1 Lucas (1978) - Tree Model 1.

1 Assignment
Consumption Based Asset Pricing 2.1 The family unit receives an income et and et+1 in these respec-
Say we are solving for an agent i. Population is Lt . Consump- tive periods, with et+1 being a random variable. A risk-free bond,
denoted by bt+1 , and a volatile asset, at+1 . The bond pays out a
tion is Cti . The number of trees is Kt (Total). Fruits from each single consumption unit in the future and has a current value of
tree are dt . qt , whereas the return from the asset is captured by the stochas-
Assumption: Each tree produces the same number of fruits tic variable xt+1 = pt+1 + dt+1 and has a current valuation of pt .
dt . Initially, the household owns neither bonds nor assets. Future ex-
Lt · Cti = Kt · dt pectations are set based on the state (et+1 , xt+1 ) ∈ S, where S
encapsulates all potential outcomes for the state st+1 .
Cti : Consumption per person. The agent’s budget constraints read
Price of a tree is Pt (ex-dividend, after payment of any divi- ct = et − qt bt+1 − pt at+1
dend).
Total resources of a person: ct+1 (st+1 ) = et+1 + bt+1 + xt+1 at+1 , ∀st+1 ∈ S.
Kti · dt + Pt · Kti After reducing consumption, lifetime utility reads
(et − qt bt+1 − pt at+1 )1−θ
Cti + Kt+1
i
· Pt = dt · Kti + Pt · Kti U (bt+1 , at+1 ) = +
1−θ
(et+1 +bt+1 +xt+1 at+1 )1−θ
h i
Cti
 
i Pt + d t βEt , and the associated first-order
⇒ Kt+1 = · Kti − 1−θ
Pt Pt optimality conditions are given by
"  −θ #
Total endowment (resources) at time t: −θ
h
−θ
i ct+1 (st+1 )
−ct qt +βEt ct+1 (st+1 ) =0 ⇐⇒ qt = Et β
ωti = (Pt + dt )Kti ct
"  (1) −θ
Maximization Problem −c−θ p +βE
h
c (s )−θ
x
i
= 0 ⇐⇒ p = E β
ct+1 (st+1 )
t t t t+1 t+1 t+1 t t
ct
Value function V (ωti ):
"∞ # (2)
X 2.2 The investor’s Euler equation is
V (ωti ) = max Et β t
u(Cti ) ( −2 )
Cti
t=0 ct+1
1 = βE Rt+1
" ∞
# ct
X
= max Et u(Cti ) +β β t i
u(Ct+1 ) For the riskless rate, we have Rt+1 = Rf in both states. Using
Cti
t=1
given probabilities and dividend growth data, we get
h i
= max Et u(Cti ) + βV (ωt+1
i
1 = 0.99 0.9(1.03)−2 + 0.1(0.80)−2 Rf

)
Cti
Solving this gives Rf ≈ 1.0055.
subject to Let x be the disaster state rate of return. Substituting in the
given information gives us the following equation for x
Cti
 
i Pt + d t
Kt+1 = · Kti −
1 = 0.99 0.9(1.03)−2 (1.17) + 0.1(0.80)−2 (x)

Pt Pt
and Solving for x gives x ≈ 0.1127, implying a rate of return of
i i about −88.7%!
ωt+1 = (Pt+1 + dt+1 ) · Kt+1

∂ω i
 2 Portfolio Choice under CARA and Normally
u′ (Cti ) + βEt V ′ (ωt+1
i
) t+1 =0 Distributed Returns
∂Cti
Under CARA utility and normal distribution of returns (x̃ ∼
   N (µ, σ 2 )), the optimization problem simplifies to:
′ i ′ i 1
⇒ u (Ct ) + βEt V (ωt+1 ) (Pt+1 + dt+1 ) − =0 max V (θ) = E [− exp(−A(W0 + θx̃))] . (1)
Pt θ
This can be alternatively represented as a minimization prob-
V ′ (ωt+1
i
 
) lem:
⇒ Pt = βEt (Pt+1 + d t+1 )
u′ (Cti ) min E [exp(−A(W0 + θx̃))] . (2)
θ
Envelope Theorem The expression inside the expectation is lognormally dis-
V ′ (ωt+1
i
) = u′ (Ct+1
i
) tributed.
Jensen’s Inequality for lognormal distributions implies:
1
u′ (Ct+1
i
 
