, we can finally state a precise notion of bounded rationality in the context of MPCapproximations.
Definition 4.
Thanks to the MPC framework we can define the degree of rationality θ ∈ [0 , of an agent by: θ = 1 − exp (cid:26) T − T P ∆ t T (cid:27) , T P = p ∆ t, p = 1 , ..., (cid:22) T ∆ t (cid:23) , and T (cid:29) ∆ t, where θ = 1 corresponds to a fully rational agent and θ = 0 corresponds to a fully myopicagent in the MPC setting. Here, we assume that the model is at the beginning of the consideredtime period t = 0 . In fact a rational agent is obtained for θ = 1, which is only the case if T P = T and ∆ t → θ = 0) is obtained for T P = ∆ t .7 emark 1. As the previous definition reveals, the MPC method introduces two differentkinds of errors. A discretization error due to the numerical approximation of the optimalcontrol and a prediction error due to the approximated control horizon. There are manycontributions which derive perfomance bounds of the MPC method in order to quantify theimpact of a limited time horizon [21]. The impact of time varying control horizons T C onthe performance of the MPC method in comparison to the exact closed-loop solution has beendiscussed by Gr¨une et al. [20]. Optimality system
As pointed out previously, we want to solve the MPC problem in agame theoretic setting. We want to search for Nash equilibria. In this setting, each agentassumes that the strategies of the other players are fixed and optimal. Thus, we get N opti-mization problems which need to be solved simultaneously. Hence, we have a N -dimensionalLagrangian L ∈ R N . The i-th entry L i corresponds to the i-th player and reads: L i ( x i , y i , S, u i , λ x i , λ y i , λ S ) = ¯ t + p ∆ t (cid:90) ¯ t (cid:18) ν ∆ t u i ( t ) + Ψ i ( t ) (cid:19) dt (4)+ ¯ t + p ∆ t (cid:90) ¯ t ˙ λ x i x i + λ x i κ ED N x i + λ x i DS x i + λ x i u i dt − λ x i ¯ x i + ¯ t + p ∆ t (cid:90) ¯ t ˙ λ y i y i + λ y i r y i − λ y i u i dt − λ y i ¯ y i , + ¯ t + p ∆ t (cid:90) ¯ t ˙ λ S S + λ S κ ED N S dt − λ S ¯ S, with Lagrange multiplier λ x i , λ y i , λ S . Notice that the quantities ( x ∗ j , y ∗ j , u ∗ j ) , j = 1 , ..., i − , i +1 , ..., N are assumed to be optimal in the i-th optimization and therefore only enter as param-eters in the i-th Lagrangian L i . We assume λ x i (¯ t + p ∆ t ) = λ y i (¯ t + p ∆ t ) = λ S (¯ t + p ∆ t ) = 0.The optimal control can be obtained by solving the corresponding necessary optimality con-ditions. The number of optimality conditions depends on the size of the prediction horizon T P = p ∆ t . Thus, for each agent one needs to solve an optimality system consisting of n = p Fully myopic agent
In the further discussion of our model we confine the study on the case p = 1. This choice has several reasons: First of all this simplification enables us to computethe optimal control explicitly and thus to derive the macroscopic limit of our microscopicdynamics. As a direct consequence, we obtain a three dimensional ODE system which we areable to analyze analytically. Finally, the simulations of our resulting ODE model reveals that8ur model shares many similarities with macroeconomic financial market models in literature.In the case of a fully myopic agent ( θ = 0), which corresponds to p = 1 we can even computethe optimal control explicitly. We still have to define our objective function Ψ i that deter-mines the agent’s actions. We assume that the agent minimizes a quantity proportional tothe expected missed revenues in each portfolio. The quantity K , which we define in detaillater, is a return estimate of the stock return over the bond return. One can expect that K depends on the current or past stock prices. If stocks are believed to be better ( K > y i >
