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Thou Shalt Still Buy and Hold
Xunyu Zhou/Oxford
3rd September 2008/Kyoto
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
This Talk Based on ...
A. Shiryaev, Z. Xu, and X. Zhou, “Thou shalt buy and hold”,Quantitative Finance, 2008
M. Dai, H. Jin and X. Zhou, “Buy on the lows and sell on thehighs”, working paper, 2008
M. Dai, H. Jin and X. Zhou, “Thou shalt still buy and hold”,working paper, 2008.
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Classical Continuous-Time Portfolio Selection Models
Merton’s model
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Classical Continuous-Time Portfolio Selection Models
Merton’s model
Markowitz’s model
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Classical Continuous-Time Portfolio Selection Models
Merton’s model
Markowitz’s model
Behavioral model (recently)
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Classical Continuous-Time Portfolio Selection Models
Merton’s model
Markowitz’s model
Behavioral model (recently)
No transaction costs
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Classical Continuous-Time Portfolio Selection Models
Merton’s model
Markowitz’s model
Behavioral model (recently)
No transaction costs
Optimal portfolios are to “trade all the time” so as to keepcertain “constant proportions”
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Portfolio Selection with Transaction Costs
“Continuous Trading” clearly incurs infinite costs
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Portfolio Selection with Transaction Costs
“Continuous Trading” clearly incurs infinite costs
In the presence of transaction costs: sell region, buy regionand no trade region
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Portfolio Selection with Transaction Costs
“Continuous Trading” clearly incurs infinite costs
In the presence of transaction costs: sell region, buy regionand no trade region
Trade only necessary
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Portfolio Selection with Transaction Costs
“Continuous Trading” clearly incurs infinite costs
In the presence of transaction costs: sell region, buy regionand no trade region
Trade only necessary
Liu and Loewenstein (2002), Rev Fin Studies, CRRA investor,transaction costs and finite horizon
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Portfolio Selection with Transaction Costs
“Continuous Trading” clearly incurs infinite costs
In the presence of transaction costs: sell region, buy regionand no trade region
Trade only necessary
Liu and Loewenstein (2002), Rev Fin Studies, CRRA investor,transaction costs and finite horizon
An investor might optimally never buy stock if the horizon isshort
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Portfolio Selection with Transaction Costs
“Continuous Trading” clearly incurs infinite costs
In the presence of transaction costs: sell region, buy regionand no trade region
Trade only necessary
Liu and Loewenstein (2002), Rev Fin Studies, CRRA investor,transaction costs and finite horizon
An investor might optimally never buy stock if the horizon isshortAn investor should largely buy and hold if the horizon is long
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Conventional Wisdom: Buy and Hold
Buy and Hold: Buy a good stock and leave it alone for longtime
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Conventional Wisdom: Buy and Hold
Buy and Hold: Buy a good stock and leave it alone for longtime
“Never sell a security unless you need money”(EfficientMarket.ca)
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Conventional Wisdom: Buy and Hold
Buy and Hold: Buy a good stock and leave it alone for longtime
“Never sell a security unless you need money”(EfficientMarket.ca)
“Trading is hazardous to your wealth” (Barber and Odean,2000, J Fin)
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Conventional Wisdom: Buy and Hold
Buy and Hold: Buy a good stock and leave it alone for longtime
“Never sell a security unless you need money”(EfficientMarket.ca)
“Trading is hazardous to your wealth” (Barber and Odean,2000, J Fin)
Grounds
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Conventional Wisdom: Buy and Hold
Buy and Hold: Buy a good stock and leave it alone for longtime
“Never sell a security unless you need money”(EfficientMarket.ca)
“Trading is hazardous to your wealth” (Barber and Odean,2000, J Fin)
Grounds
Efficient market hypothesis (EMH): every security is fairlyvalued at all times, so there is no point to trade (you tradebecause something happens to you, not because somethinghappens to the market)
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Conventional Wisdom: Buy and Hold
Buy and Hold: Buy a good stock and leave it alone for longtime
“Never sell a security unless you need money”(EfficientMarket.ca)
“Trading is hazardous to your wealth” (Barber and Odean,2000, J Fin)
Grounds
Efficient market hypothesis (EMH): every security is fairlyvalued at all times, so there is no point to trade (you tradebecause something happens to you, not because somethinghappens to the market)
Pure costs: bid/offer spread, brokerage, capital gain tax etc.(Warren Buffett is a buy-and-hold advocate rejecting EMH)
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Conventional Wisdom: Buy and Hold
Buy and Hold: Buy a good stock and leave it alone for longtime
“Never sell a security unless you need money”(EfficientMarket.ca)
“Trading is hazardous to your wealth” (Barber and Odean,2000, J Fin)
Grounds
Efficient market hypothesis (EMH): every security is fairlyvalued at all times, so there is no point to trade (you tradebecause something happens to you, not because somethinghappens to the market)
Pure costs: bid/offer spread, brokerage, capital gain tax etc.(Warren Buffett is a buy-and-hold advocate rejecting EMH)
However, no well-established portfolio models produced purebuy-and-hold strategy
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
If I Were an Innocent Investor ...
Let’s say I just bought a stock, and must sell it in one year
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
If I Were an Innocent Investor ...
Let’s say I just bought a stock, and must sell it in one year
Need to decide when to sell
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
If I Were an Innocent Investor ...
Let’s say I just bought a stock, and must sell it in one year
Need to decide when to sell
Criterion?
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
If I Were an Innocent Investor ...
Let’s say I just bought a stock, and must sell it in one year
Need to decide when to sell
Criterion?
Mean-Variance? Expected utility?
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
If I Were an Innocent Investor ...