) log E(z̃) = E [log z̃] + Var [log z̃] . (3)
⇒ Pt = βEt (Pt+1 + dt+1 ) 2
u′ (Cti )
1 2 2 2
min log E [exp(−A(W0 + θx̃))] ≡ −A(W0 + θµ) + A θ σ (4)
Consumption and Dividends 2
This is equivalent to:
Dividend Process 1 2 2 2
max A(W0 + θµ) − A θ σ . (5)
Assume the dividend dt follows a stochastic process. For instance, 2
let it be driven by an exogenous shock process such as: Recall that the optimization of the log transformation of a func-
dt+1 = µdt + ϵt+1 tion will give the same optimal results. The lognormal property
simplifies the optimization further, leading to the solution:
where µ is the growth rate of the dividend and ϵt+1 is an i.i.d. µ
shock with mean zero. θ∗ = . (6)
Aσ 2

1
3 Conditional CAPM • βexpansion = 2
3

The Conditional CAPM accounts for varying economic conditions, Stock B:

such as recessions and expansions.
• βrecession = 1
2
• Economic Conditions:
• βexpansion = 7
– Recessions: 25% of the time (βrecession ) 6

– Expansions: 75% of the time (βexplosion ) Average Betas:

• Expected Market Risk Premium: 5

• Stock A: β̄A = 1
4
·2+ 3
4
· 2
3
=
6
– Recessions: 12%
– Expansions: 4%
11
• Stock B: β̄B = 1
4
· 1
2
+ 3
4
· 7
6
=
This results in an average market risk premium of 6%, which 8
aligns with observed data.
Example: Two Stocks with Different Betas E(βs · Rm ) = E(βs ) · E(Rm ) + Cov(βs , Rm ) (7)
Consider two stocks, A and B, with different betas in recessions
and expansions: The covariance between stock beta and market risk premium
Stock A: is given by:
• βrecession = 2 Cov(βs , Rm ) = Corr(βs , Rm ) · σ(βs ) · σ(Rm ) (8)
4 Options Pricing 5 Stochastic Calculus
4.1 1. Up and Down Factors 5.1 Brownian Motion with Drift
√ The simplest generalization of the Wiener process is Brownian mo-
u = eσ ∆t
(9) tion with drift:
√
d = e−σ ∆t
(10)
dx = αdt + σdz (1)
(Alternatively, if given directly as percentages: u = 1 +
up percentage and d = 1 − down percentage) Where:

4.2 2. Risk-neutral Probability • dz: Increment of the Wiener process.

• α: Drift parameter.
er∆t − d
p= (11)
u−d
• σ: Variance parameter.
4.3 3. Stock Prices at Each Node
Changes in x, denoted as ∆x, are normally distributed with:
After one period:

Su = S0 × u (12) E[∆x] = α∆t

Sd = S0 × d (13) Var[∆x] = σ 2 ∆t

After two periods:

5.2 Generalized Brownian Motion: Ito’s Process
Suu = Su × u = S0 × u2 (14)
Sud = Su × d = Sd × u = S0 × u × d (15) An extension of the Wiener process:
2
Sdd = Sd × d = S0 × d (16)
dx = a(x, t)dt + b(x, t)dz (2)
4.4 4. Option Payoffs at Maturity
where a(x, t) and b(x, t) are known (non-random) variables, state-
dependent. Equation (2) is known as Ito’s Process.
Cuu = max(Suu − K, 0) (17)
Cud = max(Sud − K, 0) (18) • E[dx] = a(x, t)dt as E[dz] = 0
Cdd = max(Sdd − K, 0) (19)
• Var[dx] = E[x2 ] − [E[x]]2
4.5 5. Option Value at Intermediate Nodes
At the end of the first period: • The right side will have constant terms in dt, (dt)2 , and
(dt)(dz) (of order (dt)3/2 ).
Cu = e−r∆t [p × Cuu + (1 − p) × Cud ] (20)
Cd = e−r∆t [p × Cud + (1 − p) × Cdd ] (21)
• For small dt, we can ignore (dt)2 and (dt)3/2 T heref ore, Var[dx] =
b2 (x, t)dt.
4.6 6. Option Value at the Initial Node
a(x, t): instantaneous drift rate
C0 = e−r∆t [p × Cu + (1 − p) × Cd ] (22) b2 (x, t): instantaneous variance rate

2
5.3 Ito’s Lemma Substituting this back into equation (1), we get:
Let F (x) = log x, then dF (x) = F ′ (x)dx + 12 F ′′ (x)(dx)2 . Here,
r(t) = µ + u(t)e−at
F ′ (x) = x1 , F ′′ (x) = − x12 . (dx)2 = σ 2 x2 dt (Approximated by
ignoring higher-order terms of dt). Z t