0) is bad, and vice versa. Then | K | x i for K < i by having invested in stocks but not in bonds. Equivalently, | K | y i for K > i by having invested in bonds but notin stocks. Then we weight the expected missed revenue by the wealth in the correspondingportfolio and define the running cost byΨ i := | K | x i , K < , , K = 0 , | K | y i , K > . The weighed missed revenue is larger, the larger the estimated difference between returns K .We still need to define the precise shape of the return estimate K .As done in many asset pricing models, we consider a chartist and a fundamental tradingstrategy. A trading strategy refers to an estimate of future stock return in order to evaluatethe profit of the portfolio.Fundamentalists believe in a fundamental value of the stock price denoted by s f > K f := U γ (cid:18) ω s f − SS (cid:19) − r. Here, U γ is a value function in the sense of Kahnemann and Tversky [26] which depends onthe risk tolerance γ of an investor. A typical example is U γ ( x ) = sgn ( x ) | x | γ with 0 < γ < sgn . The constant ω > s f . We want to point out that this stock return estimate is a rateand thus ω needs to scale with time.Chartists assume that the future stock return is best approximated by the current or paststock return. They estimate the return rate of stocks over bonds by K c := U γ (cid:32) ˙ S + DS (cid:33) − r. Both estimates are aggregated into one estimate of stock return over bond return by a convexcombination K = χ K f + (1 − χ ) K c . This idea has been previously applied to a kinetic model of opinion formation [1]. The weight χ is determined from an instantaneous comparison as modeled in [33]. We let χ = W ( K f − K c ) , W : R → [0 ,
1] is a continuous function. If for example, W = tanh + , the investoroptimistically believes in the higher estimate. Together, if K >
0, the investor believes thatstocks will perform better and if
K < x ( t ) = κ ED N ( t ) x i ( t ) + D ( t ) S ( t ) x i + u i , x i (¯ t ) = ¯ x i , ˙ y i ( t ) = ry i ( t ) − u i ( t ) , y i (¯ t ) = ¯ y i , ˙ S ( t ) = κ ED N ( t ) S ( t ) , S (¯ t ) = ¯ S,ν ∆ t u i ( t ) = − λ x i ( t ) − λ x i ( t ) κN x i ( t ) + λ y i ( t ) − κN S ( t ) λ S ( t ) , ˙ λ x i ( t ) = − κ ED N ( t ) λ x i ( t ) − D ( t ) S ( t ) λ x i ( t ) − ∂ x i Ψ i ( t ) , λ x i (¯ t + ∆ t ) = 0˙ λ y i ( t ) = − rλ y i ( t ) − ∂ y i Ψ i ( t ) , λ y i (¯ t + ∆ t ) = 0˙ λ S ( t ) = λ x i ( t ) D ( t ) S ( t ) x i − κ ED N ( t ) λ S ( t ) − ∂ S Ψ i ( t ) , λ S (¯ t + ∆ t ) = 0 . Here, we neglect for the purpose of readability the dependence of the ED N ( t ) = ED N ( t, x i , y i , S )and Ψ i ( t ) = Ψ i ( t, x i , y i , S ) on the wealth and stock price. Then we apply a backward Eulerdiscretization to the adjoint equations and get. λ x i ( t + ∆ t ) − λ x i (¯ t )∆ t = − κ ED N ( t + ∆ t ) λ x i ( t + ∆ t ) − D ( t + ∆ t ) S ( t + ∆ t ) λ x i ( t + ∆ t ) − ∂ x i Ψ i ( t + ∆ t ) ,λ y i ( t + ∆ t ) − λ y i (¯ t )∆ t = − rλ y i ( t + ∆ t ) − ∂ y i Ψ i ( t + ∆ t ) ,λ S ( t + ∆ t ) − λ S (¯ t )∆ t = λ x i ( t + ∆ t ) D ( t + ∆ t ) S ( t + ∆ t ) x i − κ ED N ( t + ∆ t ) λ S ( t + ∆ t ) − ∂ S Ψ i ( t + ∆ t ) . Then, we insert the final conditions of the costates and obtain. λ x i (¯ t ) = ∆ t ∂ x i Ψ i (¯ t + ∆ t ) ,λ y i (¯ t ) = ∆ t ∂ y i Ψ i (¯ t + ∆ t ) ,λ S (¯ t ) = ∆ t ∂ S Ψ i (¯ t + ∆ t ) . Hence, the optimal strategy is given by u ∗ N ( x i , y i , S ) = ν ( K y i − κN S ( ∂ S K ) y i ) , K > , , K = 0 , ν ( K x i + K κN x i + κN S ( ∂ S K ) x i ) , K < . Instantaneous controlled model
Thank to our simplified setting we could compute thecontrol explicitly. In the engineering literature such a control is frequently called instantaneouscontrol and our model reads:˙ x i ( t ) = κ ED N ( t ) x i ( t ) + D ( t ) S ( t ) x i ( t ) + u ∗ N ( t, x i , y i , S )˙ y i ( t ) = r y i ( t ) − u ∗ N ( t, x i , y i , S )˙ S ( t ) = κ ED N ( t ) S ( t ) . In order to derive the macroscopic portfolio model we average the microscopic quantitiesand consider the limit of infinitely many agents. This limit of infinitely many particles isknown in physics as mean field limit [44, 18]. We define the average wealth invested in stocks,respectively bonds by: X N ( t ) := 1 N N (cid:88) i =1 x i ( t ) , Y N ( t ) := 1 N N (cid:88) i =1 y i ( t ) . Furthermore, we assume that the limitslim N →∞ X N ( t ) = X ( t ) < ∞ , lim N →∞ Y N ( t ) = Y ( t ) < ∞ , exist for all times. The only non-linearity in x i , y i is the average of investment strategies.This is nothing else than our excess demand we considered before ED N = 1 N N (cid:88) i =1 u ∗ i = ν (cid:16) K N N (cid:80) i =1 y i − κN S ( ∂ S K ) N N (cid:80) i =1 y i (cid:17) , K > , , K = 0 , ν (cid:16) K x i + K κN N N (cid:80) i =1 x i + κN S ( ∂ S K ) N N (cid:80) i =1 x i (cid:17) , K < . In the limit of infinitely many agents we obtainlim N →∞ ED N = ED ( t, X, Y, S, ˙ S ) := ν K ( t, S, ˙ S ) X ( t ) , K ( t, S, ˙ S ) < , , K ( t, S, ˙ S ) = 0 , ν K ( t, S, ˙ S ) Y ( t ) , K ( t, S, ˙ S ) > . Here, we have used that the quadratic terms1 N N (cid:88) i =1 x i ( t ) , N N (cid:88) i =1 y i ( t ) , vanish in the limit, which can be made evidend by analyzing the order of: ∞ > (cid:32) N N (cid:88) i =1 x i (cid:33) = 1 N N (cid:88) i =1 x i + 1 N (cid:88) ≤ k,j ≤ N, k (cid:54) = j x k x j . Remark 2.
A rigorous derivation of the macroscopic portfolio model can be performed by theuse of mean field theory. We refer to [46] for details. acroscopic portfolio model Then in order to obtain the limit equation we simply sumover the number of agents and consider the limit N → ∞ . Thus, the macroscopic portfoliomodel reads. ddt X ( t ) = ˙ S ( t ) + D ( t ) S ( t ) X ( t ) + ED ( t, X, Y, S, ˙ S ) ddt Y ( t ) = r Y ( t ) − ED ( t, X, Y, S, ˙ S ) ddt S ( t ) = κ ED ( t, X, Y, S, ˙ S ) S ( t ) . The macroscopic wealth variable W ( t ) > W ( t ) = X ( t ) + Y ( t ) , X ( t ) , Y ( t ) > This section is devoted to analyze the rich dynamics of the ODE system (6). We discussexistence and uniqueness of solution, steady states and their stability and even computeexplicit solutions in special cases.
Macroscopic steady states
In order to obtain steady states, the equations0 = κ ED ( X, Y, S ) X + DS X + ED ( X, Y, S )0 = rY − ED ( X, Y, S )0 = κ ED ( X, Y, S ) S, need to be fulfilled. Besides the trivial solution the following steady state configurations arepossible.i) X = 0, Y = 0, S arbitraryii) K ( S ) = 0, Y = 0, D = 0, X arbitraryiii) K ( S ) = 0, r = 0, D = 0, X and Y arbitraryiv) K ( S ) > Y = 0, D = 0, X arbitraryv) K ( S ) < X = 0, r = 0, Y arbitraryThe case i ) corresponds to the situation when all investors are bankrupt. In the cases ii ) and iii ), the investors expect to have no benefit of shifting the capital between both portfolios.This means that the expected return K ( S ) is zero, which is equivalent to U γ (cid:18) ω s f − SS (cid:19) χ + (1 − χ ) U γ (0) = r. If we choose the value function U γ to be the identity, we observe S ∞ = χ ω s f r ,
12s the equilibrium stock price. One might assume that the reference point of the valuefunction is not zero. This means that the financial agent has a fixed bias towards potentialgains or losses. Mathematically, U γ (0) (cid:54) = 0 holds and thus the steady state would be shiftedby the reference point. Hence, psychological misperceptions of investors lead to changes ofthe equilibrium price. The case iv ) corresponds to the situation that the investor wants toshift wealth from the bond portfolio into the stock portfolio. In fact, no transaction takesplace, since there is no wealth left in the bond portfolio. Thus K ( S ) > U γ (cid:18) ω s f − SS (cid:19) χ + (1 − χ ) U γ (0) > r. In the simple case of the identity function as utility function, we obtain: ω χ s f r + ω χ > S. (7)In this equilibrium case, the amount of transactions have been too low to push the price abovea certain threshold defined by inequality (7). The reason for the steady state is the bankruptcyin the bond portfolio. Such a situation does not reflect a usual situation in financial markets.In case v ), we face the opposite situation. Here, the investor wants to shift wealth from stocksto bonds although there is no wealth left in the stock portfolio. In order to give a detailed characterizations of the complex dynamics we assume during therest of the section that the weight W ∈ [0 ,
1] is constant and that the value function U γ isgiven by the identity. In this setting it is possible to obtain local Lipschitz continuity directly. Proposition 1.
