Introduction Asset Pricing
Max Batstra
March 29, 2026
1
, 2 Portfolio Choice
In this chapter we look at the investment decision of a single agent when prices are given. When we know
the decisions that will be made by agents at given prices, then we can determine supply and demand in the
market, and we can use this to determine prices.
The most frequently used utility functions are:
• Constant Absolute Risk Aversion (CARA)
u(x) = −αe−x/α where α > 0
• Constant Relative Risk Aversion (CRRA)
x1−γ − 1
u(x) = where γ ≥ 0 and γ ̸= 1
1−γ
x1−γ − 1
Important: lim = ln(x) [proof]
γ→1 1−γ
Overview of popular utility functions
Type Domain u(x) Parameter bounds u′ (x) v(x) R(x) θ
1 1 1 1
Log-utility (0, ∞) ln(x) none x x x c· ωag
ωag
x1−γ
−γ − γ1 γ − kτ
Power-utility (0, ∞) 1−γ γ > 0, γ ̸= 1 x x x c·e
− τx x
Exponential-utility (−∞, ∞) τc τ >0 c− τ τ ln(x) 1
τ
−γ
c · ωag
How the terms in the θ column are derived will be shown later.
Notation
• Outcomes are called ”states of the world”, or simply ”states”. i = 1, . . . , n states
• Single-period situation t = 0, 1 (static model).
t = 0 period in which investment takes place, information about states in t = 1 is unknown.
t = 1 period in which actual states are revealed.
• State prices πi for state contract i. This is a contract that pays 1 if state i materializes, and 0
otherwise.
• pi denotes the probability that state i materializes. Note E[xi ] = 1 · pi + 0 · (1 − pi ) = pi , so the pi ’s
denote the expected payoffs of the state contract.
• u(x) utility function of the agent, with u′ (x) > 0 (increasing), u′′ (x) < 0 (concave)
• W budget
2
,The investor aims to maximize it’s expected utility. This yields the maximization problem:
n
X
max pi u(xi )
x1 ,...,xn
i=1
Xn
s.t. πi xi = W
i=1
From [derivation 1], it follows that
πi
• xi = v λ for i = 1, . . . , n , where v(·) = u′ (·)−1 .
pi
Pn Pn
• λ follows from δ
δλ L(xi , λ) = 0 ⇐⇒ i=1 πi xi − W = 0 ⇐⇒ π
i=1 i v λ πi
pi =W
Time Effect
If we buy every asset, we pay π1 + . . . + πn , and we are sure to receive 1 at time 1 (no uncertainty), there-
fore, there is only a time effect, which can be described in terms of an interest rate r. r is the simply
compounded interest rate and is defined as follows:
If you buy every asset at time t = 0, you receive 1 at time t = 1 (1 state materializes) with probability 1, if
we have compound interest rate r. Therefore we have that:
n n
X X 1
(1 + r) πi = 1 ⇐⇒ πi =
i=1 i=1
1+r
It is often convenient to work with normalized state prices (qi for i = 1, . . . , n) which are defined by:
(
qi = (1 + r)πi π qi
=⇒ qi = Pn and πi =
(1 + r) = Pn1 πi i=1 π i 1 +r
i=1
Implied / Risk-neutral Probabilities (qi )
qi ’s are called risk neutral, because under these probabilities, asset prices behave as if investors are indifferent
to risk and are only concerned with expected payoffs, discounted at the risk free rate.
πi
qi = (1 + r)πi or qi = Pn
i=1 πi
qi
Using that πi = 1+r we can rewrite the portfolio choice problem in terms of the qi ’s:
n
X n
X n
X
max pi u(xi ) max pi u(xi ) max pi u(xi )
x1 ,...,xn x1 ,...,xn x1 ,...,xn
i=1 i=1 i=1
n
⇐⇒ n
⇐⇒ n
X X qi X
s.t. πi xi = W s.t. xi = W s.t. qi xi = W (1 + r)
i=1 i=1
1+r i=1
3
,Similarly to [derivation 1] it follows that the solution to this optimization problem can be written as:
n
λqi λqi X λqi
u′ (xi ) = =⇒ xi = v and λ is derived from qi v = (1 + r)W
pi pi i=1
pi
2.3 Interpretations & The Normalized Pricing Kernel
λqi
From xi = v pi it follows that the amount of money to be invested in the i-th state contract depends on
qi qi
the quotient pi . pican be viewed as a price/performance ratio, because qi is the normalized price of one unit
of state contract i and pi is the expected payoff of state contract i, (note E[xi ] = 1 · pi + 0 · (1 − pi ) = pi ).
From the following interpretations follow:
qi
• If < 1 state contract i is undervalued
pi
qi
• If > 1 state contract i is overvalued
pi
From the two observations:
• From u′ (·) is decreasing, it follows that v(·) is also decreasing
• From pi u′ (xi ) = λqi , pi > 0 , qi > 0 , u(xi ) > 0 (i = 1, . . . , n), it follows that λ > 0
and that xi = v( λq
pi ),
i
it follows that:
qi qj
xi < xj ⇐⇒ v λ >v λ
pi pj
Interpretation: The largest number of state contracts will be bought for the state j that has the lowest
q
value of pjj .
