New Formulations of Ambiguous Volatility with an Application to Optimal Dynamic Contracting
NNew Formulations of Ambiguous Volatility with anApplication to Optimal Dynamic Contracting ∗ Peter G. Hansen † February 1, 2021
Abstract
I introduce novel preference formulations which capture aversion to ambiguityabout unknown and potentially time-varying volatility. I compare these preferenceswith Gilboa and Schmeidler’s maxmin expected utility as well as variational formu-lations of ambiguity aversion. The impact of ambiguity aversion is illustrated ina simple static model of portfolio choice, as well as a dynamic model of optimalcontracting under repeated moral hazard. Implications for investor beliefs, optimaldesign of corporate securities, and asset pricing are explored.
JEL Classification:
D81, D86, G11, G12, G32
Keywords: ambiguity, stochastic volatility, moral hazard, capital structure, assetpricing ∗ I acknowledge helpful comments and suggestions from Anne Balter, Hui Chen, Sharada Dharmasankar,Leonid Kogan, Andrey Malenko, Jianjun Miao, Jian Sun, Xiangyu Zhang, three anonymous referees, andparticipants at the MIT Finance lunch seminar and the Becker Friedman Institute mini-conference onAmbiguity and Robustness. I am especially grateful to Lars Hansen, Andrey Malenko, and Tom Sargent(editor) for their support and feedback which greatly improved the paper. † Sloan School of Management, Massachusetts Institute of Technology, 100 Main St, Cambridge, MA02142. Email: [email protected] a r X i v : . [ ec on . T H ] J a n Introduction
There is ample evidence that time-varying stochastic volatility exists and has important ef-fects on real macroeconomic variables and is important in understanding empirical featuresof financial markets. The empirical evidence suggests that volatility follows complicatednonlinear dynamics, which often leads model builders to write down complicated parametricmodels of the evolution of volatility as well as its correlation with other economic quantitiesof interest. An obvious concern with this approach is whether it is possible for economicagents to learn or estimate these models precisely, or through repeated observation developconfidence that a particular parametric model is correct. While one may argue that thisconcern is unwarranted in financial markets with high-frequency observations which makevolatility effectively observable, there is no convincing reason to dismiss these concerns asthey pertain to real variables which are often observed at low frequencies.Motivated by these concerns, I propose new preferences that capture nonparametricmodel uncertainty about an unknown, possibly time-varying volatility process. These pref-erence formulations build on existing models of ambiguity aversion, notably being a specialcase of the variational preferences proposed by Maccheroni et al. (2006a). These prefer-ences, which I call moment-constrained variational preferences , are first formulated in astatic setting of decision-making under uncertainty. I illustrate the impact of these staticpreferences in a simple portfolio choice problem, where I show how the degree of ambi-guity aversion affects both the implied worst-case beliefs of the investor and the optimalportfolio weight. Then, I explore dynamic counterparts to these preferences and derive acontinuous-time limit in which the decision-maker is uncertain about an unknown, time-varying volatility process. The impact of these preferences is then illustrated in a model ofrepeated moral hazard based on papers by DeMarzo and Sannikov (2006) and Biais et al.(2007). Using this model, I explore the implications of ambiguity aversion for optimalsecurity design and asset prices, and compare and contrast the implications of dynamicvariational preferences with those of G -expectations.Ambiguity aversion leads the principal to design a contract that is robustly optimal givenuncertainty about the volatility process. Under the optimal contract, belief heterogeneityemerges between the principal and the agent. The agent trusts the benchmark volatilitymodel, whereas the principal forms expectations as if volatility is strictly higher and state-dependent. As in DeMarzo and Sannikov (2006), the optimal contract can be interpretedas featuring a line of credit between the principal and the agent. I show how ambiguityaversion increases the optimal credit limit, while reducing the reliance on long-term debt.This is important since credit lines are a commonly used corporate security. Additionally,I derive asset pricing implications of volatility ambiguity under the optimal contract. Even in high-frequency settings, direct statistical measurement of volatility may be significantly con-founded by microstructure and liquidity effects. See for example Zhang et al. (2005). See Maccheroni et al. (2006b) for axiomatic approaches to dynamic variational preferences. .1 Related literature This paper builds on a large literature on ambiguity aversion and model uncertainty inthe context of economic decisions. The preference formulations I introduce fit within theframework of variational preferences introduced by Maccheroni et al. (2006a) which neststhe maxmin expected utility of Gilboa and Schmeidler (1989) and the “multiplier” for-mulation of ambiguity due to Hansen and Sargent (2001). Dynamic models of ambiguityand robustness can be broadly thought of as belonging to one of three categories, namelythe “recursive multiple priors” model proposed by Epstein and Schneider (2003), the “re-cursive smooth ambiguity” model proposed by Klibanoff et al. (2009), and the “dynamicvariational preferences” model described in Maccheroni et al. (2006b) and Hansen and Sar-gent (2018) as a generalization of the “multiplier preferences” introduced by Hansen andSargent (2001). My paper adds to this literature by proposing a new form of preferencesthat captures ambiguity or uncertainty about volatility in continuous time. To my knowl-edge, the only other model of volatility ambiguity is the “G-expectations” model of Peng(2007), which can be interpreted as a recursive multiple priors model. The continuous-timepreferences developed in this paper can be thought of as a particular continuous-time limitof the discrete-time preferences of Maccheroni et al. (2006b) which nest the G-expectationsmodel.Models of financial contracting typically assume that all economic actors fully under-stand the model environment, and that such understanding is common knowledge. This issimilar to (but stronger than) an assumption of rational expectations, and has been criti-cized as overly restrictive in models with strategic interaction by Harsanyi (1967), Wilson(1987), Bergemann and Morris (2005), Woodford (2010), Hansen and Sargent (2012), andothers. This paper attempts to relax the assumption that economic actors fully understandtheir model environment and study the corresponding effect on financial contracting. Inparticular, I study a long-term contracting problem where economic actors have ambiguousbeliefs about the possibly time-varying volatility of future cash flows.My paper builds on the large literature studying models of long-term financial contract-ing. DeMarzo and Fishman (2007), DeMarzo and Sannikov (2006), and Biais et al. (2007)show that in stationary environments with risk-neutral economic agents, the optimal long-term financial contract can be implemented by an interpretable capital structure. I buildon these papers by introducing uncertainty about the volatility of the cash flow processand study how this affects the optimal contract. As with many of these papers, I rely onthe martingale approach to dynamic contracting problems developed by Sannikov (2008)and Williams (2008).Particularly relevant are papers that take robust approaches to incentive problems, suchas Bergemann and Morris (2005), Carroll (2015), Zhu (2016), and Malenko and Tsoy (2018).The closest paper to this one is Miao and Rivera (2016) who characterize the optimalcontract in continuous time when the principal faces ambiguity about expected cash flows.As I will demonstrate, my model produces substantially different optimal security designyet has qualitatively similar asset pricing implications. Szydlowski (2012) and Prat andJovanovic (2014) study related problems where the principal is uncertain about the details3f the agency problem. Adrian and Westerfield (2009) characterize optimal contractingwhen the principal and the agent disagree about the underlying dynamics of the cash flowprocess and both learn through time. By focusing on uncertainty about second moments,my paper is similar in spirit to Wolitzky (2016) who studies a static mechanism designproblem.
