Risk Aversion and Pricing Kernel Dynamics
Enzo Giacomini
Wolfgang Härdle
CASE - Center for Applied
Statistics and Economics
Humboldt-Universität zu Berlin
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Motivation
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1.1
Figure 1: Empirical (red) and theoretical (blue) pricing kernel, DAX 19990205, τ
= 10
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Risk Aversion and Pricing Kernels Dynamics
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3
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days.
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Motivation
1-2
Price at time t from a payoff ψ(ST )
0
u (ST )
ψ(ST )
zt = E
u 0 (St )
where
1. St is value at time t from wealth, consumption, asset
2. ψ(ST ) is a payoff dependent on ST
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Risk Aversion and Pricing Kernels Dynamics
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3. u(x) is utility function representing investors preferences
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Motivation
1-3
Pricing Kernel
Pricing Kernel (PK) at time t and maturity τ = T − t
Mτ (ST ) =
u 0 (ST )
u 0 (St )
under risk aversion
1. utility function u(x) concave, increasing
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Risk Aversion and Pricing Kernels Dynamics
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3
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2. pricing kernel M(x) monotone decreasing
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Motivation
1-4
1. absolute risk aversion (ARA)
α(x) = −
u 00 (x)
u 0 (x)
2. relative risk aversion (RRA)
u 00 (x)
u 0 (x)
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Risk Aversion and Pricing Kernels Dynamics
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3
4
ρ(x) = −x
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1.1
Motivation
1-5
CRRA / CARA Utility Functions
CARA utility
1
u(x) = − e −αx
α
α > 0 is the absolute risk aversion coefficient.
CRRA utility (power utility)
u(x) =
x 1−γ − 1
1−γ
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Risk Aversion and Pricing Kernels Dynamics
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3
4
γ ∈ (0, 1) is the relative risk aversion coefficient.
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5
Motivation
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1
2
3
4
5
X
CRRA, u(x) =
xγ
γ
(red), α(x) =
1−γ
x
(blue), ρ(x) = γ (green), γ = 0.5.
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Risk Aversion and Pricing Kernels Dynamics
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3
4
Figure 2:
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1-7
-20
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20
40
Motivation
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0.85
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0.95
1
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1.1
X/F
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Empirical RRA, DAX 19990205, τ = 10 days.
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Risk Aversion and Pricing Kernels Dynamics
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Figure 3:
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1.1
Motivation
1-8
4.00
98.77
3.20
72.95
2.40
47.13
1.60
21.31
0.80
-4.51
10.00
32.00
54.00
76.00
98.00
Figure 4:
0.88
0.96
1.03
1.11
1.19
10.00
32.00
54.00
76.00
98.00
0.88
0.95
1.18
1.10
1.03
Empirical PK Mτ (κ) and RRA ρτ (κ) across maturities τ and future mon-
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Risk Aversion and Pricing Kernels Dynamics
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4
eyness κ, DAX 19990225.
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1.1
Motivation
1-9
Empirical pricing kernels (EPK)
1. do not reflect risk aversion across all strikes
2. vary across time to maturity τ and time t
M(x) = Mt,τ (x)
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Risk Aversion and Pricing Kernels Dynamics
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How to explain pricing kernel and risk aversion dynamics ?
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Motivation
1-10
Outline
1. Motivation
X
2. Risk Measures
3. Pricing Kernels
4. Estimation
5. Empirical Results
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Risk Aversion and Pricing Kernels Dynamics
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6. References
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1.1
Risk Measures
2-1
Risk Measures
Preference order on a set P. For µ, ν ∈ P:
µ ν: µ is preferred to ν
µ ∼ ν: indifference between µ and ν
µ ν: ν is not preferred to µ
Numerical representation U : P → R of µ ν ⇔ U(µ) ≥ U(ν)
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Risk Aversion and Pricing Kernels Dynamics
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3
4
U is unique up to affine transformations.
0.9
1
X/F
1.1
Risk Measures
2-2
Von Neumann-Morgenstern representation
Z
U(µ) = u(x)µ(dx)
1. P set of probability measures on R
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Risk Aversion and Pricing Kernels Dynamics
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4
2. u elementary utility function.