We assume that the states are numbered in order of attractiveness, so that:
q1 q2 qn
≤ ≤ ··· ≤
p1 p2 pn
q
From xi < xj ⇐⇒ v λ pqii > v λ pjj it then follows that:
x1 ≥ x2 ≥ . . . ≥ xn
Ranking of alternatives on the basis of expected utility is not changed when a constant is added to the
utility function, or when the utility function is multiplied by a positive constant. That means that taking
decision on the basis of u(x) is the same as taking decision on the basis of ũ(x) = a · u(x) + b where a > 0.
In this case, the utility functions u(·) and ũ(·) are said to be equivalent.
4
, Coefficient of absolute risk aversion:
u′′ (x)
R(x) = −
u′ (x)
Theorem
u′′ (x) u′′ (x)
Utility functions 1, 2 are equivalent ⇐⇒ ∀x: R1 (x) = − u1′ (x) = − u2′ (x) = R2 (x)
1 2
[Proof]
For v(·) = ln(·), we have that
Z xi
′ λqi qi qj
u (xi ) = ⇐⇒ R(x) d x = ln − ln
pi xj pi pj
Interpretation:
• When risk aversion increases and other things remain the same, the xi ’s will move closer together.
• When the ratios pqii are spread out more widely and other things remain the same, then the difference
between the xi ’s will become larger
• When the budget increases and other things remain the same all xi ’s will become larger.
Normalized Pricing Kernel
Define the following variables:
• for all i = 1, . . . , n : θi = qi
pi
• θ that takes value θ = θi if state i materializes.
• X that takes value X = xi if state i materializes.
n
X
• π(X) for the price of the collection of state contracts represented by X, i.e.: π(X) = πi xi
i=1
Using that qi = (1 + r)πi , we can rewrite π(X) as;
n n
X qi 1 X 1
π(X) = xi = θi pi xi = E[θX]
i=1
1+r r + 1 i=1 r+1
We call θ the normalized pricing kernel, because E[θ] = 1
Optimality Condition
u′ (X) ∝ θ ⇐⇒ u′ (X) = c · θ for some c ∈ R
The proportionality factor is chosen to fulfill the budget constraint.
Interpretation: this condition implies that there exists some Lagrange multiplier λ such that u′ (X) = λθ.
The proportionality factor λ is determined by solving the budget constraint in the Lagrange optimization
problem.
5
Max Batstra
March 29, 2026
1
, 2 Portfolio Choice
In this chapter we look at the investment decision of a single agent when prices are given. When we know
the decisions that will be made by agents at given prices, then we can determine supply and demand in the
market, and we can use this to determine prices.
The most frequently used utility functions are:
• Constant Absolute Risk Aversion (CARA)
u(x) = −αe−x/α where α > 0
• Constant Relative Risk Aversion (CRRA)
x1−γ − 1
u(x) = where γ ≥ 0 and γ ̸= 1
1−γ
x1−γ − 1
Important: lim = ln(x) [proof]
γ→1 1−γ
Overview of popular utility functions
Type Domain u(x) Parameter bounds u′ (x) v(x) R(x) θ
1 1 1 1
Log-utility (0, ∞) ln(x) none x x x c· ωag
ωag
x1−γ
−γ − γ1 γ − kτ
Power-utility (0, ∞) 1−γ γ > 0, γ ̸= 1 x x x c·e
− τx x
Exponential-utility (−∞, ∞) τc τ >0 c− τ τ ln(x) 1
τ
−γ
c · ωag
How the terms in the θ column are derived will be shown later.
Notation
• Outcomes are called ”states of the world”, or simply ”states”. i = 1, . . . , n states
• Single-period situation t = 0, 1 (static model).
t = 0 period in which investment takes place, information about states in t = 1 is unknown.
t = 1 period in which actual states are revealed.
• State prices πi for state contract i. This is a contract that pays 1 if state i materializes, and 0
otherwise.
• pi denotes the probability that state i materializes. Note E[xi ] = 1 · pi + 0 · (1 − pi ) = pi , so the pi ’s
denote the expected payoffs of the state contract.
• u(x) utility function of the agent, with u′ (x) > 0 (increasing), u′′ (x) < 0 (concave)
• W budget
2
,The investor aims to maximize it’s expected utility. This yields the maximization problem:
n
X
max pi u(xi )
x1 ,...,xn
i=1
Xn
s.t. πi xi = W
i=1
From [derivation 1], it follows that
πi
• xi = v λ for i = 1, . . . , n , where v(·) = u′ (·)−1 .