The outline of this paper is as follows. Section 2 defines moment-constrained variationalpreferences in the context of static decision problems and then illustrates the impact ofthese preferences in a static model of portfolio choice under quadratic utility. Section 3extends these preferences to dynamic problems in a discrete-time setting and explores aninteresting continuous-time limit under which the more general ambiguity about the prob-ability distribution of state evolution reduces to ambiguity about an unknown stochasticvolatility process. I compare the continuous-time limit to the G -expectations model of Peng(2007). Section 4 applies the continuous-time preferences to a model of optimal contractingunder repeated moral hazard and illustrates the implications of ambiguous volatility forsecurity design and asset pricing. Section 5 concludes. Let P denote the decision-maker’s benchmark probability measure, and let E P r¨s denotethe expectation operator under P . Let a P A denote the action of the decision-maker andlet (cid:15) denote a vector of payoff-relevant shocks. I would like to capture the notion that thedecision-maker is uncertain about the entire distribution P , but is certain about certainmoments or functionals of the distribution P . More precisely, let g p¨q be any functionsuch that E P r g p (cid:15) qs “
0. The decision-maker allows for absolutely continuous probabilisticdistortions of P , which I parameterize by likelihood ratio random variables M which satisfy M ě P -probability 1 and E P r M s “
1. The decision-maker’s certainty about therandom variable g p (cid:15) q being mean zero is captured by restricting the set of likelihood ratiosto those that satisfy E P r M g p (cid:15) qs “ V p a ; P, θ q as V p a ; P, θ q “ inf M ě , E P r M s“ E P r M U p a, (cid:15) qs ` θ Φ p M q (1)where Φ p M q “ E P r c p M qs if E P r M g p (cid:15) qs “ otherwisewhere c p¨q is a convex function with c p q “ If the infimum in equation (1) is attained,I refer to the M that attains it as the worst-case belief distortion. Note that the penalty Note that the penalization function Φ p¨q is equivalent to an f -divergence on the set of M ’s which satisfythe moment restriction. E P r M g p (cid:15) qs “ M will never violate the moment restriction. Since the penaltyfunction Φ p¨q is convex in M , these preferences fit into the variational preference frameworkaxiomatized by Maccheroni et al. (2006a). I therefore refer to preferences defined by (1) asmoment-constrained variational preferences.The decision-maker’s problem can then expressed concisely asmax a P A V p a ; P, θ q . A particularly tractable choice of divergence function c p¨q is given by c p M q “ M log M . This divergence gives particularly tractable expressions for the worst-case likelihood ratio M ˚ . In particular, by standard convex duality arguments, it is possible to show that M ˚ has the following exponential tilting form M ˚ p a, θ q “ exp p´ θ ´ U p a, (cid:15) q ´ λ p a, θ q g p (cid:15) qq E P r exp p´ θ ´ U p a, (cid:15) q ´ λ p a, θ q g p (cid:15) qqs (2)where the Lagrange multiplier λ p a, θ q is chosen to satisfy the moment restriction E P r M ˚ p a, θ q g p (cid:15) qs “ . It is worth comparing the expression for the likelihood ratio in equation (2) to the corre-sponding expression with the constraint E P r M g p (cid:15) qs “ M ˚ p a, θ q “ exp p´ θ ´ U p a, (cid:15) qq E P r exp p´ θ ´ U p a, (cid:15) qqs . (3)In equation (3), we see that the worst-case likelihood ratio distorts probabilities towardsstates in which the decision-maker’s utility is low. Such a distortion will generally violatethe moment condition. By contrast, the likelihood ratio in equation (2) must distort prob-abilities in such a way that the moment restriction remains satisfied. In particular, forconcave utility functions, this intuitively corresponds to a likelihood ratio that increasesthe dispersion of (cid:15) by overweighting states in which (cid:15) takes values that are far from itsmean. This will be seen in the example presented in the next section. This corresponds to an f -divergence known as relative entropy or Kullback-Leibler divergence, and hasnumerous statistical interpretations. Certain members of the larger family of power divergences proposedby Cressie and Read (1984) including Hellinger and logarithmic divergences, as well as the related notionof Chernoff entropy, can lead to degenerate solutions to (1). See Chen et al. (2021) for further discussion. .2 Example: Portfolio choice under quadratic utility I illustrate the impact of these preferences in a static portfolio choice problem. Consideran investor with quadratic utility U ´Ă W ¯ “ ´ ´Ă W ´ b ¯ over period-1 wealth Ă W where b is the investor’s bliss-point wealth level. Investor has initialwealth W which can be invested at a gross risk-free rate R f or a vector of risky assets withexcess return vector r R .While quadratic utility has the well-known and arguably undesirable feature that theinvestor’s utility can be decreasing in wealth, I include this example for two reasons. First,it leads to quasi-analytic solutions which facilitate intuition. Second, in continuous-timediffusion limits, local quadratic approximations become exact. Therefore, much of theintuition obtained from the static linear-quadratic Gaussian model examined in this sectionwill carry over to more general continuous-time diffusion models studied later. Under the investor’s subjective probability measure P , the vector of excess returns R isdistributed as R „ Normal p µ, Σ q . Formally, the investor’s portfolio optimization problemcan be written as max φ P R k E P ” U ´Ă W ¯ı s.t. Ă W “ W R f ` φ r R. One can verify that the optimal portfolio weight vector φ ˚ is given by φ ˚ “ r µµ ` Σ s ´ µ p b ´ W R f q . Consider the same portfolio choice problem as before, but now the investor treats theirsubjective probability measure P under which r R „ Normal p µ, Σ q as an approximation.They are willing to entertain other probability measures as possible, but treat the expectedreturn vector E r r R s “ µ as certain. Then we can model the investor’s preferences as V p φ ; P, θ q “ inf M ě , E P r M s“ E P ” M U ´Ă W ¯ı ` θ Φ p M q where Φ p M q “ E P r M log M s if E P ” M ´ r R ´ µ ¯ı “ otherwise.6he investor’s portfolio choice problem is thenmax φ P R k V p φ ; P, θ q . As in the expected utility case, the investor’s problem can be solved in closed form. First,for fixed φ P R k we can solve the infimization problem for V p φ ; P, θ q . Using standardduality results, it can be shown that the worst-case M is unique and given by M ´ r R ; φ ¯ “ exp ˆ ´ θ ´ r R ´ µ ¯ r´ φφ s ´ r R ´ µ ¯˙ E P „ exp ˆ ´ θ ´ r R ´ µ ¯ r´ φφ s ´ r R ´ µ ¯˙ for choices of φ for which the objective function is finite. This shows that for any φ the resulting penalized worst-case probability distribution is also a multivariate normaldistribution. As expected, we have E P ” M ´ r R ; φ ¯ı “ E P ”´ r R ; φ ¯ r R ı “ µ , so there isno mean distortion. Additionally, we can see that the worst-case M distorts the variance-covariance matrix of the returns from Σ to r Σ ´ ´ θ ´ φφ s ´ .Due to the regularity of the problem, it can be shown that the orders of minimizationand maximization can be exchanged at the optimal M ˚ and θ ˚ . Exchanging the order ofmaximization and minimization, we can solve for the optimal φ as a function of M . Weobtain easily that φ p M q “ E ” M r R r R ı ´ µ p b ´ W R f q . Next observe that at the optimal M ˚ must satisfy M ˚ “ M ´ r R ; φ p M ˚ q ¯ . Write S “ E ” M ˚ r R r R ı . By our previous observation, we must have that S is a positive-definite solution of the equation S ´ µµ “ “ Σ ´ ´ θ ´ p b ´ W R f q µ S ´ µ p b ´ W R f q ‰ ´ . where this equation can be obtained by simply writing the penalized worst-case distortedvariance of r R in two ways. Unfortunately, this equation is a third-order matrix polynomialin S so we cannot easily express S in closed form. Nonetheless, we know that under thepenalized worst-case distribution of r R we have r R „ Normal p µ, S ´ µµ q . and that φ ˚ p θ q “ S ´ µ p b ´ W R f q . This need not be the case. For some choices for φ the adversarial nature’s choice of M can make theinvestor’s utility arbitrarily negative. .2.3 Scalar risky asset To facilitate intuition further, I focus on the case where the risky return ˜ R is a scalarrandom variable, so that the portfolio weight φ is also a scalar. Assume that ˜ R has mean µ and benchmark variance σ . Under expected utility, the optimal portfolio weight is φ ˚ “ µ ` σ µ p b ´ W R f q . To characterize the solution under ambiguity aversion, it is helpful to define the scalar s p θ q “ E r M ˚ r R s . Note that the optimal portfolio weight φ ˚ p θ q can be written in terms of s p θ q as φ ˚ p θ q “ s µ p b ´ W R f q . By the arguments in the previous section, we can write s p θ q as the solution of theminimization problem of the adversarial nature, which simplifies tomin s ´ p b ´ W R f q ` µ p b ´ W R f q s ` θ „ s ´ µ σ ´ ´ log ˆ s ´ µ σ ˙ . (4)Note that the objective function in (4) is strictly convex in s , so s p θ q is uniquely defined.Additionally, it is easy to see that for θ ą
0, we must have s p θ q ą µ ` σ . It followsimmediately that φ ˚ p θ q P p , φ ˚ q .While it is possible to obtain closed-form expressions for s p θ q and correspondingly φ ˚ p θ q as roots of the cubic polynomial under appropriate inequality restrictions, these expressionsare complicated and convey little intuition. Instead I characterize the comparative staticsof these quantities in propositions 2.1 and 2.2 below. The solutions s p θ q and φ ˚ p θ q arecharacterized more explicitly in the appendix in terms of the unique positive root of aparticular cubic polynomial. Proposition 2.1.
