The Value Function - Equilibrium Prices

The Basic Model

3.3 Equilibrium Prices

3.3.1 The Value Function

Each rational investor chooses an asset holding strategy that maximizes his ex-pected utility (i.e. present value) of his lifetime consumption. Since each indi-vidual lives infinitely, a continuous and infinite consumption process has to be modeled by considering a search and matching process. At an arbitrary time t, the investor’s utility depends only on his current type, σ(t) ∈ ^Γ, and wealth or money W_t, which he has in his bank account. The infinite horizon expected utility-maximization problem for all investor types, who are risk-neutral and measure their lifetime consumption with a utility function, can be derived by means of dynamic programming. The optimal value function J(·), the optimum value of the utility-maximization problem, is stated as follows:⁶⁵

J(W_t,σ(t),t) =sup

C,θ E_t



 Z∞

e⁻^rvdC_t+v



, (3.17)

given the dynamics

dWt =rWtdt−^dCt +θ_t(D−^δ1{^σ^θ⁽^t⁾⁼^lo})dt−^P^ˆ(t)dθt, (3.18) with the expectation Et, conditioned on Ft. An investor can freely decide over his consumption and asset holding, so that the two control processes are: (1) a cumulative consumption process C_t, and (2) a feasible asset holding process

65 In general, it is not clear that the maximum actually exists for these processes until it is known that J(Wt,σ(t),t)is bounded. Therefore, the supremum for a precise formulation of the utility-maximization problem is applied first. Verification that the maximum is actually attained follows in a second step, i.e. the verification that the value function is bounded. If a value function is unbounded, it can go to infinity, but it can never attain it. In such a case, the supremum is suitable. See Bellman (1954), p. 507.

θ_t ∈ {^{0, 1}}. The other input parameters are: (i) σ^θ, the type process induced by θ, and (ii) ˆP(t) ∈ {^P(t),A(t),B(t)}which is the trade price at time t, dependent on the agent’s counterparty. J(W_t,σ(t),t) is called a value function or indirect utility function. It differs from a normal utility function as it always implies an optimization process.

The core of the dynamic programming theory is Bellman’s ‘principle of optimal-ity’:

“An optimal policy has the property that whatever the initial state and initial decisions are, the remaining decisions must constitute an opti-mal policy with regard to the state resulting from the first decisions.”⁶⁶ A recursive application of Bellman’s ‘principle of optimality’ on equation (3.17) leads to an iterative optimization problem

J(W_t,σ(t),t) =sup withk=t+v. Dynamic programming approaches a dynamic optimization prob-lem by a recursive solution technique, translating a probprob-lem composed of multi-stages into a sequence of separate states.⁶⁷ This recursive solution implies the consideration of all possible states within a final period by weighting the corre-sponding payoffs with the probability of their occurrence. Working backward in time leads to the optimal equilibrium path.

The first part of equation (3.19) can be approximated by the mean value the-orem of integral calculus. The second part can be derived by a Taylor se-ries expansion of function J(·) around the point (Wt,σ(t),t) to approximate J(Wt+dWt,σ(t) +dσ(t),t+dt). Inserting both parts into equation (3.19), sub-tractingJ(Wt,σ(t),t)on both sides, dividing everything bydt, and lettingdt →^0, leads to the Hamilton–Jacobi–Bellman (HJB) equation. The optimal value func-tion in continuous time dynamic programming is the solufunc-tion to the HJB equa-tion, which is in general a partial differential equation. This equation acts as a

66 Bellman (1954), p. 504.

67 See Bellman (1954), p. 503.

necessary and sufficient condition to ensure optimality.⁶⁸

Since agents are risk-neutral by assumption, the value function (3.17) describing investors’ lifetime utility is a linear function in wealthW_t. I show this by inserting equation (3.18) into equation (3.17), leading to

J(W_t,σ(t),t) = sup

The utility-maximization problem is thus changed from deciding over both op-timal consumption and opop-timal asset holding to only choosing the opop-timal as-set holding. The transversality condition (also called the no-bubble condition)

xlim→^∞E_t[e⁻^rxmax{^P(x),A(x),B(x)}] = 0 ensures that the value function is well defined.

Relating this finding to continuous time dynamic programming, the value func-tionV_σ(t) can be calculated by applying Bellman’s ‘principle of optimality’ and focusing on a particular agent at a particular timet. Since agents are risk-neutral

68 See Schöbel (1995), ch. 3.3 and Björk (2009), ch. 19.

by assumption and the value function J(W_t,σ(t),t) is linear in holding cashW_t, the HJB equation happens to be a system of ordinary differential equations, and is derived in the following passage.

