I. Introduction

Almost always, it seems, capital assets occur as one of two kinds: either real or fiat. Real assets are characterized by a limited rate of increase (possibly zero or negative) in amounts outstanding, whereas fiat assets have a potentially unlimited rate of increase. Real assets (e.g. physical assets) are subject to varying marginal product, whereas fiat assets (e.g. a contract with an obligor) are subject to constant marginal product (assuming that investor creditworthiness ceilings restrict the aggregate obligations). In some sense, therefore, real assets underlying a financial asset do not occupy identical states within a financial system, whereas fiat assets do.

We shall crystallize these considerations in the form of an axiom of allowed residency of identical states, namely, for real assets, single residency at most, and for fiat assets, unlimited residency. For the sake of making a concrete application, we shall consider stocks as an exemplar of a financial asset supported by real assets, and corporate bonds of one supported by fiat assets; a unified theory for the two will be developed on the basis of optimizing behaviour in frictionless markets.

The financial system that we shall consider consists of a set of assets of fixed types, comprising a stock market and a bond market (including a money market deposit), in equilibrium with other asset markets. In order to abstract from issues of term structure, however, bonds are considered to be of indefinite maturity and subject only to credit risk. Investors (the treatment of whom will be largely implicit) optimally manage their portfolios through time, exercising short-run rational expectations such that prices evolve deterministically. However, because portfolio cashflows are discontinuous (for instance, discrete dividend or interest payments), they trade in finite (as distinct from marginal) quantities at discrete points in time. Thus, the optimal path through microfinancial state space—the asset holdings by investor—experiences discrete jumps. For that reason, we choose to regard a macrofinancial description of the system—a macrofinancial state—to be a probability distribution over all the microfinancial states; and macrofinancial equilibrium to be a probability distribution at a point in time that is time invariant (with respect to endogenous variables).

Within this probabilistic framework, definite answers can be established if we assume that the number of individual assets, assumed indivisible, is very large. It is this assumption which replaces that of infinitely divisible time and continuous finite trading in models that postulate a price process. In this paper, the random distribution is not postulated, but derived. Further, the random distribution refers to not a process over time, but rather a distribution at a point in time. Even in the absence of exogenous uncertainty, equilibrium is a dynamic equilibrium, characterized by a determinate distribution (effectively, risk neutral) of expected prices around a mean.

Determining the dynamic equilibrium for stocks and bonds (Section 2 and Appendix) requires some notion of what constitutes distinct microfinancial states of the system. To that end, we adopt an axiom of indistinguishability, meaning an inability to distinguish marginal redistributions of portfolio assets through trade. Underlying this notion is the observation that financial markets display a determinate-price equilibrium in which optimal portfolio composition is indeterminate (when prices evolve deterministically). Indistinguishability combined with allowed residency determines the enumeration of states and the assignment of a probability measure. Accordingly, one can calculate (Section 3) the equilibrium probability distributions and macrofinancial quantities, that is, probability averaged financial quantities and their distributions.

The effect of risk on probability distribution averaging (Section 4) is introduced through a set of exogenous risk parameters, fixed in number, whose value affects the value of the stocks or bonds. The parameters undergo fluctuations, assumed to be small. Additionally, investors may make random reallocations of cash between the stock and bond markets and other markets. Even though almost nothing is specified about the risk parameters, the resulting distributions are determinate and depend critically on the dimensionality of risk, that is, the number of parameters. This number is regarded as a quantity to be determined empirically.

Empirical findings (for instance, Jackwerth and Rubinstein (1996) and Kim (1999)) show that the distribution of stock prices is approximately lognormal, with systematic deviations such as left skewness and leptokurtosis; and that the distribution of corporate bond prices is approximately double exponential, interpreting short-run changes in credit quality as a realization of the distribution. Prima facie, the analytical expressions that we derive for the equilibrium probability distributions offer the promise of an excellent fit to the empirical data.

II. Dynamic Equilibrium for Stocks and Bonds with No Risk

Let there be given a financial system 𝔉, consisting of a set of assets chosen through time t by a given set of investors, denoted by index μ (μ = 1,…, M), so as to optimize consumption c (t) ≡ c ≡ (c¹,…, c^M). 𝔉 is implicitly part of an economy E in which firms issue and redeem liabilities that are the counterparts to the assets.