2
= µ + (r(0) − µ + σ eas dzs )e−at
σ 0
dF (x) = α − dt + σdz (3)
2 Z t

Over any finite interval T , changes in log x are normally distributed = (r(0) − µ)e−at + µ + σe−at eas dzs
2 0
with mean (α − σ2 )T and variance σ 2 T . Integrating equation (3): R t as
Given that 0
e dzs follows a normal distribution
Z t Z t 2
 t Z R t 2as
σ N (0, 0 e dzs ), using this, we get:
d(log x) = α− du + σdz
2
0 0 0 E[r(t)] = (r0 − µ)e−at + µ
σ2
 
⇒ log xt − log x0 = α − t + σzt As t → ∞, e−at → 0:
2
⇒ E[r(t)] = µ
σ2
 
St
= exp α − t + σzt
S0 2 This means the mean value of the interest rate is mean-
reverting. The interest rate cannot increase forever; eventually,
σ2
 
all business activities will cease.
St = S0 exp α − t + σzt
2 Z t
Var(r(t)) = Var(σe−at eas dzs )
Suppose dx(t) = a(x, t)dt + b(x, t)dz. 0
Ito’s Lemma is a Taylor’s Expansion Z t 
Consider another function F (x, t) that is at least twice differ-
= σ 2 e−2at Var e2as ds
entiable in x and once in t. 0
In usual calculus:
Z t

dF =
∂F
dx +
∂F
dt = σ 2 e−2at e2as ds
∂x ∂t 0

Add higher-order terms to this: σ 2 −2at 2at

= e e −1
2a
∂F ∂F 1 ∂2F 1 ∂3F
dF = dx + dt + 2
(dx)2 + (dx)3 + . . . σ2
∂x ∂t 2 ∂x 6 ∂x3 = (1 − e−2at )
2a
In ordinary calculus, higher terms vanish when we take limits. 2
In stochastic calculus: As t → ∞, Var[r(t)] = σ2a
 
σ2
(dx)2 = b2 (x, t)dt Hence, r(t) ∼ N (r0 − µ)e−at + µ, 2a
(1 − e−2at )
2
As t → ∞, r(t) ∼ N (µ, σ2a )
As dt approaches 0, (dt)3/2 and (dt)2 go to zero faster than dt.
For large values of t, the distribution of r(t) does not depend
on time t.
⇒ (dx)2 = b2 (x, t)dt One problem with this model is that r(t) can be negative for
small values of σ.
So, we can write:

∂F ∂F 1 ∂2F ∂F 6 Diamond Dybvig Model

dF = dt + dx + b2 (x, t) 2 dt + b(x, t) dz
∂t ∂x 2 ∂x ∂x
6.1 Autarky
5.4 Mean-Reverting Process - The Ornstein- Each consumer wants to maximize her expected utility:
Uhlenbeck Process (OUP)
max θU (c1 ) + (1 − θ)ρU (c2 )
c1 ,c2
Most financial prices are expected to return to the mean, but trend
upward/downward as per the marginal cost of production: A consumer’s expected utility is therefore:
dr = −a(r − µ)dt + σdzt (4) θU (1) + (1 − θ)ρU (R)

Where a, µ, σ are parameters of the model. Generally, a, µ, σ > 0. where U (1) represents the utility from consuming in period 1
and U (R) represents the utility from consuming in period 2 after
investing in period 0.
r(t) = µ + ce−at (5)

For a > 0, as t → ∞, r(t) → µ. This is called a stable solution to 6.2 Social Planner
the equation r′ = −a(r − µ). She can observe the realized ”type” of the individuals.
One can think of other variations of this equation: Type 1 individuals get no utility from consumption in period
2, so c12 = 0.
dx = −a(x − µ)dt + σxdz Type 2 individuals are indifferent, but they get higher utility
dx = −ax(x − µ)dt + σxdz by consuming in period 2, so c21 = 0.
So, 1 − θc11 is the fraction of projects held to period 2, and this
Solving OUP: should equal to (1 − θ)c22 .
Z t
(1 − θc11 )R
u(t) = r(0) − µ + σ eas dzs ⇒ c22 =
0 1−θ