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1
X/F
1.1
Risk Measures
2-3
Mean and Variance
Z
mµ =
σµ2 =
xµ(dx)
Z
{x − mµ }2 µ(dx)
Example
µ = δa
Z
mµ =
xδa (dx) = a
Z
σµ2 =
(x − a)2 δa (dx) = 0
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1
Risk Aversion and Pricing Kernels Dynamics
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3
4
where δa is the Dirac measure on a.
0.9
1
X/F
1.1
Risk Measures
2-4
Definition: Preference order is
1. risk averse
µ 6= δmµ : δmµ µ
2. risk proclive
µ 6= δmµ : µ δmµ
3. risk neutral
δmµ ∼ µ
4. monotone
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Risk Aversion and Pricing Kernels Dynamics
0
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3
4
a, b ∈ R, a > b : δa δb
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1
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1.1
Risk Measures
2-5
Preference order on P with von Neumann-Morgenstern
representation
Z
U(µ) = u(x)µ(dx)
risk averse iff u strictly concave
risk proclive iff u strictly convex
risk neutral iff u linear
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Risk Aversion and Pricing Kernels Dynamics
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3
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monotone iff u strictly increasing
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1.1
Risk Measures
2-6
2
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3
4
u(x)
0
0.5
Figure 5: Monotone utility functions:
1
X
1.5
risk averse u(x) =
2
√
x (red), risk neutral u(x) =
2
1
Risk Aversion and Pricing Kernels Dynamics
0
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3
4
x (blue) and risk proclive u(x) = x 2 (green).
0.9
1
X/F
1.1
Risk Measures
2-7
Example
1. USD 4 or 9 with probability 0.5:
µ = 0.5(δ4 + δ9 )
mµ = 6.5
2. USD 6.5 with probability 1:
ν = δmµ = δ6.5
2
1
Risk Aversion and Pricing Kernels Dynamics
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3
4
How do µ and ν relate under risk neutrality / aversion / proclivity ?
0.9
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1.1
Risk Measures
2-8
Linear utility function u(x) = x (risk neutrality)
Z
4+9
= 6.5
U(µ) =
xµ(dx) =
2
Z
U(ν) =
xδ6.5 (dx) = 6.5
Therefore
U(ν) = U(µ)
hence,
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Risk Aversion and Pricing Kernels Dynamics
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3
4
δmµ ∼ µ
0.9
1
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1.1
Risk Measures
2-9
Concave utility function u(x) =
√
x (risk aversion)
√
√
Z
√
4+ 9
xµ(dx) =
U(µ) =
= 2.5
2
Z
√
√
xδ6.5 (dx) = 6.5 = 2.5495
U(ν) =
Therefore
U(ν) > U(µ)
hence,
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Risk Aversion and Pricing Kernels Dynamics
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3
4
δmµ µ
0.9
1
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1.1
Risk Measures
2-10
Convex utility function u(x) = x 2 (risk proclivity)
U(ν) =
42 + 92
= 48.5
2
Z
x 2 µ(dx) =
Z
x 2 δ6.5 (dx) = 6.52 = 42.25
U(µ) =
Therefore
U(ν) < U(µ)
hence,
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Risk Aversion and Pricing Kernels Dynamics
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3
4
µ δmµ
0.9
1
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1.1
Risk Measures
2-11
Some Definitions
Certainty equivalent cµ :
δ cµ ∼ µ
Risk premium πµ
πµ = m µ − c µ
1. insurance buyer: πµ maximal price to exchange µ for δmµ
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Risk Aversion and Pricing Kernels Dynamics
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4
2. insurance seller: πµ minimal price to exchange δmµ for µ
0.9
1
X/F
1.1
Risk Measures
2-12
Risk aversion: positive risk premium
δmµ
µ ∼ δ cµ
u(mµ ) > U(µ) = u(cµ )
mµ > cµ
πµ > 0