Let’s say I just bought a stock, and must sell it in one year
Need to decide when to sell
Criterion?
Mean-Variance? Expected utility? I don’t know what you aretalking about
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
If I Were an Innocent Investor ...
Let’s say I just bought a stock, and must sell it in one year
Need to decide when to sell
Criterion?
Mean-Variance? Expected utility? I don’t know what you aretalking about
Let’s sell higher ...
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
If I Were an Innocent Investor ...
Let’s say I just bought a stock, and must sell it in one year
Need to decide when to sell
Criterion?
Mean-Variance? Expected utility? I don’t know what you aretalking about
Let’s sell higher ... say, at the maximum price?
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
If I Were an Innocent Investor ...
Let’s say I just bought a stock, and must sell it in one year
Need to decide when to sell
Criterion?
Mean-Variance? Expected utility? I don’t know what you aretalking about
Let’s sell higher ... say, at the maximum price?
Selling at the maximum price is a “mission impossible”
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
If I Were an Innocent Investor ...
Let’s say I just bought a stock, and must sell it in one year
Need to decide when to sell
Criterion?
Mean-Variance? Expected utility? I don’t know what you aretalking about
Let’s sell higher ... say, at the maximum price?
Selling at the maximum price is a “mission impossible”
How about selling at the price closest to the maximum?
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
If I Were an Innocent Investor ...
Let’s say I just bought a stock, and must sell it in one year
Need to decide when to sell
Criterion?
Mean-Variance? Expected utility? I don’t know what you aretalking about
Let’s sell higher ... say, at the maximum price?
Selling at the maximum price is a “mission impossible”
How about selling at the price closest to the maximum?
...or sell at the time when the expected relative error betweenthe current price and the maximum price is minimised
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
The Model
A Black–Scholes market with a stock and a saving account
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
The Model
A Black–Scholes market with a stock and a saving account
The discounted stock price follows, on (Ω,F , P ):
dPt = (a − r)Ptdt + σPtdBt, or Pt = eµt+σBt
where µ = a − r − 12σ2
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
The Model
A Black–Scholes market with a stock and a saving account
The discounted stock price follows, on (Ω,F , P ):
dPt = (a − r)Ptdt + σPtdBt, or Pt = eµt+σBt
where µ = a − r − 12σ2
Let Mt = max06s6t Ps, 0 6 t 6 T
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
The Model
A Black–Scholes market with a stock and a saving account
The discounted stock price follows, on (Ω,F , P ):
dPt = (a − r)Ptdt + σPtdBt, or Pt = eµt+σBt
where µ = a − r − 12σ2
Let Mt = max06s6t Ps, 0 6 t 6 T
Consider the following optimal stopping problem
min06τ6T
E
[
MT − Pτ
MT
]
where τ ∈ [0, T ] is a Bt-stopping time
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
The Model
A Black–Scholes market with a stock and a saving account
The discounted stock price follows, on (Ω,F , P ):
dPt = (a − r)Ptdt + σPtdBt, or Pt = eµt+σBt
where µ = a − r − 12σ2
Let Mt = max06s6t Ps, 0 6 t 6 T
Consider the following optimal stopping problem
min06τ6T
E
[
MT − Pτ
MT
]
where τ ∈ [0, T ] is a Bt-stopping time
... or equivalently
max06τ6T
E
[
Pτ
MT
]
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Related (Probabilistic) Literature
Graversen, Peskir, Shiryaev (2000), Theory Prob Appl, studied
min0≤τ≤T
E(B0τ − S0
T )2
where S0t := max06s6t B0
s , and obtained optimal
τ∗ = inf
0 6 t 6 T∣
∣
∣
S0t − B0
t√T − t
> z∗
where z∗ = 1.12...
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Related (Probabilistic) Literature
Graversen, Peskir, Shiryaev (2000), Theory Prob Appl, studied
min0≤τ≤T
E(B0τ − S0
T )2
where S0t := max06s6t B0
s , and obtained optimal
τ∗ = inf
0 6 t 6 T∣
∣
∣
S0t − B0
t√T − t
> z∗
where z∗ = 1.12...
Pedersen (2003), Stoch Stoch Rep, considered
min0≤τ≤T
E|B0τ − S0
T |p, 0 6 p < +∞
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
The Model Rewritten
Assume σ = 1 (otherwise rescale the time)
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
The Model Rewritten
Assume σ = 1 (otherwise rescale the time)
Rewrite Bµt = µt + Bt, Sµ
t = max06s6t Bµs , 0 6 t 6 T
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
The Model Rewritten
Assume σ = 1 (otherwise rescale the time)
Rewrite Bµt = µt + Bt, Sµ
t = max06s6t Bµs , 0 6 t 6 T
The problem is
max06τ6T
E
[
eBµτ
eSµT
]
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
The Model Rewritten
Assume σ = 1 (otherwise rescale the time)
Rewrite Bµt = µt + Bt, Sµ
t = max06s6t Bµs , 0 6 t 6 T
The problem is
max06τ6T
E
[
eBµτ
eSµT
]
Not a standard optimal stopping problem: SµT is not
Bt-adapted!