3
6.3 Banks 8 Consumer Optimization in a Small Open
Optimal Contract: Let’s first look for an equilibrium in which Economy
Type 2’s believe other Type 2’s will abide by the contract. The
bank’s (implicit) contract design problem is as follows: In a small open economy, consumers aim to maximize their life-
time utility from consumption, considering borrowing and saving
max θU (d1 ) + (1 − θ)U (d2 ) options. This can be modeled as an optimization problem:
x,d1 ,d2
max {u(c1 ) + βu(c2 )} (23)
c1 ,c2 ,d1
subject to:
1. θN d1 = xN 2. (1 − θ)N d2 = (1 − x)RN 3. d1 ≤ d2 subject to the following constraints:
Where x is the proportion of projects that are terminated early. c 1 = y1 + d 1 (24)
Note that we can substitute the first constraint into the second
and get (after canceling the N ’s): c2 + d1 (1 + r∗ ) = y2 (25)
R θR d1 ≤ κy1 (26)
d2 = − d1 The present-value budget constraint:
1−θ 1−θ
c2 y2
 
R θR
 c1 + = y1 + ≡ yP DV (27)
max θU (d1 ) + (1 − θ)U − d1 1 + r∗ 1 + r∗
d1 1−θ 1−θ The Euler equation:
Using the chain rule, we get the following first-order optimality u′ (c1 ) = β(1 + r∗ )u′ (c2 ) (28)
condition:
  Assume that β(1 + r∗ ) = 1 The unconstrained solution is:
1−θ
U ′ (d1 ) = RU ′ Rd1 1 + r∗
θ c1 = c2 = yP DV (29)
2 + r∗
Notice that this guarantees that d2 > d1 , so the incentive con- Now: no borrowing and lending but households can invest in
straint is satisfied. an asset at
Fixed asset supply in each period as = 1 (one unit)
6.4 Bank Runs Household Optimization Problem:
θN Type 1’s will withdraw in period 1. (Type 1’s always with- max {u(c1 ) + βu(c2 )}
draw). This means that if the bank liquidates all its projects, c1 ,c2 ,a1
the most that can be made available for Type 2’s in period 1 is s.t. c1 + p1 a1 = y1 + p1 a0
N (1 − θd1 ). Now, since there are (1 − θ)N Type 2’s and d1 > 1, c2 = y2 + Da1
the available funds are less than the potential demands of the Type
2’s. In particular, as long as at least Equilibrium Condition:
a1 = a2 = 1
1 − θd1 Equilibrium Consumption Levels:
1−θ
• c1e = y1 + p1
Type 2’s show up early, there won’t be anything left for those
who wait, and it will be optimal to run to the bank. • c2e = y2 + D

7 Credit Rationing p1 u′ (c1e ) = βDu′ (c2e ) (30)

′ ′
p1 u (y1 ) = βDu (y2 + D) (31)
7.1 Firms’ Profits
βu′ (y2 + D)
Projects deliver a sum of R but this is uncertain and the distri- ⇒ p1 = D (32)
u′ (y1 )
bution of outcomes varies across borrowers. Borrowers of type A
have a return distribution given by the probability density function Household Optimization Problem:
f (R, θ).
The return to the firm is given by: max {u(c1 ) + βu(c2 )}
c1 ,c2 ,a1 ,d1

s.t. c1 + p1 a1 = y1 + p1 a0 + d1
P (R, r) = max(R − (1 + r)B, −C)
c2 + d1 (1 + r∗ ) = y2 + Da1
Who borrows and who doesn’t? You borrow if: d1 ≤ κp1 a0
d1 = κp1 a0 = κp1 , c1 = y1 + κp1 , c2 = y2 + D − κp1 (1 + r∗ )
Z ∞
E[P (R, r)] = P (R, r)f (R, θ) dR > 0 The equilibrium asset price is determined by:
0
βu′ (y2 + D − κp1 (1 + r∗ ))
p1 = D (33)
Bank Profits u′ (y1 + κp1 )
• The payoff to the bank is min(R + C, (1 + r)B). Also assume κβD < 1.
βDy1 y1
Solving gives: p1 = 1−βDkappa
and c1 = 1−βDκ
• If the bank knew that it was lending to type θ, then its ex-
pected return would be:
Z ∞ The Capital Asset Pricing Model (CAPM)
ρ(θ, r) = min(R + C, (1 + r)B)f (R, θ) dR Portfolio Adjustments and Risk Implications
0
Changing Portfolio Weights:
• But the bank can’t tell who is risky or not. Its expected • Adjusting asset weights impacts expected return and portfo-
payoff can be calculated by averaging across all types that lio risk.
look for loans at interest rate r:
dE[Rp ]
R∞ • Effect on mean return: dwi
= Ri − Rf .
θ̂(r)
ρ(θ, r)g(θ) dθ
E[ρ(θ, r)] = dV ar(Rp )
1 − G(θ̂(r)) • Effect on portfolio variance: dwi
= 2Cov(Ri , Rp ).