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1
Risk Aversion and Pricing Kernels Dynamics
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3
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Risk neutrality: zero risk premium
Risk proclivity: negative risk premium
0.9
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1.1
Risk Measures
1. u(x) =
√
2-13
x - risk aversion
U(µ) = 2.5 = U(δcµ ) =
√
cµ
2
cµ = 2.5 = 6.25
πµ = 6.5 − 6.25 = 0.25
2. u(x) = x - risk neutrality
U(µ) = 6.5 = U(δcµ ) = cµ
πµ = 6.5 − 6.5 = 0
3. u(x) = x 2 - risk proclivity
U(µ) = 48.5 = U(δcµ ) = cµ2
√
cµ =
48.5 = 6.964
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Risk Aversion and Pricing Kernels Dynamics
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3
4
πµ = 6.5 − 6.964 = −0.4642
0.9
1
X/F
1.1
Risk Measures
2-14
Measuring Risk Aversion
Taylor expansion at c = cµ around m = mµ
u(c) ≈ u(m) + u 0 (m)π
Z
u(c) =
u(x)µ(dx)
Z
1
{u(m) + u 0 (m)(x − m) + u 00 (m)(x − m)2 }µ(dx)
2
1 00
≈ u(m) + u (m)σµ2
2
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Risk Aversion and Pricing Kernels Dynamics
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3
4
≈
0.9
1
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1.1
Risk Measures
2-15
Thus
πµ ≈ −
1 u 00 (m) 2
σ
2 u 0 (m) µ
Arrow-Pratt measures of
absolute risk aversion (ARA)
α(x) = −
u 00 (x)
u 0 (x)
relative risk aversion (RRA)
ρ(x) = −x
u 00 (x)
u 0 (x)
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1
Risk Aversion and Pricing Kernels Dynamics
0
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3
4
where x is the expected value from a distribution.
0.9
1
X/F
1.1
Risk Measures
2-16
Absolute Risk Aversion
Constant (CARA): γ ∈ R, α(x) = γ
⇒ ∃a, b ∈ R : u(x) = a − be γx
Decreasing (DARA): x1 > x2 ⇒ α(x1 ) < α(x2 )
Increasing (IARA): x1 > x2 ⇒ α(x1 ) > α(x2 )
γ −1 x γ , γ ∈ (0, 1)
log(x) , γ = 0
2
1
Risk Aversion and Pricing Kernels Dynamics
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4
Hyperbolic (HARA): x ∈ (0, ∞)
1−γ
α(x) =
⇒ u(x) =
x
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1.1
Risk Measures
2-17
Relative Risk Aversion
Constant (CRRA): ρ ∈ R, ρ(x) = ρ
Decreasing (DRRA): x1 > x2 ⇒ ρ(x1 ) < ρ(x2 )
Increasing (IRRA): x1 > x2 ⇒ ρ(x1 ) > ρ(x2 )
Special cases:
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Risk Aversion and Pricing Kernels Dynamics
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HARA utilities are CRRA
CARA utilities are IRRA.
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X/F
1.1
Risk Measures
2-18
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0
1
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3
4
5
Power Utility / HARA / CRRA
0
1
2
3
4
5
X
(red), α(x) =
1−γ
x
HARA (blue), ρ(x) = γ CRRA (green),
Y
3
4
γ = 0.5.
Risk Aversion and Pricing Kernels Dynamics
2
xγ
γ
1
u(x) =
0
Figure 6:
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1.1
Risk Measures
2-19
5
0
Y
10
Quadratic Utility / IARA / IRRA
2γ
β−2γ
for x ≤
β
,
2γ
IARA (blue), ρ(x) =
3
4
IRRA (green), γ = −1.9, β = 4.
Risk Aversion and Pricing Kernels Dynamics
2
u(x) = βx + γx 2 (red), α(x) =
3
Y
x
2γ
,
β−2γ
2
X
1
Figure 7:
1
0
0
0.9
1
X/F
1.1
Risk Measures
2-20
Preference order 1 more risk averse than 2 :
π1 (µ) > π2 (µ), ∀µ ∈ P
α1 (x) > α2 (x), ∀x ∈ I
u1 = F ◦ u2
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1
Risk Aversion and Pricing Kernels Dynamics
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3
4
where F is strictly increasing and concave.
0.9
1
X/F
1.1
Risk Measures
2-21
Comparing Distributions
µ is uniformly preferred over ν
µ ∗ ν
if
Z
Z
u(x)µ(dx) ≥
u(x)ν(dx)
2
1
Risk Aversion and Pricing Kernels Dynamics
0
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3
4
for ALL strictly concave and increasing functions u.
