Expected utility models. and optimal investments. Lecture III

Transcription

1 Expected utility models and optimal investments Lecture III 1

2 Market uncertainty, risk preferences and investments 2

3 Portfolio choice and stochastic optimization Maximal expected utility models Preferences are given exogeneously Methods Primal problem (HJB eqn under stringent model assumptions) Dual problem (Linearity under market completeness) Optimal policies: consumption and portfolios 3

4 Maximal expected utility models Market uncertainty (Ω, F, P), W =(W 1,...,W d ) d-dim Brownian motion Trading horizon: [0,T], (0, + ) Asset returns: dr t = µ t dt + σ t dw t µ L 1 (R m ), σ L 2 (R d m ) riskless asset Wealth process: dx t = π t dr t C t dt Control processes: consumption rate C t, asset allocation π t 4

5 Maximal expected utility models Preferences: U : R R increasing, concave, asymptotically elastic... U(x) = 1 γ xγ, log x, e γx Objective: maximize intermediate utility of consumption and utility of terminal wealth ( T V (x, t) = sup E P (C,π) t U 1 (C s ) ds + U 2 (X T )/X t = x ) Generalizations: infinite horizon, long-term average, ergodic criteria... Recall that U 1, U 2 are not related to the investment opportunities 5

6 Primal maximal expected utility problem V solves the Hamilton-Jacobi-Bellman eqn V t + F (x, V x,v xx ; U 1 )=0 V (x, T )=U 2 (x) Viscosity theory (Crandall-Lions) Z., Soner, Touzi, Duffie-Z., Elliott, Davis-Z., Bouchard Optimal policies in feedback form C s = C((V 1 x ) (X s,s)), π s = π(v x (X s,s),v xx (X s,s)) Degeneracies, discontinuities, state and control constraints 6

7 Dual maximal expected utility problem in complete markets Dual utility functional U (y) =max(u(x) xy) x Dual problem becomes linear direct consequence of market completeness and representation, via risk neutrality, of replicable contingent claims Problem reduces to an optimal choice of measure intuitive connection with the so-called state prices Cox-Huang, Karatzas, Shreve, Cvitanic, Schachermayer, Zitkovic, Kramkov, Delbaen et al, Kabanov, Kallsen,... 7

8 Extensions Recursive utilities and Backward Stochasticc Differential Equations (BSDEs) Kreps-Porteus, Duffie-Epstein, Duffie-Skiadas, Schroder-Skiadas, Skiadas, El Karoui-Peng-Quenez, Lazrak and Quenez, Hamadene, Ma-Yong, Kobylanski Ambiguity and robust optimization Ellsberg, Chen-Epstein, Epstein-Schneider, Anderson et al., Hansen et al, Maenhout, Uppal-Wang, Skiadas 8

9 Mental accounting and prospect theory Discontinuous risk curvature Huang-Barberis, Barberis et al., Thaler et al., Gneezy et al. Large trader models Feedback effects Kyle, Platen-Schweizer, Bank-Baum, Frey-Stremme, Back, Cuoco-Cvitanic Social interactions Continuous of agents Propagation of fronts Malinvaud, Schelling, Glaesser-Scheinkman, Horst-Scheinkman, Foellmer Fund management and fee structure Non-zero sum stochastic differential games Huggonier-Kaniel 9

10 Optimal portfolios HJB equation yields the optimal policy in feedback form πs = π(xs,s) π(x, t) = (x, V x,v xx,...) Duality yields the optimmal policy via a martingale representation theorem or via replicating strategies of a dual pseudo-claim These representations, albeit general, offer very little intuition and are of very low practical importance, if any 10

11 Incomplete markets Duality breaks down HJB equation too complex and stringent assumptions are needed Portfolios consist of the myopic and the non-myopic component Myopic portfolio is the investment as if the Sharpe ratio were constant Non-myopic component is the excess risky demand, known as the hedging component Notion of hedging opaque 11

12 An example with myopic and non-myopic portfolios 12

13 Optimal investments under CRRA preferences Market environment ds s = M(Y s,s)s s ds +Σ(Y s,s)s s dws 1 dy s = B(Y s,s) ds + A(Y s,s) dw s riskless bond of zero interest rate Preferences U(x) = xα α (α <0, 0 <α<1) 13

14 Value function V (x, y, t) =sup π E ( X α T α X t = x, Y t = y State controlled wealth process dx s = M(Y s,s)π s ds +Σ(Y s,s)π s dw 1 s X t = x, x 0 Objective Characterize the optimal investment process π s Feedback controls π s = π (X s,y s,s) (Wachter, Campell and Viciera, Liu,... ) ) 14

15 The Hamilton-Jacobi-Bellman equation V t +max π ( 1 2 Σ2 (y, t)π 2 V xx + π(rσ(y, t)a(y, t)v xy + M(y, t)v x )) A2 (y, t)v yy + B(y, t)v y =0 V (x, y, T )= xα α ; (x, y, t) D = R+ R [0,T] 15