4
Mean-Variance Trade-off: or Mean-Variance Efficiency 10 Black-Litterman Model: Optimization with
Criterion: Investor’s Views
dRp /dwi Ri − Rf
= 10.1 Black-Litterman Model: Optimization
dV ar(Rp )/dwi 2Cov(Ri , Rp )
with Investor’s Views
– Objective: Minimize the deviation from CAPM-implied
Capital Asset Pricing Model (CAPM) Foundations
returns while considering the views: minRe 12 (Re −
e T −1
Portfolio Variance and Mean-Variance Efficiency RCAP M) Σ (Re − RCAP
e T e
M ) subject to P R = Q

Ri − Rf Rj − Rf – Solution integrates the investor’s views with market

−1
= (34) equilibrium: Re = RCAP e T
M + ΣP (P ΣP ) (Q −
2Cov(Ri, Rp) 2Cov(Rj, Rp) T e
P RCAP M )
Ri−Rf
The ratio 2Cov(Ri,Rp) is equal across all assets for a mean-variance – Ensures minimal deviation from CAPM when investor’s
efficient portfolio. views are aligned with the market.

Capital Asset Pricing Model (CAPM) Derivation 11 Black-Litterman Model: Portfolio

Weights
Portfolio Efficiency (equation (4)) gives
– Portfolio weights adjust based on investor’s views and
Ri − Rf = βip (Rp − Rf ) (35) risk aversion.
– Weights are obtained from the expression: w =
where βip = Cov(Ri,Rp)
V ar(Rp)
is the regression coefficient of asset i’s re- θΣ−1 Re with the adjustment factor
for views:
turn on portfolio p’s return. θ δwm + P (P T ΣP )−1 (Q − P T RCAP e
M ) where θ is the
The CAPM framework assumes that the market portfolio is scale factor tied to risk aversion.
mean-variance efficient. – Investor’s portfolio combines market portfolio with
cash, or leverages by borrowing (as w ̸= 1)
Market Portfolio Return: or Security Market Line (SML) – For Q − P T RCAP
e
M = 0 The scale factor θδ indicates
investment stance:
R̄i − Rf = βim (R̄m − Rf ) (36) – θδ = 1: Market portfolio of risky assets.
where βim is asset i’s sensitivity to the market portfolio’s move- – θδ < 1: Conservative stance with cash and market port-
ments. folio.
Portfolio CAPM – θδ > 1: Leveraged position with borrowed cash.
E(Rp ) = Rf + βp (E(Rm ) − Rf ) (37)
12 Consumption CAPM
The beta of the portfolio, βp , is given by:
n
X 12.1 Optimal Asset Purchase and Consumption
βp = wi βi (38)
i=1
" ∞
#
X
or u′ (Ct )pit = Et β k u′ (Ct+k )Di,t+k (41)
E(Rp ) − Rf k=1
βp = (39)
E(Rm ) − Rf
u′ (Ct ) = βEt (1 + rit+1 )u′ (Ct+1 )
 
for all i. (42)

9 Jensen’s Alpha and the Security Market Line

(SML) 12.2 Individual Optimization with Covariance

• Measures the intercept of regression of excess returns against u′ (Ct ) = β {Et [1 + rit+1 ]
the market excess return. Et [u′ (Ct+1 )] + Covt (1 + rit+1 , u′ (Ct+1 )) (43)
• Rit − Rf t = αi + βim (Rmt − Rf t ) + ϵit
u′ (Ct ) = β Et [1 + rit+1 ] Et u′ (Ct+1 )
  

• where αi = R̄i − Rf − βim (R̄m − Rf ) is Jensen’s Alpha: - a Covt (1 + rit+1 , Ct+1 ) (44)
• α indicates if assets are priced correctly relative to the CAPM
and measures deviation from SML: R̄i = Rf + βim (R̄m − Rf ) 12.3 Understanding the Consumption CAPM
• A positive Jensen’s alpha (αi ) suggests an undervalued asset.
" ∞
#
X u′ (Ct+k )
k
pit = Et β Di,t+k . (45)
• A negative Jensen’s alpha suggests an overvalued asset. u′ (Ct )
k=1