With the previously stated assumptions we can ensure existence and unique-ness of a solution (at least for short times [ t , t + (cid:15) ] ). In addition, we are interested in the stability of the steady state S ∞ = χ ω s f r characterizedin iii ). From economic perspective this is the only reasonable stationary point. Proposition 2.
In addition to the previously stated assumptions we assume that D = r = 0 holds. Then S ∞ = χ ω s f is a unique asymptotically stable steady state. For the proof of both results we refer to the appendix. Furthermore, we state the explicitsolutions of the stock price and portfolio dynamics in the appendix A. In the subsequentparagraph a qualitative discussion of the observed dynamics is given.
Booms, crashes and oscillatory solutions
We want to study whether the stock pricesatisfies the most prominent features of stock markets. These are crashes, booms and oscilla-tory solutions. Mathematically, a boom or crash is described by exponential growth or decayof the price. • Fundamentalists merely ( χ = 1) influence the price by their fundamental value s f .The price is driven to the steady state S ∞ = ω s f ω + r exponentially. Interestingly, theconvergence speed depends on the market depth κ , the interest rate r , the expectedspeed of mean reversion ω and the amount of wealth invested.13 Chartists merely ( χ = 0) build their investment decision on the current stock return.The price gets driven exponentially to the equilibrium stock price S ∞ = Dr or awayfrom the equilibrium stock price. This behavior is determined by the average wealthinvested in stocks or bonds. In general, we observe exponential growth or decay of thestock price (e.g. D ≡ • In our last case, we consider a mix of chartist and fundamental return expectationswith a constant weight χ ∈ (0 , S ∞ = χ ω s f +(1 − χ ) Dχ ω + r which is a combination of the previous equilibrium prices.Thus, the weight χ heavily influences the price dynamic. Furthermore, we can expectto observe oscillatory solutions if we consider a non constant weight χ ( t, S ). Wealth evolution
We can analyze the wealth evolution in the same manner as previouslythe stock price equation. We consider each portfolio separately. The computation can befound in the appendix A, as well. • We have exponential growth in the stock portfolio, if the wealth gets transferred frombonds to stocks. In the opposite case, the decay of wealth is described by an exponentialas well. • In the bond portfolio, we also observe an exponential increase if the wealth gets shiftedinto the bond portfolio. If stocks are assumed to perform substantially better ( K ( S ) >r )), we have exponential decay in the bond portfolio.So far we have only discussed the simplified case of a constant weight and the identityfunction as value function. The previous discussion indicates that a nonlinear interplay of afundamental and chartist strategy may cause oscillatory behavior. A rigorous quantificationof this behavior is left open for further research. We want to provide insights into the portfolio dynamics of the model. Furthermore, we intendto determine the influence of each parameter in the model. We will verify the existence ofoscillatory solutions of the model. For simulations we choose the value function U γ and theweight function W as follows: W ( K f − K c ) := β (cid:18)
12 tanh (cid:18) K f − K c α (cid:19) + 12 (cid:19) + (1 − β ) (cid:18)
12 tanh (cid:18) − K f − K c α (cid:19) + 12 (cid:19) ,α > , β ∈ [0 , ,U γ ( x ) := (cid:40) x γ +0 . , x > , − ( | x | ) γ − . , x ≤ , γ ∈ [0 . , . . The weight function W models the instantaneous comparison of the fundamental and chartistreturn estimate. The constant β ∈ [0 ,
1] determines if the investor trusts in the higher ( β = 1)or lower estimate ( β = 0) and we thus call this constant the trust coefficient. The constant α > time -2-1.5-1-0.500.511.52 c h i Weight -5 -4 -3 -2 -1 0 1 2 3 4 5 time -3-2.5-2-1.5-1-0.500.51 c h i Weight
Figure 2: Example of Value functions with different reference points.The value function U γ models psychological behavior of an investor towards gains andlosses. In order to derive the value function, one needs to measure the attitude of an individualas a deviation from a reference point. We have chosen the reference point to be zero, since U γ (0) = 0 holds. In Figure 2 we have plotted U γ and ¯ U γ := U γ − U γ isan example of a value function with a negative reference point. This choice of value functionsatisfies the usual assumptions: the function is concave for gains and convex for losses, whichcorresponds to risk aversion and risk seeking behavior of investors. Furthermore, the valuefunction is steeper for losses than for gains, which models the psychological loss aversion offinancial agents (see Figure 2).We have solved the moment system with a simple forward Euler discretization. The time stephas been chosen sufficiently small to exclude stability problems due to stiffness. We verifiedthe results with the ode15i Matlab solver. In fact one time step ∆ t may correspond to onetrading day. Then our simulation with 30 ,
000 time steps may correspond to approximately120 years of trading.We have chosen a trust coefficient β = 0 .
