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Article Excerpt
Summary. In a three-period finite exchange economy with incomplete financial markets and retrading, we study the effects of the degree of incompleteness and of changes in the financial structure on asset price volatility. In what are essentially no aggregate risk economies, asset price volatility is a sunspot-like phenomenon. If markets are completed by financial innovation, asset price volatility reduction is generic. With aggregate risk, changes in the financial structure affect asset price volatility through a pecuniary externality. Financial innovation which decreases equilibrium price volatility can be crafted under conditions of sufficient market incompleteness. Numerical examples illustrate the role of risk aversion for volatility changes and show that, with or without aggregate risk, reducing the degree of incompleteness per se is not necessarily associated with a volatility reduction.
Keywords and Phrases: Incomplete markets, Financial innovation, Volatility.
JEL Classification Numbers: C60, D52, G10.
1 Introduction
How do we design financial assets to reduce asset price volatility? Under what conditions can such assets be constructed? Do the degree of market incompleteness and the financial structure affect asset price volatility? In what direction? In this paper we find some sufficient conditions to answer robustly these intimately related questions by studying how the financial structure affects price volatility. (4)
We use the standard model of discrete-time dynamic trading of assets and multiple goods in a finite economy (see Arrow [1], Radner [30] and Duffie and Shafer [9]). In our setup, asset design is not the result of optimizing behavior, and asset markets are exogenously incomplete. We concentrate on the equilibrium effects of having different financial structures. We impose time separability and use von Neumann-Morgenstern expected utilities. Any weaker version of separability (such as time nonseparability, or habit formation) can be easily accommodated. The model covers any finite time horizon trading economy, even though we focus on the three-period case. Although a new financial asset generally plays a role as a hedging device and as an information vehicle, the results in this paper do not address differential information economies. Rather, we concentrate on the spanning role of financial assets. The term price volatility refers to the variance of asset prices at each date. 'Excess' volatility is defined as volatility over and above its complete markets value. With these maintained general specifications, we carry on the analysis by separating economies in two classes, those with and those without aggregate risk.
In economies with no aggregate risk, i.e., invariant aggregate resources, and stationary conditional expected asset payoffs, asset prices show zero volatility with complete markets, so market-completing financial innovation never increases price volatility. In Theorem 1, we prove that within this class of economies any financial innovation is generically volatility-reducing, if: i) we go from one asset to complete markets; ii) there is one commodity. Theorem 1 also suggests that no qualitative role for this result is played by the convexity of marginal utility, or the precautionary motive. Numerical examples indicate that the type of risk aversion, e.g., CRRA or CARA utility, may determine in what dimension--utility or endowment--heterogeneity matters for genericity, and that the level of risk aversion affects the volatility level.
In economies with aggregate risk, Theorem 2 establishes that, generically, one can design a new asset which reduces price volatility. For this to occur the dispersion in the cross-sectional distribution of traders' characteristics must be positive but at least less than the maximal degree of disagreement in individual pricing kernels. (5) If the new asset can be retraded, as in Theorem 3, a comparison between incomplete and (dynamically) complete markets is possible, with one initial asset and a binomial-tree structure for uncertainty. However, in the case of impossibility of retrading (Theorem 2) the condition linking the number of states, traders and assets is weaker than in the case of retrading (Theorem 3).
1.1 Discussion and related literature
In economies with no aggregate risk, of widespread use in the literature, price volatility implies an efficiency loss relative to a complete market benchmark. But lack of efficiency--or the fact that one cannot build a representative agent--per se is not enough to imply volatility in excess of its complete market value. Excess volatility is a phenomenon akin to sunspots, in that asset prices and allocations are state-dependent while aggregate resources are not. Indeed, for Theorem 1 we extend techniques used in two-period sunspot economies by Pietra ([28, 29]). (6) We show that there is endogenous aggregate (i.e., asset price) uncertainty due to market incompleteness even with one commodity, complementing results on pure sunspots by Gottardi and Kajii [13] and Hens [16], for real assets with no retrading. The basic idea is well-known: when financial markets are incomplete, risk averse individuals cannot perfectly smooth consumption across time and states. Heterogeneity across individuals makes wealth distribution matter even though aggregate wealth does not vary across states. This causes fluctuations in aggregate endogenous variables across states of the world, or over time, and asset prices show excess volatility. (7,8) Theorem 1 shows, within the limits of its assumptions, the genericity of this fact and that it is not caused by the precautionary motive.
In addressing economies with aggregate risk, asset price volatility can reflect aggregate differences across states, even when markets are complete. Theorem 2 shows that nothing so general can be deduced as that volatility is lower whenever incompleteness is reduced, if not eliminated. (9) Without controlling for asset specifications, more incompleteness may be associated with lower volatility. However, we show how to design volatility-reducing assets. (10) Numerical examples show that changes in volatility may be quantitatively significant, depending on the level of market incompleteness. They illustrate that asset payoffs generally depend on the equilibrium and the type of economy specified. The intuition here is, along the lines of what is known in the constrained suboptimality literature (see, e.g., Geanakoplos and Polemarchakis [11] and Citanna, Kajii and Villanacci [6]), that price effects induced by payoff changes--the so-called 'pecuniary externality'--may allow modifying asset prices in such a way to control volatilities at leisure. Theorems 2 and 3 in fact extend to a multi-period setting the logic of the results on financial innovation of Cass and Citanna [4] (see also Elul [10]). However, the restrictions that naturally arise on the payoff matrix representing financial markets with dynamic trading are not encompassed by the previous theorems. Also, while using the framework of [4], extended to multi-period trading, it is possible to control for volatility and welfare with the same asset, this requires a very large degree of incompleteness to start with. When controlling only for price volatility, our results show for instance that volatility increases do not necessarily imply a loss of efficiency. Efficiency and volatility are generally not comonotonic.