1  ′
9.1 Compact SML Representation Et [1 + rit+1 ] = u (Ct ) +
Et [u′ (Ct+1 )]
• Rie = βim Rm
e
, where a Covt (1 + rit+1 , Ct+1 ).(46)
• Excess return on asset: Rie = R̄i − Rf
12.4 Consumption CAPM and Risk-Free Rate
• Excess return on market: Rm
e
= R̄m − Rf
u′ (Ct )
1 + rt+1 = . (47)
9.2 CML βEt [u′ (Ct+1 )]
 
E(Rm ) − Rf aCovt (1 + rit+1 , Ct+1 )
E(Rp ) = Rf + σp (40) Et [rit+1 ] − rt+1 = . (48)
σm Et [u′ (Ct+1 )]

5
13 SDF Riskless Asset and Risk-Neutral Probabili-
ties
Law of One Price:
A riskless asset has a payoff of 1 in every state. Its price,
S
X Pf , is the expected value of the stochastic discount factor
P (X) = q(s)X(s) (49) (SDF), M :
s=1 XS
Pf = q(s) = E[M ]
Introducing probabilities for each state, π(s): s=1

S
The riskless interest rate, Rf , is derived from the price of
X q(s) the riskless asset:
P (X) = π(s) X(s)
s=1
π(s) 1 1
1 + Rf = =
Pf E[M ]
PS
= s=1 π(s)M (s)X(s) = E[M X](50)
For practical application, a riskless rate of 2% implies
where M (s) = q(s)
is the stochastic discount factor (SDF). E[M ] ≈ 0.98 (for US).
π(s)

Utility Maximization:
Risk-neutral Probabilities with Power Util-
ity
S
!
X
max u(C0 ) + βπ(s)u(C(s)) (51)
s=1
In macro-finance models, payoffs in good states are worth
subject to the budget constraint: less than payoffs in bad states. This can be illustrated with
Bernoulli risks:
S
X z = log c1 − log c0 = log g
C0 + q(s)C(s) = W0 (52)
s=1 which takes on two values:

z = δδ1 > δ with probability 1 − ω

n
First-order conditions for utility maximization: 2 1 with probability ω
βπ(s)u′ (C(s)) = q(s)u′ (C0 ) (53) The pricing kernel is βg −γ = βe−γz .
The risk-neutral probability of state 1 is:
The SDF:
q(s) βu′ (C(s)) (1 − ω)βe−γδ1
M (s) = = (54) π ∗ (z = δ1 ) =
π(s) u′ (C0 ) (1 − ω)βe−γδ1 + ωβe−γδ2
1−ω
=
Riskless Asset and Risk-Neutral Probabili- (1 − ω) + ωe−γ(δ2 −δ1 )
ties >1−ω

A riskless asset has a payoff of 1 in every state. Its price,

Pf , is the expected value of the stochastic discount factor 14 Choice Under Uncertainty
(SDF), M :
S – A
Eu(W0 + x̃) ≤ u(W0 )
X
Pf = q(s) = E[M ]
s=1 where x̃ is a zero-mean risk.
The riskless interest rate, Rf , is derived from the price of –
u′′ (W0 )
the riskless asset: A(W0 ) = −
u′ (W0 )
1 1
1 + Rf = = – The risk premium :
Pf E[M ]
Eu(W0 + x̃) = u(W0 − π)
For practical application, a riskless rate of 2% implies
E[M ] ≈ 0.98 (for US). – The certainty equivalent Ce :
Eu(W0 + µ + x̃) = u(W0 + Ce )
Risk-neutral Probabilities
– This implies:
Risk-neutral probabilities, π ∗ (s), are defined using the SDF
and the actual probability of each state, π(s): Ce (W0 , µ, x̃) = µ − π

M (s) Arrow-Pratt Approximation

π ∗ (s) = (1 + Rf )q(s) = π(s)
E[M ] E[u(W0 + kx̃)] = u(W0 − g(k)).
P ∗
where s π (s) = 1. The price of an asset can be written as E[x̃u′ (W0 + kx̃)] = −g ′ (k)u′ (W0 − g(k)).
if investors were risk-neutral:
At k = 0, this implies:
S
1 X g ′ (0) = 0
P (X) = π ∗ (s)X(s)
1 + Rf s=1 since E[x̃] = 0.
1
= E ∗ [X] E[x̃2 u′′ (W0 + kx̃)] = g ′ (k)2 u′′ (W0 − g(k))
1 + Rf

6
- g”(k) u’(W0 − g(k)). Average Duration of a Bubble:
1
u′′ (W0 ) Average duration = 1−π
′′
g (0) = − ′ E[x̃2 ] = A(W0 )E[x̃]
u (W0 ) The fundamental value of the stock is captured by:
∞
1 2 ′′ X 1
g(k) ≈ g(0) + kg ′ (0) + k g (0). p∗t = πi
2 i=0
(1 + r)i+1

1 1 where π is the probability of war continuing each period.

