The Basic Model
3.1 Model Setup
A probability space (Ω,F,P) and an information filtration Ft : t>0 in a risk-neutral world with continuous time (t ≥ 0) are fixed in advance. Ω is the set of all possible states in the world. F is the filtration of sub-σ-algebras. It describes the revealing of information to investors over time. Pis the probability measure on F. It is assumed that probability space and the information filtration satisfy the usual hypotheses as defined by Protter (2005, p. 3).
Let us assume the economy is populated by two kinds of agents—investors and market makers—who are both risk-neutral and live infinitely. All agents have a constant and known time preference rater >0, with which they discount the future. A single nonstorable consumption good, consumed by all agents (e.g.
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‘cash’), is used as a numéraire.59 All agents have access to a (risk free) bank account offering interest rater, and to the OTC market for a special asset. The bank account is comparable with a liquid security, which can be traded without frictions. To avoid unlimited borrowing, the value Wt of the bank account is bounded from below. The asset traded in the OTC market is illiquid, as it can only be traded when a potential buyer and a potential seller can find each other.
The illiquid asset pays a constant dividend rateDof consumption per time unit (e.g. one year), like a consol bond.60 Time runs forever. Each agent can only hold a maximum of (illiquid) assets at a time, which is normalized to one. As the utility function of risk-neutral agents is linear, asset holding corresponds to either zero or one unit in equilibrium. Initially, a fractionsof all investors is endowed with one unit of this asset, implying a fixed asset supply. Short selling is not allowed.
The population of investors is segmented into four different groups: Investors can either own the asset (o), or not (n), and they all have either an intrinsic type that is high (h) or low (l). These intrinsic types can be interpreted as the investor’s marginal utility from the asset. Duffie, Gârleanu, and Pedersen (2005, pp. 1818.) give some possible explanations for low-type investors: “(i) low liquidity (that is, a need for cash), (ii) high financing costs, (iii) hedging reasons to sell, (iv) a relative tax disadvantage, or (v) a low personal use of the asset.” The full set of investor types isΓ={ho,hn,lo,ln}.
The intrinsic type of an investor is modeled as a Markov chain. A low investor receives an exogenous idiosyncratic preference or funding shock causing a type switch from low to high with an intensity λu >0. A high investor who suffers such a preference (or funding) shock switches from high to low with intensity λd >0. The switching processes are random and are assumed to be pairwise in-dependent for any two investors. Type switches generate a need for change in asset holdings, since investors’ valuation, i.e. marginal utility, towards the asset changes over time and depends on investor’s type. Because only low-type own-ers want to sell their asset, whereas high-type non-ownown-ers want to buy one in equilibrium, type switches generate trade. Lnagents and hoagents do not trade.
Consequently, low owners (lo) are called potential sellers and high non-owners (hn) are called potential buyers.
High owners who are affected by an idiosyncratic preference shock switch to a
59 See Weill (2007), p. 1332.
60 Duffie, Gârleanu, and Pedersen (2005, 2007) normalize this dividend toD=1.
low intrinsic type. Those low-type agents are exposed to a holding cost for the asset ofδper unit of time, withδ >0, leading to a utility flow ofD−δ. Holding costs only occur for low-type investors owning the asset, as it reflects the negative impact of an idiosyncratic liquidity shock. Since there can be gains from trade due to different utility and due to costs of holding assets, low owners want to sell their asset.
Duffie, Gârleanu, and Pedersen (2005) define a unit mass continuum of investors with measure normalized to one. Then,µσ(t)denotes for eachσ∈ Γthe fraction at timetof type-σinvestors in the total population. These fractions must add up to one at any time and must be nonnegative, leading to
µlo(t) +µhn(t) +µln(t) +µho(t) = 1, (3.1)
µσ(t) ≥0. (3.2)
By assumption, only a fractions ∈ [0, 1] of investors owns one unit of the asset.
This prerequisite defines the market clearing condition, which implies for every timetthat
µlo(t) +µho(t) = s. (3.3) Agents who want to trade in an OTC market must search for each other, since no central trading device is available. Assumeλ ∈ [0,∞) is the exogenous and constant intensity of a homogenous Poisson process. This intensity λdescribes the random contact of one investor with a counterparty, and reflects search abil-ity or efficiency in the OTC market. The search technology in this market is as follows: Assume that any agent, which is chosen from the set of all agents, is of type σ1. The probability for being of type σ1 is equal toµσ1(t). The probability of any agent, which is not of typeσ1, being matched with an agent of type σ1 is thenλµσ1(t). All agents of, say, a setσ2, where setσ2is distinct from setσ1, search for agents of setσ1withλµσ1(t)µσ2(t). Simultaneously, agents of setσ1search for agents of setσ2. The matching function derived in equation (2.13) models exactly this independent search and matching process for potential buyers and sellers.61 With application to the prevailing notation, the appropriate matching function isMλ(t) = 2λµlo(t)µhn(t), whereM(t)is the number of successful matches per unit of time. As soon as two agents meet, they start bargaining over the price,
61 The law of large numbers is assumed to hold throughout. See Duffie and Sun (2007, 2012) and footnote 43.
according to a bargaining process described in chapter 2.2. After completing the transaction, the lo agent (seller) becomes an ln agent, the hn agent (buyer) be-comes anhoagent, and both part ways.
This search model contains, according to Vayanos and Wang (2007, p. 75), a natu-ral liquidity measure. As investors are prevented from immediate trading, costs of delay accrue. These costs can be measured by the expected time it takes until an investor finds an adequate counterparty. This expected time is nothing else but the inverse of the measure of liquidity. Anhn agent (potential buyer) meets lo agents (potential sellers) at the rate 2λµlo(t). The average time it takes until a potential buyer meets potential sellers is thn(t) =1/(2λµlo(t)). In the same way, the expected meeting time for a potential seller to meet potential buyers is tlo(t) =1/(2λµhn(t)).62
Additionally, it is assumed that there are independent nonatomic market makers in this OTC market who are of unit mass and who maximize their profit. In-vestors and market makers search for each other and meet with exogenous and constant intensityρ ≥0. It is the sum of investors’ intensity of searching for mar-ket makers and marmar-ket makers’ intensity of searching for investors. This intensity captures the availability of market makers in the market. Investors and market makers also start bargaining over the price as soon as they meet. It is assumed that market makers have no inventory but immediately unload their asset on a frictionless interdealer market. Without bearing any inventory risk, market mak-ers are matchmakmak-ers. They ration either the buy side or the sell side, depending on which one is higher. The matching function isMρ(t) = ρmin{µlo(t),µhn(t)}.
For tractability, it is assumed that all described Poisson processes are indepen-dent. The flow diagram in figure 3.1, which is in the style of Chiu and Koeppl (2011, p. 7), illustrates search and bargaining in the just-specified OTC market.
62 See chapter 2.1.3, Vayanos and Wang (2007), p. 75, and Duffie, Gârleanu, and Pedersen (2007), p. 1878.
Buyer (high non-owner)
Non-Owner (low non-owner) Owner
(high owner)
Seller (low owner)
Market Maker
switch with intensityλd
switch with intensityλu
trade with
intensity λ switch with intensityλd
switch with intensityλu
trade
withintensity ρ
trade with intensityρ
change upon trade
changeupontrade
Figure 3.1:Flow diagram.