0.9
1
X/F
1.1
Risk Measures
2-22
Example
N(a, σ1 ) ∗ N(b, σ2 ) ⇔ a ≥ b, σ1 ≤ σ2
N(2, 1) ∗ N(1, 1) ∗ N(1, 2)
N(1, 1) ∗ N(0, 1)
2
1
Risk Aversion and Pricing Kernels Dynamics
0
Y
3
4
N(1, 2) ∗ N(0, 1)
0.9
1
X/F
1.1
Risk Measures
2-23
Example
log N(a1 , b1 ) ∗ log N(a, b2 ) ⇔

 a1 + 12 b12 ≥ a2 + 21 b22

b12 ≤ b22
log N(5, 1) ∗ log N(1.5, 2)
2
1
Risk Aversion and Pricing Kernels Dynamics
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3
4
log N(1, 1) ∗ log N(1, 2)
0.9
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1.1
Pricing Kernels
3-1
Pricing Kernels
Stock prices follow
dSt
= µdt + σdBt
St
where
1. Bt is standard Browinan motion
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Risk Aversion and Pricing Kernels Dynamics
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3
4
2. p(ST ) is the conditional distribution of ST given information
up to time t
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1.1
Pricing Kernels
3-2
Representative investor with
1. interest rates r ,
2. utility function u(x)
3. wealth proces {Ws }
4. stock proces {Ss }
5. consumption process {Cs }, Cs = 0, t < s < T
6. adjusts amounts {qs } invested in Ss at times t ≤ s ≤ T
7. consumes all wealth at T , CT = WT
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Risk Aversion and Pricing Kernels Dynamics
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3
4
chooses qs via
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1
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1.1
Pricing Kernels
3-3
Merton Optimization Problem
max
{qs ,t≤s≤T }
E [u(WT )]
subject to
Ws ≥ 0
dWs = {rWs + qs (µ − r )}ds + qs σdBs
1. in equilibrium, t ≤ s ≤ T : Ws = Ss
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1
Risk Aversion and Pricing Kernels Dynamics
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3
4
2. at the end consume all wealth, i.e. CT = WT = ST
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1
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1.1
Pricing Kernels
3-4
Terminal condition at s = T :
u 0 (WT )
= e −r τ ζT
u 0 (Wt )
where
"Z
ζs = exp
t
s
µ−r
σ
1
dBu −
2
Z
t
s
µ−r
σ
#
2
du
Defining state-price density as
2
1
Risk Aversion and Pricing Kernels Dynamics
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3
4
q(ST ) = p(ST )ζT
0.9
1
X/F
1.1
Pricing Kernels
3-5
Pricing Kernel
Price zt of payoff ψ(ST )
u 0 (ST )
= E
ψ(ST )
u 0 (St )
zt
Pricing kernel
u 0 (ST )
q(ST )
= e −r τ
u 0 (St )
p(ST )
2
1
Risk Aversion and Pricing Kernels Dynamics
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3
4
Mt (ST ) =
0.9
1
X/F
1.1
Pricing Kernels
3-6
Prices zt can be written as
zt
= E [Mt (ST )ψ(ST )]
Z
=
Mt (ST )ψ(ST )p(ST )dST
Z
−r τ
= e
ψ(ST )q(ST )dST
2
1
Risk Aversion and Pricing Kernels Dynamics
0
Y
3
4
= e −r τ E ∗ [ψ(ST )]
0.9
1
X/F
1.1
Pricing Kernels
3-7
Utility Functions and Pricing Kernels
From the pricing kernel
Mt (ST ) =
u 0 (ST )
q(ST )
= e −r τ
0
u (St )
p(ST )
we obtain the utility funtion
Z
u (St )
q(ST )
dST
p(ST )
2
1
Risk Aversion and Pricing Kernels Dynamics
0
Y
3
4
u(ST ) = e
−r τ 0
0.9
1
X/F
1.1
Pricing Kernels
3-8
u 0 (ST )
u 0 (St )
M(ST ) =
u
@
@
@
@
@
@
@
p(ST )
@
@u q(ST )
Pricing kernel, utility function, risk neutral and objective measures.