≈ A(W0 )k2 E[x̃2 ] = A(W0 )E[ỹ 2 ].
2 2 Consider the emergence of a bubble bt+i dependent on war
events:
15 Bubbles 
1+r
bt+i−1 if war at t + i
bt+i = π
0 if no war at t + i
Properties of Asset Pricing:
The asset price pt must satisfy the difference equation:
– The expected excess return over the risk-free rate is
zero:
E(pt+1 |Ωt ) − pt + dt = rpt pt = yt + βpt+1 (∗)
∞
– The current price of the asset can be expressed as: X β2 β3
pt = β j yt+j = yt + βyt+1 + yt+2 + yt+3 + . . .
j=0
2 3
E(pt+1 |Ωt )
pt = d t +
1 + rt Claim: The general solution to (∗) is:
To obtain the fundamental value, we iterate the asset pricing
equation forward: ∞
X
pt = β j yt+j (fundamental value)+cβ −t (bubble component)
∞
X 1 j=0
p∗t = E(dt+i |Ωt )
i=0
(1 + r)i+1
for any constant c.
This process employs the law of iterated expectations: In words: Price = present discounted value (PDV) of future
dividends.
E(E(·|Ωt+i )|Ωt ) = E(·|Ωt ) Another solution is:

Here, p∗t represents the ”fundamental” value, calculated as pt = cβ −t

the expected net present value of future dividends.
The law of motion for the price of an asset pt can be satisfied for a constant c.
by any solution of the form:

pt = p∗t + bt
16 Term Structure of Interest Rates

where: PN,t is the current price of an N-period Zero Coupon Bond

at time t, pN,t = log PN,t . Say the future price of a Z-
– p∗t is the fundamental component of the asset price: bond = 1 unit, then future continuous compounding gives
PN,t eN r = 1, rN = − log(PN,t ). Also, Yield (fixed, constant,
∞
X 1 known, annual IR on all bond cash flow) PN,t = 1−[1−YN,t ]N
p∗t = E(dt+i |Ωt )
(1 + r)i+1 or Taking log both sides, yN,t = − N1 pN,t .
i=0
Yield satisfies: P = N J
P
j=0 CFj Y , CFj =Cash flows at j-th
– bt is the bubble component, satisfying: time period.
E(bt+1 |Ωt ) = (1 + r)bt Holding period Return (hold N-period bond and sell it in 1
period) hpr = P (T,t+)−P
P (T,t)
(T,t)
. Taking bond pricing function
This allows for the existence of a bubble component in the that fixes maturity, hpr = dP (N,t)
− 1 P (N,t)
dt.
asset price that evolves independently of the fundamental P P N
value. Forward Rates: is defined as the rate at which one can
contract today to borrow or lend money starting at period
Modeling Bubbles: Consider the stochastic process for bt :
N , to be paid back at time (N + 1). Take (1 + r(2))2 =
 1+r (1 + r(1))(1 + re (1)) where, r(2) is the cumulative IR for 2
bt = π
bt−1 + µt with probability π
periods, r(1) is the IR for 1st period, re (1) is the IR for 2nd
µt with probability (1 − π)
period.
where E(µt |Ωt ) = 0. Arbitrage strategy involving zero-coupon bonds: Buy one N -
period zero-coupon bond and simultaneously sell x(N + 1)-
Interpreting the Stochastic Bubble Process: (N →N +1)
period zero-coupon bonds. Then, x = PP t (N )
t (N +1)
= Ft .
– With probability π:  
1
∗ The bubble persists. Important relation: Pt (N ) = (j→j+1) − 1 or
jFt
∗ Returns on the asset exceed the risk-free rate r. [Price today = Cumulative gross return].
– With probability 1 − π: (N →N +1)
yN,t = − N1 pN,t so, ft = −N yt,N + (N + 1)yt,N +1
∗ The bubble collapses. (Forward rate is above the yield curve if the yield curve is
∗ Asset prices revert to their fundamental values. rising).

