Prospect Theory: A New Paradigm for Portfolio Choice

Transcription

1 Prospect Theory: A New Paradigm for Portfolio Choice

2 1 Prospect Theory Expected Utility Theory and Its Paradoxes Prospect Theory 2 Portfolio Selection Model and Solution Continuous-Time Market Setting Portfolio Selection Model Ill-posedness Solution Flow: Divide and Conquer Example: Two-Piece Power Utilities Single Period Problem 3 Conclusions

3 Expected Utility Theory and Its Paradoxes Prospect Theory Expected Utility Theory Expected Utility Theory (EUT): dominant model for decision making under uncertainty

4 Expected Utility Theory and Its Paradoxes Prospect Theory Expected Utility Theory Expected Utility Theory (EUT): dominant model for decision making under uncertainty Underlying assumptions: rational behavior in particular risk aversion

5 Expected Utility Theory and Its Paradoxes Prospect Theory Expected Utility Theory Expected Utility Theory (EUT): dominant model for decision making under uncertainty Underlying assumptions: rational behavior in particular risk aversion Basic tenets in the context of asset allocation: Investors evaluate assets according to final asset positions Investors are (globally) risk averse Investors are able to objectively evaluate probabilities

6 Expected Utility Theory and Its Paradoxes Prospect Theory Anomalies in Human Behaviors Substantial evidences suggest systematic violation of EUT

7 Expected Utility Theory and Its Paradoxes Prospect Theory Anomalies in Human Behaviors Substantial evidences suggest systematic violation of EUT People evaluate assets according to gains and losses (people compare)

8 Expected Utility Theory and Its Paradoxes Prospect Theory Anomalies in Human Behaviors Substantial evidences suggest systematic violation of EUT People evaluate assets according to gains and losses (people compare) People are not globally risk averse, and distinctively more sensitive to losses than to gains (people behave differently on gains than on losses)

9 Expected Utility Theory and Its Paradoxes Prospect Theory Anomalies in Human Behaviors Substantial evidences suggest systematic violation of EUT People evaluate assets according to gains and losses (people compare) People are not globally risk averse, and distinctively more sensitive to losses than to gains (people behave differently on gains than on losses) People overweights small probabilities (people are subjective)

10 Expected Utility Theory and Its Paradoxes Prospect Theory Experiments on Risk Attitude Experiment 1: compare the following two options A1: 90% chance to win $1000, 10% chance to win nothing B1: win $900 for sure

11 Expected Utility Theory and Its Paradoxes Prospect Theory Experiments on Risk Attitude Experiment 1: compare the following two options A1: 90% chance to win $1000, 10% chance to win nothing B1: win $900 for sure Risk averse (people don t take chance when gaining)

12 Expected Utility Theory and Its Paradoxes Prospect Theory Experiments on Risk Attitude (Cont d) Experiment 2: compare the following two options A2: 90% chance to lose $1000, 10% chance to lose nothing B2: lose $900 for sure

13 Expected Utility Theory and Its Paradoxes Prospect Theory Experiments on Risk Attitude (Cont d) Experiment 2: compare the following two options A2: 90% chance to lose $1000, 10% chance to lose nothing B2: lose $900 for sure Risk seeking (people take chance when losing)

14 Expected Utility Theory and Its Paradoxes Prospect Theory Experiments on Risk Attitude (Cont d) Experiment 3 (Samuelson 1963): compare the following two options A3: 50% chance to win $2000, 50% chance to lose $1000 B3: do nothing

15 Expected Utility Theory and Its Paradoxes Prospect Theory Experiments on Risk Attitude (Cont d) Experiment 3 (Samuelson 1963): compare the following two options A3: 50% chance to win $2000, 50% chance to lose $1000 B3: do nothing Loss aversion (people are more sensitive to losses)

16 Expected Utility Theory and Its Paradoxes Prospect Theory Paradoxes/Puzzles with EUT Allais paradox: Allais (1953)

17 Expected Utility Theory and Its Paradoxes Prospect Theory Paradoxes/Puzzles with EUT Allais paradox: Allais (1953) Ellesberg paradox: Ellesberg (1961)

18 Expected Utility Theory and Its Paradoxes Prospect Theory Paradoxes/Puzzles with EUT Allais paradox: Allais (1953) Ellesberg paradox: Ellesberg (1961) Friedman and Savage puzzle: Friedman and Savage (1948)

