An Introduction to Moral Hazard in Continuous Time

Transcription

1 An Introduction to Moral Hazard in Continuous Time Columbia University, NY Chairs Days: Insurance, Actuarial Science, Data and Models, June 12th, 2018

2 Outline 1 2 Intuition and verification 2BSDEs 3 Control reinterpretation Extensions

3 Motivation B. Salanié, The economics of contracts Customers know more about their tastes than firms, firms know more about their costs than the government and all agents take actions that are at least partly unobservable. Vast economic literature revisiting general equilibrium theory by incorporating incitations and asymmetry of information (Holmström, Hart, Milgrom, Mirrlees, Picard, Tirole, Rogerson, Salanié, Laffont, Martimort, Bolton, Dewatripoint, Ekeland, Rochet, Choné, Carlier, Williams, Sung, Cvitanić, Zhang, Sannikov,... ).

4 Motivation B. Salanié, The economics of contracts Customers know more about their tastes than firms, firms know more about their costs than the government and all agents take actions that are at least partly unobservable. Vast economic literature revisiting general equilibrium theory by incorporating incitations and asymmetry of information (Holmström, Hart, Milgrom, Mirrlees, Picard, Tirole, Rogerson, Salanié, Laffont, Martimort, Bolton, Dewatripoint, Ekeland, Rochet, Choné, Carlier, Williams, Sung, Cvitanić, Zhang, Sannikov,... ). Moral hazard: situation where an Agent can benefit from an action (unobservable), whose cost is incurred by others a type of externality

5 Motivation B. Salanié, The economics of contracts Customers know more about their tastes than firms, firms know more about their costs than the government and all agents take actions that are at least partly unobservable. Vast economic literature revisiting general equilibrium theory by incorporating incitations and asymmetry of information (Holmström, Hart, Milgrom, Mirrlees, Picard, Tirole, Rogerson, Salanié, Laffont, Martimort, Bolton, Dewatripoint, Ekeland, Rochet, Choné, Carlier, Williams, Sung, Cvitanić, Zhang, Sannikov,... ). Moral hazard: situation where an Agent can benefit from an action (unobservable), whose cost is incurred by others a type of externality How can one design optimal contracts?

6 Relevance to insurance Avent of massively available data on insured clients is changing deeply the field.

7 Relevance to insurance Avent of massively available data on insured clients is changing deeply the field. Better identification of clients actuarial risk is paving the way for innovative contracts, with targeted prevention and virtuous incentive mechanisms.

8 Relevance to insurance Avent of massively available data on insured clients is changing deeply the field. Better identification of clients actuarial risk is paving the way for innovative contracts, with targeted prevention and virtuous incentive mechanisms. Crux of the matter: information sharing auto insurance (pay how you drive), health insurance (connected objects)...

9 Relevance to insurance Avent of massively available data on insured clients is changing deeply the field. Better identification of clients actuarial risk is paving the way for innovative contracts, with targeted prevention and virtuous incentive mechanisms. Crux of the matter: information sharing auto insurance (pay how you drive), health insurance (connected objects)... How to quantify and exploit optimally the information shared by the clients?

10 Modelisation Contract between an Agent and a Principal.

11 Modelisation Contract between an Agent and a Principal. Principal has the initiative of the contract.

12 Modelisation Contract between an Agent and a Principal. Principal has the initiative of the contract. Agent can accept or refuse it, but is then fully committed.

13 Modelisation Contract between an Agent and a Principal. Principal has the initiative of the contract. Agent can accept or refuse it, but is then fully committed. When he accepts, Agent has to perform a costly task (examples to come).

14 Modelisation Agent chooses his action (or effort): process ν.

15 Modelisation Agent chooses his action (or effort): process ν. Choice of Agent impacts the distribution of an output process X t t X t = X 0 + λ s (X s, ν s )ds + σ s (X s, ν s )dws ν, t [0, T ], 0 0 where W ν is a P ν Brownian motion.

