Prospect Theory Model and Solution Conclusions
Prospect Theory: A New Paradigm for Portfolio
Choice
Xun Yu Zhou/Oxford
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
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
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Expected Utility Theory and Its Paradoxes Prospect Theory
Expected Utility Theory
Expected Utility Theory (EUT): dominant model for decision
making under uncertainty
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
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
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
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
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Expected Utility Theory and Its Paradoxes Prospect Theory
Anomalies in Human Behaviors
Substantial evidences suggest systematic violation of EUT
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
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)
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
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)
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
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)
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
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
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
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)
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
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
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
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)
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
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
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
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)
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Expected Utility Theory and Its Paradoxes Prospect Theory
Paradoxes/Puzzles with EUT
Allais paradox: Allais (1953)
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Expected Utility Theory and Its Paradoxes Prospect Theory
Paradoxes/Puzzles with EUT
Allais paradox: Allais (1953)
Ellesberg paradox: Ellesberg (1961)
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
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)
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
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)
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
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
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
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
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
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
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
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
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Expected Utility Theory and Its Paradoxes Prospect Theory
S-shaped Function
u(x)
o
Xun Yu Zhou/Oxford
x
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Expected Utility Theory and Its Paradoxes Prospect Theory
Probability Distortion Function
T(s)
1
0.5
0
0.5
1
p
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Expected Utility Theory and Its Paradoxes Prospect Theory
KT’s Utility and Distortions
Kahneman and Tversky (1992) suggest the following
Utility function
u(x) =
xα ,
−k(−x)β ,
x ≥ 0,
x<0
where α = β = 0.88, k = 2.25
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Expected Utility Theory and Its Paradoxes Prospect Theory
KT’s Utility and Distortions
Kahneman and Tversky (1992) suggest the following
Utility function
u(x) =
xα ,
−k(−x)β ,
x ≥ 0,
x<0
where α = β = 0.88, k = 2.25
Probability distortion functions
T+ (p) =
pγ
(pγ +(1−p)γ )1/γ
T− (p) =
(pδ +(1−p)δ )1/δ
pδ
where γ = 0.61, δ = 0.69
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
A Continuous-Time Economy
An economy in which m + 1 securities traded continuously
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
A Continuous-Time Economy
An economy in which m + 1 securities traded continuously
Market randomness described by a complete filtered
probability space (Ω, F , {Ft }t≥0 , P ) along with an
IRm -valued, Ft -adapted standard Brownian motion
W (t) = (W 1 (t), · · · , W m (t))′ with {Ft }t≥0 generated by
W (·)
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
A Continuous-Time Economy
An economy in which m + 1 securities traded continuously
Market randomness described by a complete filtered
probability space (Ω, F , {Ft }t≥0 , P ) along with an
IRm -valued, Ft -adapted standard Brownian motion
W (t) = (W 1 (t), · · · , W m (t))′ with {Ft }t≥0 generated by
W (·)
A bond (or a bank account) whose price process S0 (t) satisfies
dS0 (t) = r(t)S0 (t)dt; S0 (0) = s0
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
A Continuous-Time Economy
An economy in which m + 1 securities traded continuously
Market randomness described by a complete filtered
probability space (Ω, F , {Ft }t≥0 , P ) along with an
IRm -valued, Ft -adapted standard Brownian motion
W (t) = (W 1 (t), · · · , W m (t))′ with {Ft }t≥0 generated by
W (·)
A bond (or a bank account) whose price process S0 (t) satisfies
dS0 (t) = r(t)S0 (t)dt; S0 (0) = s0
m stocks whose price processes S1 (t), · · · Sm (t) satisfy
stochastic differential equation (SDE)
m
X
σij (t)dW j (t) ; Si (0) = si
dSi (t) = Si (t) µi (t)dt +
j=1
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Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
Tame Portfolios
Let
σ(t) := (σij (t))m×m
B(t) := (µ1 (t) − r(t), · · · , µm (t) − r(t))′
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
Tame Portfolios
Let
σ(t) := (σij (t))m×m
B(t) := (µ1 (t) − r(t), · · · , µm (t) − r(t))′
An Ft -progressively measurable process
π(t) = (π1 (t), · · · , πm (t))′ represents a (monetary) portfolio,
where πi (t) is the capital amount invested in stock i
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
Tame Portfolios
Let
σ(t) := (σij (t))m×m
B(t) := (µ1 (t) − r(t), · · · , µm (t) − r(t))′
An Ft -progressively measurable process
π(t) = (π1 (t), · · · , πm (t))′ represents a (monetary) portfolio,
where πi (t) is the capital amount invested in stock i
A portfolio π(·) is admissible if
Z
T
′
2
|σ(t) π(t)| dt < +∞,
0
Z
T
|B(t)′ π(t)|dt < +∞, a.s.
0
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
Tame Portfolios
Let
σ(t) := (σij (t))m×m
B(t) := (µ1 (t) − r(t), · · · , µm (t) − r(t))′
An Ft -progressively measurable process
π(t) = (π1 (t), · · · , πm (t))′ represents a (monetary) portfolio,
where πi (t) is the capital amount invested in stock i
A portfolio π(·) is admissible if
Z
T
′
2
|σ(t) π(t)| dt < +∞,
0
Z
T
|B(t)′ π(t)|dt < +∞, a.s.
0
An agent has an initial endowment x0 and an reference point
0 (for simplicity)
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
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) = x0
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
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) = x0