pi
Pn Pn
• λ follows from δ
δλ L(xi , λ) = 0 ⇐⇒ i=1 πi xi − W = 0 ⇐⇒ π
i=1 i v λ πi
pi =W
Time Effect
If we buy every asset, we pay π1 + . . . + πn , and we are sure to receive 1 at time 1 (no uncertainty), there-
fore, there is only a time effect, which can be described in terms of an interest rate r. r is the simply
compounded interest rate and is defined as follows:
If you buy every asset at time t = 0, you receive 1 at time t = 1 (1 state materializes) with probability 1, if
we have compound interest rate r. Therefore we have that:
n n
X X 1
(1 + r) πi = 1 ⇐⇒ πi =
i=1 i=1
1+r
It is often convenient to work with normalized state prices (qi for i = 1, . . . , n) which are defined by:
(
qi = (1 + r)πi π qi
=⇒ qi = Pn and πi =
(1 + r) = Pn1 πi i=1 π i 1 +r
i=1
Implied / Risk-neutral Probabilities (qi )
qi ’s are called risk neutral, because under these probabilities, asset prices behave as if investors are indifferent
to risk and are only concerned with expected payoffs, discounted at the risk free rate.
πi
qi = (1 + r)πi or qi = Pn
i=1 πi
qi
Using that πi = 1+r we can rewrite the portfolio choice problem in terms of the qi ’s:
n
X n
X n
X
max pi u(xi ) max pi u(xi ) max pi u(xi )
x1 ,...,xn x1 ,...,xn x1 ,...,xn
i=1 i=1 i=1
n
⇐⇒ n
⇐⇒ n
X X qi X
s.t. πi xi = W s.t. xi = W s.t. qi xi = W (1 + r)
i=1 i=1
1+r i=1
3
,Similarly to [derivation 1] it follows that the solution to this optimization problem can be written as:
n
λqi λqi X λqi
u′ (xi ) = =⇒ xi = v and λ is derived from qi v = (1 + r)W
pi pi i=1
pi
2.3 Interpretations & The Normalized Pricing Kernel
λqi
From xi = v pi it follows that the amount of money to be invested in the i-th state contract depends on
qi qi
the quotient pi . pican be viewed as a price/performance ratio, because qi is the normalized price of one unit
of state contract i and pi is the expected payoff of state contract i, (note E[xi ] = 1 · pi + 0 · (1 − pi ) = pi ).
From the following interpretations follow:
qi
• If < 1 state contract i is undervalued
pi
qi
• If > 1 state contract i is overvalued
pi
From the two observations:
• From u′ (·) is decreasing, it follows that v(·) is also decreasing
• From pi u′ (xi ) = λqi , pi > 0 , qi > 0 , u(xi ) > 0 (i = 1, . . . , n), it follows that λ > 0
and that xi = v( λq
pi ),
i
it follows that:
qi qj
xi < xj ⇐⇒ v λ >v λ
pi pj
Interpretation: The largest number of state contracts will be bought for the state j that has the lowest
q
value of pjj .
We assume that the states are numbered in order of attractiveness, so that:
q1 q2 qn
≤ ≤ ··· ≤
p1 p2 pn
q
From xi < xj ⇐⇒ v λ pqii > v λ pjj it then follows that:
x1 ≥ x2 ≥ . . . ≥ xn
Ranking of alternatives on the basis of expected utility is not changed when a constant is added to the
utility function, or when the utility function is multiplied by a positive constant. That means that taking
decision on the basis of u(x) is the same as taking decision on the basis of ũ(x) = a · u(x) + b where a > 0.
In this case, the utility functions u(·) and ũ(·) are said to be equivalent.
4
, Coefficient of absolute risk aversion:
u′′ (x)
R(x) = −
u′ (x)
Theorem
u′′ (x) u′′ (x)
Utility functions 1, 2 are equivalent ⇐⇒ ∀x: R1 (x) = − u1′ (x) = − u2′ (x) = R2 (x)
1 2
[Proof]
For v(·) = ln(·), we have that
Z xi
′ λqi qi qj
u (xi ) = ⇐⇒ R(x) d x = ln − ln
pi xj pi pj
Interpretation:
• When risk aversion increases and other things remain the same, the xi ’s will move closer together.
• When the ratios pqii are spread out more widely and other things remain the same, then the difference
between the xi ’s will become larger
• When the budget increases and other things remain the same all xi ’s will become larger.
Normalized Pricing Kernel
Define the following variables:
• for all i = 1, . . . , n : θi = qi
pi
• θ that takes value θ = θi if state i materializes.
• X that takes value X = xi if state i materializes.
n
X
• π(X) for the price of the collection of state contracts represented by X, i.e.: π(X) = πi xi
i=1
Using that qi = (1 + r)πi , we can rewrite π(X) as;
n n
X qi 1 X 1
π(X) = xi = θi pi xi = E[θX]
i=1
1+r r + 1 i=1 r+1
We call θ the normalized pricing kernel, because E[θ] = 1
Optimality Condition
u′ (X) ∝ θ ⇐⇒ u′ (X) = c · θ for some c ∈ R
The proportionality factor is chosen to fulfill the budget constraint.
Interpretation: this condition implies that there exists some Lagrange multiplier λ such that u′ (X) = λθ.
The proportionality factor λ is determined by solving the budget constraint in the Lagrange optimization
problem.
5
