Sensitivity analysis of the expected utility maximization problem with respect to model perturbations

Transcription

1 Sensitivity analysis of the expected utility maximization problem with respect to model perturbations Mihai Sîrbu, The University of Texas at Austin based on joint work with Oleksii Mostovyi University of Connecticut Mathematical Finance Colloquium, University of Southern California, January 22nd, 2018

2 Outline Overview Expected utility maximization Existence and uniqueness Stability and asymptotics The model of perturbed markets Structure of perturbation/relation to random endowment Abstract version Analysis 1-d duality for a 2-d problem First order Second order Risk-tolerance wealth process Definition and basic properties Connection to the second-order asymptotics Summary

3 The starting point consider the perturbation analysis in Larsen, Mostovyi and Žitković do a similar analysis with a general rather than power utility

4 The mathematics present a method to approximate 1. value functions to second order 2. optimizers to the first order stochastic control problems which are convex, but not convex with respect to a parameter abstract version (over random variables) back to the original model, write approximation of strategies as Kunita-Watanabe decomposition under risk tolerance wealth process as numeraire

5 Utility Maximization Problem Agent initial wealth utility stock bank account Market no arbitrage frictionless Initial wealth x Controlling investment H Wealth at time t x + t 0 H uds u Value function: u(x) max X X (x) E [U(X T )].

6 Utility Maximization Problem Utility function U: (0, ) R: strictly increasing, strictly concave, C 1, Satisfies the Inada conditions: lim x 0 U (x) = and lim U (x) = 0. x Standard results in the literature: existence and uniqueness of solutions, properties of the value function, properties of the solutions. Merton, Cox, Huang, Karatzas, Lehoczky, Shreve, Xu, Kramkov, Schachermayer... Under certain (quite weak) conditions, the optimal Ĥ and X exist and are unique.

7 Stability and asymptotics stability: continuous dependence on parameters (goes back to Hadamard) asymptotics: higher order dependence (needs differentiability structure on parameters)

8 Stability and asymptotics: literature Existing results (small fraction): dependence on x: Kramkov and Schachermayer (1999, 2003) dependence on U (and/or P): Jouini and Napp (2004), Carasus and Rasonyi (2005), Larsen (2006), Kardaras and Žitković (2011), dependence on the parametrization of the stock price: Prigent (2003), Larsen and Žitković (2007), Larsen, Mostovyi, and Žitković (2014). on random endowment: Henderson (2002), Kramkov and S. (2006, 2007), Kallsen, Muhle-Karbe, and Vierthauer (2014).

9 Our model: the family of markets (from Larsen, Mostovyi, Žitković) A family of markets is parametrized by δ. Every market consist of a stock and a bond. (Return of) the stock prince process evolves as ds δ t (λ s + δν s ) d M s + dm t (see Hulley and Schweizer (2010), Delbaen and Schachermayer (1995)); The price process of the bond equals to 1 at all times. Goal: study dependence on δ.

10 Primal problem Define X (x, δ) { X : X t = x + } t 0 H udsu, δ t [0, T ] and X 0, x > 0. A utility function U : (0, ) R is strictly increasing, strictly concave, two times continuously differentiable on (0, ) and there exist positive constants c 1 and c 2, such that c 1 A(x) U (x)x U c 2, (x) and define the value function as: u(x, δ) sup E [U(X T )], (x, δ) (0, ) R. X X (x,δ)

11 Mathematical goal How to establish an expansion with respect to δ of the value function u(x, δ) (second order), the corresponding trading strategy (first order)? Remark Dual problem can be helpful.