16 Modelisation Agent chooses his action (or effort): process ν. Choice of Agent impacts the distribution of an output process X t t X t = X 0 + λ s (X s, ν s )ds + σ s (X s, ν s )dws ν, t [0, T ], 0 0 where W ν is a P ν Brownian motion. Profit of Principal depends on X, which he observes. But ν is inaccessible! = hidden action = asymmetry of information.

17 Examples (from finished or on-going works) Optimal remuneration of managers (Élie, Mastrolia, Réveillac and Villeneuve) { X value of the firm. ν work of the manager. t X t = X 0 + ν s ds + σws ν. 0

18 Examples (from finished or on-going works) Optimal remuneration of managers (Élie, Mastrolia, Réveillac and Villeneuve) { X value of the firm. ν work of the manager. t X t = X 0 + ν s ds + σws ν. 0 Contract salary of the manager.

19 Examples (from finished or on-going works) Delegated portfolio management (Cvitanić and Touzi). { X value of the portfolio. ν investment choices. t X t = X 0 + ν s (b s ds + σ s dws ν ). 0

20 Examples (from finished or on-going works) Delegated portfolio management (Cvitanić and Touzi). { X value of the portfolio. ν investment choices. t X t = X 0 + ν s (b s ds + σ s dws ν ). 0 Contract remuneration of the fund manager.

21 Examples (from finished or on-going works) Electricity demand response management (Aïd, Élie, Hubert, Mastrolia and Touzi). { X deviation from baseline consumption. ν = (α, β) effort on the mean and the volatility of consumption. t t X t = X 0 α s 1ds + diag(σ) β s dws ν. 0 0

22 Examples (from finished or on-going works) Electricity demand response management (Aïd, Élie, Hubert, Mastrolia and Touzi). { X deviation from baseline consumption. ν = (α, β) effort on the mean and the volatility of consumption. t t X t = X 0 α s 1ds + diag(σ) β s dws ν. 0 0 Contract compensation of the consumer.

23 Examples (from finished or on-going works) Optimal securitization of mortgage loans (Pagès, Hernández Santibáñez and Zhou). { X number of observed defaults of the I loans. ν bank monitoring actions (number of loans non-monitored). X Poisson process with intensity λ ν = α I X (I X + εν).

24 Examples (from finished or on-going works) Optimal securitization of mortgage loans (Pagès, Hernández Santibáñez and Zhou). { X number of observed defaults of the I loans. ν bank monitoring actions (number of loans non-monitored). X Poisson process with intensity λ ν = α I X (I X + εν). Contract remuneration of the bank, and liquidation procedure.

25 Examples (from finished or on-going works) Insurance contracts (Kazi-Tani, Hernández Santibáñez and Zhou). { X number of accidents. ν = (λ, α, β) effort to reduce frequency and severity of accidents. X process with jump compensator ρ ν s (dx, ds) = λ s δ Ys (dx)ds, with dy s = α s ds + σ β s dw ν s.

26 Examples (from finished or on-going works) Insurance contracts (Kazi-Tani, Hernández Santibáñez and Zhou). { X number of accidents. ν = (λ, α, β) effort to reduce frequency and severity of accidents. X process with jump compensator ρ ν s (dx, ds) = λ s δ Ys (dx)ds, with dy s = α s ds + σ β s dw ν s. Contract premium charged to the insureds, and deductible.

27 Principal proposes a contract to Agent at t = 0. This corresponds to a salary/price/premium ξ given at T, and contingent on the observable X.

28 Principal proposes a contract to Agent at t = 0. This corresponds to a salary/price/premium ξ given at T, and contingent on the observable X. Agent solves then V A 0 (ξ) := sup ν [ E Pν ξ(x T ) }{{} Salary T 0 c t (X t, ν t ) }{{} Cost ] dt.

29 Principal proposes a contract to Agent at t = 0. This corresponds to a salary/price/premium ξ given at T, and contingent on the observable X. Agent solves then V A 0 (ξ) := sup ν [ E Pν ξ(x T ) }{{} Salary T 0 c t (X t, ν t ) }{{} Cost ] dt. Dependence of ξ on the whole trajectory of X is crucial.