19 Expected Utility Theory and Its Paradoxes Prospect Theory Paradoxes/Puzzles with EUT Allais paradox: Allais (1953) Ellesberg paradox: Ellesberg (1961) Friedman and Savage puzzle: Friedman and Savage (1948) Equity premium puzzle: Mehra and Prescott (1985)

20 Expected Utility Theory and Its Paradoxes Prospect Theory Key Elements of Prospect Theory Kahneman and Tversky (1979, 1992): (cumulative) prospect theory (CPT); incorporate human behaviors and psychology into decision-making process; Nobel prize 2002

21 Expected Utility Theory and Its Paradoxes Prospect Theory Key Elements of Prospect Theory Kahneman and Tversky (1979, 1992): (cumulative) prospect theory (CPT); incorporate human behaviors and psychology into decision-making process; Nobel prize 2002 A reference point (or neutral outcome/benchmark/breakeven/status quo) in wealth that defines gains and losses

22 Expected Utility Theory and Its Paradoxes Prospect Theory Key Elements of Prospect Theory Kahneman and Tversky (1979, 1992): (cumulative) prospect theory (CPT); incorporate human behaviors and psychology into decision-making process; Nobel prize 2002 A reference point (or neutral outcome/benchmark/breakeven/status quo) in wealth that defines gains and losses A value (utility) function, concave for gains, convex for losses, and steeper for losses than for gains

23 Expected Utility Theory and Its Paradoxes Prospect Theory Key Elements of Prospect Theory Kahneman and Tversky (1979, 1992): (cumulative) prospect theory (CPT); incorporate human behaviors and psychology into decision-making process; Nobel prize 2002 A reference point (or neutral outcome/benchmark/breakeven/status quo) in wealth that defines gains and losses A value (utility) function, concave for gains, convex for losses, and steeper for losses than for gains A probability distortion that is a nonlinear transformation of the probability scale

24 Expected Utility Theory and Its Paradoxes Prospect Theory S-shaped Function u(x) o x

25 Expected Utility Theory and Its Paradoxes Prospect Theory Probability Distortion Function 1 T(s) p

26 Expected Utility Theory and Its Paradoxes Prospect Theory KT s Utility and Distortions Kahneman and Tversky (1992) suggest the following Utility function u(x) = where α = β = 0.88, k = 2.25 { x α, x 0, k( x) β, x < 0

27 Expected Utility Theory and Its Paradoxes Prospect Theory KT s Utility and Distortions Kahneman and Tversky (1992) suggest the following Utility function u(x) = where α = β = 0.88, k = 2.25 Probability distortion functions { x α, x 0, k( x) β, x < 0 T + (p) = T (p) = where γ = 0.61, δ = 0.69 p γ (p γ +(1 p) γ ) 1/γ p δ (p δ +(1 p) δ ) 1/δ

28 A Continuous-Time Economy An economy in which m + 1 securities traded continuously

29 A Continuous-Time Economy An economy in which m + 1 securities traded continuously Market randomness described by a complete filtered probability space (Ω, F, {F t } t 0, P) along with an IR m -valued, F t -adapted standard Brownian motion W(t) = (W 1 (t),, W m (t)) with {F t } t 0 generated by W( )

30 A Continuous-Time Economy An economy in which m + 1 securities traded continuously Market randomness described by a complete filtered probability space (Ω, F, {F t } t 0, P) along with an IR m -valued, F t -adapted standard Brownian motion W(t) = (W 1 (t),, W m (t)) with {F t } t 0 generated by W( ) A bond (or a bank account) whose price process S 0 (t) satisfies ds 0 (t) = r(t)s 0 (t)dt; S 0 (0) = s 0

31 A Continuous-Time Economy An economy in which m + 1 securities traded continuously Market randomness described by a complete filtered probability space (Ω, F, {F t } t 0, P) along with an IR m -valued, F t -adapted standard Brownian motion W(t) = (W 1 (t),, W m (t)) with {F t } t 0 generated by W( ) A bond (or a bank account) whose price process S 0 (t) satisfies ds 0 (t) = r(t)s 0 (t)dt; S 0 (0) = s 0 m stocks whose price processes S 1 (t), S m (t) satisfy stochastic differential equation (SDE) ds i (t) = S i (t) { µ i (t)dt + m σ ij (t)dw j (t) } ; S i (0) = s i j=1

