what could be possible explanations for this puzzle?
Essentially, one can argue from three point of views: the economics point, the behavioral finance point and the mathematical finance point.
The classical economist could say that the model simply overlooks market friction, e.g., transaction costs. Considering such costs, an active continuous-time trading structure is not feasible, and it might be better to stick to some simpler (but less general) portfolio.
The advocate of behavioral finance could argue that investment decisions and markets are far away from being rational. Investors would therefore exhibit preferences that are not covered by the Mutual Fund Theorem, in particular they might be risk-seeking for some wealth levels, have reference points, overweight small probabilities and are – in essence – too messy to fit into the neat model of Merton. Moreover, due to these irrationalities, markets show anomalies that make them deviate from the models we have studied so far. In particular, future returns might not always be independent of past returns.
The devotee ofmathematical financefinally would partially agree to both and start improving his models in order to capture transaction costs on the one hand and more realistic stock market processes on the other hand.
What dowesay on this matter? Probably that all three are right: they observe weak points in the model that have to be addressed one way or the other. We have already seen, e.g., that the utility function implied by option prices is not everywhere concave (see Sect.4.6.3). Moreover, we will collect some more evidence for the need for more complicated processes (see Sect.8.8).
To sum up: the puzzle gives rise to improving the theory, and some directions for an improvement we will discuss in the next sections.
8.7 Market Equilibria in Continuous Time
To formulate an equilibrium model in continuous time requires more complicated tools than in discrete time, although the fundamental ideas are very similar. To attain the existence of an equilibrium, generally we need very strong assumptions. Yet, unfortunately, we might not get continuous and diffusion prices from a general equilibrium theory although this is assumed in most of the literature of finance and Arbitrage Pricing Theory. As in the multi-period equilibrium model, prices are determined by the state prices with the next period payoff structure. In complete markets, these state prices are unique, non-negative adapted processes.
In this part, we try to show briefly how the equilibrium in continuous time would look like and what possible implications for asset prices would be. For this, we
adopt the general model of Duffie and Zame [DZ89]. According to this approach, the endowments are stochastic processes and each agent has time additive differentiable expected utility functions. The model has a very crucial implication that unlike the most of the general equilibrium models, under this model, we can have continuous stochastic asset prices.
As in the previous sections of this chapter, we work with a probability space .˝;F;P/equipped with a standard filtration fFt; 0 t Tg. We assume the consumption processctis square integrable and the cumulative dividend process is defined as
dDnt DD.t/dtCD.t/dB.t/; for allnD1; : : : ;K;
such that it has a finite variance and for then D 0;D0t D 0, for allt < T and fort D T;D0T D 1. Thus, for the asset pricesS0;S1; : : : ;SK the gain process is expressed asGDDCSfrom which we can express the gain process as geometric Brownian motion. With this, we can define the cumulative gains with predictable13 squared integrable portfolio process D.0; : : : ; K/. This portfoliois generally defined as piecewise constant function over time intervals, which is very reasonable when we consider the transaction costs.
Definition 8.10 A feasible consumption-portfolio plan .cit; ti/ for agent i D 1; : : : ;Iis defined as a pair for timetsuch that
tStD Z t
0 sdG.s/C Z t
0 ps.!sicis/ds:
Definition 8.11 A feasible consumption-portfolio plan.cit; ti/is optimal if there is no other feasible consumption-portfolio plan.cQit; ti/such thatUi.cQi/ > Ui.ci/for agenti.
Definition 8.12 An equilibrium for an economy ED
.S;p/; .c1; 1/; : : : ; .cI; I/
is a collection such that given the security price and commodity spot price processes
• For every agenti, the consumption-portfolio plan is optimal, i.e. maximizes their utility,
• Markets clear, i.e.P
iiD0andP
i.ci!i/D0.
13For technical definition, e.g. predictable process, see [Duf96]
8.7 Market Equilibria in Continuous Time 311 For the sufficient conditions for the existence of such an equilibrium, one should refer to Duffie [Duf86]. We continue to give the equilibrium condition for the model.
Assumption 8.13 Each agent has the following utility function Uiwith the follow- ing representation:
Ui.c/DEZ T
0 ui.ct;t/dt
;
where uiW RC !Ris differentiable and concave and ctdenotes the non-negative consumption process. Moreover, to avoid unbounded spot prices for consumption, limc#0u0i.c;t/D C1.
Assumption 8.14 The aggregate endowment process ! D PI
iD1!i follows a stochastic process which satisfies the following differential equation:
d!tD!.t/dtC!.t/dB.t/;
such that the volatility term satisfies E
Z T 0 !2.t/dt
<1:
Assumption 8.15 Equivalent assumption for spanning in discrete time to show any feasible consumption plan is attainable by trading in financial markets is that the processes M1;M2; : : : ;MNare martingales, i.e.
MtkDEŒDktjFt; for all t2Œ0;T:
The representative agent model for a given equilibrium economy E D .S;p/; .c1; 1/; : : : ; .cI; I/
is a single agent .U; !/ maximizing the following utility form:
U.c/D sup
c1;:::;cI
X
i
iUi.ci/
subject toP
icicfor some coefficient vectors2RICand the equilibrium prices are the same.S;p/. By the equilibrium functional form, we can express the single representative agent’s utility as
U.c/DE Z T
0 u.ct;t/dt
;
where
u.c;t/D sup
c1;:::;cI/2RIC
X
i
iui.ci;t/
subject toP
icic.
Theorem 8.16 Under the Assumptions 8.13 and 8.15, on the defined economy, there exists an equilibrium with a representative agent.U; !/, such that the real security pricesSOtsatisfy the following representative agent pricing formula for any time t:
SOtD 1 u0.!t;t/E
Z T t
u0.!s;s/dDOsjFt
;
for all t2Œ0;T/.
For the proof one should refer to Duffie and Zame [DZ89].
The last expression is analogous to discrete time, because it has the same characteristics regarding the no-arbitrage condition of the equilibrium asset price formulation. Under some dividend and utility specifications, one can derive a C- CAPM model in a straightforward way. Moreover, the model obviously produces stochastic asset prices under some utility specifications. With the right utility choice, the Black-Scholes model can be supported by an equilibrium model. However, for the Black-Scholed model to hold one does not need to know the equilibrium allocations, the only thing is to estimate the market price of risk as long as the underlying asset price functional is given by the no-arbitrage condition. In fact, we also know that the no arbitrage condition is also a necessary condition for an equilibrium. Black-Scholes can be considered as a partial equilibrium model of prices.