30 Principal proposes a contract to Agent at t = 0. This corresponds to a salary/price/premium ξ given at T, and contingent on the observable X. Agent solves then V A 0 (ξ) := sup ν [ E Pν ξ(x T ) }{{} Salary T 0 c t (X t, ν t ) }{{} Cost ] dt. Dependence of ξ on the whole trajectory of X is crucial. Agent faces a non Markovian stochastic control problem.

31 Principal looks for a Stackelberg equilibrium and proceeds in 2 steps. (i) Compute best reaction function of Agent to ξ ν (ξ) P (ξ).

32 Principal looks for a Stackelberg equilibrium and proceeds in 2 steps. (i) Compute best reaction function of Agent to ξ ν (ξ) P (ξ). (ii) Feedback optimisation on the contracts V P 0 [ := sup E P (ξ) U(X T, ξ(x T ))], ξ Ξ R U: utility function of Principal. Ξ R : contracts such that V A (ξ) R (participation constraint).

33 The program is clear, one has to tackle the two previous steps : non Markovian control (2)BSDEs.

34 The program is clear, one has to tackle the two previous steps : non Markovian control (2)BSDEs. : non standard optimisation over random variables necessary reinterpretation to pave the way to explicit solutions and, at least, to efficient numerical methods!

35 Intuition and verification 2BSDEs Outline 1 2 Intuition and verification 2BSDEs 3 Control reinterpretation Extensions

36 Intuition and verification 2BSDEs Intuition and verification Dynamic writes T Vt A (ξ) := essup E [ξ(x T Pν ) ν t c s (X s, ν s )ds F t ], t [0, T ].

37 Intuition and verification 2BSDEs Intuition and verification Dynamic writes T Vt A (ξ) := essup E [ξ(x T Pν ) ν t c s (X s, ν s )ds F t ], t [0, T ]. Hamiltonian naturally associated to this control problem is H(t, x, z, γ) := sup h(t, x, z, γ, v), v h(t, x, z, γ, v) := λ t (x, v) z Tr [ γ(σσ ) t (x, v) ] c t (x, v).

38 Intuition and verification 2BSDEs Intuition and verification If V A t (ξ) =: v A (t, X t ) is a smooth function of (t, X t ), where smooth means that an Itō formula is valid dv A (t, X ) = t v A (t, X t )dt + Z t dx t Tr [Γ td X t ], with the correspondence Z t x v A (t, X t ), Γ t xx v A (t, X t ).

39 Intuition and verification 2BSDEs Intuition and verification If V A t (ξ) =: v A (t, X t ) is a smooth function of (t, X t ), where smooth means that an Itō formula is valid dv A (t, X ) = t v A (t, X t )dt + Z t dx t Tr [Γ td X t ], with the correspondence Z t x v A (t, X t ), Γ t xx v A (t, X t ). Z and Γ only supposed integrable Sobolev type regularity.

40 Intuition and verification 2BSDEs Intuition and verification If V A t (ξ) =: v A (t, X t ) is a smooth function of (t, X t ), where smooth means that an Itō formula is valid dv A (t, X ) = t v A (t, X t )dt + Z t dx t Tr [Γ td X t ], with the correspondence Z t x v A (t, X t ), Γ t xx v A (t, X t ). Z and Γ only supposed integrable Sobolev type regularity. Dynamic programming and martingale optimality principle = t v A (t, X t ) + H ( t, X t, Z t, Γ t ) = 0.

41 Intuition and verification 2BSDEs Intuition and verification To sum up T V0 A (ξ) = ξ+ H ( ) 1 T T t, X t, Z t, Γ t dt Tr [Γ t d X t ] Z t dx t

42 Intuition and verification 2BSDEs Intuition and verification To sum up T V0 A (ξ) = ξ+ H ( ) 1 T T t, X t, Z t, Γ t dt Tr [Γ t d X t ] Z t dx t It is a non linear representation theorem for ξ! In addition, optimal effort of Agent is given by ν t argmax {h(t, X, Z t, Γ t, v)}. v

43 Intuition and verification 2BSDEs Intuition and verification The converse statement is true! It is a simple verification theorem Theorem If, for an admissible pair (Z, Γ) and V 0 R, the contract ξ V0,Z,Γ T, with ξ V0,Z,Γ t t := V 0 H ( ) 1 t t s, X, Z s, Γ s ds + Tr [Γ s d X s ] Z s dx s, is offered to Agent, then his utility is V 0 and his optimal effort ν t argmax {h(t, X, Z t, Γ t, v)}. v

44 Intuition and verification 2BSDEs Interpretation Interpretation of the contracts Linear use of the path of X Z is a first order sensitivity remuneration using stocks.