25 for the simulations in Figure 3 and 4. We refer tothe appendix B for further settings. The oscillations of the stock price is caused by oscillationsin the excess demand. The stock price is always less than or equal to the fundamental price.In addition, the oscillations get translated to the wealth evolution of the portfolios. Increasingwealth in the stock portfolio leads to decreasing wealth in the bond portfolio. Furthermore,we can observe on average a small positive slope of the wealth invested in bonds (see Figure4). This is caused by the positive interest rate r . In the next simulations (Figure 5), we havealtered the trust coefficient to study the impact on the price behavior. As Figure 5 reveals,the trust coefficient β influences the amplitude and frequency of the oscillations. In fact theoscillations obtained for the values β = 0 .
25 and β = 0 .
85 can be interpreted as businesscycles since they last approximately for 10 years. In addition, β determines the location ofthe oscillatory stock price evolution with respect to the fundamental value s f . A low trustcoefficient leads to oscillations located below the fundamental price and a high trust coefficientto oscillations above the fundamental value. 15 time P r i ce Stock PriceFundamental Price time -2024 E D Excess Demand time c h i Weight
Figure 3: Stock price, excess demand and weight χ . time M o n e y Money in Stocks time M o n e y Money in Bonds
Figure 4: Wealth evolution.16 time P r i ce Stock PriceFundamental Price time P r i ce StockPriceFundamentalPrice
Figure 5: Stock price with trust coefficient β = 0 . β = 0 . γ, κ, ω, α and β . Remark 3.
The parameters influence the price dynamics as follows: • A larger risk tolerance γ leads to smaller wave periods and smaller amplitudes. Ahigh risk tolerance heavily changes the price characteristics. We could thus observeconvergence of the price to the fundamental value. • The market depth κ influences the amplitude of the oscillations. A bigger κ value leadsto a larger amplitude. • The speed of mean reversion ω , the scale parameter α influence the wave period andamplitude. The wave period and amplitude decrease with increasing ω , respectively α . In order to quantify the findings of the previous Remark 3 we analyze the asymptoticbehavior of the stock price with respect to the parameters ω, γ . In fact we have simulatedthe model for 700 ,
000 time steps. In Figure 6 we have plotted the parameter value againstthe stock price ranges of the last 200 ,
000 time steps. A dot corresponds to a converged stockprice (steady state), whereas a line to oscillatory stock prices. Hence, Figure 6 reveals thatan increase in ω and γ leads to a decrease of the wave amplitude.Figure 6: Asymptotic stock price behavior for varying ω (left hand side) and γ (right handside). For further parameter settings we refer to table 10 in appendix B. Random fundamental price
Although the deterministic model can reproduce booms andcrashes, the periodic behavior is very unrealistic. In order to obtain reasonable stock pricedata we introduce a random fundamental price. The fundamental price is defined as thesolution of the SDE ds f = s f dW, s f (0) = s f , which needs to be interpreted in the Itˆo sense. For our numerical investigations we havesimply added the SDE to the macroscopic portfolio model. In fact it would be also possibleto add the previously introduced SDE to the microscopic model. Then one needs to repeatthe MPC formalism in the stochastic setting, which is in general possible.18he logarithmic return distribution of the stochastic process s f is well fitted by a Gaus-sian distribution (see Figure 7). The Figure 8 reveals that we obtain realistic stock price data Standard Normal Quantiles -0.0500.05 Q u a n t il es o f I npu t S a m p l e QQ Plot of Sample Data versus Standard Normal
Figure 7: Random fundamental price s f (left hand side) and the corresponding logarithmicreturn (right hand side) in a quantile-quantile plot fitted by a Gaussian distribution. Initialfundamental price is set to s f = 5 . rng(767) was chosen.with a random fundamental price. The quantile-quantile plot in Figure 8 clearly illustratesthe existence of fat tails in the logarithmic stock price return distribution. In addition, wemay note that the stock price usually follows the fundamental price, but it is also possibleto obtain market regimes where the stock price is more volatile than the fundamental price.Furthermore, the Figure 9 shows that a decreasing risk tolerance leads to slower chasing ofthe fundamental price by the stock price and a less volatile price behavior. Standard Normal Quantiles -0.14-0.12-0.1-0.08-0.06-0.04-0.0200.02 Q u a n t il es o f I npu t S a m p l e QQ Plot of Sample Data versus Standard Normal
Figure 8: The simulation was conducted with the parameters γ = 0 . , ω = 160 , β = 0 . Remark 4.