32 Tame Portfolios Let σ(t) := (σ ij (t)) m m B(t) := (µ 1 (t) r(t),, µ m (t) r(t))

33 Tame Portfolios Let σ(t) := (σ ij (t)) m m B(t) := (µ 1 (t) r(t),, µ m (t) r(t)) An F t -progressively measurable process π(t) = (π 1 (t),, π m (t)) represents a (monetary) portfolio, where π i (t) is the capital amount invested in stock i

34 Tame Portfolios Let σ(t) := (σ ij (t)) m m B(t) := (µ 1 (t) r(t),, µ m (t) r(t)) An F t -progressively measurable process π(t) = (π 1 (t),, π m (t)) represents a (monetary) portfolio, where π i (t) is the capital amount invested in stock i A portfolio π( ) is admissible if T 0 σ(t) π(t) 2 dt < +, T 0 B(t) π(t) dt < +, a.s.

35 Tame Portfolios Let σ(t) := (σ ij (t)) m m B(t) := (µ 1 (t) r(t),, µ m (t) r(t)) An F t -progressively measurable process π(t) = (π 1 (t),, π m (t)) represents a (monetary) portfolio, where π i (t) is the capital amount invested in stock i A portfolio π( ) is admissible if T 0 σ(t) π(t) 2 dt < +, T 0 B(t) π(t) dt < +, a.s. An agent has an initial endowment x 0 and an reference point 0 (for simplicity)

36 Tame Portfolios (Cont d) Wealth process x( ) follows the wealth equation { dx(t) = [r(t)x(t) + B(t) π(t)]dt + π(t) σ(t)dw(t) x(0) = x 0

37 Tame Portfolios (Cont d) Wealth process x( ) follows the wealth equation { dx(t) = [r(t)x(t) + B(t) π(t)]dt + π(t) σ(t)dw(t) x(0) = x 0 An admissible portfolio π( ) is called to be tame if the corresponding wealth process x( ) is uniformly lower bounded

38 Market Assumptions Market assumptions: (i) c IR such that T 0 r(t)dt c, a.s.

39 Market Assumptions Market assumptions: (i) c IR such that T 0 r(t)dt c, a.s. (ii) T 0 [ m i=1 b i(t) + m i,j=1 σ ij(t) 2 ]dt < +, a.s.

40 Market Assumptions Market assumptions: (i) c IR such that T 0 r(t)dt c, a.s. (ii) T 0 [ m i=1 b i(t) + m i,j=1 σ ij(t) 2 ]dt < +, a.s. (iii) Rank (σ(t)) = m, a.e.t [0, T], a.s.

41 Market Assumptions Market assumptions: (i) c IR such that T 0 r(t)dt c, a.s. (ii) T 0 [ m i=1 b i(t) + m i,j=1 σ ij(t) 2 ]dt < +, a.s. (iii) Rank (σ(t)) = m, a.e.t [0, T], a.s. (iv) There exists an IR m -valued, uniformly bounded, F t -progressively measurable process θ( ) such that σ(t)θ(t) = B(t)

42 Pricing Kernel Define the pricing kernel { ρ(t) := exp t 0 [r(s) + 12 θ(s) 2 ] ds t 0 } θ(s) dw(s)

43 Pricing Kernel Define the pricing kernel { ρ(t) := exp t Denote ρ := ρ(t), and 0 [r(s) + 12 θ(s) 2 ] ds t 0 } θ(s) dw(s) ρ esssup ρ := sup {a IR : P {ρ > a} > 0}, ρ essinf ρ := inf {a IR : P {ρ < a} > 0}

44 Pricing Kernel Define the pricing kernel { ρ(t) := exp t Denote ρ := ρ(t), and 0 [r(s) + 12 θ(s) 2 ] ds t 0 } θ(s) dw(s) ρ esssup ρ := sup {a IR : P {ρ > a} > 0}, ρ essinf ρ := inf {a IR : P {ρ < a} > 0} We assume that ρ admits no atom

45 Behavioral Portfolio Selection Model Define the behavioral criterion for a random payoff X: V (X) = V + (X + ) V (X ) with V + (Y ) = + T 0 + (P {u + (Y ) > y})dy, V (Y ) = + T 0 (P {u (Y ) > y})dy where u ± ( ) : IR + IR + are strictly increasing, concave and u ± (0) = 0 T ± ( ) : [0, 1] [0, 1] are strictly increasing, T ± (0) = 0, T ± (1) = 1 and T ± (p) > p when p close to 0 u ± ( ), T ± ( ) all twice differentiable. Furthermore, u +(0) = +, u +(+ ) = 0 (Inada s condition)