12 Dual problem V (y) sup (U(x) xy), y > 0, x>0 V (y)y V (y) = 1 A(x), if y = U (x). Let Y(y, δ) be a set of nonnegative supermartingales such that: 1. Y 0 = y, 2. (X t Y t ) t [0,T ] is a supermartingale for every X X (1, δ). The dual value function is v(y, δ) inf E [V (Y T )], (y, δ) (0, ) R. Y Y(y,δ)

13 Structural lemma Lemma For every δ R, we have Y(1, δ) = Y(1, 0)E ( δν S 0), X (1, δ) = 1 X (1, 0) E( δν S 0 ). Remark Looks like a multiplicative (and non-linear) random endowment.

14 Abstract theorems In the spirit of Kramkov-Schachermayer (99), consider the sets C and D polar in L 0 + : Assumption Both C and D contain a stricly positive element and as well as ξ C iff E [ξη] 1 for every η D, η D iff E [ξη] 1 for every ξ C.

15 Primal and dual problems for 0-model We set C(x, 0) xc and D(x, 0) xd, x > 0. Now we can state the abstract primal and dual problems as u(x, 0) sup E [U(ξ)], x > 0, ξ C(x,0) v(y, 0) inf E [V (η)], y > 0. η D(y,0)

16 Abstract version for δ-models For some random variables F and G 0, we set ( ) L δ exp (δf δ2 G), C(x, δ) C(x, 0) 1 L δ and D(y, δ) D(y, 0)L δ, δ R. The abstract versions of the perturbed optimization problems: [ u(x, δ) U (ξ 1L )] δ, x > 0, δ R, v(y, δ) sup E [U(ξ)] = ξ C(x,δ) sup E ξ C(x,0) [ ( inf E [V (η)] = inf E V ηl δ)], y > 0, δ R. η D(y,δ) η D(y,0)

17 The approach Follows Henderson (2002). find lower bound up to second order for u upper bound up to second order for v "match" them Matching one-sided bounds can be found using quadratic optimization problems: (Kramkov, S. (2006))

18 Lack of convexity in δ The value functions u and v are convex in x, y. u v But for the parametrized family of markets, we do not have convexity in δ.

19 No-convexity in δ, cont d Can use convexity only in direction of x, y. For y = u x (x, δ), u(x, δ) xy = v(y, δ) Even if we fix x and vary δ alone, y depends on δ: need to approximate at least v in both directions (y, δ). Summary: better provide joint expansion for both (x, δ) for u (y, δ) for v

20 The 0-model: If u is finite at some point (i) u(x, 0) <, for every x > 0, and v(y, 0) >, for every y > 0. The functions u and v are Legendre conjugate v(y, 0) = sup (u(x, 0) xy), y > 0, x>0 u(x, 0) = inf (v(y, 0) + xy), y>0 x > 0. (ii) The functions u and v are continuously differentiable on (0, ), strictly concave, strictly increasing and satisfy the Inada conditions lim u x (x, 0) x 0 =, lim( v x (y, 0)) y 0 =, lim x(x, 0) x = 0, lim ( v y (y, 0)) y = 0. (iii) For every x > 0 and y > 0, the solutions X(x, 0) and Ŷ (y, 0) exist and are unique and, if y = u (x), we have Ŷ T (y) = U ( XT ) (x), P-a.s.

21 Assumption on perturbations First, we set: dr(x, 0) dp X T (x, 0)ŶT (y, 0). xy Let x > 0 be fixed. There exists c > 0, such that [ ( )] E R(x,0) exp c( ν ST 0 + ν S0 T ) <.

22 Assumption under P and original numéraire Let us assume that c 1 > 1, i.e. that relative-risk aversion of U is strictly greater than 1 (relative risk aversion uniformly exceeds 1) A sufficient condition for the previous slide Assumption to hold is the existence of some positive exponential moments under P

23 First-order analysis Theorem (Envelope) Let x > 0 be fixed and assumptions above hold. Then we have There exists δ 0 > 0 such that for every δ ( δ 0, δ 0 ), we have u(z, δ) R and v(z, δ) R, z > 0. The first-order derivatives are u δ (x, 0) = v δ (y, 0) = xye R(x,0) [ ν S 0 T ], y = u x (x, 0). The value functions u and v are continuous at (x, 0) and (y, 0), respectively. Remark u δ (x, 0) and v δ (y, 0) are linear in ν.