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
A Few Good Steps ... Towards a Standard Problem
For any Bt-stopping time 0 6 τ 6 T , we have
E
[
eBµτ
eSµT
]
=E
[
eBµτ
maxeSµτ , emaxτ6t6T B
µt
]
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
A Few Good Steps ... Towards a Standard Problem
For any Bt-stopping time 0 6 τ 6 T , we have
E
[
eBµτ
eSµT
]
=E
[
eBµτ
maxeSµτ , emaxτ6t6T B
µt
]
=E
[
min
e−(Sµτ −B
µτ ), e
− maxτ6t6T
(Bµt −B
µτ )
]
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
A Few Good Steps ... Towards a Standard Problem
For any Bt-stopping time 0 6 τ 6 T , we have
E
[
eBµτ
eSµT
]
=E
[
eBµτ
maxeSµτ , emaxτ6t6T B
µt
]
=E
[
min
e−(Sµτ −B
µτ ), e
− maxτ6t6T
(Bµt −B
µτ )
]
=E
[
E
[
min
e−(Sµτ −B
µτ ), e
− maxτ6t6T
(Bµt −B
µτ )
∣
∣
∣
∣
Fτ
]]
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
A Few Good Steps ... Towards a Standard Problem
For any Bt-stopping time 0 6 τ 6 T , we have
E
[
eBµτ
eSµT
]
=E
[
eBµτ
maxeSµτ , emaxτ6t6T B
µt
]
=E
[
min
e−(Sµτ −B
µτ ), e
− maxτ6t6T
(Bµt −B
µτ )
]
=E
[
E
[
min
e−(Sµτ −B
µτ ), e
− maxτ6t6T
(Bµt −B
µτ )
∣
∣
∣
∣
Fτ
]]
=E
[
E[
min
e−x, e−SµT−τ
]∣
∣
∣
x=Sµτ −B
µτ
]
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
A Few Good Steps ... Towards a Standard Problem
For any Bt-stopping time 0 6 τ 6 T , we have
E
[
eBµτ
eSµT
]
=E
[
eBµτ
maxeSµτ , emaxτ6t6T B
µt
]
=E
[
min
e−(Sµτ −B
µτ ), e
− maxτ6t6T
(Bµt −B
µτ )
]
=E
[
E
[
min
e−(Sµτ −B
µτ ), e
− maxτ6t6T
(Bµt −B
µτ )
∣
∣
∣
∣
Fτ
]]
=E
[
E[
min
e−x, e−SµT−τ
]∣
∣
∣
x=Sµτ −B
µτ
]
=E [G(τ, Xτ )]
where G(t, x) = E[
min
e−x, e−SµT−t
]
, Xt = Sµt − Bµ
t
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
A Few Good Steps ... Towards a Standard Problem
For any Bt-stopping time 0 6 τ 6 T , we have
E
[
eBµτ
eSµT
]
=E
[
eBµτ
maxeSµτ , emaxτ6t6T B
µt
]
=E
[
min
e−(Sµτ −B
µτ ), e
− maxτ6t6T
(Bµt −B
µτ )
]
=E
[
E
[
min
e−(Sµτ −B
µτ ), e
− maxτ6t6T
(Bµt −B
µτ )
∣
∣
∣
∣
Fτ
]]
=E
[
E[
min
e−x, e−SµT−τ
]∣
∣
∣
x=Sµτ −B
µτ
]
=E [G(τ, Xτ )]
where G(t, x) = E[
min
e−x, e−SµT−t
]
, Xt = Sµt − Bµ
t
Xt – drawdown process – is Bt-adapted!
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Function G
Assuming µ 6= 12 , then
G(t, x) =2(µ − 1)
2µ − 1e−(µ− 1
2)(T−t)Φ
(−x + (µ − 1)(T − t)√T − t
)
+1
2µ − 1e−(1−2µ)xΦ
(−x − µ(T − t)√T − t
)
+ e−xΦ
(
x − µ(T − t)√T − t
)
,
where Φ is the CDF of standard normal distribution
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Function G
Assuming µ 6= 12 , then
G(t, x) =2(µ − 1)
2µ − 1e−(µ− 1
2)(T−t)Φ
(−x + (µ − 1)(T − t)√T − t
)
+1
2µ − 1e−(1−2µ)xΦ
(−x − µ(T − t)√T − t
)
+ e−xΦ
(
x − µ(T − t)√T − t
)
,
where Φ is the CDF of standard normal distribution
If µ = 12 , then G has a different (explicit) expression
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Dynamic Programming
Define Xxt+s = x ∨ Sµ
s − Bµs , s ≥ 0
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Dynamic Programming
Define Xxt+s = x ∨ Sµ
s − Bµs , s ≥ 0
Xxt+s is Markovian under P , ∀(t, x)
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Dynamic Programming
Define Xxt+s = x ∨ Sµ
s − Bµs , s ≥ 0
Xxt+s is Markovian under P , ∀(t, x)
Value function
V (t, x) = sup06τ6T−t
E[
G(t + τ, Xxt+τ )
]
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Dynamic Programming
Define Xxt+s = x ∨ Sµ
s − Bµs , s ≥ 0
Xxt+s is Markovian under P , ∀(t, x)
Value function
V (t, x) = sup06τ6T−t
E[
G(t + τ, Xxt+τ )
]
In particular
V (0, 0) = sup06τ6T
E
[
eBµτ
eSµT
]
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Variational Inequalities
Dynamic programming equation (Variational Inequalities)
min−LV, V − G = 0, (t, x) ∈ [0, T ) × (0,∞) (1)
V (T, x) = G(T, x), x ∈ (0,∞) (2)
Vx(t, 0+) = 0, t ∈ [0, T ) (normal reflection) (3)
where
LV = Vt − µVx +1
2Vxx (4)
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Holding and Selling Region
Holding region
C = (t, x) ∈ [0, T ] × [0,∞) : V (t, x) > G(t, x)
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Holding and Selling Region
Holding region
C = (t, x) ∈ [0, T ] × [0,∞) : V (t, x) > G(t, x)
Selling region
D = (t, x) ∈ [0, T ] × [0,∞) : V (t, x) = G(t, x)
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Holding and Selling Region