Assume that b ą W R f and µ ą . Then the optimal portfolio weight φ ˚ p θ q has the following properties:(i) φ ˚ p θ q ą .(ii) φ ˚ p θ q is strictly increasing in θ .(iii) As θ Ñ 8 we have φ ˚ p θ q Ñ φ ˚ .(iv) As θ Ñ we have φ ˚ p θ q Ñ . The results of proposition 2.1 are illustrated in figure 1. Result p i q shows that the in-vestor will always invest a strictly positive amount of their wealth in the risky asset. Thismakes intuitive sense because µ ą
0. Result p ii q shows that the amount the investor’sportfolio weight on the risky asset is increasing in their model confidence θ or equivalentlydecreasing in their ambiguity aversion 1 { θ . Result p iii q shows that as the investor’s model8igure 1: φ ˚ p θ q as a function of θ plotted in red. Horizontal line at φ ˚ shown in dashedblue. Parameter values are b “ W “ µ “ . σ “ .
5. Note that φ ˚ p q “ p iv q shows that as the investor becomesinfinitely ambiguity averse, they will invest none of their wealth in the risky asset. This isbecause they perceive the risky asset as becoming infinitely risky.Write ν p θ q as the ratio of the worst-case volatility to the benchmark volatility, formally ν p θ q ” s p θ q ´ µ σ “ E r M ˚ p θ q r R s ´ µ E r r R s ´ µ Then we have the following continuity result:
Proposition 2.2.
Under the conditions of proposition 2.1 we have the following limits:(i) As θ Ñ 8 we have ν p θ q Ñ .(ii) As θ Ñ we have ν p θ q Ñ 8 . The results of proposition 2.2 can be seen in figure 2.2.3. This result shows that the in-vestor’s implied beliefs converge to the benchmark model as their model confidence becomesinfinite. Together with result p iii q of the previous proposition, this formally establishes thesubjective expected utility model without ambiguity aversion as a limit of the problem withambiguity aversion. It is natural to compare the solution to the portfolio choice problem in the previous sec-tion to the corresponding “unconstrained” problem where we ignore the mean restriction9igure 2: Worst-case volatility ratio ν p θ q as a function of θ . E P r M p r R ´ µ qs “
0. This corresponds to the portfolio choice of an ambiguity-averse investorwith quadratic utility and “multiplier” preferences.Section A.2 of the appendix describes the solution to the portfolio choice problem inthe previous section ignoring the mean restriction. The unrestricted problem has similarfeatures to the restricted problem. In particular, under the implied worst-case M , thereturn on wealth is still normally distributed. However, both the mean and variance aredistorted and will depend on the penalty parameter θ . For regions of the penalty parameter θ where a solution exists, the distorted mean µ u p θ q is increasing in θ , while the distortedvariance σ u p θ q is decreasing in θ . The optimal portfolio weight on the risky asset is φ u p θ q “ µ u p θ q ` σ u p θ q µ u p θ qp b ´ W R f q . Results analogous to proposition 2.1 hold for φ u p θ q , albeit for slightly different reasons.Under moment-constrained, increasing θ decreases the implied variance of the return underthe worst-case distribution. In the unconstrained problem, both the mean and variancechange with θ . Next, I present a dynamic extension to the static preferences defined by equation (1).Consider a J ` t is discrete and satisfies t P t , ..., J ∆ u . I assume that the decision-maker’s utilityin period t ` t and t ` t X t u t Pt ,...,J ∆ u . For simplicity, I abstract from modelling how the decision-maker’s choices10ffect X and simply define utility taking X to be an exogenous, time homogeneous Markovprocess with Gaussian increments under the benchmark probability measure P , i.e. X t ` ∆ ´ X t “ µ p X t q ∆ ` σ p X t q (cid:15) t ` ∆ (5)where (cid:15) j ∆ iid „ Normal p , ∆ q under P for all j P t , ..., J u . As in the previous section,we consider absolutely continuous changes of measure parameterized by positive randomvariables M with unit expectation. For each M , define M t “ E t r M s so that M t is a positive martingale relative to the filtration generated by the process t X t u Tt “ .Additionally, define M t,t ` ∆ “ M t ` ∆ M t . Note that M t,t ` ∆ is a positive random variable with time- t conditional expectation 1. Ob-serve that M t,t ` ∆ can be interpreted as a conditional change-of-measure or conditionallikelihood ratio.While the decision maker does not have full confidence in P , she is certain about specificconditional moments of the data-generating process. Analogous to the moment restrictionin the static model, the decision maker will only consider models generated by martingaledistortions M which satisfy the moment restriction E t r M t,t ` ∆ p X t ` ∆ ´ X t ´ µ p X t q ∆ qs “ , @ t. (6)Equation (6) captures the decision-maker’s certainty about the expected one-period changein the process X t . Thus, the decision-maker’s uncertainty is limited to uncertainty abouthigher moments of the distribution, such as second moments or variances. Note thatequation (6) is satisfied by the constant random variable M “ V p X ; P, θ q “ inf M ě , E r M s“ E « M p j ` q ∆ J ÿ j “ e ´ ρj ∆ u p X j ∆ , X p j ` q ∆ q ∆ ` c p M q ff (7)where c p M q “ θ p ∆ q M p j ` q ∆ ř Jj “ e ´ ρj ∆ log p M j ∆ , p j ` q ∆ q if (6) holds for all j P t , ..., J u8 otherwiseIt can easily be shown that the ambiguity index c p¨q is convex in M . Therefore, this pref-erence specification fits into the very general framework of dynamic variational preferencesproposed and axiomatized by Maccheroni et al. (2006b). The particular form of the ambi-guity index considered here is the same as the discounted relative entropy penalty used in11he multiplier preferences of Hansen and Sargent (2001) and followup papers but with anadded sequence of conditional moment restrictions on M .Chen et al. (2020) study a mathematically similar problem that arises when an econome-trician is interested in bounded implied subjective expectations of economic agents subjectto a vector of conditional moment conditions, meant to convey partial information aboutasset prices or survey data on subjective expectations. In particular, their problem can bethought of as an Ergodic control problem that arises in the limit as ρ Ñ 8 . Next, I give a heuristic derivation of a continuous-time limit of (7). The corresponding limitwill be used as preferences in subsequent sections where I consider models of repeated moralhazard. In these settings, the use of continuous-time diffusion models greatly improvestractability of the optimal contracting problems.We will take the limit of the discrete-time problem in the previous section as ∆ Ñ X t will converge in law to a continuous-time diffusionprocess with evolution equation dX t “ µ p X t q dt ` σ p X t q dZ t where Z t is a standard Brownian motion.Let V t p X t q denote the time- t continuation value function of the decision-maker in thediscrete-time problem. Note that we have the following Bellman-type equation for V t , V t p X t q “ inf m ě , E t r m s“ E t “ mu p X t ` ∆ , X t q ` θ p ∆ q m log m ` e ´ ρ ∆ V t ` ∆ p X t ` ∆ q ‰ subject to E t r m p X t ` ∆ ´ X t ´ µ p X t q ∆ qs “ m is the conditional one-period likelihood ratio chosen by theadversarial nature. Under mild regularity conditions, the functions u p¨ , ¨q and V t p¨q will betwice continuously differentiable. Therefore, we can approximate them as local quadraticfunctions in the increment X t ` ∆ ´ X t . Under this approximation, we then have that theworst-case one-period likelihood ratio m will have the form m ˚ “ m p (cid:15) t ` ∆ , X t q “ exp ` ´ p (cid:15) t ` ∆ q ω p X t q ˘ E t “ exp ` ´ p (cid:15) t ` ∆ q ω p X t q ˘‰ where ω p¨q is a function of X t that depends implicitly on ∆ , θ, ρ , the curvature of u p¨ , X t q ,and V t ` ∆ p¨q . As before, this is easily obtained from convex duality results. We see that theimplied m ˚ will change the variance of the Gaussian shock (cid:15) t,t ` ∆ from 1 to some unknownfunction ν p X t q . We can then equivalently think of the choice of likelihood ratio m as beinga choice of change-of-volatility ν .As a function of ν , the one-period relative entropy of a volatility distortion is given by E r m p ν q log m p ν qs “ (cid:32) ν ´ ´ log ν ( Ñ
0, and the number of periods J Ñ 8 so that J ∆ Ñ T and letting θ p ∆ q “ θ ∆, we obtain that time-0 lifetime utility will be given by V p X ; P, θ q “ inf t ν t u Tt “ E „ż T e ´ ρt ˆ u p X t q ` θ (cid:32) ν t ´ ´ log ν t (˙ dt (8)where the infimum is subject to the constraint dX t “ µ p X t q dt ` σ p X t q ν t dZ t . I make two important observations. First, the infimum problem in equation (8) is a differentproblem from the infimization in the discrete-time problem in equation (7). In discrete-time, preferences were defined as an infimum over probability distributions, whereas now weare representing preferences as an infimum over a controlled process. Thus the continuous-time limit here is best thought of as an equivalent problem that produces the same valuefunction. Second, the linear scaling θ p ∆ q “ θ ∆ is important for the limit. The standardcontinuous-time limit for multiplier preferences would let θ p ∆ q be constant as ∆ Ñ
0. Thisimposes absolute continuity of measures in the continuous-time limit, which by Girsanov’stheorem restricts all probabilistic distortions to conditional drift distortions. By contrast,the limit I consider allows violations of absolute continuity in the continuous-time setup. Ifit weren’t for the conditional moment restrictions, this would allow the adversarial natureto choose arbitrarily large drift distortions at zero cost. I refer to any process t ν t u Tt “ thatattains the infimum in (8) as a worst-case change-of-volatility process.Additionally, observe that for the function V t p X t ; P, θ q we will have the following PDE,0 “ min ν u p X q ` θ (cid:32) ν ´ ´ log ν ( ` B V B t ´ ρV ` B V B X µ p X q ` B V B X σ p X q ν . For simplicity, I will consider the infinite-horizon limit as T Ñ 8 so that the problem willbecome stationary and therefore B V B t “ θ The penalty parameter θ captures the degree of confidence that the economic actor hasin their benchmark model of volatility, with θ “ 8 corresponding to complete modelconfidence, i.e. expected utility.It is well-known that continuous-time diffusion models imply that volatility is directlyobservable via the quadratic variation. One might conclude that this implies that the onlyreasonable value for θ is infinity. I give several reasons why this is not the case in manydomains of interest:(i) Across empirical domains, despite numerous proposed models of stochastic volatil-ity, there is generally no accepted consensus on the “correct” parametric model forstochastic volatility. It seems sensible therefore that economic actors would take anyparametric model of volatility as an approximation. Similar considerations arise for continuous-time multiplier preferences. θ by either adopting a fixed ∆ ą θ as “reasonable” if the implied worst-case model issubjectively reasonable. G -expectations An alternate approach to modelling ambiguous volatility in continuous-time was intro-duced by Peng (2007). This approach, known as G -expectations, can be thought of asa continuous-time counterpart to the max-min expected utility of Gilboa and Schmeidler(1989) applied to an unknown volatility parameter. The implications of this approach areexplored by Epstein and Ji (2013) in an asset pricing setting. As will be demonstrated,the G -expectations approach is closely related to the approach described in the previoussection.Consider preferences defined by the following generalization of the continuous-time pref-erences defined by (8) U p X q “ inf t ν t u t “ E „ż e ´ ρt p u p X t q ` ξ p ν t qq dt where ξ p¨q is a convex function of ν which I refer to as the penalty function. Note thatthese preferences imply the following PDE for U ,0 “ min ν u p X q ` ξ p ν q ´ ρU ` B U B X µ p X q ` B U B X σ p X q ν .
14f course, the preferences in (8) can be seen to be a special case by taking ξ p ν q “ θ t ν ´ ´ log p ν qu . G -expectations can also be thought of as a special case, but now bytaking ξ p ν q to be a convex indicator function. In the case where the benchmark volatility σ p X q “ σ is constant, this corresponds to ξ p ν q “ ν P r σ { σ, σ { σ s8 otherwise.Note that here the convex indicator function restricts the instantaneous volatility σν t underthe worst-case model to be in the interval r σ, σ s . The differences between these two ap-proaches are analogous to the differences between max-min expected utility and variationalor multiplier formulations of ambiguity aversion.Obviously these two approaches are mathematically similar. Nonetheless there willbe differences in implications of the two models, some of which will be explored in thesubsequent application. In particular, while the two models will have qualitatively similarimplications for the optimal contract, they will have markedly different implications forthe worst-case volatility model and asset prices. For the optimal contracting problem Iconsider in the following section, the G -expectations model imply a constant worst-casevolatility of σν t “ σ whereas the relative entropy model will imply a worst-case volatilityprocess that is state-dependent and higher in states that are worse for the decision-maker. Remark 3.1.
As Peng (2007) and Nutz (2013) demonstrate, ambiguity about volatility indiffusion environments can be represented via a mathematically convenient control theoryimplementation. The implied value function and HJB equation are the same as one inwhich the unknown volatility is treated as a controlled process. This paper does not formallyextend their equivalence results, and sidesteps this by simply studying the equivalent controlproblem. Rigorous development of nonlinear expectation theory extending the equivalenceresults of Peng (2007) and Nutz (2013) to cover the applications considered here is wellbeyond the scope of this paper.
To illustrate the effect of the continuous-time preferences derived in the previous section,I apply them to a model of optimal contracting under repeated moral hazard based onpapers by DeMarzo and Sannikov (2006) and Biais et al. (2007).