24 Second-order analysis Let S X(x,0) be the price process of the traded securities under the numéraire X(x,0) x, i.e. ( ) S X(x,0) x = X(x, 0), xs 0. X(x, 0) For every x > 0, let H 2 0 (R(x, 0)) denote the space of square integrable martingales under R(x, 0), such that M 2 (x, 0) { M H 2 0 (R(x, 0)) : M = H SX(x,0)}, N 2 (y, 0) { N H 2 0 (R(x, 0)) : MN is R(x, 0)-martingale for every M M 2 (x, 0) }, y = u x (x, 0).

25 Auxiliary minimization problems (for u xx and v yy ) Let us set a(x, x) b(y, y) where y = u x (x, 0), [ inf E R(x,0) A( X T (x, 0))(1 + M T ) 2], M M 2 (x,0) [ inf E R(x,0) B(ŶT (y, 0))(1 + N T ) 2], N N 2 (y,0) A(x) = U (x)x U (x) and B(y) = V (y)y V (y) = 1 A(x).

26 Second-order derivatives with respect to x and y Proved in Kramkov and S. (2006): auxiliary minimization problems admit unique solutions M 0 (x, 0) and N 0 (y, 0); the value functions are two-times differentiable and u xx (x, 0) = y x a(x, x), v yy (y, 0) = x y b(y, y); u xx and v yy are linked via u xx (x, 0)v yy (y, 0) = 1, a(x, x)b(y, y) = 1; the optimizers to auxiliary problems satisfy A( X T (x, 0))(1 + MT 0 (x, 0)) = a(x, x)(1 + N0 T (y, 0)).

27 Auxiliary minimization problem (for u δδ and v δδ ) With F ν S 0 T and G ν 2 M T, we consider the following minimization problems. a(d, d) inf E [A( X R(x,0) T (x, 0))(M T + xf ) 2 M M 2 (x,0) 2xFM T x 2 (F 2 + G) ], b(d, d) inf E [B(ŶT R(x,0) (y, 0))(N T yf ) 2 N N 2 (y,0) + 2yFN T y 2 (F 2 G) ].

28 Structure of u xδ and v yδ Denoting by M 1 (x, 0) and N 1 (y, 0) the unique solutions the auxiliary problems above, we set a(x, d) E [A( X R(x,0) T (x, 0))(1 + MT 0(x, 0))(xF + M1 T (x, 0)) xf(1 + MT 0(x, 0))], b(y, d) E R(x,0) [B(ŶT (y, 0))(1 + N 0 T (y, 0))(N1 T (y, 0) yf ) +yf(1 + N 0 T (y, 0))].

29 Theorem (Mostovyi., S.) Let x > 0 be fixed. Let the assumptions above hold and y = u x (x, 0). Define H u (x, 0) y ( ) a(x, x) a(x, d), x a(x, d) a(d, d) H v (y, 0) x y ( ) b(y, y) b(y, d). b(y, d) b(d, d) Then, the value functions u and v admit the second-order expansions around (x, 0) and (y, 0), respectively, u(x + x, δ) = u(x, 0) + ( x δ) u(x, ( 0) ) + 1 x 2 ( x δ)h u(x, 0) + o( x δ 2 + δ 2 ), v(y + y, δ) = v(y, 0) + ( y δ) v(y, ( 0) ) + 1 y 2 ( y δ)h v(y, 0) + o( y δ 2 + δ 2 ).