Holding region
C = (t, x) ∈ [0, T ] × [0,∞) : V (t, x) > G(t, x)
Selling region
D = (t, x) ∈ [0, T ] × [0,∞) : V (t, x) = G(t, x)
An optimal selling time is
τ∗ = inft ∈ [0, T ] : (t, Sµt − Bµ
t ) ∈ D
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Holding and Selling Region
Holding region
C = (t, x) ∈ [0, T ] × [0,∞) : V (t, x) > G(t, x)
Selling region
D = (t, x) ∈ [0, T ] × [0,∞) : V (t, x) = G(t, x)
An optimal selling time is
τ∗ = inft ∈ [0, T ] : (t, Sµt − Bµ
t ) ∈ D
So it boils down to finding V - which is hard
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Optimal Stopping with Running Payoff
To get around: turn terminal payoff into running payoff
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Optimal Stopping with Running Payoff
To get around: turn terminal payoff into running payoff
Xxt+· = |Y·| in law, where
dYt = −µsign(Yt)dt + dBt, Y0 = x
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Optimal Stopping with Running Payoff
To get around: turn terminal payoff into running payoff
Xxt+· = |Y·| in law, where
dYt = −µsign(Yt)dt + dBt, Y0 = x
By Tanaka’s formula
|Ys| = x−µ
∫ s
0I(Yu 6= 0)du+
∫ s
0sign(Yu)I(Yu 6= 0)dBu+ℓ0
s(Y )
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Optimal Stopping with Running Payoff (Cont’d)
Ito’s formula then implies
G(t + s, Xxt+s) = G(t, x) +
∫ s
0H(t + u, Xx
t+u)du + Ms
provided that Gx(t, 0+) = 0 (so as to kill the local timeterm), where M is a martingale and
H(t, x) = LG(t, x) = Gt(t, x) − µGx(t, x) +1
2Gxx(t, x)
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Optimal Stopping with Running Payoff (Cont’d)
Ito’s formula then implies
G(t + s, Xxt+s) = G(t, x) +
∫ s
0H(t + u, Xx
t+u)du + Ms
provided that Gx(t, 0+) = 0 (so as to kill the local timeterm), where M is a martingale and
H(t, x) = LG(t, x) = Gt(t, x) − µGx(t, x) +1
2Gxx(t, x)
Hence
V (t, x) = G(t, x) + sup06τ6T−t
E
[∫ τ
0H(t + u, Xx
t+u)du
]
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Optimal Stopping with Running Payoff (Cont’d)
Ito’s formula then implies
G(t + s, Xxt+s) = G(t, x) +
∫ s
0H(t + u, Xx
t+u)du + Ms
provided that Gx(t, 0+) = 0 (so as to kill the local timeterm), where M is a martingale and
H(t, x) = LG(t, x) = Gt(t, x) − µGx(t, x) +1
2Gxx(t, x)
Hence
V (t, x) = G(t, x) + sup06τ6T−t
E
[∫ τ
0H(t + u, Xx
t+u)du
]
Need to calculate H and show Gx(t, 0+) = 0
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Function H
Write G(t, x) = 2(µ−1)2µ−1 A(t, x) + 1
2µ−1B(t, x) + C(t, x) where
A = e−(µ− 12)(T−t)Φ
(−x + (µ − 1)(T − t)√T − t
)
B = e−(1−2µ)xΦ
(−x − µ(T − t)√T − t
)
C = e−xΦ
(
x − µ(T − t)√T − t
)
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Function H
Write G(t, x) = 2(µ−1)2µ−1 A(t, x) + 1
2µ−1B(t, x) + C(t, x) where
A = e−(µ− 12)(T−t)Φ
(−x + (µ − 1)(T − t)√T − t
)
B = e−(1−2µ)xΦ
(−x − µ(T − t)√T − t
)
C = e−xΦ
(
x − µ(T − t)√T − t
)
Lengthy calculations show for ϕ = A, B, C,
Lϕ = (µ − 12)ϕ − ϕx
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Function H
Write G(t, x) = 2(µ−1)2µ−1 A(t, x) + 1
2µ−1B(t, x) + C(t, x) where
A = e−(µ− 12)(T−t)Φ
(−x + (µ − 1)(T − t)√T − t
)
B = e−(1−2µ)xΦ
(−x − µ(T − t)√T − t
)
C = e−xΦ
(
x − µ(T − t)√T − t
)
Lengthy calculations show for ϕ = A, B, C,
Lϕ = (µ − 12)ϕ − ϕx
Hence H = LG = (µ − 12)G − Gx
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Function H
Write G(t, x) = 2(µ−1)2µ−1 A(t, x) + 1
2µ−1B(t, x) + C(t, x) where
A = e−(µ− 12)(T−t)Φ
(−x + (µ − 1)(T − t)√T − t
)
B = e−(1−2µ)xΦ
(−x − µ(T − t)√T − t
)
C = e−xΦ
(
x − µ(T − t)√T − t
)
Lengthy calculations show for ϕ = A, B, C,
Lϕ = (µ − 12)ϕ − ϕx
Hence H = LG = (µ − 12)G − Gx
Moreover Gx = B − C
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Function H
Write G(t, x) = 2(µ−1)2µ−1 A(t, x) + 1
2µ−1B(t, x) + C(t, x) where
A = e−(µ− 12)(T−t)Φ
(−x + (µ − 1)(T − t)√T − t
)
B = e−(1−2µ)xΦ
(−x − µ(T − t)√T − t
)
C = e−xΦ
(
x − µ(T − t)√T − t
)
Lengthy calculations show for ϕ = A, B, C,
Lϕ = (µ − 12)ϕ − ϕx
Hence H = LG = (µ − 12)G − Gx
Moreover Gx = B − C
So Gx(t, 0+) = B(t, 0+) − C(t, 0+) = 0
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
The Good...