2
1
Risk Aversion and Pricing Kernels Dynamics
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3
4
Figure 8:
u
0.9
1
X/F
1.1
Pricing Kernels
3-9
Example
Obtain u(ST ) under Black-Scholes assumptions
1. under objective measure P : St ∼ log N{τ (µ −
2. under risk neutral measure Q : St ∼ log N{τ (r
3. market price of risk λ =
4. c =
σ2
2
2 ), σ τ }
2
− σ2 ), σ 2 τ }
µ−r
σ
λ
σ
5. a = exp
n
λ(λ−σ)τ
2
o
2
1
Risk Aversion and Pricing Kernels Dynamics
0
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3
4
6. b = e −r τ u 0 (St )
0.9
1
X/F
1.1
Pricing Kernels
q(ST )
p(ST )
3-10
=
ST
St
− µ−r
2
σ
(µ − r )(µ − r − σ 2 )τ
exp
2σ 2
λ(λ − σ)τ
exp
2
λ
ST − σ
=
St
−c
ST
= a
St
Hence
Z
a
−c
dST
2
1
Risk Aversion and Pricing Kernels Dynamics
0
Y
3
4
u(ST ) = b
ST
St
0.9
1
X/F
1.1
Pricing Kernels
3-11
Therefore, up to a constant either we get
1. HARA / CRRA utility functions:

1−c

T
 S1−c
, c=
6 1
u(ST ) =


log(ST ) , c = 1
2. or risk neutral utility function
u(ST ) = kST , c = 0
2
1
Risk Aversion and Pricing Kernels Dynamics
0
Y
3
4
where k is a constant.
0.9
1
X/F
1.1
Pricing Kernels
3-12
Example
Obtain q(ST ) given power utility and
dSt
St
= µdt + σdWt
1. under objective measure P : St ∼ log N{τ (µ −
σ2
2
2 ), σ τ }
1−c
2. power utility function u(x) = ka x1−c
3. λ = µ − r , c = σλ
n
o
4. a = exp λ(λ−σ)τ
2
−r τ
5. k = u 0 (St ) eS −c
t
= e −r τ a
ST
St
−c
2
1
Risk Aversion and Pricing Kernels Dynamics
0
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3
4
6.
u 0 (ST )
u 0 (St )
0.9
1
X/F
1.1
Pricing Kernels
u 0 (ST )
erτ 0
p(ST )
u (St )
3-13
=
a
√
ST πσ 2 τ
ST
St
− σλ
2 n ST
6 log St − (µ −
exp 4−
2
σ2
)
2
2σ τ
o2 3
7
5
2
n S log ST − (µ −
λ
ST
6
t
exp 4− log
−
2
=
a
√
ST πσ 2 τ
=
1
6 log
√ 2 exp 4−
ST πσ τ
=
q(ST )
σ
2 n
2σ τ
S T
St
− (r −
2σ 2 τ
σ2
)
2
o2 3
7
5
o2 3
7
5
σ2
2
2 ), σ τ }
2
1
Risk Aversion and Pricing Kernels Dynamics
0
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3
4
Hence under Q : ST ∼ log N{τ (r −
St
σ2
)
2
0.9
1
X/F
1.1
Pricing Kernels
3-14
Relative Risk Aversion
2
1
Risk Aversion and Pricing Kernels Dynamics
0
Y
3
4
From p(ST ) and q(ST ) we obtain the relative risk aversion
0
u 00 (ST )
p (ST ) q 0 (ST )
ρ(ST ) = −ST 0
= ST
−
u (ST )
p(ST )
q(ST )
0.9
1
X/F
1.1
Pricing Kernels
3-15
Black Scholes
Asset St , call option ψ(ST ) = (ST − K )+ , time to maturity
τ = T − t, interest rate rt , q(ST ) risk neutral density:
Z ∞
−r τ
C (St , K , τ, rt , σt ) = e
ψ(ST )q(ST )dST
0
= St Φ(d1 ) − Ke −r τ Φ(d2 )
where
d1 =
St
K
+ (r +
√
σ τ
σ2
2 )τ
√
d2 = d1 − σ τ
2
1
Risk Aversion and Pricing Kernels Dynamics
0
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3
4
and
log
0.9
1
X/F
1.1
Pricing Kernels
3-16
The risk-neutral density may be obtained as
 n 2 log SSTt − (r −
1

rτ ∂ C √ 2 exp −
e =
∂K 2 2σ 2 τ