7
Stochastic Discount Factor and Risk Premia Deriving the Bound
Given the non-negativity of covariance under lognormal
Fundamental Asset Pricing Equation returns, we have:
σ2
Et [ri,t+1 ] − rf,t+1 + 2i
Pit = Et [Mt+1 Xi,t+1 ] = Et [Mt+1 ] Et [Xi,t+1 ]+Covt (Mt+1 , Xi,t+1 ) , σmt ≥ . (66)
(55) σit
Calculating Risk Premium Using the SDF Example: The volatility of a non-dividend-paying
stock whose price is $78, is 30%. The risk-free rate is 3%
Divide the equation (1) by Pit and replace Xi,t+1 = Pit (1 + per annum (continuously compounded) for all maturi-
Ri,t+1 ), we get ties. Calculate values for u, d, and p when a 2-month
time step is used. What is the value a 4-month Eu-
Et [1 + Ri,t+1 ] = (1 + Rf,t+1 )(1 − Covt (Mt+1 , Ri,t+1 )). (56) ropean call option with a strike price of $80 given by
a two-step binomial tree. Suppose a trader sells 1,000
options (10 contracts). What position in the stock is
Expected Return and Risk Premium necessary to hedge the trader’s position at the time of
the trade? 1. Time and Parameters:
4 1 T 1
T = 4 months = = year, n = 2, t = = years
Et [Ri,t+1 − Rf,t+1 ] = −(1+Rf,t+1 )Covt (Mt+1 , Ri,t+1 −Rf,t+1 ), 12 3 n 6
(57) 2. Up and Down Factors:
1
Et [Ri,t+1 − Rf,t+1 ] = βiMt λMt , (58) u = 1.1303, d = 1/u = = 0.8847
1.1303
where 3. Risk-neutral Probability:
Covt (Mt+1 , Ri,t+1 )
βiMt = (59) ert − d
Vart (Mt+1 ) p=
u−d
and where r = 3% per annum, t = 1
year.
6
λMt ≡ −(1 + Rf,t+1 )Vart (Mt+1 ). (60)
0.03× 1
e − 0.8847
6
p=
βiMt reflects the sensitivity of asset i to the SDF, indicating 1.1303 − 0.8847
its risk level.
p ≈ 48.98%
4. Stock Prices at Each Node:
Riskless Interest Rate and Log Risk Premium Su = S0 × u = 78 × 1.1303 = 88.16
Suu = u × Su = 1.1303 × 88.16 = 99.65
2
σm t
σi2 Similarlyf orothers
0 = Et mt+1 + Et ri,t+1 + + t + σimt (61)
2 2 5. Option Payoffs at Maturity:
Cuu = max(Suu − K, 0) = max(99.65 − 80, 0) = 19.65
2
σm t
σi2
⇒ Et ri,t+1 = −Et mt+1 − − t − σimt (62)
2 2 6. Option Value at Intermediate Nodes:
For a risk-free asset: Variance and covariances are zero. So, Cu = [p × Cuu + (1 − p) × Cud ]e−rt
Cd = [p × Cud + (1 − p) × Cdd ]e−rt
2
σm t
rf,t+1 = −Et mt+1 − , (63) 7. Option Value at Time t = 0:
2
Value of the option = [p × Cu + (1 − p) × Cd ]e−rt
where mt+1 = log(Mt+1 ), ri,t+1 = log(1 + Ri,t+1 ), and σimt 8. Hedge Position: To hedge the trader’s position,
is the covariance between ri,t+1 and mt+1 . we calculate the delta and then determine the number
Adjusting for Jensen’s Inequality, the log risk premium in- of shares to buy.
corporates the variance of the asset’s log return: Cu − Cd 9.58 − 0
Delta = = = 0.4
Su − Sd 88.16 − 69.01
σi2 9.58−0
Et ri,t+1 − rf,t+1 + = −σimt . (64) Hence, the trader needs to buy 88.16−69.01
× 1000 = 500
2 shares to hedge their position.
Hedge Portfolio:
Given the equations:
Simple Volatility Bound
φn un xn + ψn (1 + r)Bn = a
Given the correlation between the SDF and any excess re- φn dn xn + ψn (1 + r)Bn = b
turn, we have:
Solving the Set of Simultaneous Equations:
σt (Mt+1 ) Et [Ri,t+1 − Rf,t+1 ] For φn xn :
≥ . (65) a−b
Et [Mt+1 ] σt (Ri,t+1 − Rf,t+1 ) φn xn =
u−d
For ψn Bn :
– The Sharpe ratio for an asset puts a lower bound on the
SDF’s volatility. bu − ad 1
ψn Bn = ×
u−d 1+r
– The tightest lower bound is found by maximizing the
Sharpe ratio across assets. article amsmath amsfonts We can now calculate φn xn and ψn Bn for each stock
amssymb price node at each time step.