If µ ≥ 12 :
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
The Good...
If µ ≥ 12 :
Gx 6 0 by the definition G(t, x) = E[
min
e−x, e−Sµ
T−t
]
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
The Good...
If µ ≥ 12 :
Gx 6 0 by the definition G(t, x) = E[
min
e−x, e−Sµ
T−t
]
H = (µ − 12 )G − Gx > 0
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
The Good...
If µ ≥ 12 :
Gx 6 0 by the definition G(t, x) = E[
min
e−x, e−Sµ
T−t
]
H = (µ − 12 )G − Gx > 0
So the optimal τ∗ = T
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
The Good...
If µ ≥ 12 :
Gx 6 0 by the definition G(t, x) = E[
min
e−x, e−Sµ
T−t
]
H = (µ − 12 )G − Gx > 0
So the optimal τ∗ = T
The stock is “good”
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
... and The Bad
If µ ≤ −12 :
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
... and The Bad
If µ ≤ −12 :
exG(t, x) = E[
min
1, e−Sµ
T−t+x
]
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
... and The Bad
If µ ≤ −12 :
exG(t, x) = E[
min
1, e−Sµ
T−t+x
]
∂(exG)∂x > 0 so ex(Gx + G) > 0 or Gx + G > 0
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
... and The Bad
If µ ≤ −12 :
exG(t, x) = E[
min
1, e−Sµ
T−t+x
]
∂(exG)∂x > 0 so ex(Gx + G) > 0 or Gx + G > 0
H = (µ − 12 )G − Gx = (µ + 1
2 )G − (Gx + G) 6 0
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
... and The Bad
If µ ≤ −12 :
exG(t, x) = E[
min
1, e−Sµ
T−t+x
]
∂(exG)∂x > 0 so ex(Gx + G) > 0 or Gx + G > 0
H = (µ − 12 )G − Gx = (µ + 1
2 )G − (Gx + G) 6 0
So the optimal τ∗ = 0
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
... and The Bad
If µ ≤ −12 :
exG(t, x) = E[
min
1, e−Sµ
T−t+x
]
∂(exG)∂x > 0 so ex(Gx + G) > 0 or Gx + G > 0
H = (µ − 12 )G − Gx = (µ + 1
2 )G − (Gx + G) 6 0
So the optimal τ∗ = 0
The stock is “bad”
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Finding More Good Guys
If µ = 0:
V (t, x) > Et,x[G(T, XxT )] > G(t, x), ∀ t ∈ [0, T ), x > 0
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Finding More Good Guys
If µ = 0:
V (t, x) > Et,x[G(T, XxT )] > G(t, x), ∀ t ∈ [0, T ), x > 0
On the other hand ∀µ ∈ R:
∂
∂µ
e12µ2(T−t) (Et,x[G(T, Xx
T )] − G(t, x))
> 0
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Finding More Good Guys
If µ = 0:
V (t, x) > Et,x[G(T, XxT )] > G(t, x), ∀ t ∈ [0, T ), x > 0
On the other hand ∀µ ∈ R:
∂
∂µ
e12µ2(T−t) (Et,x[G(T, Xx
T )] − G(t, x))
> 0
Hence ∀µ > 0:
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Finding More Good Guys
If µ = 0:
V (t, x) > Et,x[G(T, XxT )] > G(t, x), ∀ t ∈ [0, T ), x > 0
On the other hand ∀µ ∈ R:
∂
∂µ
e12µ2(T−t) (Et,x[G(T, Xx
T )] − G(t, x))
> 0
Hence ∀µ > 0:
V (t, x) > Et,x[G(T,XxT )] > G(t, x), ∀ t ∈ [0, T ), x ≥ 0
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Finding More Good Guys
If µ = 0:
V (t, x) > Et,x[G(T, XxT )] > G(t, x), ∀ t ∈ [0, T ), x > 0
On the other hand ∀µ ∈ R:
∂
∂µ
e12µ2(T−t) (Et,x[G(T, Xx
T )] − G(t, x))
> 0
Hence ∀µ > 0:
V (t, x) > Et,x[G(T,XxT )] > G(t, x), ∀ t ∈ [0, T ), x ≥ 0
So the optimal τ∗ = T
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Finding More Good Guys
If µ = 0:
V (t, x) > Et,x[G(T, XxT )] > G(t, x), ∀ t ∈ [0, T ), x > 0
On the other hand ∀µ ∈ R:
∂
∂µ
e12µ2(T−t) (Et,x[G(T, Xx
T )] − G(t, x))
> 0
Hence ∀µ > 0:
V (t, x) > Et,x[G(T,XxT )] > G(t, x), ∀ t ∈ [0, T ), x ≥ 0
So the optimal τ∗ = TThe stock is “good”
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Optimal Seeling Times
Define a goodness index of the stock
α =a − r
σ2
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Optimal Seeling Times
Define a goodness index of the stock
α =a − r
σ2
Theorem
(Shiryaev, Xu and Zhou, Quantitative Finance 2008) Optimalselling time τ∗ = T if α ≥ 0.5, and τ∗ = 0 if α ≤ 0
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Optimal Seeling Times
Define a goodness index of the stock
α =a − r
σ2
Theorem
(Shiryaev, Xu and Zhou, Quantitative Finance 2008) Optimalselling time τ∗ = T if α ≥ 0.5, and τ∗ = 0 if α ≤ 0
Theorem
(Dai, Jin, and Zhou 2008) Optimal selling time τ∗ = T ifα ≥ 0.5, and τ∗ = 0 if α < 0.5.