Assets are either of two kinds, stocks and bonds, denoted by respective indices S and B; we shall develop the theory in parallel for both S and B, differentiating the two only as needed. (It seems that a similar approach as for the aggregate market could be applied to an individual firm’s stocks or bonds, with similar results.) Assets are of K types, denoted by index κ (κ = 1,…, K), each corresponding to a liability of a particular firm; excepting that κ = 1 denotes the money market deposit (potentially, the liability of any and all firms), which is a capital-certain asset paying a nominal rate of interest ρ. Assets κ = 1, …, K_B are bonds, and κ = K_B + 1,…,K are stocks (K_B < K). d_κ denotes the current return on asset κ (the dividend or coupon rate, with d₁ = ρ). All assets are denominated in currency units h > 0, assumed to be indivisible. Asset holdings by type and investor are specified by quantity n ≡ (n) ≡ (n¹¹,…, n^KM) ≡ {n^Kμ}, of nominal value nh, and by corresponding marginal market price p ≡ {p_Kμ}; p_1μ = 1, the numeraire, and p_−1μ denotes the price of c^μ. 𝔘= {n}, the set of all possible (n), is called the state space; 𝔙 = {p, n}, the phase space; and $𝔐=\left\{t,n,\dot{n}\right\},\dot{n}\equiv \mathit{dn}/\mathit{dt}$, we shall call the configuration space. Each consumer’s unit portfolio, (1^μ) ≡ (l^1μ,…, 1^Kμ), is assumed to constitute a linearly independent basis set. The formalism admits the possibility of a local market structure in 𝔉, across which p_κμ may differ for fixed κ = (≠1).

Implicitly in E, profit maximizing firms produce and invest in such a way as to follow their optimal path of investment, and ρ is determined endogenously by the equating of aggregate savings and investment. Accordingly, firms’ net cashflow on securities D—issuance less servicing of liabilities:

—is determined as to size (by the aggregate budget constraint of either firms or investors), though not as to composition, since we shall maintain the assumption that firms are indifferent as to the composition of these cashflows (meaning, for instance, Modigliani-Miller indifference to debt/equity ratios). The $n^{\kappa }={\displaystyle {\sum }_{\mu }{n}^{\kappa \mu }}$ will vary since firms may create or redeem liabilities at will (subject to an investor creditworthiness ceiling).

Asset markets are assumed to be frictionless (apart from the indivisibility of assets) with, in particular, no transaction costs, no restrictions on short-selling, and no taxes. Investors are assumed to be price takers, and markets are assumed to clear. Investors maximize the present value of utility flows (with heterogeneous rates of time preference) through the exercise of short-run rational expectations alone. The system is nonetheless determinate, because we regard not only initial quantities n (t_A), but also initial prices p (t_A), as given empirical parameters.

Hence, asset prices evolve deterministically and there is no risk, in the sense that there is no short-run nominal uncertainty. The total rate of return ρ, comprising both income and capital gains, is assumed to be a function on the joint phase spaces {q_e, k}, the real economy shadow-price and capital stock, and {p₋₁, c}, without explicit dependence on time:

expressing the presumption that with a given static technology, the value of optimal investment is determined by (q_e, k) (for given p₋₁). Total rate of return is equalized across all assets, both financial and real; thus, ρ may be regarded as a risk-neutral return in that all returns are risk free (and the issue of completeness of securities markets does not arise).

Nevertheless, these conditions only hold almost always, because dividend and coupon payments are allowed to take discrete finite values; “almost always” means at almost all points, excepting at most a set of measure zero, on the optimal time path $O_{A{\Omega }}^{\oplus }$ (in general, not unique) in U. At points of discrete cashflow, asset market equilibrium cannot be accomplished by marginal trades, thereby necessitating finite trades. As a consequence of optimal investor behaviour, therefore, trading is marginal almost always, and finite only at discrete points. (These points constitute a set of measure zero; recall from set theory, however, that their number may be denumerably infinite, or even non-denumerably infinite.)

With optimal portfolio choice, the present value of portfolio returns (the momentary rate of return) including capital gains is given by

where “a.e.” denotes “almost everywhere”, synonymous with “almost always”. Here, the” denotes a double transformation of coordinates, from current to present value prices, and from current to total return (but we write $Y_t^{″}$ in terms of observable (p_κμ, n^κμ)).