An admissible portfolio π(·) is called to be tame if the
corresponding wealth process x(·) is uniformly lower bounded
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
Market Assumptions
Market assumptions:
(i) ∃ c ∈ IR such that
RT
0
r(t)dt ≥ c, a.s.
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
Market Assumptions
Market assumptions:
RT
(i) ∃ c ∈ IR such that 0 r(t)dt ≥ c, a.s.
RT P
Pm
2
(ii) 0 [ m
i=1 |bi (t)| +
i,j=1 |σij (t)| ]dt < +∞, a.s.
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
Market Assumptions
Market assumptions:
RT
(i) ∃ c ∈ IR such that 0 r(t)dt ≥ c, a.s.
RT P
Pm
2
(ii) 0 [ m
i=1 |bi (t)| +
i,j=1 |σij (t)| ]dt < +∞, a.s.
(iii) Rank (σ(t)) = m, a.e.t ∈ [0, T ], a.s.
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
Market Assumptions
Market assumptions:
RT
(i) ∃ c ∈ IR such that 0 r(t)dt ≥ c, a.s.
RT P
Pm
2
(ii) 0 [ m
i=1 |bi (t)| +
i,j=1 |σij (t)| ]dt < +∞, a.s.
(iii) Rank (σ(t)) = m, a.e.t ∈ [0, T ], a.s.
(iv) There exists an IRm -valued, uniformly bounded,
Ft -progressively measurable process θ(·) such that
σ(t)θ(t) = B(t)
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Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
Pricing Kernel
Define the pricing kernel
Z t
Z t
1
r(s) + |θ(s)|2 ds −
ρ(t) := exp −
θ(s)′ dW (s)
2
0
0
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
Pricing Kernel
Define the pricing kernel
Z t
Z t
1
r(s) + |θ(s)|2 ds −
ρ(t) := exp −
θ(s)′ dW (s)
2
0
0
Denote ρ := ρ(T ), and
ρ̄ ≡ esssup ρ := sup {a ∈ IR : P {ρ > a} > 0} ,
ρ ≡ essinf ρ := inf {a ∈ IR : P {ρ < a} > 0}
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
Pricing Kernel
Define the pricing kernel
Z t
Z t
1
r(s) + |θ(s)|2 ds −
ρ(t) := exp −
θ(s)′ dW (s)
2
0
0
Denote ρ := ρ(T ), and
ρ̄ ≡ 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
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
Behavioral Portfolio Selection Model
Define the behavioral criterion for a random payoff X:
V (X) = V+ (X + ) − V− (X − ) with
where
R +∞
V+ (Y ) = 0 T+ (P {u+ (Y ) > y})dy,
R +∞
V− (Y ) = 0 T− (P {u− (Y ) > y})dy
u± (·) : IR+ 7→ IR+ are strictly increasing, concave and
u± (0) = 0
T± (·) : [0, 1] 7→ [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)
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Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
Behavioral Portfolio Selection Model (Cont’d)
V (X) = Eu(X) if T± (x) = x and
u(x) := u+ (x+ )1x≥0 − u− (x− )1x≤0
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
Behavioral Portfolio Selection Model (Cont’d)
V (X) = Eu(X) if T± (x) = x and
u(x) := u+ (x+ )1x≥0 − u− (x− )1x≤0
Consider the portfolio selection problem with behavioral
criterion
Max V (x(T ))
s.t.
(x(·), π(·)) a tame admissible wealth-portfolio pair.
(1)
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Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
History
Burgeoning research interests in incorporating PT into
portfolio choice: mainly single period
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
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
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Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
A Backward Approach
Consider a static optimization problem in terminal wealth
Maximize V (X) = V+ (X + ) − V− (X − )
subject to E[ρX] = x0
X is lower bounded and FT -measurable.
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(2)
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
A Backward Approach
Consider a static optimization problem in terminal wealth
Maximize V (X) = V+ (X + ) − V− (X − )
subject to E[ρX] = x0
X is lower bounded and FT -measurable.
(2)
Theorem
X ∗ solves (2) iff its replicating portfolio π ∗ (·) solves (1).
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Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
Major Difficulties
An overall S-shaped “utility” function
the problem may not even be well-posed
non-convex optimization even if well-posed
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Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
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
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
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
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
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)
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Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
Ill-posedness
A maximization problem is called ill-posed if its supremum is
+∞
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Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
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
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Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
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
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
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)
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Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
Ill-posedness Case 1
Theorem
If there exists a nonnegative FT -measurable random variable X
such that E[ρX] < +∞ and V+ (X) = +∞, then (2) is ill-posed.
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Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
Ill-posedness Case 1
Theorem
If there exists a nonnegative FT -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) := t1/4 on
[0, 1/2], u+ (x) := x1/2 . Take X := (F (ρ))−1/2 − 1. Then
E[ρX] ≤ 42/3 (Eρ3 )1/3 < +∞,
R +∞
V+ (X) ≥ 2 (2y 2 )−1/2 dy = +∞.
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Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
Ill-posedness Case 1
Theorem
If there exists a nonnegative FT -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) := t1/4 on
[0, 1/2], u+ (x) := x1/2 . Take X := (F (ρ))−1/2 − 1. Then
E[ρX] ≤ 42/3 (Eρ3 )1/3 < +∞,
R +∞
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.
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Market Model Ill-posedness Solution Example Single Period
Ill-posedness Case 2
Theorem
If u+ (+∞) = +∞, ρ̄ = +∞, and T− (x) = x, then (2) is ill-posed.
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Market Model Ill-posedness Solution Example Single Period
Ill-posedness Case 2
Theorem
If u+ (+∞) = +∞, ρ̄ = +∞, 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.
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Deriving Optimal Solution: Divide and Conquer
We do “divide and conquer”
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Market Model Ill-posedness Solution Example Single Period
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
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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
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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+ ):
R +∞
Maximize V
+ (X) = 0 T+ (P {u+ (X) > y})dy
 E[ρX] = x+ ,
(3)
subject to
X ≥ 0, a.s.,