2 The model
We consider a standard model of an intertemporal, competitive, pure-exchange economy with incomplete financial markets. A tree structure of date-events represents uncertainty in this economy. Let t denote the time period, with t = 0, 1,..., T, where t = is today, and t = T is the terminal date. Although the formalization and the results encompass any finite-horizon economy, we will focus on the three-period case, i.e., T = 2. The successors of each date-event before t = 2 are 1 < [bar.S] < [infinity], representing uncertainty. The total number of date-events in the economy is therefore given by [[summation].sub.t=0.sup.2] [bar.S.sup.t]. We will also index date-events all together as states s, setting s = for today, and s = [bar.s][bar.S] + [bar.s'] for the future states which have [bar.s] as predecessor and where [bar.s] [member of] {0,..., S} and [bar.s'] [member of] {1,..., S}. We write S + 1 = [[summation].sub.t=0.sup.T] [bar.S.sup.t]. We assume that all the information in the economy is publicly available.
At each date and state, there are C commodities or goods indexed by c, with C [greater than or equal to] 1 (14). The commodity (and endowment) space is taken to be [R.sub.++.sup.G], where G = C(S + 1). There are H [greater than or equal to] 2 traders (or 'households') indexed by h. We also interpret H as the number of types in the economy, and therefore consider H the degree of dispersion in the traders' exogenous characteristics. Each trader has endowment [e.sub.h] [member of] [R.sub.++.sup.G], and preferences represented by the utility function [u.sub.h] : [R.sub.++.sup.G] [right arrow] R, which is assumed to be smooth, differentially strictly increasing and differentially strictly concave, and to have closure of indifference surfaces contained in [R.sub.++.sup.G]. Moreover, we consider von Neumann-Morgenstern preferences, with objective probabilities and time separable utility, (12)
[u.sub.h]([x.sub.h]) = [S.summation over (s=0)] [[pi].sup.s][v.sub.h]([x.sub.h.sup.s]), (1)
with [[pi].sup.s] > 0. After appropriate normalization, one can interpret [[pi].sup.s] as derived from a (stationary) transition probability measure [bar.[pi]] on the states s > in the following standard way:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where [bar.[pi]]([bar.s']|[bar.s]) is the conditional probability of state [bar.s'] in period t = 2 after state [bar.s] occurs in t = 1, and [bar.s], [bar.s'] [member of] {1,..., [bar.S]}; [[pi].sup.0] is interpreted instead as a simple intertemporal preference parameter.
We assume that C spot commodity markets are open at each state s, with [p.sup.s,c] > being the price of commodity c in state s, and [p.sup.s] = ([p.sup.s,c])[.sub.c[member of]C].
At t = 0, 1, J financial instruments are tradable. We posit that [bar.S] > J [greater than or equal to] 1, so financial markets are incomplete, even dynamically. These instruments are long-term securities that can be held until the terminal date T = 2. It is notationally convenient also to represent the retraded instruments as independent assets i, where i = 1,..., I, and I = J[[summation].sub.t=0.sup.T-1] [bar.S.sup.t]. We denote by [b.sub.h.sup.s,j] trader h's holdings of the j-th asset in state s [less than or equal to] S. However, in using the natural identification of asset i with a pair (s, j) for s = 0, 1,..., [bar.S], each asset j will generate [bar.S] + 1 'different' assets, and [b.sub.h.sup.i] will denote the holdings of asset i for trader h. An asset j [member of] J in this economy promises to deliver [y.sup.bar.s,j](0) units of the numeraire good -which we take to be the last at each spot s, i.e., c = C- in state [bar.s] in period t = 1, and [y.sup.bar.s',j]([bar.s]) units of the numeraire in state [bar.s'] in period t = 2 after [bar.s] occurred in period t = 1, with [bar.s], [bar.s'] [member of] {1,..., S}. Hence [y.sup.bar.s'] (s) = ([y.sup.bar.s',j](s))[.sub.j[member of]J], is the [bar.s']-th row and [y.sup.j](s) the j-th column of Y (s), an [bar.S] x J matrix of payoffs for the traded securities, for s = 0, 1,..., [bar.S]. We will assume that each Y (s) be in general position, in that every submatrix of row or column dimension m [less than or equal to] J has full rank. In the one-asset case (J = 1) this condition becomes Y (s) = Y [much greater than??] 0, which is consistent with intertemporal models of stock or bond trading. Denote by [GAMMA] [subset] [R.sup.[bar.S]J([bar.S]+1)] the space of such matrices. Each asset j [member of] J is exchanged at price [q.sup.0,j] at t = 0, and at price [q.sup.s,j] in state s and period t = 1, and [q.sup.s] = ([q.sup.s,j])[.sub.j[member of]J], for s = 0, 1,..., [bar.S] is endogenously determined in equilibrium. Let Q to be the [bar.S] x J matrix with [q.sup.s] as its s-th row, for s = 1,..., [bar.S].
Let [[PSI].sup.C] be an S + 1-dimensional square matrix, with the price of the numeraire commodity in state s on the diagonal at the s-th row, all s, and zeros elsewhere. The financial structure is then represented by an (S+1) x I-dimensional matrix of prices and payoffs R, a special case of the usual standard incomplete market model where R assumes the following form
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
With [x.sub.h] [member of] [R.sub.++.sup.G], [b.sub.h] [member of] [R.sup.I], p [member of] [R.sub.++.sup.G] and q [member of] [R.sup.I] we denote the consumption bundle and the asset portfolio of trader h, the commodity price vector and the asset price vector, respectively. It will be convenient to take quantity vectors as columns, and price vectors as rows.
The space of traders' endowments is E...
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