45 Intuition and verification 2BSDEs Interpretation Interpretation of the contracts Linear use of the path of X Z is a first order sensitivity remuneration using stocks. Linear use of X Γ is a second order sensitivity remuneration using stock options.

46 Intuition and verification 2BSDEs Interpretation Interpretation of the contracts Linear use of the path of X Z is a first order sensitivity remuneration using stocks. Linear use of X Γ is a second order sensitivity remuneration using stock options. Hamiltonian utility that Agent obtains by optimizing = subtracted from salary.

47 Intuition and verification 2BSDEs Interpretation Interpretation of the contracts Linear use of the path of X Z is a first order sensitivity remuneration using stocks. Linear use of X Γ is a second order sensitivity remuneration using stock options. Hamiltonian utility that Agent obtains by optimizing = subtracted from salary. However, quid of the existence of Z and Γ for general contracts?

48 Intuition and verification 2BSDEs 2BSDEs Existence of Z is classical (standard representation theorem), but less clear for Γ. Introduce { 1 F (t, x, z, v) := sup γ 2 Tr[γ(σσ ) t (x, v)] H ( t, x, z, γ ) }.

49 Intuition and verification 2BSDEs 2BSDEs Existence of Z is classical (standard representation theorem), but less clear for Γ. Introduce { 1 F (t, x, z, v) := sup γ 2 Tr[γ(σσ ) t (x, v)] H ( t, x, z, γ ) }. We have for any control ν T V0 A (ξ) = ξ F ( ) T t, X, Z t, ν t dt Z t dx t + KT ν, 0 0 where K ν is non decreasing with K ν T := T 0 F ( t, X, Z t, ν t ) dt 1 2 Tr[Γ td X t ] + H ( t, X, Z t, Γ t ) dt.

50 Intuition and verification 2BSDEs 2BSDEs Relaxation of existence of Γ K ν not necessarily absolutely continuous.

51 Intuition and verification 2BSDEs 2BSDEs Relaxation of existence of Γ K ν not necessarily absolutely continuous. The triplet (V A (ξ), Z, (K ν ) ν ) solves then exactly a 2BSDE with terminal condition ξ and generator F!

52 Intuition and verification 2BSDEs 2BSDEs Relaxation of existence of Γ K ν not necessarily absolutely continuous. The triplet (V A (ξ), Z, (K ν ) ν ) solves then exactly a 2BSDE with terminal condition ξ and generator F! Study of wellposedness of these equations and their generalisations with Élie, Kazi-Tani, Piozin, Mastrolia, Matoussi, Sabbagh, Soner, Tan, Touzi, Zhang and Zhou.

53 Intuition and verification 2BSDEs 2BSDEs 2BSDEs allow for a general probabilistic representation of the value function of Agent, but do not give in general access to optimal effort!

54 Intuition and verification 2BSDEs 2BSDEs 2BSDEs allow for a general probabilistic representation of the value function of Agent, but do not give in general access to optimal effort! Nonetheless, one can prove that the contracts ξ of the form ξ V0,Z,Γ are dense in an appropriate class of admissible contracts.

55 Intuition and verification 2BSDEs 2BSDEs 2BSDEs allow for a general probabilistic representation of the value function of Agent, but do not give in general access to optimal effort! Nonetheless, one can prove that the contracts ξ of the form ξ V0,Z,Γ are dense in an appropriate class of admissible contracts. No loss of generality in restricting to them!