The derivation of this expression is given in the Appendix. There, the point is made that the inclusion of capital gains so as to pass from current to total return can be justified on the basis of a time-dependent transformation of coordinates $(n,\dot{n})$ in 𝔚; and such time-dependent coordinate transformations are necessarily allowable if one makes the assumption of the invariance (scalar nature with respect to its arguments) of the Lagrangian of the portfolio choice problem:

where investors act as if to maximize ${\int }_{t_A}^{t_{{\Omega }}}$ Ldt, the present value of net cashflows (that is, act as if discounting at the common observable rate ρ).

Because ∂ρ/∂t = 0, $Y_t^{″}$ represents a constant of the dynamical evolution of W (equivalently 𝔙),

(as shown in the Appendix), and in general the only determinate constant. Consequently, a probability density over 𝔙,

must take the form

without explicit dependence of Ψ on (t, p, n) (equivalently, $\left(t,n,\dot{n}\right)$), if it is to represent macrofinancial equilibrium. ${\Psi }(Y_t^{″})\equiv {\Psi }(Y_t^{″}[p,n])$ then equals the (relative) probability of occurrence of the microfinancial state (p, n) at time t. We now proceed to construct such a macrofinancial equilibrium state Ψ.

Elsewhere (Johannes, 2001), I have shown on the basis of essentially the same assumptions as will be made here, that a heuristic argument concerning the most probable distribution leads to a Ψ of the form

for bonds and zero-return money. The first term refers to zero-return money and will not concern us further here. The second term (which we shall derive here by a different route) refers to bonds: Y ≡ Y_t = ρ^*hP denotes the current value $Y=\exp \left(+{\displaystyle {\int }_{t_A}^t\rho \mathit{\text{d}}\mathit{\tau }}\right)Y^{″}$ of an individual financial asset, and P denotes its price; α^* denotes the larger systems (market, economy) of which 𝔉 and E form a small component; α^* is the Lagrangian multiplier associated with Y^* (and is also a measure of expected nominal output in E); and ν^* is the Lagrangian multiplier associated with expected volume of financial assets in E (equivalently, volume of financial assets in (E^*). In this paper, using methods largely known from the mathematical and scientific literature, we shall derive the corresponding result for stocks and bonds, starting out with a brief heuristic argument and then proceeding to a rigorous treatment of the case of N assets.

Consider two composite assets 1 and 2 that are part of 𝔉, each consisting of all the liabilities of one kind (S or B) of respective firms 1 and 2, and with corresponding Y₁ and Y₂. The subscript t on Y is henceforth omitted, since we shall be concerned exclusively with momentary equilibrium at a point in time. Assume that these distinguishable assets form a small part of such that Y₁, Y₂ ≪ Y for 𝔉 in aggregate. In the idealized limit of many firms (identical in type though not in size) in which Y₁/Y, Y₂/Y → 0, we would expect the probability densities for the two components, Ψ (Y₁) and Ψ (Y₂) to be statistically independent; note that this expectation asserts nothing about the assets’ returns, only something about the probability of occurrence of assets bearing Y₁ and Y₂. Statistical independence implies that Ψ satisfies the functional equation

Let us assume that Ψ is continuous. Then we can apply the mathematical theorem that the only solutions of this equation for functions continuous on the real line are the exponential functions:

and such functions are necessarily analytic, a fact that will be used in Section 4. Further, it is clear that first, the common discount factor $\exp \left(-{\displaystyle {\int }_{t_A}^t\rho \mathit{\text{d}}\mathit{\tau }}\right)$ in Y” may be absorbed in the conjugate variable α, thereby justifying expressing Ψ in terms of the observable current value Y; second, we require α > 0, so as to ensure bounded probabilities for Y ≥ 0; third, the value of a must be the same for component systems that are in macrofinancial equilibrium; fourth, given the commonality of the value of a, we may confine our considerations in equilibrium to Y ≥ 0, consistent with the limited liability of financial asset ownership; fifth, the constant c plays the role of the normalizing factor, and therefore may be dropped, with normalization by $1/{\displaystyle {\int }_{V^T}{\Psi }d{V}}$ when required, where ${\int }_{V^T}$ denotes an integral over the domain of (p, n) ∈ V.