46 Behavioral Portfolio Selection Model (Cont d) V (X) = Eu(X) if T ± (x) = x and u(x) := u + (x + )1 x 0 u (x )1 x 0

47 Behavioral Portfolio Selection Model (Cont d) V (X) = Eu(X) if T ± (x) = x and u(x) := u + (x + )1 x 0 u (x )1 x 0 Consider the portfolio selection problem with behavioral criterion Max V (x(t)) s.t. (x( ), π( )) a tame admissible wealth-portfolio pair. (1)

48 History Burgeoning research interests in incorporating PT into portfolio choice: mainly single period

49 History Burgeoning research interests in incorporating PT into portfolio choice: mainly single period Behavioral portfolio choice in continuous time: nil except one paper (Berkelaar, Kouwenberg and Post 2004) where a very special S-shaped utility function is considered, but no probability distortion

50 A Backward Approach Consider a static optimization problem in terminal wealth Maximize V (X) = V + (X + ) V (X ) subject to E[ρX] = x 0 X is lower bounded and F T -measurable. (2)

51 A Backward Approach Consider a static optimization problem in terminal wealth Maximize V (X) = V + (X + ) V (X ) subject to E[ρX] = x 0 X is lower bounded and F T -measurable. (2) Theorem X solves (2) iff its replicating portfolio π ( ) solves (1).

52 Major Difficulties An overall S-shaped utility function the problem may not even be well-posed non-convex optimization even if well-posed

53 Major Difficulties An overall S-shaped utility function the problem may not even be well-posed non-convex optimization even if well-posed Probability distortions T ± P is a capacity, a non-additive measure as opposed to probability the definition of V involves Choquet integrals nonlinear expectations

54 Major Difficulties An overall S-shaped utility function the problem may not even be well-posed non-convex optimization even if well-posed Probability distortions T ± P is a capacity, a non-additive measure as opposed to probability the definition of V involves Choquet integrals nonlinear expectations Conventional approaches (stochastic control, dynamic programming, convex duality, martingale method...) fall apart

55 Major Difficulties An overall S-shaped utility function the problem may not even be well-posed non-convex optimization even if well-posed Probability distortions T ± P is a capacity, a non-additive measure as opposed to probability the definition of V involves Choquet integrals nonlinear expectations Conventional approaches (stochastic control, dynamic programming, convex duality, martingale method...) fall apart Lack of study in literature: not because the problem is uninteresting or unimportant; it is because the problem is difficult (we thought)

56 Ill-posedness A maximization problem is called ill-posed if its supremum is +

57 Ill-posedness A maximization problem is called ill-posed if its supremum is + An ill-posed problem is mis-formulated: trade-off (or incentive) is not set right and hence one can always push the objective value to arbitrarily high

58 Ill-posedness A maximization problem is called ill-posed if its supremum is + An ill-posed problem is mis-formulated: trade-off (or incentive) is not set right and hence one can always push the objective value to arbitrarily high Well-posedness is a modeling issue; can be technically challenging; has not received adequate attention in literature

59 Ill-posedness A maximization problem is called ill-posed if its supremum is + An ill-posed problem is mis-formulated: trade-off (or incentive) is not set right and hence one can always push the objective value to arbitrarily high Well-posedness is a modeling issue; can be technically challenging; has not received adequate attention in literature In classical portfolio selection literature the utility function is nice and the expectation is nice, so the problem is well-posed... in most cases (Jin, Xu and Zhou, Math Finance 2008, for exceptions)

60 Ill-posedness Case 1 Theorem If there exists a nonnegative F T -measurable random variable X such that E[ρX] < + and V + (X) = +, then (2) is ill-posed.

61 Ill-posedness Case 1 Theorem If there exists a nonnegative F T -measurable random variable X such that E[ρX] < + and V + (X) = +, then (2) is ill-posed. Example. Let ρ be lognormal with pdf F, T + (t) := t 1/4 on [0, 1/2], u + (x) := x 1/2. Take X := (F(ρ)) 1/2 1. Then E[ρX] 4 2/3 (Eρ 3 ) 1/3 < +, V + (X) + 2 (2y 2 ) 1/2 dy = +.