I first describe a benchmark model without ambiguity based on DeMarzo and Sannikov(2006) and Biais et al. (2007). At each instant t , agent chooses an effort level a t P r , s .15iven an effort choice, the cumulative cash-flow process t Y t u obeys the law of motion dY t “ µa t dt ` σdZ t (9)where µ, σ ą
0, and Z t a standard Brownian motion.The agent can derive private benefits λµ p ´ a t q from the action a t where λ P p , q .Due to linearity, it is without loss of generality to take a t P t , u . At any time t ě L . The principal and the agentare both assumed to be risk neutral. The principal discounts cash flows at a rate r ą γ ą r . In selecting an optimal contract, the principal chooses a cumulative compensation pro-cess C for the agent, a liquidation stopping time τ , and a suggested effort process a for theagent. The benchmark model optimal contracting problem is given as follows. Problem 4.1 (benchmark model) . max p C,τ,a q E P a „ż τ e ´ rs p dY s ´ dC s q ` e ´ rτ L (10) subject to E P a „ż τ e ´ γs p dC s ` λµ p ´ a s q ds q ě E P p a „ż τ e ´ γs p dC s ` λµ p ´ p a s q ds q (11) E P a „ż τ e ´ γs p dC s ` λµ p ´ a s q ds q “ W . (12) Assume for simplicity that only the principal is ambiguity-averse. This will turn out to bewithout loss of generality. The principal takes the evolution equation (9) as an approximatebenchmark model, but allows for alternate models where the cumulative cash flow processevolves as dY t “ µa t dt ` σν t dZ t . (13)To avoid a degenerate effect of ambiguous volatility, it is necessary to assume that realizedvolatility is not directly contractible. Otherwise it would be possible for the agent to fullyinsure the principal’s uncertainty about volatility without any subjective welfare loss. Iformalize this with the following restriction. Restriction 4.2.
Under any feasible contract p C, τ, a q , the process M t “ E P,at „ż τ e ´ γs p dC s ` λµ p ´ a s q ds (14) This assumption means that the agent is impatient relative to the principal, and avoids degeneracy. dmits the martingale representation M t “ M ` ż t φ s p dY s ´ µa s ds q . (15) under P . If there were no uncertainty about volatility, then equation (15) in restriction 4.2 wouldsimply follow from the martingale representation theorem. However, since the principal andagent have potentially different beliefs about volatility, the principal and agent could derivesubjective welfare improvements from writing contracts in which the agent’s compensationis contingent on the realized volatility. This is disallowed by restriction 4.2.The optimal contracting problem is given by:
Problem 4.3. sup p C,τ,a q inf ν E ν „ż τ e ´ rt p dY t ´ dC t q ` e ´ rτ L ` E ν „ż τ e ´ rt ξ p ν t q dt (16) subject to equations (11) , (12) , (13) , and restriction 4.2. Problem 4.3 can be thought of as a two-player, zero-sum stochastic differential game between the principal and an adversarial nature. Nature chooses the time-varying changeof volatility process ν t to minimize the welfare of the agent, but choosing ν t different fromone has cost proportional to the instantaneous relative entropy.Let W t denote the time- t continuation payoff of the agent. It follows from restriction4.2 that dW t “ γW t dt ´ dC t ´ λµ t p ´ a t q dt ` φ t p dY t ´ λµ p ´ a t q dt q (17)Observe that in view of equation (13), the principal and the agent perceive the evolutionof W t differently. The principal perceives it as dW t “ γW t dt ´ dC t ´ λµ t p ´ a t q dt ` φ t σν t dZ t (18)whereas the agent perceives it as dW t “ γW t dt ´ dC t ´ λµ t p ´ a t q dt ` φ t σdZ t . (19) The first-best contract is the same as the first-best contract with no ambiguity aversion inDeMarzo and Sannikov (2006). This is intuitively obvious since the first-best value functionis linear, hence there are no volatility effects. rF p W q “ sup c ě ,φ inf ν µ ´ c ` ψ p ν q ` p γW ´ c q F p W q ` φ ν σ F p W q . (20) See Fleming and Souganidis (1989) for further discussion.
17t is easy to verify that at the optimum, we have φ “ F p W q “ µr ´ γr W, which can be implemented by the principal paying the agent a constant wage of c “ γW .Of course, this can be improved if we allow time-0 lump sum transfers in which case theprincipal can simply give a one-time transfer of W to the agent which gives F p W q ´ W “ µr . Thus with no moral hazard, volatility ambiguity produces no reduction in welfare.
It is a simple extension of lemma 3 of DeMarzo and Sannikov (2006) to show that for anychange-of-volatility process ν t , the agent’s incentive compatibility constraint can be writtenas φ t ě λ (21)The HJBI equation for the optimal contract with agency is given by rF p W q “ sup c ě ,φ ě λ inf ν µ ´ c ` θ (cid:32) ν ´ ´ log p ν q ( `p γW ´ c q F p W q` φ ν σ F p W q . (22)A simple calculation shows that the worst-case change of volatility ν is given by ν “ θθ ` φ σ F p W q . (23)Plugging in the our expression for ν , the HJBI reduces to the following nonlinear HJBequation rF p W q “ sup c ě ,φ ě λ µ ´ c ´ θ p θ q ` p γW ´ c q F p W q ` θ p θ ` φ σ F p W qq . (24)Consider the region r , W q for which F p W q ą ´ c “ φ ě λ θ ˆ ` φ σ θ F p W q ˙ “ rF p W q ´ µ ´ θ ´ γW F p W q Now I apply rF p W q ´ µ ď γW which comes from the second-best value function being lessthan or equal to the first-best value function without lump-sum transfers, and F p W q ą ´ φ ě λ θ ˆ ` φ σ θ F p W q ˙ ă F p W q ă
0. This shows that F is strictly concave on r , W s .Intuitively this holds because unnecessarily exposing the agent to cash flow shocks is costlyto the principal, since it increases the probability of inefficient liquidation.Thus we have shown the following. On the interval r , W s , the principal’s value functionsatisfies the ODE rF p W q “ µ ` γW F p W q ` θ ˆ ` λ σ θ F p W q ˙ .F is strictly concave so the worst-case change of volatility given by ν ˚ p W q “ θθ ` λ σ F p W q (25)is strictly greater than 1. Additionally, the strict concavity of the value function impliesthat the incentive constraint always binds, i.e. φ ˚ p W q “ λ. While this is the same as DeMarzo and Sannikov (2006), it stands in contrast to Miao andRivera (2016).The next proposition characterizes the optimal contract under the assumption that higheffort is always optimal. The optimal contract with partial shirking can be described usingmethods similar to Zhu (2013), but such a characterization is beyond the scope of thispaper.
Proposition 4.4.
Assume that L ă µr and that implementing high effort is optimal. As-sume further that there exists a unique twice differentiable solution F to the ODE rF p W q “ µ ` γW F p W q ` θ ˆ ` λ σ θ F p W q ˙ (26) with boundary conditions F p q “ L, , F p W q “ ´ and F p W q ă for all W P r , W q where W is defined by F p W q “ . Then:(i) When W P r , W s , F p W q is the principal’s value function for problem 4.3, the optimalcash flow sensitivity is φ ˚ p W q “ λ and the worst case change of volatility ν ˚ p W q isgiven by (25) . The contract delivers value W to the agent whose continuation value W t evolves according to dW t “ γW t dt ´ dC ˚ t ` φ ˚ p W t q σν ˚ p W t q dZ t where dC ˚ t is 0 in r , W q and causes W t to reflect at W . The contract terminates attime τ “ inf t t ě W t “ u when the project is liquidated. ii) When W ą W , the principal’s value function is F p W q “ F p W q ´ p W ´ W q . Theprincipal immediately pays W ´ W to the agent and contracting continues with theagent’s new initial value W . Observe that as the degree of model confidence θ Ñ 8 , the quantity θ log ´ ` λ σ θ F p W q ¯ converges to λ σ F p W q for any value of F p W q . Hence (26) converges to the ordinarydifferential equation of DeMarzo and Sannikov (2006), i.e. the benchmark model with noambiguity aversion. Proposition 4.5.