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Optimal Relative Error - Good Stock
Let Pt = eµt+σBt , Mt = max06s6t Ps
Assume α ≥ 0.5
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Optimal Relative Error - Good Stock
Let Pt = eµt+σBt , Mt = max06s6t Ps
Assume α ≥ 0.5
Optimal selling time τ∗ = T
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Optimal Relative Error - Good Stock
Let Pt = eµt+σBt , Mt = max06s6t Ps
Assume α ≥ 0.5
Optimal selling time τ∗ = T
Need to determine E[
MT−PTMT
]
: the optimal relative error
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Optimal Relative Error - Good Stock
Let Pt = eµt+σBt , Mt = max06s6t Ps
Assume α ≥ 0.5
Optimal selling time τ∗ = T
Need to determine E[
MT−PTMT
]
: the optimal relative error
The joint density function of (PT , MT ) is
fT (p, m) =2
σ3√
2πT 3
ln(m2/p)
pme−
ln2(m2/p)
2σ2T+ β
σln(p)− 1
2β2T
where 0 < p 6 m, m > 1, β = µσ≡
(
α − 12
)
σ
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Optimal Relative Error - Good Stock (Cont’d)
We have
E
[
PT
MT
]
=
∫ 1
0
P
(
PT
MT> y
)
dy
=
∫ 1
0
∫
∞
y
∫ p/y
p∨1
fT (p,m)dmdpdy
= · · ·
=
(
1 − 1
2α
)
Φ
(
(α − 1
2)σ√
T
)
+
(
1 +1
2α
)
eασ2T Φ
(
−(α +1
2)σ√
T
)
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Optimal Relative Error - Good Stock (Cont’d)
We have
E
[
PT
MT
]
=
∫ 1
0
P
(
PT
MT> y
)
dy
=
∫ 1
0
∫
∞
y
∫ p/y
p∨1
fT (p,m)dmdpdy
= · · ·
=
(
1 − 1
2α
)
Φ
(
(α − 1
2)σ√
T
)
+
(
1 +1
2α
)
eασ2T Φ
(
−(α +1
2)σ√
T
)
Optimal relative error r∗(α, σ) decreases in α and increases inσ
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Optimal Relative Error - Good Stock (Cont’d)
We have
E
[
PT
MT
]
=
∫ 1
0
P
(
PT
MT> y
)
dy
=
∫ 1
0
∫
∞
y
∫ p/y
p∨1
fT (p,m)dmdpdy
= · · ·
=
(
1 − 1
2α
)
Φ
(
(α − 1
2)σ√
T
)
+
(
1 +1
2α
)
eασ2T Φ
(
−(α +1
2)σ√
T
)
Optimal relative error r∗(α, σ) decreases in α and increases inσ
0 ≤ r∗(α, σ) < 12α
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Optimal Relative Error - Bad Stock
Assume α < 0.5
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Optimal Relative Error - Bad Stock
Assume α < 0.5
Optimal selling time τ∗ = 0
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Optimal Relative Error - Bad Stock
Assume α < 0.5
Optimal selling time τ∗ = 0
Optimal relative error
r∗(α, σ)
=1 − 2α − 1
2(α − 1)Φ
(
(1
2− α)σ
√T
)
− 2α − 3
2(α − 1)e(1−α)σ2T Φ
(
(α − 3
2)σ√
T
)
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
The Messages
If one accepts the investment criterion in our model, then
The stock is said to be “good” if α ≥ 0.5
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
The Messages
If one accepts the investment criterion in our model, then
The stock is said to be “good” if α ≥ 0.5
In this case one should apply a pure buy-and-hold policy
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
The Messages
If one accepts the investment criterion in our model, then
The stock is said to be “good” if α ≥ 0.5
In this case one should apply a pure buy-and-hold policyThe better the stock the smaller relative error; the latterdiminishes to zero as α goes to infinity
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
The Messages
If one accepts the investment criterion in our model, then
The stock is said to be “good” if α ≥ 0.5
In this case one should apply a pure buy-and-hold policyThe better the stock the smaller relative error; the latterdiminishes to zero as α goes to infinity
The stock is said to be “bad” if α < 0.5
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
The Messages
If one accepts the investment criterion in our model, then
The stock is said to be “good” if α ≥ 0.5
In this case one should apply a pure buy-and-hold policyThe better the stock the smaller relative error; the latterdiminishes to zero as α goes to infinity
The stock is said to be “bad” if α < 0.5
In this case one should not buy in the first place, or shouldshort if possible
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
S&P 500
Mehra & Prescott (1985): a − r = 6.18%, σ = 16.67% basedon S&P 500 (1889-1978)
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
S&P 500
Mehra & Prescott (1985): a − r = 6.18%, σ = 16.67% basedon S&P 500 (1889-1978)
α = 2.2239 > 0.5 (by large margin)!
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
S&P 500
Mehra & Prescott (1985): a − r = 6.18%, σ = 16.67% basedon S&P 500 (1889-1978)
α = 2.2239 > 0.5 (by large margin)!
Taking T = 1, r∗(α, σ) = 10.15%
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
S&P 500
Mehra & Prescott (1985): a − r = 6.18%, σ = 16.67% basedon S&P 500 (1889-1978)
α = 2.2239 > 0.5 (by large margin)!
Taking T = 1, r∗(α, σ) = 10.15%
Buy and hold an S&P 500 index fund for one year:Statistically expected to achieve almost 90% of the maximumpossible return!
