Fin 501: Asset Pricing
Lecture 03: One Period Model:
Pricing
Prof. Markus K. Brunnermeier
10:59 Lecture 02
One Period Model
Slide 22--1
Fin 501: Asset Pricing
O
Overview:
i
P
Pricing
i i
1.
2.
3.
4.
5.
6.
LOOP
LOOP,, No arbitrage
Parity relationship between options
No arbitrage and existence of state prices
Market completeness and uniqueness of state prices
Pricing kernel q*
Four pricing formulas (state prices, SDF, EMM, state
pricing)
7. Recovering state prices from options
10:59 Lecture 02
One Period Model
Slide 22--2
Fin 501: Asset Pricing
V t N
Vector
Notation
t ti
• Notation:
N t ti
y,x ∈ Rn
‰ y ≥ x ⇔ yi ≥ xi for each i=1,…,n.
‰ y > x ⇔ y ≥ x and y ≠ x.
x
‰ y >> x ⇔ yi > xi for each i=1,…,n.
• Inner product
‰ y · x = ∑i yx
• Matrix multiplication
10:59 Lecture 02
One Period Model
Slide 22--3
Fin 501: Asset Pricing
Th Forms
Three
F
off N
No-ARBITRAGE
No1. Law of one price (LOOP)
If h’X = k’X then p· h = p· k.
2. No strong arbitrage
There exists no portfolio h which is a strong
arbitrage, that is h’X ≥ 0 and p · h < 0.
3 No arbitrage
3.
There exists no strong arbitrage
nor portfolio k with kk’ X > 0 and p · k ≤ 0.
0
10:59 Lecture 02
One Period Model
Slide 22--4
Fin 501: Asset Pricing
Th Forms
Three
F
off N
No-ARBITRAGE
No• Law of one price is equivalent to
every
y portfolio
p
with zero payoff
p y has zero price.
p
• No arbitrage
⇒ no strong arbitrage
No strong arbitrage ⇒ law of one price
10:59 Lecture 02
One Period Model
Slide 22--5
Fin 501: Asset Pricing
O
Overview:
i
P
Pricing
i i
1.
2.
3.
4.
5.
6.
7.
LOOP
LOOP,, No arbitrage
Forwards
Parity relationship between options
No arbitrage and existence of state prices
Market completeness and uniqueness of state prices
Pricing kernel q*
Four pricing formulas (state prices, SDF, EMM, state
pricing)
8 Recovering state prices from options
8.
10:59 Lecture 02
One Period Model
Slide 22--6
Fin 501: Asset Pricing
Alternative
Alt
ti ways to
t buy
b a stock
t k
•
Four different payment and receipt timing combinations:
‰
‰
‰
‰
•
Outright purchase: ordinary transaction
Fully leveraged purchase: investor borrows the full amount
Prepaid forward contract: pay today, receive the share later
Forward contract: agree on price now, pay/receive later
Payments, receipts, and their timing:
Slide 22--7
Fin 501: Asset Pricing
P i i prepaid
Pricing
id forwards
f
d
• If we can price the prepaid forward (FP), then we can calculate
the price for a forward contract:
F = Future
F
value
l off FP
• Pricing by analogy
‰ In the absence of dividends
dividends, the timing of delivery is irrelevant
‰ Price of the prepaid forward contract same as current stock price
‰ FP0, T = S0
(where the asset is bought at t = 0, delivered at t = T)
• Pricing by discounted preset value (α: risk-adjusted discount rate)
−αT
‰ If expected tt=T
T stock price at tt=0
0 is E0(ST),
) then FP0,
0 T = E0(ST) e
‰ Since t=0 expected value of price at t=T is E0(ST) = S0 eαT
Slide 22--8
‰ Combining the two, FP0, T = S0 eαT e−αT = S0
Fin 501: Asset Pricing
P i i prepaid
Pricing
id forwards
f
d (cont.)
• Pricingg by
y arbitrage
g
‰ If at time t=0, the prepaid forward price somehow exceeded the
stock price, i.e., FP0, T > S0 , an arbitrageur could do the following:
Slide 22--9
Fin 501: Asset Pricing
P i i prepaid
Pricing
id forwards
f
d (cont.)