The same argument applies to 𝔉 as a whole (that is, the markets S + B), if we assume that 𝔉 is a small component of a larger set of asset markets, not only the financial assets in F^*, but also the real asset markets in E and E^*. We conclude, therefore, that the distribution of financial assets in 𝔉 may be described by the normalized probability density

where α^* is determined outside of 𝔉 (by equilibrium in 𝔉^* and E^*). In the event that Y takes not continuous, but discrete values Y_λ, λ = 1,2,…, the probability law assumes the form

On this basis, let us turn to the problem of determining Ψ for N individual assets (distinguishing N^s and N^B as needed). Hidden in the foregoing argument has been the presumption that the composite assets are distinguishable, that is, that each asset may be specified precisely as to its constituent types κ = κ₁, κ₂,… and quantities $n^{{\kappa }_1},n^{{\kappa }_2},\dots$. For individual assets, that presumption may be doubted. For, optimal investor behaviour in 5” precisely determines the path of prices p (t), but leaves indeterminate the path of portfolio composition n (t) (as shown in the Appendix). Such behaviour accords with the notion that financial assets are subject to a no arbitrage principle, whereby deviations from precise equilibrium prices are eliminated through large portfolio purchases and sales of indeterminate quantity.

It seems reasonable to suppose, therefore, that at equilibrium prices, one cannot distinguish marginal differences arising from trade between investor portfolios. To do so would require information on market transactions that could not be obtained without disturbing the equilibrium prices concerned (with the presumption that investors eschew voluntary disclosure of their transactions). We shall embody this supposition in the form of the following axiom of indistinguisuability.

The economic interpretation is that for the aggregate portfolio n in 𝔉, marginally differing states n₁ and n₂ of equal trader utility, as evidenced by agreed marginal prices p_κμ are regarded as indistinguishable. The application to our 𝔉 is that the p_κμ evolve in a determinate way, and so a portfolio n is regarded as one and the same state regardless of the marginal redistribution of individual assets (that is, assets of nominal quantity h). Redistribution refers to not only ownership μ, but also type κ, since firms may issue and retire liabilities. In other words, the axiom implies that we cannot attach a label (κ, μ) to an individual asset.

Accordingly, individual assets are identified simply by price P_ι, and the aggregate portfolio n is completely specified by the set of numbers of individual assets N^ι at each price, of total number N = ∑N^ι. We denote a specific set by $(N_{\lambda }^1,N_{\lambda }^2,\dots )$. We shall start by considering discrete single-asset price levels P_ι,ι=1,2,…, assumed fixed, and then pass to the quasi-continuous case P_ι → P (at which point, the density of states or state price density dV/dP will be introduced).

Consider a market (S or B) of N individual assets with observed current return Ŷ, where a  denotes an observed quantity (empirical datum). By virtue of the axiom of indistinguishability, a single state of the market is specified by (N¹, N²,) with corresponding Y = ρ^*h∑_ι P_ιN^ι, where ρ^* is determined in E^*. The potential values Y_λ of Y, each of which is defined by a set of values $(N_{\lambda }^1,N_{\lambda }^2,\cdots )$,

for given ρ^* and {P_ι}, will follow the distribution law ${\Psi }(Y_{\lambda })=\frac{\exp (-Y_{\lambda }/{\alpha }^{*})}{{\displaystyle {\sum }_{\lambda }\exp (-Y_{\lambda }/{\alpha }^{*})}}=\frac{\exp (-[{\rho }^{*}h/{\alpha }^{*}]{\displaystyle {\sum }_{\iota }{P}_{\iota }{N}_{\lambda }^{\iota }})}{{\displaystyle {\sum }_{\lambda }\exp (-[{\rho }^{*}h/{\alpha }^{*}]{\displaystyle {\sum }_{\iota }{P}_{\iota }{N}_{\lambda }^{\iota }})}}$.

To comprehend the empirical significance of Ψ, we must express it in terms of accessible quantities through direct calculation.