X = 0, a.s. on AC ,
where x+ ≥ x+
0 and A ∈ FT with P (A) ≤ 1
Define its optimal value to be v+ (A, x+ )
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Divide and Conquer (Cont’d)
Negative Part Problem (NPP): A problem with parameters
(A, x+ ):
R +∞
Minimize V
− (X) = 0 T− (P {u− (X) > y})dy

 E[ρX] = x+ − x0 ,

X ≥ 0, a.s.,
subject to
X = 0, a.s. on A,



X is bounded a.s.,
(4)
where x+ ≥ x+
0 and A ∈ FT with P (A) ≤ 1
Define its optimal value to be v− (A, x+ )
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Divide and Conquer (Cont’d)
Negative Part Problem (NPP): A problem with parameters
(A, x+ ):
R +∞
Minimize V
− (X) = 0 T− (P {u− (X) > y})dy

 E[ρX] = x+ − x0 ,

X ≥ 0, a.s.,
subject to
X = 0, a.s. on A,



X is bounded a.s.,
(4)
where x+ ≥ x+
0 and A ∈ FT 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 ∈ FT , x+ ≥ x0 ,
subject to
x+ = 0 when P (A) = 0,

x+ = x0 when P (A) = 1.
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(5)
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Market Model Ill-posedness Solution Example Single Period
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∗+ ).
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Market Model Ill-posedness Solution Example Single Period
Solution Flow
Solve PPP for any parameter (A, x+ ), getting optimal
solution X+ (A, x+ ) and optimal value v+ (A, x+ )
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Market Model Ill-posedness Solution Example Single Period
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+ )
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Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
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∗+ )
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Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
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
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Market Model Ill-posedness Solution Example Single Period
Simplification
Step 2 problem (5) optimizes over a set of random events A: hard
to handle
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Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
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, ρ̄]
such that Ā := {ω : ρ ≤ c} satisfies
v+ (Ā, x+ ) − v− (Ā, x+ ) ≥ v+ (A, x+ ) − v− (A, x+ ).
Xun Yu Zhou/Oxford
(6)
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
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, ρ̄]
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
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Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
Simplification (Cont’d)
Problem (5) is equivalent to
Maximize
subject to
v+ (c, x+ ) − v− (c, x+ )
+
 ρ ≤ c ≤ ρ̄, x+ ≥ x0 ,
x+ = 0 when c = ρ,

x+ = x0 when c = ρ̄.
Xun Yu Zhou/Oxford
(7)
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
Solving PPP
Difficulty: Non-concavity due to distortion
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Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
Solving PPP
Difficulty: Non-concavity due to distortion
Way out: change variable from X (r.v.) to its quantile G−1
(function)
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Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
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
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Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
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)
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Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
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 − Fρ (ρ)) where G−1 is the quantile of X ∗ .
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Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
Solving NPP
Difficulty: use the same quantile trick, but minimizing a
concave function is annoying!
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Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
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”
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Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
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?
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
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!
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Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
Grand Solution
Introduce an auxiliary problem
x+ −x0
v+ (c, x+ ) − u− ( E[ρ1
)T− (1 − F (c))
ρ>c ]
+
ρ ≤ c ≤ ρ̄, x+ ≥ x0 ,
subject to
x+ = 0 when c = ρ, x+ = x0 when c = ρ̄,
Maximize
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(8)
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
Grand Solution
Introduce an auxiliary problem
x+ −x0
v+ (c, x+ ) − u− ( E[ρ1
)T− (1 − F (c))
ρ>c ]
+
ρ ≤ c ≤ ρ̄, x+ ≥ x0 ,
subject to
x+ = 0 when c = ρ, x+ = x0 when c = ρ̄,
Maximize
(8)
Theorem
Suppose that u− (·) is strictly concave at 0.
(i) If (c∗ , x∗+ ) is optimal for (8), then
x∗+ − x0
λρ
∗
′ −1
∗ −
1
1ρ>c∗
X := (u+ )
ρ≤c
T+′ (F (ρ))
E[ρ1ρ>c∗ ]
solves (2).
(ii) If (8) admits no optimal solution then (2) admits no optimal
solution.
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Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
A Combined Binary Option
The final solution is beautifully simple:
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
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∗
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Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
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)
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Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
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
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
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
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Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
Example: Two-Piece Power Utilities
Take u+ (x) = xα , u− (x) = kxα with α ∈ (0, 1) and k > 0
(taken in Kahneman and Tversky 1992)
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Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
Example: Two-Piece Power Utilities
Take u+ (x) = xα , u− (x) = kxα with α ∈ (0, 1) and k > 0
(taken in Kahneman and Tversky 1992)
ln ρ ∼ N (µ, σ) with σ > 0
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Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
Example: Two-Piece Power Utilities
Take u+ (x) = xα , u− (x) = kxα with α ∈ (0, 1) and k > 0
(taken in Kahneman and Tversky 1992)
ln ρ ∼ N (µ, σ) with σ > 0
u′+ (x) = αxα−1 , (u′+ )−1 (y) = (y/α)1/(α−1), u+ ((u′+ )−1 (y)) =
(y/α)α/(α−1), ρ = 0, ρ̄ = +∞, and F (x) = N ((ln x − µ)/σ)
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
Example: Two-Piece Power Utilities
Take u+ (x) = xα , u− (x) = kxα with α ∈ (0, 1) and k > 0
(taken in Kahneman and Tversky 1992)
ln ρ ∼ N (µ, σ) with σ > 0
u′+ (x) = αxα−1 , (u′+ )−1 (y) = (y/α)1/(α−1), u+ ((u′+ )−1 (y)) =
(y/α)α/(α−1), ρ = 0, ρ̄ = +∞, and F (x) = N ((ln x − µ)/σ)
Solution to PPP is
′
T+ (F (ρ)) 1/(1−α)
x+
∗
X+ (c, x+ ) = j(c)
1ρ≤c
ρ
v+ (c, x+ ) = j(c)1−α xα+
where
j(c) := E
"
T+′ (F (ρ))
ρ
1/(1−α)
Xun Yu Zhou/Oxford
#
ρ1ρ≤c > 0, 0 < c ≤ +∞
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
Example: Two-Piece Power Utilities (Cont’d)