62 Ill-posedness Case 1 Theorem If there exists a nonnegative F T -measurable random variable X such that E[ρX] < + and V + (X) = +, then (2) is ill-posed. Example. Let ρ be lognormal with pdf F, T + (t) := t 1/4 on [0, 1/2], u + (x) := x 1/2. Take X := (F(ρ)) 1/2 1. Then E[ρX] 4 2/3 (Eρ 3 ) 1/3 < +, V + (X) + 2 (2y 2 ) 1/2 dy = +. Remark. This case bears no relevance whatsoever with the negative part (i.e., the part on the loss side). It is a case where, on the gain side, personal taste (utility function), psychology (probability distortion) and investment opportunities (market) do not coordinate well.

63 Ill-posedness Case 2 Theorem If u + (+ ) = +, ρ = +, and T (x) = x, then (2) is ill-posed.

64 Ill-posedness Case 2 Theorem If u + (+ ) = +, ρ = +, and T (x) = x, then (2) is ill-posed. Remark. A probability distortion on losses is necessary for the well-posedness if the utility on gains can go arbitrarily large.

65 Deriving Optimal Solution: Divide and Conquer We do divide and conquer

66 Deriving Optimal Solution: Divide and Conquer We do divide and conquer Step 1: divide into two problems: one concerns the positive part of X and the other the negative part of X

67 Deriving Optimal Solution: Divide and Conquer We do divide and conquer Step 1: divide into two problems: one concerns the positive part of X and the other the negative part of X Step 2: combine them together via solving another problem

68 Deriving Optimal Solution: Divide and Conquer We do divide and conquer Step 1: divide into two problems: one concerns the positive part of X and the other the negative part of X Step 2: combine them together via solving another problem Positive Part Problem (PPP): A problem with parameters (A, x + ): Maximize V + (X) = + 0 T + (P {u + (X) > y})dy E[ρX] = x +, subject to X 0, a.s., X = 0, a.s. on A C, (3) where x + x + 0 and A F T with P(A) 1 Define its optimal value to be v + (A, x + )

69 Divide and Conquer (Cont d) Negative Part Problem (NPP): A problem with parameters (A, x + ): Minimize V (X) = + 0 T (P {u (X) > y})dy E[ρX] = x + x 0, X 0, a.s., subject to X = 0, a.s. on A, X is bounded a.s., where x + x + 0 and A F T with P(A) 1 Define its optimal value to be v (A, x + ) (4)

70 Divide and Conquer (Cont d) Negative Part Problem (NPP): A problem with parameters (A, x + ): Minimize V (X) = + 0 T (P {u (X) > y})dy E[ρX] = x + x 0, X 0, a.s., subject to X = 0, a.s. on A, X is bounded a.s., where x + x + 0 and A F T with P(A) 1 Define its optimal value to be v (A, x + ) Then, in Step 2 we solve Maximize v + (A, x + ) v (A, x + ) A F T, x + x + 0, subject to x + = 0 when P(A) = 0, x + = x 0 when P(A) = 1. (4) (5)

71 Yes It Works Theorem We have the following conclusions. (i) Problem (2) is ill-posed iff Problem (5) is ill-posed. (ii) Given X, define A := {ω : X 0} and x + := E[ρ(X ) + ]. Then X is optimal for Problem (2) iff (A, x +) are optimal for Problem (5) and (X ) + and (X ) are respectively optimal for Problems (3) and (4) with parameters (A, x +).

72 Solution Flow Solve PPP for any parameter (A, x + ), getting optimal solution X + (A, x + ) and optimal value v + (A, x + )

73 Solution Flow Solve PPP for any parameter (A, x + ), getting optimal solution X + (A, x + ) and optimal value v + (A, x + ) Solve NPP for any parameter (A, x + ), getting optimal solution X (A, x + ) and optimal value v (A, x + )

74 Solution Flow Solve PPP for any parameter (A, x + ), getting optimal solution X + (A, x + ) and optimal value v + (A, x + ) Solve NPP for any parameter (A, x + ), getting optimal solution X (A, x + ) and optimal value v (A, x + ) Solve Step 2 problem and get optimal (A, x +)

75 Solution Flow Solve PPP for any parameter (A, x + ), getting optimal solution X + (A, x + ) and optimal value v + (A, x + ) Solve NPP for any parameter (A, x + ), getting optimal solution X (A, x + ) and optimal value v (A, x + ) Solve Step 2 problem and get optimal (A, x +) Then X + (A, x +) X (A, x +) solves the behavioral model