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Market Timing
Buy-and-hold rule believed to be the antithesis of markettiming
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Market Timing
Buy-and-hold rule believed to be the antithesis of markettiming
You can’t really enter on lows and sell on highs so you mightas well just buy and hold
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Market Timing
Buy-and-hold rule believed to be the antithesis of markettiming
You can’t really enter on lows and sell on highs so you mightas well just buy and hold
Yet, our model indeed based on market timing (attempting tosell high)
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Market Timing
Buy-and-hold rule believed to be the antithesis of markettiming
You can’t really enter on lows and sell on highs so you mightas well just buy and hold
Yet, our model indeed based on market timing (attempting tosell high)
It actually leads to buy and hold!
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Market Timing
Buy-and-hold rule believed to be the antithesis of markettiming
You can’t really enter on lows and sell on highs so you mightas well just buy and hold
Yet, our model indeed based on market timing (attempting tosell high)
It actually leads to buy and hold!So the market timing is consistent with buy-and-hold
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Insensitivity to Market Parameters
Our optimal solutions insensitive to the market parameters
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Insensitivity to Market Parameters
Our optimal solutions insensitive to the market parameters
Definition of good/bad stock involves a range of parameters,instead of specific values
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Insensitivity to Market Parameters
Our optimal solutions insensitive to the market parameters
Definition of good/bad stock involves a range of parameters,instead of specific values
As with S&P 500, α ≥ 0.5 satisfied by a large margin whichwould accommodate sufficient level of (estimation) errors
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Insensitivity to Market Parameters
Our optimal solutions insensitive to the market parameters
Definition of good/bad stock involves a range of parameters,instead of specific values
As with S&P 500, α ≥ 0.5 satisfied by a large margin whichwould accommodate sufficient level of (estimation) errors
In statistical terms, verifying whether α ≥ 0.5 much easierthan estimating α itself
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Insensitivity to Market Parameters
Our optimal solutions insensitive to the market parameters
Definition of good/bad stock involves a range of parameters,instead of specific values
As with S&P 500, α ≥ 0.5 satisfied by a large margin whichwould accommodate sufficient level of (estimation) errors
In statistical terms, verifying whether α ≥ 0.5 much easierthan estimating α itself
Notorious mean–blur problem hardly an issue in our model
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Insensitivity to Market Parameters
Our optimal solutions insensitive to the market parameters
Definition of good/bad stock involves a range of parameters,instead of specific values
As with S&P 500, α ≥ 0.5 satisfied by a large margin whichwould accommodate sufficient level of (estimation) errors
In statistical terms, verifying whether α ≥ 0.5 much easierthan estimating α itself
Notorious mean–blur problem hardly an issue in our model
On the other hand: an interesting problem to test hypothesisα ≥ 0.5
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
How Essential is Bang–Bang Policy?
Consider
max06τ6T
E
[
U
(
Pτ
MT
)]
(5)
We have shown that if U is linear then optimal stopping isbang–bang (either τ∗ = T or τ∗ = 0)
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
How Essential is Bang–Bang Policy?
Consider
max06τ6T
E
[
U
(
Pτ
MT
)]
(5)
We have shown that if U is linear then optimal stopping isbang–bang (either τ∗ = T or τ∗ = 0)
If U is logarithm or power then optimal stopping times areexactly the same
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
How Essential is Bang–Bang Policy?
Consider
max06τ6T
E
[
U
(
Pτ
MT
)]
(5)
We have shown that if U is linear then optimal stopping isbang–bang (either τ∗ = T or τ∗ = 0)
If U is logarithm or power then optimal stopping times areexactly the same
What about a general U?
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Thou Shalt Still Buy and Hold
Let v(x) := U(ex). The problem is equivalent to
max06τ6T
E[
v(Bµτ − Sµ
T )]
Theorem
(Dai, Jin and Zhou 2008)
If v is increasing and convex, then optimal stopping isbang–bang (τ∗ = T if α ≥ 0.5, and τ∗ = 0 if α < 0.5).
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Thou Shalt Still Buy and Hold
Let v(x) := U(ex). The problem is equivalent to
max06τ6T
E[
v(Bµτ − Sµ
T )]
Theorem
(Dai, Jin and Zhou 2008)
If v is increasing and convex, then optimal stopping isbang–bang (τ∗ = T if α ≥ 0.5, and τ∗ = 0 if α < 0.5).
Assume v is increasing, C2, and |v′′(x)| ≤ ek(x2+1). If optimalstopping is bang–bang, then v must be convex.