Let F p¨q , W be defined as in proposition 4.4. Then high effort is optimalif and only if min W Pr ,W s rF p W q ´ F p W qp γW ´ λµ q ě . The argument is simple, and is consistent with proposition 8 of DeMarzo and Sannikov(2006). Next, I show how the optimal contract changes with the level of ambiguity aversion.
Proposition 4.6.
For any promised wealth level to the agent, the principal’s value function F p W q strictly increases in θ . Figure 3: Value functions F p W q for contracting problem. Value function with no ambiguity( θ “ 8 ) shown in blue. Value function with θ “ µ “ , r “ . , γ “ . , λ “ . , σ “ , L “
90. Observe that value function withno ambiguity is strictly higher than value function with θ “
5, consistent with proposition4.6.Proposition 4.6 confirms the obvious intuition that the principal’s value function isincreasing in θ i.e. decreasing in the level of ambiguity aversion. This is illustrated in20igure 4: Worst case change-of-variance ν p W q for θ “ ν “
1, i.e. change-of-variance when θ “ 8 shown in blue. Parameter values are µ “ , r “ . , γ “ . , λ “ . , σ “ , L “ F p W q for θ “ θ “ 8 ) shown in blue. Parameter values are µ “ , r “ . , γ “ . , λ “ . , σ “ , L “
90. Points for which F p W q “ ´ W .figure 1. While this result is unsurprising, it is nonetheless useful in establishing subsequentcomparative static results for the optimal contract. The following proposition shows how21he payoff boundary W changes with θ . Proposition 4.7.
The payoff boundary W is strictly decreasing in θ . Thus, higher levels of ambiguity aversion leads to a higher payoff boundary for theagent. This result is illustrated in figure 6.Figure 6: Upper payoff boundary W as a function of ambiguity aversion 1 { θ . Parametervalues are µ “ , r “ . , γ “ . , λ “ . , σ “ , L “ Up until this point, I have assumed that only the principal was ambiguity averse. It isnatural to ask what happens if the agent is ambiguity averse as well. As it turns out, solong as the agent has the same form of variational ambiguity with a strictly convex penaltyfunction, the agent’s ambiguity aversion will not affect the optimal contract.
Proposition 4.8.
Assume that the agent is ambiguity averse. Then the contract describedin proposition 4.4 remains optimal. Moreover, the agent’s implied worst-case belief is ν p W q “ . Even when the agent is ambiguity averse, the optimal contract is unaffected and theyform expectations as if they fully trust that volatility is constant at level σ . The optimal contract characterized by proposition 4.4 is the solution of a particular max-min problem between the principal and nature. A natural question to ask is whether the22ptimal contract would remain optimal if the worst-case volatility process t ν t u were specifiedexogenously. Formally, this corresponds to what is known as a Bellman-Isaacs condition.As discussed in Hansen et al. (2006), this condition is important for the interpretation ofthe solution to a robust control problem. In particular, it allows for an ex-post Bayesianinterpretation of the robust control problem.For the robust contracting problem described in this paper, the value function is in factglobally concave, and the optimal control of nature has no binding inequality constraints.One can verify (see Fan (1953), Hansen et al. (2006)) that the Bellman-Isaacs condition issatisfied. Therefore, the optimal contract described in proposition 4.4 is optimal in an ex-post Bayesian sense where the principal believes that volatility evolves according to (25), ina restricted space of contracts where changes in the agent’s continuation payoff are locallylinear in project cash flows. As such, it is reasonable to interpret my model as a model ofendogenous belief formation about the volatility process. Following DeMarzo and Sannikov (2006), I show how to implement the optimal contractwith a capital structure of equity, debt, and a credit line. The implementation is as follows: • Equity:
The agent holds inside equity for a fraction λ of the firm. Dividend paymentsare at the discretion of the agent. • Long-term debt:
Long term debt is a consol bond that pays coupons at a rate x “ µ ´ γλ W . If the firm ever defaults on a coupon payment, debt holders force liquidation. • Credit line:
The firm has a revolving credit line with credit limit C L “ Wγ . Balanceson the credit line are subject to an interest rate γ . The firm borrows and repays fundson the credit line at the discretion of the agent. If the balance ever exceeds C L , theproject is terminated.The following proposition characterizes how this implementation changes with the levelof ambiguity aversion. Proposition 4.9.
As the level of ambiguity aversion { θ increases • The optimal credit limit strictly increases. • The face value of the optimal long-term debt strictly decreases.
Note that the fraction of equity held by the agent is determined by the incentive com-patibility constraints, and does not change with θ .I consider asset prices in a representative agent setting where the principal is the rep-resentative investor who trades debt and equity, whereas the agent is an insider who is23estricted from trading in either security. I take r as the risk-free rate. Then I price secu-rities under the worst-case belief measure of the principal. This approach is analogous tothose taken in Anderson et al. (2003), Biais et al. (2007), and Miao and Rivera (2016).The value of equity is given by S t “ E ν ˚ t „ż τt e ´ r p s ´ t q λ dC ˚ t It is straightforward to obtain that the stock price is given by S t “ S p W t q where thefunction S p¨q satisfies the ODE rS p W q “ γW S p W q ` λ σ ν ˚ p W q S p W q with boundary conditions S p q “ S p W q “
1. A simple argument now shows thatthe equity premium is given by E t „ dS t S t ´ r “ ´ λ σ “ ν ˚ p W t q ´ ‰ S p W t q S p W t q (27)which is strictly positive in numerical computations. Note also that in the no-ambiguitybenchmark, the equity premium is identically zero.Define the credit yield spread ∆ t by ż t e ´p r ` ∆ t qp s ´ t q ds “ E ν ˚ t „ż τt e ´ r p s ´ t q ds (28)which when solved yields ∆ t “ rT t ´ T t where T t “ E ν ˚ t “ e ´ r p τ ´ t q ‰ is the time- t price of oneunit of consumption at the time of default. T t “ T p W t q satisfies the ODE rT p W q “ γW T p W q ` λ σ ν ˚ p W q T p W q with boundary conditions T p q “ T p W q “ W . Asset prices withoutambiguity p θ “ 8q shown in blue. Asset prices with ambiguity p θ “ q shown in red. I briefly describe an alternate capital structure implementation of the optimal contract,similar to Biais et al. (2007), using equity, debt, and cash reserves. The firm holds cashreserves M t “ W t λ which earn the risk-free interest rate r . The project payoffs dY t are putinto the firm’s cash account. Outside investors hold a fraction 1 ´ λ of equity, and debtwhich pays coupons at a state-dependent rate r µ ´ p γ ´ r q M t s dt , while the agent holds afraction λ of equity. Then the cash reserves evolve according to dM t “ rM t dt loomoon interest ` dY t loomoon project cash flows ´ λ dC t loomoon dividends ´ r µ ´ p γ ´ r q M t s dt loooooooooomoooooooooon coupon (29)with M “ W { λ . One can easily verify that equation (29) agrees with the evolution for W t { λ . Under the cash-based implementation, proposition 4.7 implies that higher levels ofambiguity aversion increase the amount of cash the firm will hold before it is willing to paydividends.We see in both the credit line implementation and the cash-based implenentation thathigher levels of ambiguity aversion increase the maximum “financial slack” that the firm isgiven under the optimal contract. In the credit line implementation this corresponds to ahigher maximum credit limit, whereas in the cash-based implementation this correspondsto a higher cash buffer that the firm accumulates before paying dividends to equity holders.25 .4 The role of commitment The optimal contract described is proposition 4.4 is not generically renegotiation-proof. Forsmall values of W , the principal’s value function F under the optimal contract is increasingin W , so the principal and the agent can both be made better off by a one-off increasein the continuation value of the agent. To be renegotiation-proof, the principal’s valuefunction F p W q must not have positive slope. However, it is possible to modify the contractdescribed in proposition 4.4 to obtain the optimal renegotiation-proof contract, which Idescribe in this section. Additionally, I show that the implied worst-case volatility underthe optimal renegotiation-proof contract is strictly decreasing in the agent’s continuationvalue.Renegotiation effectively raises the minimum payoff of the agent to a point R such that F p R q “
0. The agent’s promised value evolves on the interval r R, W s according to dW t “ γW t dt ´ dC t ´ λµdt ` λσν p W t q dZ t ` dP t (30)where the processes C and P reflect W t at endpoints W and R respectively. The project isterminated stochastically whenever W t is reflected at R . The probability that the projectcontinues at time t is P r p τ ě t q “ exp ˆ ´ P t R ˙ . (31)The optimal contract can still be implemented with equity, long-term debt and a creditline, though the level of long-term debt and the length of the credit line will be different. Proposition 4.10.