76 Simplification Step 2 problem (5) optimizes over a set of random events A: hard to handle

77 Simplification Step 2 problem (5) optimizes over a set of random events A: hard to handle Theorem For any feasible pair (A, x + ) of Problem (5), there exists c [ρ, ρ] such that Ā := {ω : ρ c} satisfies v + (Ā, x +) v (Ā, x +) v + (A, x + ) v (A, x + ). (6)

78 Simplification Step 2 problem (5) optimizes over a set of random events A: hard to handle Theorem For any feasible pair (A, x + ) of Problem (5), there exists c [ρ, ρ] such that Ā := {ω : ρ c} satisfies v + (Ā, x +) v (Ā, x +) v + (A, x + ) v (A, x + ). (6) Use v + (c, x + ) and v (c, x+) to denote v + ({ω : ρ c}, x + ) and v ({ω : ρ c}, x + ) respectively

79 Simplification (Cont d) Problem (5) is equivalent to Maximize v + (c, x + ) v (c, x + ) ρ c ρ, x + x + 0, subject to x + = 0 when c = ρ, x + = x 0 when c = ρ. (7)

80 Solving PPP Difficulty: Non-concavity due to distortion

81 Solving PPP Difficulty: Non-concavity due to distortion Way out: change variable from X (r.v.) to its quantile G 1 (function)

82 Solving PPP Difficulty: Non-concavity due to distortion Way out: change variable from X (r.v.) to its quantile G 1 (function) Maximize E [ ( u + G 1 (Z) ) T +(1 Z) ] Subject to E [ Fρ 1 (1 Z)G 1 (Z) ] = x + G 1 is a quantile function

83 Solving PPP Difficulty: Non-concavity due to distortion Way out: change variable from X (r.v.) to its quantile G 1 (function) Maximize E [ ( u + G 1 (Z) ) T +(1 Z) ] Subject to E [ Fρ 1 (1 Z)G 1 (Z) ] = x + G 1 is a quantile function where Z U(0, 1)

84 Solving PPP Difficulty: Non-concavity due to distortion Way out: change variable from X (r.v.) to its quantile G 1 (function) Maximize E [ ( u + G 1 (Z) ) T +(1 Z) ] Subject to E [ Fρ 1 (1 Z)G 1 (Z) ] = x + G 1 is a quantile function where Z U(0, 1) Ground: Any optimal solution X of PPP is represented as X = G 1 (1 F ρ (ρ)) where G 1 is the quantile of X.

85 Solving NPP Difficulty: use the same quantile trick, but minimizing a concave function is annoying!

86 Solving NPP Difficulty: use the same quantile trick, but minimizing a concave function is annoying! Minima of a concave function with constraints must be corener points

87 Solving NPP Difficulty: use the same quantile trick, but minimizing a concave function is annoying! Minima of a concave function with constraints must be corener points What are the corner points in an infinite dimension?

88 Solving NPP Difficulty: use the same quantile trick, but minimizing a concave function is annoying! Minima of a concave function with constraints must be corener points What are the corner points in an infinite dimension? Step functions!

89 Grand Solution Introduce an auxiliary problem Maximize v + (c, x + ) u ( x + x 0 E[ρ1 ρ>c ] )T (1 F(c)) { ρ c ρ, x+ x + subject to 0, x + = 0 when c = ρ, x + = x 0 when c = ρ, (8)

90 Grand Solution Introduce an auxiliary problem Maximize v + (c, x + ) u ( x + x 0 E[ρ1 ρ>c ] )T (1 F(c)) { ρ c ρ, x+ x + subject to 0, x + = 0 when c = ρ, x + = x 0 when c = ρ, (8) Theorem Suppose that u ( ) is strictly concave at 0. (i) If (c, x +) is optimal for (8), then ( ) X := (u +) 1 λρ T + (F(ρ)) 1 ρ c x + x 0 E[ρ1 ρ>c ] 1 ρ>c solves (2). (ii) If (8) admits no optimal solution then (2) admits no optimal solution.