S
πσ τ
K =ST
o2 
σ2
2 )


T
2
1
Risk Aversion and Pricing Kernels Dynamics
0
Y
3
4
= q(ST )
0.9
1
X/F
1.1
Estimation
4-1
Estimation
The following function are estimated by
1. implied volatility
Pn
τ −τi
S−Si
K
σ̂i
K
τ
i=1 S
h
hτ
S σ̂t (S, τ ) = P
n
S−Si
i
Kτ τ h−τ
i=1 KS
hS
τ
2. call prices
Ĉ (St , K , τ, rt , σ̂t )
3. state-price density
K =ST
2
1
Risk Aversion and Pricing Kernels Dynamics
0
Y
3
4
∂ Ĉ 2 (S , K , τ, r , σ̂ ) t
t
t q̂t,τ (ST ) = e r τ ∂2K
0.9
1
X/F
1.1
Estimation
4-2
1. objective distribution p̂t,τ (ST )
GARCH (1, 1) or µdt + σdWt
2. pricing kernel
M̂(ST ) = e −r τ
q̂(ST )
p̂(ST )
3. relative risk aversion
2
1
Risk Aversion and Pricing Kernels Dynamics
0
Y
3
4
ρ̂t,τ (ST ) = ST
p̂ 0 (ST ) q̂ 0 (ST )
−
p̂(ST )
q̂(ST )
0.9
1
X/F
1.1
Empirical Results
5-1
Empirical Results
0.00
0.00
0.00
0.00
0.00
10.00
32.00
54.00
76.00
98.00
Figure 9:
6550.00
5840.00
5130.00
4420.00
3710.00
Implied distribution q̂t,τ (ST ) (red) for different strikes and maturities at
2
1
Risk Aversion and Pricing Kernels Dynamics
0
Y
3
4
date t = 19990303, St = 4746.
0.9
1
X/F
1.1
Empirical Results
5-2
0.00
0.00
0.00
0.00
0.00
10.00
32.00
54.00
76.00
98.00
Figure 10:
6950.00
6160.00
5370.00
4580.00
3790.00
Subjective distribution p̂t,τ (ST ) (blue) for different strikes and maturities
2
1
Risk Aversion and Pricing Kernels Dynamics
0
Y
3
4
at date t = 19990303, St = 4746.
0.9
1
X/F
1.1
Empirical Results
5-3
1.66
1.33
1.00
0.67
0.34
30.00
48.00
66.00
84.00
102.00
Figure 11:
6450.00
5760.00
5070.00
4380.00
3690.00
Pricing kernel m̂t,τ (ST ) (green) for different strikes and maturities at date
2
1
Risk Aversion and Pricing Kernels Dynamics
0
Y
3
4
t = 19990303, St = 4746.
0.9
1
X/F
1.1
Empirical Results
5-4
rra surface
16.69
12.77
8.85
4.93
1.01
30.00
48.00
66.00
84.00
102.00
Figure 12:
5750.00
5370.00
4990.00
4610.00
4230.00
Implied relative risk aversion ρ̂t,τ (ST ) (magenta) for different strikes and
2
1
Risk Aversion and Pricing Kernels Dynamics
0
Y
3
4
maturities at date t = 19990303, St = 4746.
0.9
1
X/F
1.1
References
6-1
References
Y. Ait-Sahalia and A. Lo
Nonparametric Risk Management and Implied Risk Aversion
Journal of Econometrics, 94 (2000) 9-51.
P. Fishburn
Utility Theory
Management Science, Vol. 14, No. 5, Theory Series (Jan,
1968) 335-378.
2
1
Risk Aversion and Pricing Kernels Dynamics
0
Y
3
4
H. Föllmer and A. Schied
Stochastic Finance
Walter de Gruyter, Berlin 2002.
0.9
1
X/F
1.1
References
6-2
J. Franke, W. Härdle and C. Hafner
Statistics of Financial Markets
Springer-Verlag, Heidelberg, 2004.
W. Härdle, T. Kleinow and G. Stahl
Applied Quantitative Finance
Springer-Verlag, Heidelberg, 2002.
2
1
Risk Aversion and Pricing Kernels Dynamics
0
Y
3
4
R. Merton
Continuous-Time Finance
Blackwell Publishers, Cambridge, 1990.
0.9
1
X/F
1.1