Can Probability Weighting Help Prospect Theory Explain the Disposition Effect? Vicky Henderson University of Warwick and Oxford-Man Institute & David Hobson (Warwick) & Alex S.L. Tse (Warwick → Cambridge) 9th Bachelier World Congress, New York, July 15-19, 2016
Overview Psychologists have uncovered a wealth of behavioral biases in the way we make decisions under uncertainty → behavioral economics ◮ Thinking Fast and Slow , D Kahneman (Nobel Prize 2002) ˆ o ◦ ◮ US and UK governments have Behavioral Insights Teams/Units to provide policy recommendations Prospect theory is most well known and well supported by experimental evidence of many non-EU theories. (Tversky and Kahneman (1979, 1992))
Experimental & empirical evidence suggests..... Tend to prefer a certain $500 to a 50% chance of $1000 risk averse over gains But prefer a 50% chance of losing $1000 to a certain loss of $500 risk seeking over losses Averse to gambles such as ($110, 50%; − $100, 50% ) loss averse 1 Tend to prefer a 1000 chance of $5000 to a certain $5 1 But prefer a certain loss of $5 to a 1000 chance of losing $5000 overweight tail events Delay realization of losses (relative to gains) - disposition effect
Our Aims We study dynamic stopping models with investors who have prospect theory (PT) preferences. Goal is to show inclusion of probability weighting component of PT can: ◮ Generate more realistic trading behavior than existing PT models ◮ Assist PT in matching the magnitude of the disposition effect
The Disposition Effect ◮ The disposition effect is the stylized fact that investors have a higher propensity to sell a winner (gain relative to purchase price) than a loser (lost relative to purchase price). (Shefrin and Statman (1985)). ◮ Documented for individual investors by Odean (1998), institutional investors, in real estate and options markets. Experimental evidence (Weber and Camerer (1998) and Magnani (2015)). ◮ Shefrin and Statman (1985) propose the disposition effect can be explained by Prospect Theory
Proportion of days with sales producing a given magnitude of return. Using large US retail brokerage dataset. (Thanks to Matthew Burgess (Warwick))
Prospect Theory (Tversky & Kahneman (1979, 1992)) ◮ U a continuous, increasing S -shaped value function defined over gains and losses relative to a reference level R with U (0) = 0. ◮ utility from gains and losses rather than final wealth ◮ risk averse (concave) over gains and risk seeking (convex) over losses ◮ steeper for losses than for gains to capture loss aversion, U ( − x ) + U ( x ) < 0 or U ′ (0+) < U ′ (0 − ) ◮ w ± : [0 , 1] → [0 , 1] a pair of increasing functions with w ± (0) = 0 and w ± (1) = 1. Typically w ± concave on [0 , q ± ] and convex on [ q ± , 1] for some q ± . ◮ probability weighting function applied to CDF to overweight tails of distribution
Tversky and Kahneman (1992) suggest functional forms and calibrate to experimental data: ◮ Value function: � y α + y ≥ 0 U ( y ) = − k ( − y ) α − y < 0 where α ± ∈ (0 , 1). k > 1 governs loss aversion. TK estimate α + = α − = 0 . 88 and k = 2 . 25. Wu and Gonzales (1996) estimate α + = 0 . 5. ◮ Probability weighting function: p δ ± w ± ( p ) = ( p δ ± + (1 − p ) δ ± ) 1 /δ ± for 0 . 28 < δ ± ≤ 1. TK estimated δ + = 0 . 61, δ − = 0 . 69. Wu and Gonzales (1996) estimate δ + = 0 . 71.
3 1 w + 0.9 w − 2 0.8 0.7 1 0.6 v(x) w(p) 0 0.5 0.4 −1 0.3 0.2 −2 0.1 −3 0 −3 −2 −1 0 1 2 3 0 0.2 0.4 0.6 0.8 1 x p (a) TK (1992) value function (b) TK (1992) weighting with α ± = 0 . 5 , k = 1 . 5 functions with δ = 1, δ + = 0 . 61 and δ − = 0 . 69
Prospect Theory Prospect theory value of cont. random variable Y (reference level centred) is given by (Kothiyal, Spinu and Wakker (2011)) � � E U ( Y ) = w + ( P ( U ( Y ) > y )) dy − w − ( P ( U ( Y ) < y )) dy R + R − If w ± ( p ) = p then E U ( Y ) = E U ( Y ). Denoting by G ( . ) the quantile function of Y , a change of variable & IBP gives � 1 � P ( Y < 0) E U ( Y ) = U ( G ( q )) w ′ U ( G ( q )) w ′ + (1 − q ) dq + − ( q ) dq 1 − P ( Y > 0) 0
Formulation of the Investor’s Asset Sale Problem Investor can choose to sell asset with price P t at time t at any time in the future. Suppose asset price P = ( P t ) t ≥ 0 is a (regular, time homogeneous) diffusion on (0 , ∞ ) with initial value P 0 . Goal of the investor is to choose the best time τ to sell the asset to maximize: τ E U ( P τ − R ) sup where U is value function and R reference level. Note if w ± ( p ) = p , studied by Henderson (2012) (see also Kyle, Ou-Yang and Xiong (2006)).