Different choices of the weighting function W and value function U γ have led tosimilar oscillatory behavior. Certainly the influence and impact of parameters may changefor varied functions W and U γ . Figure 9: We have chosen the same parameter values as in the simulations in Figure 8 exceptthe risk tolerance γ . In the figure on the left hand side we have chosen a risk tolerance of γ = 0 .
45 and in the figure on the right hand side γ = 0 . In this work we have established a macroscopic portfolio model, which can be derived frommicroscopic agent dynamics. On the microscopic level we have introduced an approximationframework of the optimal control problem in order to give a precise definition of rationaland bounded rational agents. The model can be regarded as a model with fully myopicagents. The qualitative discussion has led to the conjecture that an interplay of chartist andfundamental trading behavior is essential in order to obtain oscillatory, cyclic price behavior.Our simulations confirm the existence of cyclic price behavior around the fundamental value.Thus, the model offers the mean reversion characteristic. This model behavior is similar to thestock price behavior in the models [7, 10, 31]. Interestingly, the trust coefficient β determinesthe location of the oscillations with respect to the fundamental price. This indicates that theagent’s attitude towards the currently best performing trading strategy play a major role inthe price formation. Such a trust parameter has not been introduced by any earlier modeland it may be worthwhile to study the impact of that parameter in more detail in the future.In addition, the parameter studies reveal that the risk tolerance γ and the reaction strength ω of fundamentalists heavily influence the wave speed and amplitude. In our case a higherrisk tolerance of investors leads to less volatile prices and less pronounced booms and crashes.The reason is that an increasing risk tolerance leads to a larger impact of the fundamentaltrading strategy. Similar model behavior has been reported in [11]. It has been pointed outby Odean [40] that a small risk tolerance of investors causes overconfidence and this can causehigher volatility. The reason is that trader underweight rational information which are in ourmodel given by the fundamental trading strategy. Furthermore, the simulations conductedwith random fundamental prices have led to realistic price movements. Especially, we couldobtain fat tails in logarithmic asset returns. 20part from the dynamic behavior of our macroeconomic portfolio model which coincideswith simulations of other asset pricing models, we want to stress two points. • The resulting marcoeconomic portfolio model has been derived from microscopic agentdynamics and the myopic trading rules can be seen as a simple approximation of a veryelaborated optimal control problem in the context of differential games. • Thanks to model predictive control, we were able to give a precise mathematical defini-tion of the degree of rationality of the financial agent. This has been done for a rathergeneral portfolio model and one can even expect to apply this methodology to otheroptimal control problems in finance.Nevertheless, we have to admit that we consider a simple portfolio model. Various extensionsand generalizations are possible. One may add uncertainty in the microscopic portfolio modelor study the impact of increasing rationality of the microscopic agents on the model dynamics.In addition, it seems interesting to study the impact of different cost functions on the stockprice behavior. Especially a rigorous quantification of the oscillatory stock price behavior isof major interest.This work shows that heuristic trading strategies of investors can be interpreted as approxi-mations of investments by a rational financial agent. Although a fully bounded rational agentseems to be quite unrealistic, the mathematical connection to a perfect rational agent mayhelp to discover new and more appropriate models of financial agents.