91 A Combined Binary Option The final solution is beautifully simple:

92 A Combined Binary Option The final solution is beautifully simple: The optimal terminal wealth having a gain or a loss is completely determined by the state density price being lower (good state) or higher (bad state) than a single threshold, c

93 A Combined Binary Option The final solution is beautifully simple: The optimal terminal wealth having a gain or a loss is completely determined by the state density price being lower (good state) or higher (bad state) than a single threshold, c This threshold can be obtained by solving a simple mathematical programming problem (8)

94 A Combined Binary Option The final solution is beautifully simple: The optimal terminal wealth having a gain or a loss is completely determined by the state density price being lower (good state) or higher (bad state) than a single threshold, c This threshold can be obtained by solving a simple mathematical programming problem (8) The optimal terminal wealth is the payoff of a combination of two binary options, which can be easily priced

95 A Combined Binary Option The final solution is beautifully simple: The optimal terminal wealth having a gain or a loss is completely determined by the state density price being lower (good state) or higher (bad state) than a single threshold, c This threshold can be obtained by solving a simple mathematical programming problem (8) The optimal terminal wealth is the payoff of a combination of two binary options, which can be easily priced The optimal strategy is a gambling policy, betting on a good state of the world while accepting a fixed loss on a bad state

96 Example: Two-Piece Power Utilities Take u + (x) = x α, u (x) = kx α with α (0, 1) and k > 0 (taken in Kahneman and Tversky 1992)

97 Example: Two-Piece Power Utilities Take u + (x) = x α, u (x) = kx α with α (0, 1) and k > 0 (taken in Kahneman and Tversky 1992) lnρ N(µ, σ) with σ > 0

98 Example: Two-Piece Power Utilities Take u + (x) = x α, u (x) = kx α with α (0, 1) and k > 0 (taken in Kahneman and Tversky 1992) lnρ N(µ, σ) with σ > 0 u +(x) = αx α 1, (u +) 1 (y) = (y/α) 1/(α 1), u + ((u +) 1 (y)) = (y/α) α/(α 1), ρ = 0, ρ = +, and F(x) = N((lnx µ)/σ)

99 Example: Two-Piece Power Utilities Take u + (x) = x α, u (x) = kx α with α (0, 1) and k > 0 (taken in Kahneman and Tversky 1992) lnρ N(µ, σ) with σ > 0 u +(x) = αx α 1, (u +) 1 (y) = (y/α) 1/(α 1), u + ((u +) 1 (y)) = (y/α) α/(α 1), ρ = 0, ρ = +, and F(x) = N((lnx µ)/σ) Solution to PPP is X+(c, x + ) = x + j(c) ρ v + (c, x + ) = j(c) 1 α x α + ( T ) + (F(ρ)) 1/(1 α) 1ρ c where [ (T ) j(c) := E + (F(ρ)) 1/(1 α) ρ1 ρ c] > 0, 0 < c + ρ

100 Example: Two-Piece Power Utilities (Cont d) Problem (8) specializes to Maximize { v(c, x + ) = j(c) 1 α [x α + k(c)(x + x 0 ) α ] 0 c +, x+ x + subject to 0, x + = 0 when c = 0, x + = x 0 when c = +. (9) where k(c) := kt (1 F(c)) j(c) 1 α (E[ρ1 ρ>c > 0 c > 0 ]) α

101 Example: Two-Piece Power Utilities (Cont d) Problem (8) specializes to Maximize { v(c, x + ) = j(c) 1 α [x α + k(c)(x + x 0 ) α ] 0 c +, x+ x + subject to 0, x + = 0 when c = 0, x + = x 0 when c = +. (9) where k(c) := kt (1 F(c)) j(c) 1 α (E[ρ1 ρ>c > 0 c > 0 ]) α Define ( ) T (1 F(c)) 1 k 0 := inf c>0 j(c) 1 α (E[ρ1 ρ>c ]) α

102 Solution - Case I Theorem Assume that x 0 0. (i) If k k 0, then the optimal portfolio for Problem (1) is the replicating portfolio for the contingent claim X = x 0 j(+ ) ( T ) + (F(ρ)) 1/(1 α). (ii) If k < k 0, then Problem (1) is ill-posed. ρ

103 Solution - Case I Theorem Assume that x 0 0. (i) If k k 0, then the optimal portfolio for Problem (1) is the replicating portfolio for the contingent claim X = x 0 j(+ ) ( T ) + (F(ρ)) 1/(1 α). (ii) If k < k 0, then Problem (1) is ill-posed. Remark. If investor starts with a gain situation and sufficiently loss averse, then it is optimal to spend x 0 buying a contingent claim, reminiscent of a classical utility maximizing agent (albeit with a distorted amount) ρ