56 Control reinterpretation Extensions Outline 1 2 Intuition and verification 2BSDEs 3 Control reinterpretation Extensions

57 Control reinterpretation Extensions Why is the class ξ v,z,γ T useful? becomes V P 0 = sup V0 P (v), v R V P 0 (v) := sup Z,Γ [ E P (Z,Γ) U ( X T, ξ v,z,γ ) ] T,

58 Control reinterpretation Extensions Why is the class ξ v,z,γ T useful? becomes V P 0 = sup V0 P (v), v R V P 0 (v) := sup Z,Γ [ E P (Z,Γ) U ( X T, ξ v,z,γ ) ] T, V0 P (v) is value function of a standard stochastic control problem, with state variables X and ξ v,z,γ, and with controls Z and Γ.

59 Control reinterpretation Extensions Why is the class ξ v,z,γ T useful? becomes V P 0 = sup V0 P (v), v R V P 0 (v) := sup Z,Γ [ E P (Z,Γ) U ( X T, ξ v,z,γ ) ] T, V0 P (v) is value function of a standard stochastic control problem, with state variables X and ξ v,z,γ, and with controls Z and Γ. Continuation utility of Agent is a performance index for Principal.

60 Control reinterpretation Extensions Why is the class ξ v,z,γ T useful? becomes V P 0 = sup V0 P (v), v R V P 0 (v) := sup Z,Γ [ E P (Z,Γ) U ( X T, ξ v,z,γ ) ] T, V0 P (v) is value function of a standard stochastic control problem, with state variables X and ξ v,z,γ, and with controls Z and Γ. Continuation utility of Agent is a performance index for Principal. HJB equation (possibly path dependent) associated explicit solutions or efficient numerical schemes!

61 Control reinterpretation Extensions Extensions Extension to one Principal and several Agents. Nash equilibrium between Agents characterized by a system of (quadratic) (2)BSDEs Principal problem is standard control again with 2 number of Agents state variables (with Élie)

62 Control reinterpretation Extensions Extensions Extension to one Principal and several Agents. Nash equilibrium between Agents characterized by a system of (quadratic) (2)BSDEs Principal problem is standard control again with 2 number of Agents state variables (with Élie) Can consider the mean field game limit where the number of Agents goes to infinity Principal solves a McKean Vlasov control problem HJB equation in infinite dimension (with Mastrolia and Élie)

63 Control reinterpretation Extensions Extensions Extension to one Principal and several Agents. Nash equilibrium between Agents characterized by a system of (quadratic) (2)BSDEs Principal problem is standard control again with 2 number of Agents state variables (with Élie) Can consider the mean field game limit where the number of Agents goes to infinity Principal solves a McKean Vlasov control problem HJB equation in infinite dimension (with Mastrolia and Élie) Modelization of free rider or ripple effects.

64 Control reinterpretation Extensions Perspectives Principal does not know some characteristics of Agent adverse selection problems (with Hernández Santibánez and Zhou).

65 Control reinterpretation Extensions Perspectives Principal does not know some characteristics of Agent adverse selection problems (with Hernández Santibánez and Zhou). Dynamics with jumps: Agent can impact intensity and size of the jumps of X (2)BSDEs with jumps (with Kazi-Tani and Zhou).

66 Control reinterpretation Extensions Perspectives Principal does not know some characteristics of Agent adverse selection problems (with Hernández Santibánez and Zhou). Dynamics with jumps: Agent can impact intensity and size of the jumps of X (2)BSDEs with jumps (with Kazi-Tani and Zhou). More contractual possibilities: several competing Principals, firing or promoting, limited liability, inter temporal payments, learning...

67 Control reinterpretation Extensions Perspectives Principal does not know some characteristics of Agent adverse selection problems (with Hernández Santibánez and Zhou). Dynamics with jumps: Agent can impact intensity and size of the jumps of X (2)BSDEs with jumps (with Kazi-Tani and Zhou). More contractual possibilities: several competing Principals, firing or promoting, limited liability, inter temporal payments, learning... To each extension is associated a control problem (and (2)BSDEs) with different specificities reflections, constraints, weak terminal conditions, stochastic targets,...

68 Control reinterpretation Extensions Thank you for your attention!