Acknowledgement
Torsten Trimborn was supported by the Hans-B¨ockler-Stiftung.21 ppendix A Qualitative analysis
The proof of proposition 1 reads:
Proof.
We show local Lipschitz continuity. Then existence and uniqueness directly follows bythe Picard-Lindel¨of theorem. We can rewrite the stock price equation into an explicit ODEsystem: ˙ S = (cid:40) κ ( χωs f + (1 − χ ) D − ( r + χω ) S ) Y − χ ) κ Y , K ( S ) > ,κ ( χωs f + (1 − χ ) D − ( r + χω ) S ) X − χ ) κ X , K ( S ) < . (8)Thus, we may denote the right hand side of the explicit ODE system by F ( z ) , z := ( X, Y, S ) T ∈ [0 , ∞ ) × [0 , ∞ ) × (0 , ∞ ). Notice that the excess demand ED does no longer depend on ˙ S , sincewe can insert the right hand side of the stock price equation (8). Local Lipschitz continuityis obvious except for the potential singularity in S ∗ = χ ω s f +(1 − χ ) Dr + χ ω , since K ( S ∗ ) = 0 holds.Thus, we show Lipschitz continuity on U = U X × U Y × U S , U X := [ X − (cid:15), X + (cid:15) ] , U Y :=[ Y − (cid:15), Y + (cid:15) ] , U S := [ S ∗ − (cid:15), S ∗ + (cid:15) ] , (cid:15) > z := ( X , Y , S ∗ ), where X , Y ∈ [0 , ∞ )are arbitrary but fixed. First we discuss the excess demand ED : | ED ( X, Y, S ) | ≤ κ ( r + χ ω ) max (cid:26) max X ∈ U X | X | , max Y ∈ U Y | Y | (cid:27) max S ∈ U S (cid:12)(cid:12)(cid:12)(cid:12) S (cid:12)(cid:12)(cid:12)(cid:12) | S ∗ − S | (cid:18) κ (1 − χ ) max (cid:26) max X ∈ U X (cid:12)(cid:12)(cid:12)(cid:12) X − χ ) κ X (cid:12)(cid:12)(cid:12)(cid:12) , max Y ∈ U Y (cid:12)(cid:12)(cid:12)(cid:12) Y − χ ) κ Y (cid:12)(cid:12)(cid:12)(cid:12)(cid:27)(cid:19) ≤ C | S ∗ − S | As next step we discuss each component of F = ( F , F , F ) T separately. For the stock priceevolution we obtain: | F ( Z ) − F ( Z ) | ≤ κ ( r + χω ) max (cid:26) max X ∈ U X (cid:12)(cid:12)(cid:12)(cid:12) X − χ ) κ X (cid:12)(cid:12)(cid:12)(cid:12) , max Y ∈ U Y (cid:12)(cid:12)(cid:12)(cid:12) Y − χ ) κ Y (cid:12)(cid:12)(cid:12)(cid:12)(cid:27) | S ∗ − S |≤ C | S − S ∗ | For the portfolio dynamics we get: | F ( Z ) − F ( Z ) | ≤ D max S ∈ U S (cid:12)(cid:12)(cid:12)(cid:12) S (cid:12)(cid:12)(cid:12)(cid:12) | X − X | + (1 + κ max X ∈ U X | X | ) C | S ∗ − S |≤ C | X − X | + C | S − S ∗ | , | F ( Z ) − F ( Z ) | ≤ r | Y − Y | + C | S ∗ − S | . Hence, we conclude that || F ( z ) − F ( z ) || ≤ L || z − z || , z, z ∈ U, holds on U with Lipschitz constant L := C ( C + C + C + C + r ), where the additionallyconstant C is due to the equivalence of norms.The proof of proposition 2 is given by: 22 roof. We set r = D = 0 and derive the explicit ODE system. Thus, for a continuous dif-ferentiable Lyapunov functional we can compute the Lie derivative. We define the Lyapunovfunctional as follows: ψ : R → R , ( x, y, S ) T (cid:55)→ − ( s f − S ) ( x + y − ( s f − s )). We immediatelyobtain ddt ψ (( X ( t ) , Y ( t ) , S ( t ))) ≤ , with ddt ψ (( X ( t ) , Y ( t ) , S ( t ))) (cid:12)(cid:12)(cid:12) S = S ∞ = 0 , and can conclude the asymptotic stability of S ∞ . Proposition 3.