Formulation of the Investor’s Asset Sale Problem Let s be scale function of P with s ( R ) = 0 such that X t = s ( P t ) is a (local) martingale. Then X t is transformed gains/losses relative to reference and E U ( P τ − R ) = E v ( X τ ) � s − 1 ( x ) − R � where v ( x ) := U . Suppose L := s (0) > −∞ . Let M = s ( ∞ ) ≤ ∞ . We can take X τ rather than τ as the decision variable giving: E v ( X ) sup ν ∈A where � A = { ν : y ν ( dy ) = X 0 = s ( P 0 ) } Skorokhod embedding tells us for every law ν with support on � [ s (0) , s ( ∞ )] = [ L , M ] and such that y ν ( dy ) = X 0 there is a stopping rule τ such that X τ ∼ ν .
Some Related Work ◮ Xu and Zhou (2013) study a similar problem just for gains or just for losses. ◮ We will study the combined gains/losses problem and give additional conditions under which a single point mass on losses is optimal. ◮ Probability weighting introduces time inconsistency (Barberis (2012)). ◮ We focus here on the behavior of a sophisticated investor who can precommit . ◮ Ebert and Strack (2015) study behavior of naive PT agents - with pure strategies they never stop. ◮ Henderson, Hobson and Tse (2015) show this is no longer the only behavior when naive agents can follow randomized strategies. (Saturday 2:50pm)
Two Assumptions w ± are inverse- S shaped: concave on [0 , q ± ] & convex on [ q ± , 1]. Assumption (S-shaped on v ) v is concave on [0 , M ) and convex on [ L , 0] and v ′ (0+) = ∞ Define E ( x ; f , c ) = ( x − c ) f ′ ( x ) f ′ ( x ) f ( x ) − f ( c ) = f ( x ) − f ( c ) x − c as the elasticity of a function f relative to a reference point c . Assumption (Elasticity) a. E ( x ; v , L ) is increasing in x for x ∈ [ L , 0] b. E ( p ; w − , r ) is decreasing in p for 0 ≤ p ≤ min { r , q − } for any r in [0 , 1] . Elasticity Assumption b. holds for all the popular weighting functions - TK (1992), Prelec (1998), Goldstein and Einhorn (1987)
Solving for the Optimal quantile function ◮ Adopt a sequential optimization approach (Xu and Zhou (2013), Jin and Zhou (2008)) by decomposition of X = X + − X − ◮ Optimal X + : ◮ solving a concave maximization problem via a Lagrangian approach ◮ Optimal X − : ◮ solving a convex maximization problem where solution is a 3-point distribution with atoms at L , 0 and some I ∈ [ L , 0] in general ◮ Under assumptions on v ′ (0+) and elasticity of v and w − , the optimal rv only puts one atom on losses. ◮ Combine by finding optimal allocation of probability mass over gains and losses subject to mean constraint
Theorem Under the Assumptions, optimal prospect has a distribution consisting of: ◮ a point mass in the loss regime and ◮ a point mass at some level a in the gains regime, together with a continuous distribution on the unbounded interval ( a , ∞ ) . The optimizer has quantile function G X of the form �� φ 0 dy ( v ′ ) − 1 � � � 1 λ − − X 0 u ≤ 1 − φ 1 − φ w ′ + ( ψ ∧ y ) ( v ′ ) − 1 � � λ G X ( u ) = 1 − φ < u ≤ 1 − ψ w ′ + ( ψ ) ( v ′ ) − 1 � � λ 1 − ψ < u ≤ 1 w ′ + (1 − u ) (1) for some λ > 0 , φ ∈ [0 , 1] , ψ ≤ q + ∧ φ such that � φ � � 1 λ X + dy ( v ′ ) − 1 0 ≤ ≤ X 0 − L . w ′ 1 − φ + ( ψ ∧ y ) 0
Base Model - TK value & weighting functions ◮ P t follows a GBM dP t = P t ( ρ dt + σ dB t ) (2) ◮ Define β := 1 − 2 ρ σ 2 . For ρ < σ 2 / 2 or β > 0, have s ( p ) = p β − R β and L > −∞ . ◮ TK value and weighting functions with ( α + , α − , k ) and ( γ + , γ − ). ◮ v ( x ) = U ( s − 1 ( x ) − P 0 ) is S -shaped if and only if β ≤ 1, α + < β . ◮ Assumptions on v ′ (0+) and elasticity functions of v and w − all satisfied. Proposition Under the above specification, the problem is well posed if α + /δ + < β . We will work with α + /δ + < β ≤ 1.
Optimal sale strategy involves: ◮ a fixed stop-loss threshold ◮ a random gain level supported on [( v ′ ) − 1 � � λ , ∞ ) w ′ + ( ψ ) 2.4 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Quantile function of P τ ∗ = s − 1 ( X ∗ ). Parameters are α + = 0 . 5, α − = 0 . 9, δ + = δ − = 0 . 7, k = 1 . 25, β = 0 . 9, R = 1 and P 0 = 1.
Impact of Probability Weighting Loss Threshold 3.5 Gain Distribution (lower bound and 99th percentile) 3 2.5 2 1.5 1 0.5 0 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 δ + = δ - Distributional properties of P τ ∗ = s − 1 ( X ∗ ). Vary δ ± on y-axis. Parameters are α + = 0 . 5, α − = 0 . 9, δ + = δ − = 0 . 7, k = 1 . 25, β = 0 . 9, R = 1 = P 0 .
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