16 Optimal policies π s = π (X s,y s,s) = ( ) M(Ys,s) Vx (X Σ 2 s,y s,s) (Y s,s) V xx (Xs,Y s,s) ( R A(Y ) s,s) Vxy (Xs,Y s,s) Σ(Y s,s) V xx (Xs,Y s,s) dx s = M(Y s,s)π s ds +Σ(Y s,s)π s dw 1 s 16

17 Normalized HJB Equation (Krylov, Lions) Non-compact set of admissible controls max π ( 1 1+π 2 ( V t +max π (1 2 Σ2 (y, t)π 2 V xx + π(ra(y, t)σ(y, t)v xy +M(y, t)v x )) A2 (y, t)v yy + B(y, t)v y ) =0 U(x, y, T )= xα α V is the unique constrained viscosity solution of the normalized HJB equation V is a constrained viscosity solution of the original HJB equation (Duffie-Z.) V is unique in the appropriate class (Ishii-Lions, Duffie-Z., Katsoulakis, Touzi, Z.) 17

18 Solution v t A2 (y, t)v yy + V (x, y, t) = xα α v(y, t)ε ε = ( 1 α 1 α + R 2 α B(y, t)+r α ) 1 α L(y, t)a(y, t) + 1 α 2ε1 α L2 (y, t)v =0 v y L(y, t) = M(y, t) Σ(y, t) π (x, y, t) = 1 1 α M(y, t) Σ 2 (y, t) x + R ε A(y, t) 1 α Σ(y, t) v y (y, t) v(y, t) x 18

19 Structural and characterization results on optimal policies Long-term horizon problems Logarithmic utilities, approximations for other utilities (Campbell) Finite horizon and exponential utilities The excess hedging demand (non-myopic is identified with the indifference delta of a pseudo-claim with payoff depending on risk aversion and aggregate Sharpe ratio 19

20 Other limitations 20

21 Time horizon How do we know our utility say 30 years from now? How do we manage our liabilities beyond the time the utility is prespecified? Are our portfolios consistent across different units? 21

22 Units, numeraires and expected utility 22

23 A toy incomplete model Probability space Ω={ω 1,ω 2,ω 3,ω 4 }, P {ω i } = p i, i =1,..., 4 Two risks S 0 S u Y u Y 0 S d Y d Random variables S T and Y T S T (ω 1 )=S u,y T (ω 1 )=Y u S T (ω 2 )=S u,y T (ω 2 )=Y d S T (ω 3 )=S d,y T (ω 3 )=Y u S T (ω 4 )=S d,y T (ω 4 )=Y d 23

24 Investment opportunities We invest the amount β in bond (r =0) and the amount α in stock Wealth variable X 0 = x, X T = β + αs T = x + α(s T S 0 ) Indifference price For a general claim C T, we define the value function V C T (x) =max α E( e γ(x T C T ) ) The indifference price is the amount ν(c T ) for which, V 0 (x) =V C T (x + ν(c T )) 24

25 The indifference price (MZ 2004) ν(c T )=E Q ( 1 γ log E Q(e γc(s T,Y T ) S T ) ) = E Q (C T ) Q(Y T S T )=P(Y T S T ) 25

26 Static arbitrage 26

27 Indifference prices in spot and forward units Spot units Wealth: Value function: XT s = x + α ( S T 1+r S 0 V C T (x) =sup α E P ( ) e γ(xs T C ) T 1+r ) Pricing condition: V 0 (x) =V s,c T (x + ν s (C T )) ( ) Pricing measure: E ST Q s 1+r = S0 and Q s (Y T S T )=P(Y T S T ) Indifference price: ν s ( ) (C T )=E CT Q s 1+r = EQ s ( ( 1 γ log E Q s e γ C )) T 1+r S T 27

28 Forward units Wealth: X f T = Xs T (1 + r) =f + α(f T F 0 ); f = x(1 + r) Value function: V C T (f) =supe P ( e γ(xf T C T ) ) α Pricing condition: V 0 (f) =V C T (f + ν f (C T )) Pricing measure: E Q f(f T )=F 0 and Q f (Y T F T )=P(Y T F T ) Indifference price: ν f (C T )=E Q f(c T )=E Q f ( 1γ log E Q ( e γc T F T )) 28

29 Inconsistency across prices expressed in spot and forward units Pricing measures: Q s = Q f ( ( Spot price: ν s (C T )=E 1 Q γ log E Q e γ C )) T 1+r S T Forward price: ν f (C T )=E Q ( 1γ log E Q ( e γc T S T )) ν f (C T ) (1+r)ν s (C T ) 29

30 (WWW) What went wrong? Risk preferences were not correctly specified! Risk preferences need to be consistent across units Risk aversion is not a constant 30