Under the optimal renegotiation-proof contract, the worst-case volatility ν ˚ p W q is strictly increasing in W . The renegotiation-proof implementation contract is in a sense a more robust imple-mentation than the implementation described in proposition 4.4 in that it eliminates theincentive for the principal to renegotiate the contract with the agent. However, it stillrequires the principal to commit to a stochastic (unverifiable) liquidation policy. Withoutsuch commitment, there will generally be welfare loss to the principal. In particular, if theprincipal can only commit to deterministic liquidation policies, then the Pareto frontier isgenerally characterized by a solution to the same differential equation as before, but nowwith boundary conditions F p q “ L and F p q “
0. Under this implementation, it is possi-ble to show similar comparative statics as for the optimal contract with full commitment. G -expectations Consider the “interval uncertainty” or G -expectations formulation of ambiguity aversion.Assume that the adjustment cost function ξ p ν q faced by nature is given by ξ p ν q “ ν P r σ { σ, σ { σ s8 otherwise. (32)26his is equivalent to assuming that nature is free to choose any level of volatility σ t Pr σ, σ s with no adjustment cost. This formulation of volatility ambiguity is precisely the G -expectation formulation of Peng (2007), and is similar to the κ -ignorance specificationof Chen and Epstein (2002). Proposition 4.11.
Consider the optimal contracting problem in which both the principaland the agent have interval uncertainty of the form (32) . Assume that L ă µr and im-plementing high effort is optimal. Then the optimal contract is the same as that of theoptimal contract without ambiguity aversion where both the principal and the agent believethe volatility level is σ . Proposition 4.12.
The payoff boundary W of the optimal contracting problem with intervaluncertainty is strictly increasing in σ . This paper is closely related to Miao and Rivera (2016) who study a similar dynamiccontracting problem where the principal is uncertain about the expected cash flows andis ambiguity-averse. They obtain similar asset pricing implications as I do; time-varyingrisk-premia that are generally higher for financially distressed firms. However, there aresome key differences. Firstly, the optimal contracts are quite different. In my model,the incentive compatibility constraint always binds because the principal fears inefficientliquidation and therefore does not want the agent to bear any more risk than necessary.This preserves the optimality of the simple contractual form of DeMarzo and Sannikov(2006) and Biais et al. (2007). In their model however, the principal does not like driftambiguity, and thus the the optimal contract will sometimes force the agent to bear morecash-flow sensitivity than necessary. As a result, the incentive compatibility constraint isan occasionally binding constraint, and their optimal contract is much more challenging tointerpret. Second, the value function in my model is globally concave, so the Bellman-Isaacscondition holds. This means that it is valid to interpret my model as a model of endogenousbelief formation. This is not the case in Miao and Rivera (2016). Thirdly, my model canaccommodate ambiguity aversion on the part of the agent, without any reduction in theimpact of ambiguity aversion. Miao and Rivera (2016) do not model ambiguity aversion onthe part of the agent, and in their framework, it would produce an offsetting effect whichreduces the impact of ambiguity aversion on the optimal contract.
Credit lines, also known as revolving credit facilities, are an extremely important form offirm financing. Empirically, credit lines account for more than a quarter of outstandingcorporate debt of publicly traded firms and an even larger fraction for smaller, non-publiclytraded firms. To the extent that smaller firms have more ambiguous riskiness of their cashflows, this is consistent with the predictions of my model. See Berger and Udell (1995), Sufi (2007) and DeMarzo and Sannikov (2006).
This paper developed new preference formulations which capture ambiguity aversion to-wards unknown volatility. These moment-constrained variational preferences were intro-duced in a static setting and their impact illustrated in a simple model of portfolio choiceunder quadratic utility. In this model, I showed how the degree of ambiguity aversion im-pacted the implied worst-case volatility as well as the optimal portfolio of the investor. Ithen derived a continuous-time limit in which ambiguity aversion towards unknown, poten-tially time-varying, volatility was not degenerate. These new continuous-time preferenceswere then compared with the G -expectations model of Peng (2007). The impact of thesenew continuous-time preferences was illustrated in a model of optimal security design underrepeated moral hazard. I showed how the worst-case volatility of the principal depended onthe distance to liquidation, whereas the agent always has full confidence in their benchmarkmodel. I showed how ambiguity aversion increased the dividend “hurdle” in the optimalcontract, and further showed how in a credit line implementation this corresponded to in-creasing the maximum draw on the credit line to the agent. Finally, I numerically illustratedsome of the asset pricing implications of ambiguous volatility.For pedagogical simplicity and clarity, this paper focused exclusively on ambiguity aver-sion towards unknown volatility. However, there is no deep theoretical reason for this exclu-sivity. Future work could apply moment-constrained variational preferences to modellinguncertainty about other distributional features, and the volatility penalization could beused in conjunction with other approaches for modelling ambiguity aversion.The preference formulations developed in this paper can potentially be applied to avariety of other settings. One possibility is to examine their effect in a with moral hazardand endogenous investment, similar to DeMarzo et al. (2012) or Bolton et al. (2013),and derive simultaneous implications for corporate investment and asset pricing. Anotherpossibility would be to apply them to the problem of stress testing, where a bank regulatorattempts to control the risk-taking of a bank without full confidence in a particular riskmodel. A third possibility would be to study a consumption-savings problem and investigate28he impact of volatility ambiguity on precautionary savings. I leave these and otherextensions to future research.
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Proofs and derivations for section 2
A.1 Proofs for subsection 2.2.3
Lemma A.1. s p θ q can be expressed as v p θ q ` µ ` σ where v p θ q is the unique positive rootof the cubic polynomial µ C v ` µ ` σ ` θ σ p v ` µ ` σ q ´ θ p v ` σ q ` D. where C “ p b ´ W R f q D “ ´ p b ´ W R f q ´ θ . Additionally, v p θ q is strictly decreasing in θ and we have the limits lim θ Ñ8 v p θ q “ and lim θ Ñ v p θ q “ 8 . Proof of lemma A.1
Write C “ p b ´ W R f q . Then the objective function in (4) can be written as12 µ C ` θ σ s ´ θ p s ´ µ q ` D where D is a constant that does not depend on s . We know that the minimizing s isunique and satisfies s ą µ ` σ . Now, perform the following change-of-variables. Define v “ s ´ µ ´ σ . The objective function, written now as a function of v is given by12 µ C v ` µ ` σ ` θ σ p v ` µ ` σ q ´ θ p v ` σ q ` D Clearly the minimizing v is unique and satisfies v ą
0. Since the objective function isdifferentiable, the first-order condition gives a necessary condition that the optimal v mustsatisfy. Note that this is not a sufficient condition unless there is a unique positive solution.The first-order condition for v simplies to the following cubic polynomial equation for v ,0 “ θ σ v ` θσ p µ ` σ q v ` „ θ σ p µ ` σ q ´ µ C v ´ µ C . To see that this equation has a unique positive solution, note that the terms of the cubicequation are in descending order, that the first two coefficients are positive and that thelast is negative. Hence by Descartes’ rule of signs that the cubic equation has a uniquesolution. The third coefficient has ambiguous sign.