• What if there are deterministic* dividends? Is FP0, T = S0 still
valid?
‰ No,
N bbecause th
the hholder
ld off the
th forward
f
d will
ill nott receive
i dividends
di id d that
th t will
ill
be paid to the holder of the stock Î FP0, T < S0
FP0, T = S0 – PV(all dividends paid from t=0 to t=T)
‰ For discrete dividends Dti at times ti, i = 1,…., n
• The prepaid forward price: FP0, T = S0 – Σ
n
PV0, ti (Dti)
i=1
‰ For continuous dividends with an annualized yield δ
• The prepaid forward price: FP0, T = S0 e−δT
*NB: if dividends are stochatistic, we cannot apply the one period model
Slide 22--10
Fin 501: Asset Pricing
P i i prepaid
Pricing
id forwards
f
d (cont.)
• Example 5.1
‰ XYZ stock costs $100 today and will pay a quarterly dividend of $1.25. If
the risk-free
risk free rate is 10% compounded continuously,
continuously how much does a 11
year prepaid forward cost?
‰ FP0, 1 = $100 – Σ
•
4
$1.25e−0.025i = $95.30
i=1
Example 5.2
‰ The index is $125 and the dividend yield is 3% continuously compounded.
How much does a 1-year
y prepaid
p p forward cost?
‰ FP0,1 = $125e−0.03 = $121.31
Slide 22--11
Fin 501: Asset Pricing
P i i forwards
Pricing
f
d on stock
t k
• Forward price is the future value of the prepaid forward
‰ No dividends: F0, T = FV(FP0, T ) = FV(S0) = S0 erT
n
‰ Discrete dividends: F0, T = S0 erT – Σ i=1 er(T-ti)Dti
‰ Continuous dividends: F0, T = S0 e(r-δ)T
• Forward premium
‰ The difference between current forward price and stock price
‰ Can be used to infer the current stock price from forward price
‰ Definition:
D fi iti
• Forward premium = F0, T / S0
• Annualized forward premium =: πa = (1/T) ln (F0, T / S0)
(from eπ T=F0,T / S0 )
Slide 22--12
Fin 501: Asset Pricing
C ti a synthetic
Creating
h i forward
f
d
• One can offset the risk of a forward byy creating
g a synthetic
y
forward to offset a position in the actual forward contract
• How can one do this? (assume continuous dividends at rate δ)
‰ Recall the long forward payoff at expiration: = ST - F0,
0 T
‰ Borrow and purchase shares as follows:
‰ Note
N t that
th t th
the ttotal
t l payoff
ff att expiration
i ti is
i same as forward
f
d payoff
ff
Slide 22--13
Fin 501: Asset Pricing
C ti a synthetic
Creating
h i forward
f
d (cont.)
• The idea of creating synthetic forward leads to following:
‰ Forward = Stock – zero-coupon bond
‰ Stock = Forward + zero-coupon bond
‰ Zero-coupon bond = Stock – forward
• Cash-and-carry arbitrage: Buy the index, short the forward
T bl 5.6
Table
56
Slide 22--14
Fin 501: Asset Pricing
Oth iissues in
Other
i forward
f
d pricing
i i
• Does the forward price predict the future price?
‰ According the formula F0, T = S0 e(r-δ)T the forward price conveys no
additional information beyond what S0 , r,
r and δ provides
‰ Moreover, the forward price underestimates the future stock price
• Forward pricing formula and cost of carry
‰ Forward price =
Spot price + Interest to carry the asset – asset lease rate
Cost of carry, (r-δ)S
Slide 22--15
Fin 501: Asset Pricing
O
Overview:
i
Pricing
P i i - one period
i d model
d l
1.
2.
3.
4.
5.
6.
7.
LOOP
LOOP,, No arbitrage
Forwards
Parity relationship between options
No arbitrage and existence of state prices
Market completeness and uniqueness of state prices
Pricing kernel q*
Four pricing formulas (state prices, SDF, EMM, beta
pricing)
8 Recovering state prices from options
8.