104 Solution - Case II Theorem Assume that x 0 < 0. (i) If k > k 0, then Problem (1) is well-posed. Moreover, (1) admits an optimal portfolio iff [ (kt ) 1/(1 α) (1 F(c)) argmin c 0 (E[ρ1 ρ>c ]) α j(c)] Ø. (10) Furthermore, if c > 0 is one of the minimizers in (10), then the optimal portfolio is the one to replicate X = x + ϕ(c ) ( T + (F(ρ)) ρ ) 1/(1 α) 1 ρ c x + x 0 E[ρ1 ρ>c ] 1 ρ>c, where x x + := 0 k(c ) 1/(1 α) 1 ; and if c = 0 is the unique minimizer in (10), then the unique optimal portfolio is the one to replicate X = x 0 Eρ.

105 Solution - Case II (Cont d) Theorem (cont d) (ii) If k = k 0, then the supremum value of Problem (1) is 0, which is however not achieved by any tame portfolio. (iii) If k < k 0, then Problem (1) is ill-posed.

106 Solution - Case II (Cont d) Theorem (cont d) (ii) If k = k 0, then the supremum value of Problem (1) is 0, which is however not achieved by any tame portfolio. (iii) If k < k 0, then Problem (1) is ill-posed. Remark. If investor starts with a loss, then optimal strategy is a gambling policy gambling his way out of the hole, by raising additional capital to purchase a claim that delivers a higher payoff in the case of a good state and incurs a fixed loss in the case of a bad one.

107 What about Single Period and Incomplete Market? Perversely, less general results have been obtained for the single period setting

108 What about Single Period and Incomplete Market? Perversely, less general results have been obtained for the single period setting Well-posedness: We have proved that the model is well posed if and only if the loss aversion level k is greater than a fixed quantity

109 What about Single Period and Incomplete Market? Perversely, less general results have been obtained for the single period setting Well-posedness: We have proved that the model is well posed if and only if the loss aversion level k is greater than a fixed quantity Solution: Explicit solutions obtained for two cases: reference point coincides with the risk-free return linear utilities

110 What about Single Period and Incomplete Market? Perversely, less general results have been obtained for the single period setting Well-posedness: We have proved that the model is well posed if and only if the loss aversion level k is greater than a fixed quantity Solution: Explicit solutions obtained for two cases: reference point coincides with the risk-free return linear utilities Equity premium puzzle

111 What about Single Period and Incomplete Market? Perversely, less general results have been obtained for the single period setting Well-posedness: We have proved that the model is well posed if and only if the loss aversion level k is greater than a fixed quantity Solution: Explicit solutions obtained for two cases: reference point coincides with the risk-free return linear utilities Equity premium puzzle Continuous-Time Incomplete Market has also been tackled recently

112 Conclusions Portfolio selection models with behavioral criteria (S-shaped utilities and probability distortions) are formulated and studied

113 Conclusions Portfolio selection models with behavioral criteria (S-shaped utilities and probability distortions) are formulated and studied The ill-posedness is more a rule than an exception in such a behavioral model

114 Conclusions Portfolio selection models with behavioral criteria (S-shaped utilities and probability distortions) are formulated and studied The ill-posedness is more a rule than an exception in such a behavioral model To be well-posed the investor must be sufficiently loss averse and the personal preferences and market opportunities must be well coordinated

115 Conclusions (Cont d) The continuous-time complete market model is solved thoroughly - the optimal terminal payoff is related to a combined binary option characterized by a single number

116 Conclusions (Cont d) The continuous-time complete market model is solved thoroughly - the optimal terminal payoff is related to a combined binary option characterized by a single number The work is meant to be initiating and inspiring rather than exhaustive and conclusive

117 Conclusions (Cont d) The continuous-time complete market model is solved thoroughly - the optimal terminal payoff is related to a combined binary option characterized by a single number The work is meant to be initiating and inspiring rather than exhaustive and conclusive Hopefully it will shed lights on solving many puzzles including the equity premium puzzle

118 Credits H. Jin and X.Y. Zhou, Behavioral Portfolio Selection in Continuous Time, Mathematical Finance, to appear X. He and X.Y. Zhou, Behavioral Portfolio Choice: Model, Theory, and Equity Premium Puzzle, working paper X. He and X.Y. Zhou, Behavioral Portfolio Selection in an Incomplete Market, work in progress