III. Calculation of the Probability Distribution Ψ and Macrofinancial (Ψ-Averaged) Quantities

For the calculation of Ψ and associated financial quantities, it is convenient to define the moment generating function by

where the P_ι are regarded as given implicit parameters, Φ (α^*) ≡ Φ (α^*; {ρ^*hP_ι}); the successive logarithmic derivatives of Φ with respect to α^* generate the corresponding central moments of Ψ. From log Φ can be derived observable quantities of interest pertaining to the macrofinancial state, that is, probability averaged (or “phase averaged”) quantities: for example,

the average number of assets bearing price P_ι, where 〈 〉 denotes a probability average (the mathematical expectation) and log the natural logarithm; and

the expected value of Y in 𝔉, which incidentally shows that α^* is determined by the value of Y = Y^* in 𝔉^* (and in fact can be related to nominal output in 𝔈^* on the basis of the commonality of α^* in equilibrium noted above). With the postulate that observed macrofinancial quantities are identified with probability averages, we have that 〈Y〉= Ŷ, and hence $\widehat{{\alpha }^{*}}$ can in principle be inferred from Ŷ measured in 𝔉 (given a specific model for 𝔉).

In the definition of Φ (α^*), the Σ_λ signifies the sum over all admissible sets of $(N_{\lambda }^1,N_{\lambda }^2,\cdots )$. Let us assume that the total number of individual assets N for the market (S or B) is fixed. The underlying rationale is that the sum function 〈N〉 = ∑〈N^ι〉 invariant (by construction), and for large N

fluctuations in N (the distribution around the mean) will become negligible assuming, for example, that the variance of N increases no faster than N itself. Hence N  〈N〉, a constant. Then the admissible sets $(N_{\lambda }^1,N_{\lambda }^2,\cdots )$ are those that satisfy

and permutations of individual assets within a given $N_{\lambda }^{\iota }$ are regarded not as giving rise to different states but as part of the same N–asset state λ.

The system 𝔉 will be completely specified macrofinancially once we stipulate one final quantity: the allowed residency numbers for each individual-asset state ι. Reflecting the considerations of section 1, we adopt the following axiom.

Strictly speaking, the allowed residency applies to the underlying assets and their states, since financial assets represent claims on a cross-section of all underlying assets; for our present accounting purposes, however, we may regard the financial assets as being built up by marginal additions, with corresponding additions to the underlying assets. We shall ignore the possibility of hybrid real and fiat assets, for example convertible bonds, which may be regarded as assets whose expected quantity cannot be time-invariant under dynamic optimization.

The problem of finding the distributions of 〈N^ι〉 with P_ι for S and B, that is, the equilibrium distributions of value, is thus reduced to calculation of Φ (α^*) in accordance with its defining equation (1), subject to the condition (4) and the two-part condition,

where the notation (S) and (B), it is to be hoped, is self-explanatory. In essence, the axiom on allowed residency by financial assets supported by either real or fiat assets amounts to stipulating that they follow respectively the so-called Fermi-Dirac or Bose-Einstein statistics, familiar in the scientific literature: Feller (1968) ch. II, which cites the example that a priori one might expect typographical errors on a printed page to follow Fermi-Dirac statistics.

As an illustration of the difference between the different ways of reckoning Ψ, consider the following example. Suppose that there is given a single asset, of denomination h and a certain type, that may assume either of only two states, P_HI and P_LO; and that there is an equal probability 1/2 of either outcome. Further, suppose that there is given a second single asset of the same type. When assets are distinguishable, there is a probability 1/4 of both assets residing in the same state, in accordance with the rule for independent events of multiplying probabilities. In S, the probability of both assets residing in the same state is 0; in B, it is 1/3, since there are only three independent states (P_HIP_HI, P_HIP_LO ≡ P_LOP_HI, P_LOP_LO) when assets are indistinguishable. This example verifies the assertion made earlier that the assumption of distinguishability is implicit in the argument of statistical independence.

Writing

where

we see that (4) will be satisfied provided that Φ consists solely of all terms that are homogeneous of degree N(=N^S or N^B)in the {z_ι}. Because the $N_{\lambda }^{\iota }$ appear as exponents in Φ, (5) generates as potential combinations in Σ_λ a product of geometric progressions (of which, the homogeneous terms are to be retained). In the tth factor of potential combinations, there appears for S, (1 + z_ι), and for B, $(1+z_{\iota }+z_{\iota }^2+\cdots )=1/(1-z_{\iota })$. Hence, for Φ we obtain the expressions

subject to the requirement that out of the product Π_ι one retains solely terms that are homogeneous of degree N in the {z_ι}.