Duffie, Gârleanu, and Pedersen (2005, p. 1837) defineτ_l as the next stopping time when an agent changes his intrinsic type, τ_i as the next stopping time when a search and bargaining between two investors is successfully completed,τ_m as the next stopping time when trade occurs between an investor and a market maker, andτ =min{^τl,τ_i,τ_m}. The optimal value functions result with

V_lo(t) = E_t

"Zτ t

e⁻^r⁽^u⁻^t⁾(D−^δ)du+e⁻^r⁽^τ^l⁻^t⁾V_ho(τ_l)1_{_τ_l₌_τ_} +e⁻^r⁽^τⁱ⁻^t⁾(V_ln(τ_i) +P(τ_i))1_{_τ_i₌_τ_}

+e⁻^r⁽^τ^m⁻^t⁾(V_ln(τ_m) +B(τ_m))1_{_τ_m₌_τ_}

# ,

(3.21)

V_hn(t) = Et

e⁻^r⁽^τ^l⁻^t⁾V_ln(τ_l)1_{_τ_l₌_τ_}+e⁻^r⁽^τⁱ⁻^t⁾(V_ho(τ_i)−^P(τ_i))1_{_τ_i₌_τ_}

+e⁻^r⁽^τ^m⁻^t⁾(V_ho(τm)−^A(τm))1_{_τ_m₌_τ_}

# ,

(3.22)

V_ho(t) = E_t





τl

e⁻^r⁽^u⁻^t⁾D du+e⁻^r⁽^τ^l⁻^t⁾V_lo(τ_l)



, (3.23)

V_ln(t) = E_th

e⁻^r⁽^τ^l⁻^t⁾V_hn(τ_l)ⁱ, (3.24)

where the expectation is with respect to τ_l, τ_i, τ_m and is conditional on Ft. The first term of asset owners’ value functions [V_lo(t),V_ho(t)] gives the dividend flow, possibly reduced by holding costs. The second term of owners’ value functions [V_lo(t), V_ho(t)] and the first of non-owners’ value functions [V_ln(t), V_hn(t)] de-scribes the discounted value of an intrinsic type switch, given the random stop-ping time isτ =τ_l. The second last term of potential buyers’ and potential sellers’

value functions [V_lo(t),V_hn(t)] is the discounted value of trading with an investor, given that the random stopping time isτ =τ_i. And the last part of potential buy-ers’ and potential sellbuy-ers’ value functions [V_lo(t),V_hn(t)] describes the discounted value of trading with a market maker, given that the random stopping time is τ =τ_m. As a result, investor’s utility depends on his current expected utility, e.g.

from holding the asset, and on his prospective expected utility.

I state the explicit equations for (3.21)–(3.24) and the derivation of the HJB equa-tions in appendix 3A. These HJB equaequa-tions, which are solved by the value func-tionsV_σ(t), are⁶⁹

V˙_lo(t) = rV_lo(t)−^λu(V_ho(t)−^Vlo(t))−^ρ(V_ln(t) +B(t)−^Vlo(t))

−^2λµhn(t) (V_ln(t) +P(t)−^Vlo(t))−(D−^δ), (3.25) V˙_hn(t) = rV_hn(t)−^λd(V_ln(t)−^Vhn(t))−^ρ(V_ho(t)−^A(t)−^Vhn(t))

−^2λµlo(t) (V_ho(t)−^P(t)−^Vhn(t)), (3.26) V˙_ho(t) = rV_ho(t)−^λd(V_lo(t)−^Vho(t))−^D, ^(3.27) V˙_ln(t) = rV_ln(t)−^λu(V_hn(t)−^Vln(t)). (3.28) The first part on the right hand side of equations (3.25)–(3.28) corresponds to opportunity costs. The second element characterizes value changes based on ex-pected changes in intrinsic types. For buyer (hn) and seller (lo), the third and fourth element is due to trade between investors and trade intermediated by market makers, respectively. The last term for asset owners (lo, ho) accounts for dividends and holding costs of the asset.⁷⁰

In order to consider steady state equilibria, the value changes have to be zero:

V˙_σ(t) =0. I write lim

t→^∞V_σ(t) = V_σ(ss)for steady state value functions. Since only steady state equilibria are considered, prices are time independent as well. Upon setting equations (3.25)–(3.28) to zero and rearranging them, the value functions in steady state are

V_lo(ss) = ^λ^u^V^ho(ss) + (2λµ_hn(ss) +ρ)V_ln(ss) +2λµ_hn(ss)P(ss) +ρB(ss) +D−^δ

r+λ_u+2λµ_hn(ss) +ρ ,

(3.29) V_hn(ss) = ^λ^d^V^ln(ss) + (2λµ_lo(ss) +ρ)V_ho−^2λµlo(ss)P(ss)−^ρ^A(ss)

r+λ_d+2λµ_lo(ss) +ρ , (3.30) V_ho(ss) = ^λ^d^V^lo(ss) +D

r+λ_d , (3.31)

V_ln(ss) = ^λ^u^V^hn(ss)

r+λ_u . (3.32)

69 Duffie, Gârleanu, and Pedersen (2005, pp. 1839) show optimality by verifying that under complete information any agent always trades at the stated equilibrium strategy, provided others do so. Trades are always executed at proposed equilibrium prices if gains from trade are possible with the agent in contact.

70 There is a typing error in Duffie, Gârleanu, and Pedersen (2005, p. 1823), equation (10). The Poisson arrival intensity for a buyer (hn) contacting sellers (lo) is 2λµ_lo(t).

Equation system (3.29)–(3.32) still depends on bargaining prices. The next section states the bargaining conditions for deriving these prices.

Im Dokument Liquidity Shocks in Over-the-Counter Markets (Seite 54-59)