In special cases, we can compute solutions of the stock price equation. Weassume constant weights χ and assume that the utility function is described by the identity. • Fundamentalists alone ( χ = 1 ): The stock price equation reads ˙ S = (cid:40) κ ( ω s f − ( ω + r ) S ) Y, ωs f ω + r > S,κ ( ω s f − ( ω + r ) S ) X, ωs f ω + r < S. This equation seems reasonable, so the investor shifts his capital into stocks if he expectsa positive stock return, and vice versa. The solution is given by S ( t ) = (1 − exp {− κ ( ω + r ) t (cid:82) Y ( τ ) dτ ) } ) ω s f ω + r + S (0) exp {− κ ( ω + r ) t (cid:82) Y ( τ ) dτ } , for s f ω + r > S, (1 − exp {− κ ( ω + r ) t (cid:82) X ( τ ) dτ ) } ) ω s f ω + r + S (0) exp {− κ ( ω + r ) t (cid:82) X ( τ ) dτ } , for s f ω + r < S. Hence, the price is driven exponentially fast to the steady state S ∞ = ω s f ω + r . • Chartists alone ( χ = 0 ): We get ˙ S = (cid:40) κ D Y − κ Y − r κY − κY S, for D > S, κ D X − κ X − r κX − κX S, for D < S.
The solution is given by S ( t ) = (cid:18) − exp (cid:26) − r κ t (cid:82) Y ( τ )1 − κY ( τ ) dτ (cid:27)(cid:19) Dr + S (0) exp (cid:26) − r κ t (cid:82) Y ( τ )1 − κY ( τ ) dτ (cid:27) , for κ D Y + D (1 − κ Y ) > S, (cid:18) (cid:26) − r κ t (cid:82) X ( τ )1 − κ X ( τ ) dτ (cid:27)(cid:19) Dr + S (0) exp (cid:26) − r κ t (cid:82) X ( τ )1 − κ X ( τ ) dτ (cid:27) , for κ D X + D (1 − κ Y ) < S. • Chartists and fundamentalists with a constant weight χ ∈ (0 , : The correspondingstock price equation reads ˙ S = (cid:40) κ ( χωs f + (1 − χ ) D − ( r + χω ) S ) Y − χ ) κ Y , K ( S ) > ,κ ( χωs f + (1 − χ ) D − ( r + χω ) S ) X − χ ) κ X , K ( S ) < . he solution is given by S ( t ) = (1 − exp {− κ ( χω + r ) t (cid:82) Y ( τ )1+ κ (1 − χ ) Y ( τ ) dτ ) } ) χω s f +(1 − χ ) Dχω + r + S (0) exp {− κ ( χω + r ) t (cid:82) Y ( τ )1+ κ (1 − χ ) Y ( τ ) dτ } , for K ( S ) > , (1 − exp {− κ ( χω + r ) t (cid:82) X ( τ )1+ κ (1 − χ ) X ( τ ) dτ ) } ) χω s f +(1 − χ ) Dχω + r + S (0) exp {− κ ( χω + r ) t (cid:82) X ( τ )1+ κ (1 − χ ) X ( τ ) dτ } , for K ( S ) < . Proposition 4.
For the wealth evolution, we consider the stock and bond portfolio separately. • In the stock portfolio, the wealth evolution is given by ˙ X = (cid:40) ( κ K ( S ) Y + DS ) X + K ( S ) Y, for K ( S ) > , ( κ K ( S ) X + DS ) X + K ( S ) X, for K ( S ) < . The solution is given by X ( t ) = X (0) exp (cid:26) t (cid:82) κ K ( S ) Y + DS dτ (cid:27) + (cid:18) − exp (cid:26) − t (cid:82) κ K ( S ) Y dτ (cid:27)(cid:19) κ , for K ( S ) > , X (0) exp (cid:40) t (cid:82) K ( s )+ DS dτ (cid:41) κ t (cid:82) K ( S ) exp (cid:40) ζ (cid:82) K ( S )+ DS dζ (cid:41) dτ , for K ( S ) < . • The bond portfolio is given by ˙ Y = (cid:40) r Y − K ( S ) Y, for K ( S ) > ,r Y − K ( S ) X, for K ( S ) < , with the solution Y ( t ) = Y (0) exp (cid:26) t (cid:82) ( r − K ( S )) dτ (cid:27) , for K ( S ) > , exp { r t } (cid:18) Y (0) − t (cid:82) K ( S ) X exp {− r τ } dτ (cid:19) , for K ( S ) < . Parameters of simulation
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