33t follows from Topkis’s theorem that v p θ q is decreasing in θ . The limits follow imme-diately.Note that proposition 2.2 follows immediately from lemma A.1 and the observation that ν p θ q “ v p θ q σ ` . Proposition 2.1 follows from lemma A.1 and the observation that φ ˚ p θ q “ v p θ q ` µ ` σ µ p b ´ W R f q . A.2 Portfolio choice with “multiplier” preferences
We consider the following portfolio choice problem with“unconstrained” or “multiplier”preferences: sup φ P R inf M ě , E P r M s“ E P r M U p Ă W qs ` θ Φ u p M q (33)where Φ u p M q “ E P r M log M s U p Ă W q “ ´ p Ă W ´ b q Ă W “ W R f ` φ r R r R P „ Normal p µ, σ q . As in the text, I assume that µ ą b ą W R f . Then the solution to [whatever theequation number of the problem is] has the following properties: • The minimizing M implies a Normal distribution for Ă W . • The optimal portfolio weight φ u p θ q is increasing in θ . • The worst-case mean µ u p θ q is increasing in θ , and µ u p8q “ µ . • The worst-case variance σ u p θ q is increasing in θ , and σ u p8q “ σ .I give the following argument: It follows from equation (3) that the minimizing M hasan exponential tilting form. This implies that Ă W will have a normal distribution under thechange-of-measure induced by M , with distorted mean and variance r µ and r σ respectively.Φ u p M q can then be expressed in terms of r µ and r σ as.Φ u p M q “ „ r σ σ ` σ p r µ ´ µ q ´ log ˆ r σ σ ˙ ´ φ, r µ , and r σ as fixed, we see that E ” M U p Ă W q ; φ ı “ ´ φ p r µ ` r σ q ` φ r µ p b ´ W R f q ´ p b ´ W R f q Maximizing over φ , we see that φ p M q “ r µ ` r σ r µ p b ´ W R f q Now, substituting in the optimized value of φ p M q we obtain E ” M U p Ă W q ı “ r µ r µ ` r σ p b ´ W R f q ´ p b ´ W R f q . Define L p r µ, r σ ; θ q “ r µ r µ ` r σ p b ´ W R f q ´ p b ´ W R f q ` θ „ r σ σ ` σ p r µ ´ µ q ´ log ˆ r σ σ ˙ ´ . Observe that L p r µ, r σ ; θ q “ max φ E ” M U p Ă W q ı ` θ Φ u p M q when M is restricted to implythat Ă W „ Normal p r µ, r σ q . The solution for µ u p θ q and σ u p θ q can thus be obtained by solvingmin r µ, r σ ě L p r µ, r σ ; θ q . (34)It follows directly from Topkis’ monotonicity theorem that µ u p θ q is strictly increasing in θ ,or equivalently, strictly decreasing in 1 { θ . Since µ u p8q “ µ , we see that µ u p θ q ă µ .The first-order condition for σ u p θ q implies that12 µ u p θ q µ u p θ q ` σ u p θ q p b ´ W R f q “ θ „ σ ´ σ u p θ q from which we see that r σ must be strictly greater that σ . It follows from Topkis’ theoremthat σ u p θ q is strictly increasing in θ . B Proofs for section 4
Proof of proposition 4.4
Note that F p W q is necessarily a concave solution to the HJBI equation rF p W q “ sup c ě ,φ ě λ inf ν ě µ ´ c ` ξ p ν q ` p γW ´ c q F p W q ` φ σ ν F p W q p , W q with optimal controls c “ φ “ λ , and ν “ ν ˚ p W q . It follows that for any φ ě λ we have inf ν ě µ ` ξ p ν q ` γF p W q ` φ σ ν F p W q ´ rF p W q ď G t “ ż t e ´ rs p dY s ´ dC s ` ξ p ν s q ds q ` e ´ rt F p W t q Then note that e rt dG t “ ˆ µ ` ξ p ν t q ` γW t F p W t q ` φ t σ ν t F p W t q ´ rW t ˙ dt ´p ` F p W t qq dC t ` p ` φ t F p W t qq σν t dZ t Note that F p W t q ě ´ ´p ` F p W t qq ď
0. Additionally, under the worst-case ν t we have µ ` ξ p ν t q ` γW t F p W t q ` φ t σ ν t F p W t q ´ rW t ď . Thus G t is a supermartingale. It is a martingale only if φ t “ λ, W t ď W for t ě C t is increasing only when W t ě W .Now, we can bound the principal’s time-0 payoff for an arbitrary incentive compatiblecontract. Note that F p W τ q “ L . We haveinf ν ě E „ż τ e ´ rs t dY s ´ dC s ` ξ p ν s q ds u ` e ´ rτ L “ ż ν ě E “ G t ^ τ ` t ď τ ` e ´ rs t dY s ´ dC s ` ξ p ν s q ds u ` e ´ rτ L ´ e ´ rt F p W t q ˘‰ ď E r G t ^ τ s looomooon ď G “ F p W q ` e ´ rt inf ν ě E »———– t ď τ E t „ż τt e ´ r p s ´ t q t dY s ´ dC s ` ξ p ν s q ds u ` e ´ r p τ ´ t q L loooooooooooooooooooooooooooooooomoooooooooooooooooooooooooooooooon ď µ { r ´ W t ´ F p W t q fiffiffiffifl where the second inequality follows from the first-best bound. Since F p W q ě ´ µ { r ´ W ´ F p W q ď µ { r ´ L . Letting t Ñ 8 we see thatinf ν ě E „ż τ e ´ rs t dY s ´ dC s ` ξ p ν s q ds u ` e ´ rτ L ď F p W q . roof of proposition 4.6 Applying Dynkin’s formula to write the value function as an integral of the differential gen-erator and then differentiating under the integral sign and applying the envelope theoremgives BB θ F p W q “ E „ż τ e ´ rt ` ν ˚ p W t q ´ ´ log p ν ˚ p W t q q ˘ dt ˇˇˇˇ W “ W ą . Proof of proposition 4.7
Differentiate the boundary condition rF p W q ` γW “ µ and use the smooth pasting con-dition F p W q “ ´ r „ BB θ F p W q ´ B W B θ ` γ B W B θ “ B W B θ “ ´ rγ ´ r BB θ F p W q ă . Proof of proposition 4.8
Let h p W q denote the agent’s value function under the contract described in proposition4.4. The HJBI equation for the agent is given by γh p W q “ λµ p ´ a q ` h p W qp γW ` λµ p a ´ qq ` λ σ ν h p W q ` ˜ θ (cid:32) ν ´ ´ log ν ( on r , W s with boundary conditions h p q “ h p W q “
1. Now, guess and verify that h p W q “ W is a solution with optimal controls ν p W q “ a p W q “
1. It is easy to showthat this solution must be unique.
Proof of proposition 4.9
This follows immediately from proposition 4.7
Proof of proposition 4.10