Stated thus, we see that we can apply Cauchy’s integral formula to obtain explicit expressions for Φ. Working in the complex plane, let us introduce the auxiliary multiplier β^* through the two-part definition

where β^* is a complex variable that plays a role similar to that of a Lagrange multiplier for the condition (4), with the ^* notation anticipating that β^* is determined in 𝔉^*. Then Φ is given exactly by an integral around a closed contour in the complex β^* plane that encircles the origin β^* = 0 (but no other singularity of the integrand, and is assumed to lie within the circle of convergence of w (β^*)):

The evaluation of the contour integral (6), a formidable task in general, is greatly helped by two key simplifications: that N → ∞ (by assumption), which permits the use of asymptotic methods; and that the integrand has a unique minimum (as we shall show), which permits evaluation at that point alone. Consequently, the integral can be worked out by the saddle point method (Bleistein and Handelsman, 1975, ch. 7, sect. 2), which in its more general form of the method of steepest descents was originated by B. Riemann and P. Debye.

Consider the behaviour of the integrand in (6), w (β^*)/β^*N+1, for β^* real and positive. For β^* → 0, the integrand is indefinitely large and positive, for both S and B. The logarithmic first derivatives are given by

which, for β^* → 0, are both large and negative. As β^* increases, they both increase. For B, assuming for simplicity that the lowest price level is P_ι = 0, as β^* → 1, the derivative becomes indefinitely large and positive (for fixed large N). For S, as β^* → +∞, the derivative becomes positive and remains so. To see this, note that its limiting value becomes (1/β^*) {− (N + 1) +Σ_ι 1}, and we have Σ_ι1 > (N + 1), since Σ_ι 1 represents the number of states, some of which will be empty by virtue of (5).

Therefore (since log constitutes a monotonic transformation), w (β^*)/β^*N+1 possesses a unique minimum, for both S and B. We denote the corresponding (real) value of β^* by β^*⊕, which is determined by setting d (w (β^*)/β^*N+l)/dβ^* = 0 (the first order conditions, which we shall return to shortly). The values of the second derivatives, which we shall denote by σ², are given by

and, unless qualified, we shall mean the stationary point value σ² = σ(β^*⊕)² > 0. (Case S, for which we may have β^* → +∞ and σ² < 0, illustrates the point that, in order to establish the existence of a unique minimum or maximum, it may be neither necessary—case S—nor sufficient that the second derivative be always positive or negative, respectively; a trivial example of insufficiency being log x, 0 < x < +∞, which has no maximum.)

Assuming that σ² is very large of the order of N (σ² ~ N, which we shall verify below), the saddlepoint method shows that the value of (6) consists in the contribution from that point alone, with the dominant term derived from a Taylor series expansion of log w (β^*)/β^*N+1 in the neighbourhood of β^*⊕,

to give:

where η represents the imaginary part of β^*. (In intuitive terms, at the unique saddle point β^*⊕ = β^*⊕ + i0, the modulus of the integrand attains a maximum, and its phase is stationary for a small, purely imaginary change i dη, that is, a contour that cuts the real axis orthogonally at that point. In the limit N → ∞, the contribution from the neighbourhood of β^*⊕ dominates all others, which are self-cancelling because of infinitely rapid oscillations of phase.) Hence, ${\Phi }({\alpha }^{*})=(w({\beta }^{*\oplus })/{\beta }^{*\oplus N+1})\left(1/\sqrt{2\pi {\sigma }^2}\right)$ and

Under the maintained assumption of N → ∞, we can set (N + 1)  N, and neglect the term log (2πσ²), since (1/N) log N → 0 as N → ∞. Evidently, N very large requires that N  log N. It is clear that in many circumstances, N being large cannot be regarded as a self-contained fact, but rather is a consequence of M, the number of investors, being large. We arrive at

reintroducing the observable prices P_ι (with single-asset Y_ι = ρ^*hP_ι). Φ (α^*, N) becomes a function of the parameter N by virtue of the counterpart constraining condition, and enters the expression for Φ.