31 Indifference prices in spot and forward units Spot units Wealth: Value function: XT s = x + α ( S T 1+r S 0 V s,c T (x) =sup α E P ( ) e γs (XT s C ) T 1+r ) Pricing condition: V s,0 (x) =V s,c T (x + ν s (C T )) ( ) Pricing measure: E ST Q s 1+r = S0 and Q s (Y T S T )=P(Y T S T ) Indifference price: ν s ( ) (C T )=E CT Q s 1+r = EQ s ( 1 γ s log E Q s ( e γs C T 1+r S T )) 31

32 Forward units Wealth: X f T = Xs T (1 + r) =f + α(f T F 0 ); f = x(1 + r) Value function: V f,c T (f) =supe P ( e γf (X f T C T ) ) α Pricing condition: V f,0 (f) =V f,c T (f + ν f (C T )) Pricing measure: E Q f(f T )=F 0 and Q f (Y T F T )=P(Y T F T ) Indifference price: ν f (C T )=E Q f(c T )=E Q f ( 1 γ f log E Q f (e γf C T F T )) 32

33 Consistency across spot and forward units ν f (C T )=(1+r)ν s (C T ) δ s = 1 1+r δf δ s = 1 γ s, δf = 1 : spot and forward risk tolerance γf Risk tolerance is not a number. It is expressed in wealth units. 33

34 Utility functions U s (x) = e γs x U f (x) = e γf x ; x in spot units ; x in forward units Value function representations V s,c T (x) = e γs (x ν s (C T )) H(Q P) = U s (x ν s (C T )+δ s H(Q P)) V f,c T (x) = e γf (x ν f (C T )) H(Q P) = U f ( x ν f (C T )+δ f H(Q P) ) Q = Q s = Q f 34

35 Static no arbitrage constraint Appropriate dependence across units needs to be built into the risk preference structure 35

36 The stock as the numeraire Indifference price is a unitless quantity (number of stock shares) The utility argument γ s T X T S T needs to be unitless as well Static no arbitrage constraint strongly suggests that risk aversion needs to be stochastic 36

37 Stochastic risk preferences 37

38 Indifference prices and state dependent risk tolerance γ T = γ (S T ) F S T -measurable random variable (in reciprocal to wealth units) Risk tolerance (in units of wealth) Risk tolerance (in units of wealth) δ T = 1 γ T Should γ T be allowed to be F (S,Y ) T -measurable? 38

39 Random utility and its value function Value function without the claim V 0 (x; γ T )= exp x E Q ( 1 γt ) H (Q P) Value function and utility V (x, 0; T )= e x E Q ( 1 γ T ) H(Q P) U(X T ; T )= e γ T X T 0 T 39

40 Two minimal entropy measures dq dq = δ T E Q (δ T ) E Q (S T (1 + r) S 0 )=0 E Q (γ T (S T (1 + r) S 0 )) = 0 Structural constraints between the market environment and the risk preferences 40

41 Indifference price and value function The indifference price of C T is given by The utility Value function with the claim ( ( 1 ν (C T ; γ T )=E Q log E γ Q e γ T C )) T 1+r S T T U(X T ; T )= e γ T X T ( V C T (x; γ T )= exp ( x ν (CT ; γ T ) E Q (δ T ) ) H (Q P) ) 41

42 Optimal policies for stochastic risk preferences (in the presence of the claim) α C T, = α 0, + α 1, + α 2, Optimal demand due to market incompleteness: α 0, α 0, = H (Q P) S 0 E Q (δ T ) 42

43 Optimal demand due to changes in risk tolerance: α 1, α 1, = log E Q (δ T ) x S 0 Optimal demand due to liability: α 2, α 2, = E Q (δ T ) S 0 ( ) ν (CT ; γ T ) E Q (δ T ) 43

44 Numeraire independence 44

45 Indifference prices and general numeraires The stock as the numeraire Wealth: Value function: XT S = x ( + α 1 S 0 S T V S,C T (x S )=sup α E P ) S T e γs (S T )(XT S C T S ) T Pricing condition: V S,0 (x S )=V S,C T (x S + ν S (C T )) Pricing measure: Q S (Y T S T )=P(Y T S T ); B t S t martingale w.r.t. Q S 45

46 Indifference price ν S (C T )=E Q S 1 γ S (S T ) log E Q S e γs (S T ) C T S T S T Numeraire consistency ν(c T ; γ T ) S 0 = ν S (C T ; γ S T ) δ T = δ S T S T 46

47 The term structure of risk preferences 47

48 Fundamental questions What is the proper specification of the investors risk preferences? Are risk preferences static or dynamic? Are they affected by the market environment and the trading horizon? Are there endogenous structural conditions on risk preferences? How does the choice of risk preferences affect the indifference prices and the risk monitoring policies? 48

49 Requirements for a consistent indifference pricing system (work in progress MZ) Risk preferences need to be consistent across units and trading horizons Dynamic utilities Martingality of risk tolerance process 49