Problem (8) specializes to
v(c, x+ ) = j(c)1−α [xα+ − k(c)(x+ − x0 )α ]
0 ≤ c ≤ +∞, x+ ≥ x+
0,
subject to
x+ = 0 when c = 0, x+ = x0 when c = +∞.
(9)
kT− (1−F (c))
where k(c) := j(c)1−α (E[ρ1ρ>c ])α > 0 ∀c > 0
Maximize
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Market Model Ill-posedness Solution Example Single Period
Example: Two-Piece Power Utilities (Cont’d)
Problem (8) specializes to
v(c, x+ ) = j(c)1−α [xα+ − k(c)(x+ − x0 )α ]
0 ≤ c ≤ +∞, x+ ≥ x+
0,
subject to
x+ = 0 when c = 0, x+ = x0 when c = +∞.
(9)
kT− (1−F (c))
where k(c) := j(c)1−α (E[ρ1ρ>c ])α > 0 ∀c > 0
Maximize
Define
k0 :=
T− (1 − F (c))
inf
c>0 j(c)1−α (E[ρ1ρ>c ])α
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−1
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Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
Solution - Case I
Theorem
Assume that x0 ≥ 0.
(i) If k ≥ k0 , then the optimal portfolio for Problem (1) is the
replicating portfolio for the contingent claim
x0
X =
j(+∞)
∗
T+′ (F (ρ))
ρ
1/(1−α)
.
(ii) If k < k0 , then Problem (1) is ill-posed.
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Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
Solution - Case I
Theorem
Assume that x0 ≥ 0.
(i) If k ≥ k0 , then the optimal portfolio for Problem (1) is the
replicating portfolio for the contingent claim
x0
X =
j(+∞)
∗
T+′ (F (ρ))
ρ
1/(1−α)
.
(ii) If k < k0 , then Problem (1) is ill-posed.
Remark. If investor starts with a gain situation and sufficiently
loss averse, then it is optimal to spend x0 buying a contingent
claim, reminiscent of a classical utility maximizing agent (albeit
with a distorted amount)
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Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
Solution - Case II
Theorem
Assume that x0 < 0.
(i) If k > k0 , then Problem (1) is well-posed. Moreover, (1)
admits an optimal portfolio iff
argminc≥0
"
kT− (1 − F (c))
(E[ρ1ρ>c ])α
1/(1−α)
#
− j(c) 6= 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)
1ρ≤c∗ −
x∗+ − x0
1ρ>c∗ ,
E[ρ1ρ>c∗ ]
−x0
where x∗+ := k(c∗ )1/(1−α)
; and if c∗ = 0 is the unique
−1
minimizer in (10), then the unique optimal portfolio is the one
x0
.
to replicate X ∗ = Eρ
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Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
Solution - Case II (Cont’d)
Theorem (cont’d)
(ii) If k = k0 , then the supremum value of Problem (1) is 0,
which is however not achieved by any tame portfolio.
(iii) If k < k0 , then Problem (1) is ill-posed.
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Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
Solution - Case II (Cont’d)
Theorem (cont’d)
(ii) If k = k0 , then the supremum value of Problem (1) is 0,
which is however not achieved by any tame portfolio.
(iii) If k < k0 , 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.
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
What about Single Period and Incomplete Market?
Perversely, less general results have been obtained for the
single period setting
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
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
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Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
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
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
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
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Market Model Ill-posedness Solution Example Single Period
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
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
Conclusions
Portfolio selection models with behavioral criteria (S-shaped
utilities and probability distortions) are formulated and studied
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
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
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
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
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
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
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Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
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
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
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
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice
Prospect Theory Model and Solution Conclusions
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
Xun Yu Zhou/Oxford
Prospect Theory: A New Paradigm for Portfolio Choice