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Convexity and Behavioural Finance
Experiment 1: You have been given £1000. Now choosebetween
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Convexity and Behavioural Finance
Experiment 1: You have been given £1000. Now choosebetween
A: Win £1000 with 50% chance and £0 with 50% chance
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Convexity and Behavioural Finance
Experiment 1: You have been given £1000. Now choosebetween
A: Win £1000 with 50% chance and £0 with 50% chanceB: Win £500 with 100% chance
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Convexity and Behavioural Finance
Experiment 1: You have been given £1000. Now choosebetween
A: Win £1000 with 50% chance and £0 with 50% chanceB: Win £500 with 100% chanceB was more popular
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Convexity and Behavioural Finance
Experiment 1: You have been given £1000. Now choosebetween
A: Win £1000 with 50% chance and £0 with 50% chanceB: Win £500 with 100% chanceB was more popular
Experiment 2: You have been given £2000. Now choosebetween
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Convexity and Behavioural Finance
Experiment 1: You have been given £1000. Now choosebetween
A: Win £1000 with 50% chance and £0 with 50% chanceB: Win £500 with 100% chanceB was more popular
Experiment 2: You have been given £2000. Now choosebetween
A: Lose £1000 with 50% chance and £0 with 50% chance
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Convexity and Behavioural Finance
Experiment 1: You have been given £1000. Now choosebetween
A: Win £1000 with 50% chance and £0 with 50% chanceB: Win £500 with 100% chanceB was more popular
Experiment 2: You have been given £2000. Now choosebetween
A: Lose £1000 with 50% chance and £0 with 50% chanceB: Lose £500 with 100% chance
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Convexity and Behavioural Finance
Experiment 1: You have been given £1000. Now choosebetween
A: Win £1000 with 50% chance and £0 with 50% chanceB: Win £500 with 100% chanceB was more popular
Experiment 2: You have been given £2000. Now choosebetween
A: Lose £1000 with 50% chance and £0 with 50% chanceB: Lose £500 with 100% chanceThis time: A was more popular
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Convexity and Behavioural Finance
Experiment 1: You have been given £1000. Now choosebetween
A: Win £1000 with 50% chance and £0 with 50% chanceB: Win £500 with 100% chanceB was more popular
Experiment 2: You have been given £2000. Now choosebetween
A: Lose £1000 with 50% chance and £0 with 50% chanceB: Lose £500 with 100% chanceThis time: A was more popular
Yet the two experiments have exactly the same final wealthpositions!
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Convexity and Behavioural Finance
Experiment 1: You have been given £1000. Now choosebetween
A: Win £1000 with 50% chance and £0 with 50% chanceB: Win £500 with 100% chanceB was more popular
Experiment 2: You have been given £2000. Now choosebetween
A: Lose £1000 with 50% chance and £0 with 50% chanceB: Lose £500 with 100% chanceThis time: A was more popular
Yet the two experiments have exactly the same final wealthpositions!
Reference point (Kahneman and Tversky 1979) or customarywealth (Markowizt 1952)
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Convexity and Behavioural Finance
Experiment 1: You have been given £1000. Now choosebetween
A: Win £1000 with 50% chance and £0 with 50% chanceB: Win £500 with 100% chanceB was more popular
Experiment 2: You have been given £2000. Now choosebetween
A: Lose £1000 with 50% chance and £0 with 50% chanceB: Lose £500 with 100% chanceThis time: A was more popular
Yet the two experiments have exactly the same final wealthpositions!
Reference point (Kahneman and Tversky 1979) or customarywealth (Markowizt 1952)
Risk-averse on gains, and risk-seeking on losses
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Maximum as Reference Point
max06τ6T
E[
v(Bµτ − Sµ
T )]
In this model the maximum SµT implicitly taken as reference
point
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Maximum as Reference Point
max06τ6T
E[
v(Bµτ − Sµ
T )]
In this model the maximum SµT implicitly taken as reference
point
Always in a “loss” situation
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Maximum as Reference Point
max06τ6T
E[
v(Bµτ − Sµ
T )]
In this model the maximum SµT implicitly taken as reference
point
Always in a “loss” situation
Hence risk-seeking or having convex v
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Maximum as Reference Point
max06τ6T
E[
v(Bµτ − Sµ
T )]
In this model the maximum SµT implicitly taken as reference
point
Always in a “loss” situation
Hence risk-seeking or having convex v
Bang-bang (or buy-and-hold) behaviour consistent withbehavioural theory
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Conclusions
It is only natural that an investor wishes to sell a stock closest
to the maximum price over a given planning horizon
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Conclusions
It is only natural that an investor wishes to sell a stock closest
to the maximum price over a given planning horizon
The problem is formulated as an optimal stopping problem
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Conclusions
It is only natural that an investor wishes to sell a stock closest
to the maximum price over a given planning horizon
The problem is formulated as an optimal stopping problem
We have shown that, even in the absence of transaction costs,
one should buy and hold, if the stock is good, that is
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Conclusions
It is only natural that an investor wishes to sell a stock closest
to the maximum price over a given planning horizon
The problem is formulated as an optimal stopping problem
We have shown that, even in the absence of transaction costs,
one should buy and hold, if the stock is good, that is
By a good stock we specify that its goodness index is
greater than 0.5
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Conclusions
It is only natural that an investor wishes to sell a stock closest
to the maximum price over a given planning horizon
The problem is formulated as an optimal stopping problem
We have shown that, even in the absence of transaction costs,
one should buy and hold, if the stock is good, that is
By a good stock we specify that its goodness index is
greater than 0.5
This rule is robust with respect to different “utilities” oneapplies
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Conclusions
It is only natural that an investor wishes to sell a stock closest
to the maximum price over a given planning horizon
The problem is formulated as an optimal stopping problem
We have shown that, even in the absence of transaction costs,
one should buy and hold, if the stock is good, that is
By a good stock we specify that its goodness index is
greater than 0.5
This rule is robust with respect to different “utilities” oneapplies
This, at least from one angle, reinforces the conventional
maxim of buy and hold
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
Conclusions
It is only natural that an investor wishes to sell a stock closest
to the maximum price over a given planning horizon
The problem is formulated as an optimal stopping problem
We have shown that, even in the absence of transaction costs,
one should buy and hold, if the stock is good, that is
By a good stock we specify that its goodness index is
greater than 0.5
This rule is robust with respect to different “utilities” oneapplies
This, at least from one angle, reinforces the conventional
maxim of buy and hold
That our model produces buy-and-hold rule suggests thecriterion proposed sensible and warrants further investigations
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
A Serenity Prayer: Illusion of Control
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold
A Serenity Prayer: Illusion of Control
God, grant me the serenity to accept the things I cannot
control, courage to control the things I can, and the wisdom to
know the difference.
—— Meir Statman
Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold