A Class of Explicit optimal contracts in the face of shutdown

Let $T>0$ be some fixed time horizon. The key to modeling our Principal-Agent problems under an agency-free external risk of default is the simultaneous presence over the interval $[0,T]$ of a continuous random process and a jump process as well as the ability to extend the standard mathematical techniques used for dynamic contracting to this mixed setting. Thus, we shall deal with two kinds of information : the information from the output process, denoted as $\mathbb{F}=({\cal F}_{t})_{t\geq 0}$ and the information from the default time, i.e. the knowledge of the time where the default occurred in the past, if the default has appeared. This construction is not new and occurs frequently in mathematical finance³³3We refer the curious reader to the two important references [1] and [9]..

The complete probability space that we consider will be denoted as $(\Omega,\mathcal{G},\mathbb{P}^{0})$, with two independent stochastic processes :

• •

  $B$ a standard one-dimensional $\mathbb{F}$-Brownian motion,

• •

  $N$ the right-continuous single-jump process defined as $N_{t}=\textbf{1}_{\tau\leq t}$, $t$ in $[0,T]$ where $\tau$ is some positive random variable independent of $B$ that models the default time.

$N$ will also be referred to as the default indicator process. We therefore use the standard approach of progressive enlargement of filtration by considering $\mathbb{G}=\left\{\mathcal{G}_{t},t\geq 0\right\}$ the smallest complete right-continuous extension of $\mathbb{F}$ that makes $\tau$ a $\mathbb{G}$-stopping time. Because $\tau$ is independent of $B$, $B$ is a $\mathbb{G}$-Brownian motion under $\mathbb{P}^{0}$ according to Proposition 1.21 p 11 in [1]. We also suppose that there exists a bounded deterministic compensator of $N$, $\Lambda_{t}=\int_{0}^{t}\lambda(s)\,ds$ for some bounded function $\lambda(.)$ called the intensity implying that:

$$M_{t}=N_{t}-\int_{0}^{t}\lambda(s)(1-N_{s})ds,\quad t\in[0,T]$$

is a $\mathbb{G}$-compensated martingale. Note that through knowledge of the function $\lambda,$ the principal and agent can compute at time 0 the probability of default happening over the contracting period $[0,T]$. Indeed :

$$\mathbb{P}(\tau\leq T)=1-\exp(-\Lambda_{T}).$$

We first suppose for computational ease that the intensity $\lambda$ is a constant. We will see in Section 4.3 that our results may easily be lifted to more general deterministic compensators.

We suppose that the agent agrees to work for the principal over a time period $[0,T]$ and provide up to the default time a costly action $(a_{t})_{t\in[0,T]}$ belonging to $\mathcal{A}$, where $\mathcal{A}$ denotes the set of admissible $\mathbb{F}$-predictable strategies that will be specified later on. The Principal-Agent problem models the realistic setting where the principal cannot observe the agent’s effort. As such the agent chooses his action in order to maximize his own utility. The principal must offer a wage based on the information driven by the output process up to the default time that incentivizes the agent to work efficiently and contribute positively to the output process. Mathematically, the unobservability of the agent’s behaviour is modeled through a change of measure. Under $\mathbb{P}^{0}$ , we assume that the project’s profitability evolves as

$$X_{t}:=x_{0}+\int_{0}^{t}(1-N_{s})dB_{s}.$$

Thus, $\mathbb{P}^{0}$ corresponds to the probability distribution of the profitability when the agent makes no effort over $[0,T]$. When the agent makes an effort $a=(a_{t})_{t}$, we shall assume that the project’s profitability evolves as

$$X_{t}:=x_{0}+\int_{0}^{t}a_{s}(1-N_{s})ds+\int_{0}^{t}(1-N_{s})dB^{a}_{s},$$

where $B^{a}$ is a $\mathbb{F}$-Brownian motion under a measure $\mathbb{P}^{a}$. The agent fully observes the decomposition of the production process under a measure $\mathbb{P}^{a}$ whilst the principal only observes the realization of $X_{t}$. In order for the model to be consistent, the probabilities $\mathbb{P}^{0}$ and $\mathbb{P}^{a}$ must be equivalent for all $(a_{t})_{t\in[0,T\wedge\tau]}$ belonging to $\mathcal{A}$. Therefore, we introduce the following set of actions

$${\cal B}=\Big{\{}a=(a_{t})_{t}:\quad\mathbb{F}\hbox{-predictable and taking values in }[-A,A]\text{ for some }A>0\Big{\}}.$$

The action process in ${\cal B}$ are uniformly bounded by some fixed constant $A>0$ that will be assumed as large as necessary. For $a\in{\cal B}$, we define $\mathbb{P}^{a}$ as

$$\frac{d\mathbb{P}^{a}}{d\mathbb{P}^{0}}|\mathcal{G}_{T}=\exp\left(\int_{0}^{T}a_{s}(1-N_{s})dB_{s}-\frac{1}{2}\int_{0}^{T}|{a_{s}}|^{2}(1-N_{s})ds\right):=L_{T}.$$

Because $\mathbb{E}^{0}(L_{T})=1$ , $(B^{a}_{t})_{t\in[0,T]}$ with $B^{a}_{t}=B_{t}-\int_{0}^{t}a_{s}(1-N_{s})ds,t\in[0,T]$ is a $\mathbb{G}$-Brownian motion under $\mathbb{P}^{a}$ according to Proposition 3.6 c) p 55 in [1]. It is key to note that if halt occurs, i.e. if $\tau\leq T$, then the production process is halted before $T$ meaning that : $X^{a}_{t\wedge\tau}=X_{t}^{a},\quad t\in[0,T]$. Let us then observe that an action $a=(a_{t})_{t}$ of ${\cal B}$ can be extended to a $\mathbb{G}$-predictable process $(\tilde{a}_{t})_{t\in[0,T]}$ by setting $\tilde{a}_{t}=a_{t}1\!\!1_{t\leq\tau}$.

The cost of effort for the agent is modeled through a quadratic cost function : $\kappa(a):=\kappa\frac{a^{2}}{2},$ for $\kappa>0$ some fixed parameter. As a reward for the agent’s effort, the principal pays him a wage $W$ at time $T$. $W$ is assumed to be a $\mathcal{G}_{T\wedge\tau}$ random variable which means that the payment at time $T$ in case of an early default is known at time $\tau$. The principal and the agent are considered to be risk averse and risk aversion is modeled through two CARA utility functions :

$$U_{P}(x):=-\exp(-\gamma_{P}x)\;\text{and}\;U_{A}(x):=-\exp(-\gamma_{A}x),$$

where $\gamma_{P}>0$ and $\gamma_{A}>0$ are two fixed constants modeling the principal’s and the agent’s risk aversion.

In this setting and for any given wage $W$, the agent maximizes his own utility and solves :

$$V_{0}^{A}(W)=\sup_{a\in\mathcal{B}}\mathbb{E}^{a}\left[U_{A}\left(W-\int_{0}^{T}\kappa(a_{s}(1-N_{s}))ds\right)\right].$$

(2.1)

A wage $W$ is said to be incentive compatible if there exists an action policy $a^{*}(W)\in{\cal B}$ that maximises (2.1) and thus satisfies

$$V_{0}^{A}(W)=\mathbb{E}^{a^{*}(W)}\left[U_{A}\left(W-\int_{0}^{T}\kappa(a_{s}^{*}(W)(1-N_{s}))ds\right)\right].$$

When the principal is able to offer an incentive compatible wage $W$, she knows what the agent’s best reply will be. As such the principal establishes a set $\mathcal{A}^{*}(W)\subset\mathcal{B}$ of best replies for the agent for any incentive compatible $W$. Therefore, the first task is to characterize the set of incentive-compatible wages ${\cal W}_{IC}$. Only then may the principal consider maximizing his own utility by solving :

$$\sup_{W\in\mathcal{W}_{IC}}\;\sup_{a^{*}\in\mathcal{A}^{*}(W)}\;\mathbb{E}^{{a^{*}}(W)}\left[U_{P}\left(X^{a^{*}{(W)}}_{T}-W\right)\right]$$

(2.2)

under the participation constraint

$$\mathbb{E}^{{a^{*}}(W)}\left[U_{A}\left(W-\int_{0}^{T}\kappa(a_{s}^{*}(W)(1-N_{s}))ds\right)\right]\geq U_{A}(y_{PC}),$$

(2.3)

where $y_{PC}$ is a monetary reservation utility for the agent.

A first step to optimal contracting in this first-best setting involves answering the following question: can we characterize the set ${\cal A}_{PC}$? Following the standard route, we will first establish a necessary condition. For a given pair $(W,a)\in{\cal A}_{PC}$, let us introduce the agent’s continuation utility $(U^{(W,a)}_{t})_{t}$ as follows:

$$U^{(W,a)}_{t}:=\mathbb{E}_{t}\left[U_{A}\left(W-\int_{t}^{T}\kappa(a_{s}(1-N_{s}))ds\right)\right],$$

where we use the shorthand notation : $\mathbb{E}_{t}[.]:=\mathbb{E}[.|\mathcal{G}_{t}].$ We may write the Agent’s continuation value process as the product :

$$U^{(W,a)}_{t}=\mathcal{M}_{t}^{(W,a)}\mathcal{D}_{t}^{(W,a)},$$

where :

$$\mathcal{M}_{t}^{(W,a)}:=\mathbb{E}_{t}\left[U_{A}\left(W-\int_{0}^{T}\kappa(a_{s}(1-N_{s}))ds\right)\right]\quad\text{and}\quad\mathcal{D}_{t}^{(W,a)}:=\exp\left(-\gamma_{A}\int_{0}^{t}\kappa(a_{s}(1-N_{s}))ds\right).$$

Observe that for any admissible pair $(W,a)\in{\cal A}_{PC}$, the process $\mathcal{M}=(\mathcal{M}_{t}^{(W,a)})_{t}$ is a $\mathbb{G}$-square integrable martingale. According to the Martingale Representation Theorem for $\mathbb{G}$-martingales (see [1], Theorem 3.12 p. 60), there exists some predictable pair $(z_{s},l_{s})$ in $\mathbb{H}^{2}\times\mathbb{H}^{2}$, where $\mathbb{H}^{2}$ is the set of $\mathbb{F}$-predictable processes $Z$ with $\mathbb{E}\left[\int_{0}^{T}|Z_{t}|^{2}dt\right]<+\infty$, such that :

$$\mathcal{M}_{t}^{(W,a)}:=\mathcal{M}_{0}^{(W,a)}+\int_{0}^{t}z_{s}(1-N_{s})dB_{s}+\int_{0}^{t}l_{s}(1-N_{s})dM_{s}.$$

Integration by parts yields the dynamic of $U$, noting that $\mathcal{D}$ has finite variation :

$\displaystyle dU_{t}^{(W,a)}$

$\displaystyle=-\gamma_{A}\kappa(a_{t}(1-N_{s}))U_{t}^{(W,a)}dt+\mathcal{D}_{t}^{(W,a)}z_{t}(1-N_{s})dB_{t}+\mathcal{D}_{t}^{(W,a)}l_{t}(1-N_{s})dM_{t}.$

Setting $Z_{t}^{(W,a)}:=\mathcal{D}_{t}^{(W,a)}z_{t}\in\mathbb{H}^{2}$ and $K_{t}^{(W,a)}:=\mathcal{D}_{t}^{(W,a)}l_{t}\in\mathbb{H}^{2}$, we obtain:

$\displaystyle dU_{t}^{(W,a)}$

$\displaystyle=-\gamma_{A}\kappa(a_{t}(1-N_{s}))U_{t}^{(W,a)}dt+Z_{t}^{(W,a)}(1-N_{s})dB_{t}+K_{t}^{(W,a)}(1-N_{s})dM_{t}.$

By construction, we have that $U_{T}^{(W,a)}=U_{A}(W)$. It follows that $\left(U_{t}^{(W,a)},Z_{t}^{(W,a)},K_{t}^{(W,a)}\right)$ is a solution to the BSDE:

$\displaystyle-dU_{t}^{(W,a)}=-Z_{t}^{(W,a)}(1-N_{s})dB_{t}-K_{t}^{(W,a)}(1-N_{s})dM_{t}+\gamma_{A}\kappa(a_{t}(1-N_{s}))U_{t}^{(W,a)}dt,$

(3.4)

with $U_{T}^{(W,a)}=U_{A}(W).$ Therefore, (3.2) is satisfied if and only if $U_{0}^{(W,a)}\geq U_{A}(y_{PC})$.

To obtain a sufficient condition, we introduce, for $\pi=(y_{0},a,\beta,H)\in\mathbb{R}\times{\cal B}\times\mathbb{H}^{2}\times\mathbb{H}^{2}$, the wage process $(W_{t}^{\pi})_{t}$ defined as

	$\displaystyle W_{t}^{\pi}$	$\displaystyle:=y_{0}+\int_{0}^{t}\beta_{s}(1-N_{s})dB_{s}+\int_{0}^{t}H_{s}(1-N_{s})dM_{s}+\int_{0}^{t}\left\{\frac{\gamma_{A}}{2}\beta_{s}^{2}(1-N_{s})+\kappa(a_{s}(1-N_{s}))\right.$
		$\displaystyle\left.\frac{\lambda}{\gamma_{A}}[\exp(-\gamma_{A}H_{s})-1+\gamma_{A}H_{s}](1-N_{s})\right\}ds,$		(3.5)

and consider the set

$$\Gamma:=\left\{(y_{0},a,\beta,H)\in\mathbb{R}\times{\cal B}\times\mathbb{H}^{2}\times\mathbb{H}^{2}\text{ such that }y_{0}\geq y_{PC}\text{ and }\mathbb{E}\left[\exp(-2\gamma_{A}W_{T}^{\pi})\right]<+\infty.\right\}.$$

Using Lemma 3.2, the full Risk-Sharing problem under default writes as the Markovian control problem :

$$V_{P}^{FB}:=\underset{\pi=(y_{0},a,Z,K)\in\Gamma}{\text{sup}}\mathbb{E}\left[U_{P}\left(X_{T}^{(x_{0},a)}-W_{T}^{\pi}\right)\right],$$

(3.6)

where $X_{t}^{(x_{0},a)}$ is given by :

$$dX^{(x_{0},a)}_{s}=a_{s}(1-N_{s})ds+(1-N_{s})dB_{s},$$

with $X_{0}^{(x_{0},a)}=x_{0}$ and the wage process is given by :

	$\displaystyle dW_{s}^{\pi}=Z_{s}(1-N_{s})dB_{s}$	$\displaystyle+K_{s}(1-N_{s})dM_{s}$
		$\displaystyle+\left\{\frac{\gamma_{A}}{2}Z_{s}^{2}(1-N_{s})+\kappa(a_{s}(1-N_{s}))+\frac{\lambda}{\gamma_{A}}[\exp(-\gamma_{A}K_{s})-1+\gamma_{A}K_{s}](1-N_{s})\right\}ds,$

with $W_{0}^{\pi}=y_{0}$.
We have the following key theorem for the first-best problem.

The rest of this subsection is dedicated to the proof of this Theorem. We first make the following observation. As $X$ remains constant after $\tau$, the principal has no further decision to make after the default time. Thus, its value function is constant and equal to $U_{P}(x-y)$ on the interval $[\tau,T]$.
We now focus on the control part of the problem (i.e. computation of the optimal control triplet $\tilde{\pi}=(a,Z,K)$ for a given pair $(x_{0},y_{0})$). To do so, we follow the dynamic programming approach developed in [17], Section 4 to define the value function

$$V(0,x_{0},y_{0})=\sup_{\tilde{\pi}\in\tilde{\Gamma}}\mathbb{E}\left[U_{P}(X_{T}^{a}-W_{T}^{\tilde{\pi}})(1-N_{T})+\int_{0}^{T}U_{P}(X_{t}^{a}-W_{t}^{\tilde{\pi}})\lambda e^{-\lambda t}\,dt\right],$$

(3.8)

where

$$\tilde{\Gamma}=\left\{\tilde{\pi}\in\mathcal{B}\times\mathbb{H}^{2}\times\mathbb{H}^{2}\right\},$$

Because $\Gamma\subset\mathbb{R}\times\tilde{\Gamma}$, we have

$$V_{P}^{FB}\leq\sup_{y_{0}\geq y_{PC}}V(0,x_{0},y_{0}).$$

According to stochastic control theory, the Hamilton-Jacobi-Bellman equation associated to the stochastic control problem (3.8) is the following (see [16]):

	$\displaystyle\partial_{t}v(t,x,y)+\sup_{a,Z,K}\left\{\partial_{x}v(t,x,y)a+\partial_{y}v(t,x,y)\left[\frac{\gamma_{A}}{2}Z^{2}+\kappa(a)+\frac{\lambda}{\gamma_{A}}[\exp(-\gamma_{A}K)-1]\right]\right.$
	$\displaystyle\left.+\lambda\left[U_{P}(x-y-K)-v_{0}(t,x,y)\right]+\partial_{yy}v(t,x,y)\frac{Z^{2}}{2}+\frac{1}{2}\partial_{xx}v(t,x,y)+\partial_{xy}v(t,x,y)Z\right\}=0,$		(3.9)

with the boundary condition :

$$v(T,x,y)=U_{P}(x-y).$$

The following is dedicated to our main result for the Moral Hazard problem. We shall state our main theorem with the explicit optimal contract before turning to some analysis of the effect of the shutdown on dynamic contracting. In the case of moral hazard, one is forced to make a stronger assumption about the nature of a contract. This stronger hypothesis will naturally appear to justify the martingale optimality principle. In our setting, a contract is a $\mathbb{G}_{T\wedge\tau}$ measurable random variable $W$ such that for every $\beta\in\mathbb{R}$, we have

$$\mathbb{E}\left[\exp(\beta W)\right]<+\infty.$$