From the first order conditions noted above, d (w (β^*)/(β^*N+1)/dβ^* = 0, and setting (N + 1)  N, we infer that

where we write β^*⊕ = exp[−ν^*] for notational convenience, and we set N = 〈N〉, which was tacitly the case hitherto (recall the remarks leading to (4)), Φ = Φ (α^*, 〈N〉). These equations may be thought of as determining 〈N〉 given ν^* (from 𝔉^*), or alternatively, as determining $\widehat{{\nu }^{*}}$ given observed $\widehat{N}$ in 𝔉, with $\langle N\rangle =\widehat{N}$ (and α^* and the ρ^*hP_ι considered as given parameters). Additionally, applying (2) to the expressions for log Φ we obtain

from which it is clear that (7) represents the discrete probability distributions Ψ (normalized to 〈N〉) over states defined by price P_ι, 〈N〉 = Σ_ι 〈N^ι〉.

Let us pass to the quasi-continuous limit

Here N (P) denotes the residency of state P (the analogue of N^ι). $\overline{N}\ge N$ denotes the upper bound of N, whose existence is ensured by the assumption that n is bounded by $\underset{¯}{n},\overline{n}$ (Appendix), and $\overline{N}$ may be interpreted as a counterparty creditworthiness constraint on investors. $P\le \overline{Y}/{\rho }^{*}h$ has an upper bound, given h > 0 indivisible, because when wealth is bounded, which we assume, then $Y\le \overline{Y}$ is bounded almost always.

dV (P)/dP denotes the state price density; it is this quantity that constitutes the analogue to given {P_ι}, since what matters in the quasi-continuous case is (relatively) how many states occur in a small price interval dP. dV (P)/dP is defined by way of the definition of the density of states dV° (Y)/dY as follows. V^o (Y) denotes the volume of single-asset phase space for portfolio returns y up to Y, $y\left(\tilde{N},\tilde{P}\right)=y\left(\tilde{P}\right)<Y$, where the dummy integration variable $\tilde{P}$ specifies price and $\tilde{N}$ specifies the arbitrarily assigned ordering of a single asset in a portfolio ($\tilde{N}\le \overline{N}$ is assumed to vary over a range sufficiently large as to be quasi-continuous). V° (Y) (normalized to a total number of states $\overline{N}$) is defined by

where the linearity of V° in Y is a consequence of the positively linear homogeneity (actually, for financial assets, linearity) of y in price $\tilde{P}$:

It is the degree of homogeneity interacting with the dimensionality of risk that will prove decisive in determining the effect of exogenous risk on the probability distribution (Section 4).

The equivalent state price density is defined by the circumstance V (P) = V° (Y), implying that

that is, dV/dP is constant for financial assets. The transition from discrete to continuous rests on the notion that a single-asset state is specified by an infinitesimal element $d𝔙\equiv d\tilde{N}d\tilde{P}$ of the single-asset phase space, and consequently the number of states in a given price interval is measured by (strictly speaking, proportional to) the corresponding volume. (A deeper justification is provided in Johannes (2001), namely that the volume of phase space represents the unique Haar-von Neumann invariant measure over the dynamical evolution of 𝔉)

We arrive at the quasi-continuous form of (7):

We see that 〈N〉 is directly proportional to $\overline{N}$, in accordance with the natural suppositions that 〈N〉 will vary in proportion to the size of 𝔉 in some sense, and that $\overline{N}$ is a measure of the size of 𝔉. Therefore, we can now assign a more precise meaning to the limit 〈N〉 → ∞, namely, we stipulate that as 〈N〉 → ∞, the ratio $\langle N\rangle /\overline{N}$ is subject to a fixed upper bound. Consequently, ν^* remains bounded below and β^*⊕ above, as 〈N〉 → ∞. We thus verify (from the earlier derived expression for σ²) that σ² is very large of the order of N as 〈N〉 → ∞.

We may identify 〈N〉 with ∫ Ψ dV = ∫ Ψ (dV/dP) dP (normalized to 〈N〉); that is, Ψ represents the (relative) probability weight (expected residency) of a given state:

The equilibrium distribution of value (equivalently, price) with no exogenous risk is given by the integrand in

For (S), ${\nu }_S^{*}\in (-\infty ,+\infty ),$, and for $(B),{\nu }_B^{*}\in (0,+\infty )$. For both (S) and (B), the distribution of value (or price) tends to zero as P → 0 (even if ν^* = 0), and so the integrals converge. (As a technical matter, the values of these integrals for ν^* = 0 and with the upper bound infinite can be expressed in terms of the gamma and Riemann zeta functions through integration by parts.)

Finally, we see again that the informational content of Ψ, 〈N〉, and 〈Y〉 can be deduced from knowledge of