30 Theorem (Mostovyi, S.) (i) The values of quadratic optimizations ( a(x, x) 0 a(x, d) x y ) ( b(y, y) 0 b(y, d) y x ) = ( ) 1 0, 0 1 y x a(d, d) + x b(d, d) = a(x, d)b(y, d). y (ii) The optimizers of the quadratic problems are related ( ) ( ) ( ) U ( X T (x, 0)) X T 0 M 0 (x, 0) T (x, 0) + 1 a(x, x) 0 MT 1 = (x, 0) + xf a(x, d) x ŶT 0 N 0 (y, 0) T (y, 0) + 1 y NT 1, (y, 0) yf ( ) ( ) ( ) V 1 + N 0 (ŶT (y, 0))ŶT (y, 0) T (y, 0) b(y, y) M 0 yf + NT 1 = X (y, 0) b(y, d) y T (x, 0) T (x, 0) x xf + MT 1. (x, 0) (iii) The product of any of X(x, 0), X(x, 0)M 0 (x, 0), X(x, 0)M 1 (x, 0) and any of Ŷ (y, 0), Ŷ (y, 0)N0 (y, 0), Ŷ (y, 0)N1 (y, 0) is a P-martingale.

31 Derivatives of the optimizers Theorem (Mostovyi, S.) Let us set X T (x, 0) X T (x, 0) x and X d T (x, 0) X T (x, 0) x lim x + δ 0 lim y + δ 0 Then, we have 1 x + δ 1 y + δ (1+MT 0 (x, 0)), Y T (x, 0) ŶT (y, 0) (1+NT 0 (y, 0)), y (MT 1 (x, 0)+xF), Y T d (y, 0) ŶT (y, 0) (NT 1 (y, 0) yf). y X T (x + x, δ) X T (x, 0) xx T (x, 0) δx d T (x, 0) = 0, ŶT (y + y, δ) ŶT (y, 0) yy T (y, 0) δy d T (y, 0) = 0, where the convergence takes place in P-probability.

32 Approximation of the optimal trading strategies Observation: because the "random endowment" is multiplicative, proportions work better. With M R = S 0 π(x, 0) M, let and γ 0 M R = M0 (x, 0) x and γ 1 M R = M1 (x, 0), x σ ε inf { t [0, T ] : M 0 t (x, 0) x ε or M0 (x, 0) t x ε }, τ ε inf { t [0, T ] : M 1 t (x, 0) x ε or M1 (x, 0) t x ε }, ε > 0, as well as γ 0,ε = γ 0 1 {[0,σε]} and γ 1,ε = γ 1 1 {[0,τε]}, ε > 0.

33 Approximation of the optimal trading strategies Let us set dx x,δ,ε t = X x,δ,ε t ( π t (x, 0) + xγ 0,ε t + δ(ν t + γ 1,ε t ))dst δ, X x,δ,ε 0 = x + x. Note that X x,δ,ε = (x + x) X(x, 0) x E ( ( xγ 0,ε + δγ 1,ε ) M R) E( δν S 0. ) Theorem (Mostovyi, S.) There exists a function ε = ε( x, δ), such that [ ( )] E U = u(x + x, δ) o( x 2 + δ 2 ). X x,δ,ε( x,δ) T

34 Risk-tolerance wealth process Definition For x > 0 and δ R, the risk-tolerance wealth process is a maximal wealth process R(x, δ), such that R T (x, δ) = U ( X T (x, δ)) U ( X T (x, δ)). Remark This process was introduced in Kramkov and S. (2006) in the context of asymptotic analysis of utility-based prices.

35 Theorem (Kramkov and S. (2006)) The following assertions are equivalent: (1) The risk-tolerance wealth process R(x, 0) exists. (2) The value function u admits the expansion quadratic expansion at (x, 0) and u xx (x, 0) = y x a(x, x) satisfies (u x (x, 0)) 2 u xx (x, 0) u xx (x, 0) = E [ = E U ( X T (x, 0) ( U ( X T (x, 0) ) 2 U ( X T (x, 0)), ( ) ] RT (x, 0) 2. R 0 (x, 0) (3) The value function v admits the quadratic expansion at (y, 0) and v yy (y, 0) = x y b(y, y) satisfies [ ] 2 y 2 v yy (y, 0) = E (ŶT (y, 0)) [ ] V (ŶT (y, 0)) = xye R(x,0) B(ŶT (y, 0)).

36 Theorem (..Continued) In addition, if these assertions are valid, then the initial value of R(x) is given by R 0 (x, 0) = u x(x, 0) u xx (x, 0) = x a(x, x), the product R(x, 0)Y (y, 0) = (R t (x, 0)Y t (y, 0)) t [0,T ] is a uniformly integrable martingale and X T (x + x, 0) lim X T (x, 0) = R T (x, 0) x 0 x R 0 (x, 0), Ŷ T (y + y, 0) lim ŶT (y, 0) y 0 y where the limits take place in P-probability. = ŶT (y, 0), y

37 For x > 0 and with y = u x (x, 0), let us define d R(x, 0) dp R T (x, 0)ŶT (y, 0), R 0 (x, 0)y and choose R(x,0) R 0 (x,0) as a numéraire, i.e., let us set ( S R(x,0) R0 (x, 0) R(x, 0), R ) 0(x, 0)S. R(x, 0) We define the spaces of martingales { M 2 (x, 0) M H 2 0 ( R(x, 0)) : M = H S R(x,0)}, and Ñ 2 (y, 0) it the orthogonal complement in H 2 0 ( R(x, 0)).

38 Risk-tolerance wealth process and a Kunita-Watanabe decomposition Theorem (Mostovyi, S.) Let us assume that the risk-tolerance process R(x, 0) exists. Consider the Kunita-Watanabe decomposition of the square integrable martingale [( ) ] P t E R(x,0) A( X T (x, 0)) 1 xf F t, t [0, T ]. given by P = P 0 M 1 Ñ1, where M1 M 2 (x, 0), Ñ 1 Ñ 2 (y, 0), P 0 R.

39 Theorem (..Continued) Then, the optimal solutions M 1 (x, 0) and N 1 (y, 0) of the auxiliary quadratic optimization problems for u δδ and v δδ can be obtained from the Kunita-Watanabe decomposition (above) by reverting to the original numéraire, through the identities: M 1 t = X t (x, 0) R t (x, 0) M1 t (x, 0), Ñ 1 t = x y N1 t (y, 0), t [0, T ]. In addition, the Hessian terms in the quadratic expansion of u and v can be identified as a(d, d) = R 0(x,0) x inf M M 2 (x,0) = R 0(x,0) x E R(x,0) [ ( E R(x,0) MT + xf ) ] 2 + R 0(x,0) x P0 2 + C a, [ (Ñ1 T [ ] where C a x 2 E R(x,0) F 2 A( X T (x,0)) 1 G. A( X T (x,0)) ( ) )) ] 2 A ( XT (x, 0) 1 + C a.

40 Theorem (..Continued) b(d, d) = R 0(x,0) x = R 0(x,0) x inf Ñ N 2 (y,0) ( y x E R(y,0) [ (ÑT + yf ) 2 E R(y,0) [ ( M1 T ) 2 ] + R 0(x,0) x ( ) )) ] 2 A ( XT (x, 0) 1 + C b. ( y x ) 2 P C b, [ ))] where C b y 2 E R(x,0) G + F (1 2 A ( XT (x, 0). The cross terms in the Hessians of u and v are identified as and b(y, d) is given by a(x, d) = P 0 b(y, d) = y x P 0 a(x, x). With these identifications, all the expansions of the value functions above hold.

41 Summary look at the simultaneous perturbations of the market price of risk and the initial wealth formulate quadratic optimization problems and relate the second-order approximations of both primal and dual value functions to these problems. in case when the risk-tolerance wealth process exists, we used it as a numéraire, and changed the measure accordingly, to identify solutions to the quadratic optimization problems above in terms of a Kunita-Watanabe decomposition.