10:59 Lecture 02
One Period Model
Slide 22--16
Fin 501: Asset Pricing
PutP t-Call
Put
C ll Parity
P it
• For European options with the same strike price and
time to expiration the parity relationship is:
Call – put = PV (forward price – strike price)
or
C(K, T) – P(K, T) = PV0,T (F0,T – K) = e-rT(F0,T – K)
• Intuition:
‰Buying
‰B
i a call
ll andd selling
lli a putt with
ith the
th strike
t ik equall to
t the
th
forward price (F0,T = K) creates a synthetic forward contract
and hence must have a zero price.
Slide 22--17
Fin 501: Asset Pricing
P it for
Parity
f Options
O ti
on Stocks
St k
• If underlying asset is a stock and Div is the
deterministic* dividend stream, then e-rT F0,T = S0 –
PV0,T
, (Div), therefore
C(K, T) = P(K, T) + [S0 – PV0,T (Div)] – e-rT(K)
• Rewritingg above,,
S0 = C(K, T) – P(K, T) + PV0,T (Div) + e-rT(K)
• For index options,
options S0 – PV0,T (Div) = S0e-δT, therefore
C(K, T) = P(K, T) + S0e-δT – PV0,T (K)
*allows to stick with one period setting
Slide 22--18
Fin 501: Asset Pricing
P
Properties
ti off option
ti prices
i
• American vs. European
p
‰Since an American option can be exercised at anytime,
whereas a European option can only be exercised at
expiration, an American option must always be at least as
valuable
l bl as an otherwise
h
i identical
id i l European option:
i
CAmer(S, K, T) > CEur(S, K, T)
PAmer(S, K, T) > PEur(S, K, T)
• Option price boundaries
‰Call price cannot: be negative,
negative exceed stock price,
price be less
than price implied by put-call parity using zero for put price:
S > CAmer(S, K, T) > CEur(S, K, T) >
> max [0,
[0 PV0,T(F0,T) – PV0,T(K)]
Slide 22--19
Fin 501: Asset Pricing
P
Properties
ti off option
ti prices
i
(cont.)
• Option price boundaries
‰Call price cannot:
• be
b negative
ti
• exceed stock price
• be less than price implied by put-call parity using zero for put price:
S > CAmer(S,
(S K,
K T) > CEur(S,
(S K,
K T) > max [0,
[0 PV0,T(F0,T) – PV0,T(K)]
‰Put pprice cannot:
• be more than the strike price
• be less than price implied by put-call parity using zero for call price:
K > PAmer
(S, K, T) > PEur
A
E (S, K, T) > max [0, PV0,T
0 T (K) – PV0,T
0 T(F0,T
0 T)]
Slide 22--20
Fin 501: Asset Pricing
P
Properties
ti off option
ti prices
i
(cont.)
• Early exercise of American options
‰A non-dividend paying American call option should not
b exercised
be
i d early,
l because:
b
CAmer > CEur > St – K + PEur+K(1-e-r(T-t)) > St – K
‰That means,
means one would lose money be exercising early
instead of selling the option
‰If there are dividends, it may be optimal to exercise early
‰It may be optimal to exercise a non-dividend paying put
option early if the underlying stock price is sufficiently
low
Slide 22--21
Fin 501: Asset Pricing
P
Properties
ti off option
ti prices
i
(cont.)
• Time to expiration
‰ An American option (both put and call) with more time to
expiration is at least as valuable as an American option with less
time to expiration. This is because the longer option can easily
be converted into the shorter option by exercising it early
early.
‰ European call options on dividend-paying stock and European
puts may be less valuable than an otherwise identical option with
less time to expiration.
‰A E
European call
ll option
i on a non-dividend
di id d paying
i stockk will
ill be
b
more valuable than an otherwise identical option with less time to
expiration.
‰ When the strike pprice grows
g
at the rate of interest,, European
p
call
and put prices on a non-dividend paying stock increases with
time.
• Suppose to the contrary P(T) < P(t) for T>t, then arbitrage. Buy P(T)
and sell P(t). At t if St>Kt, P(t)=0, if St<Kt, payoff St – Kt. Keep stock
r(T t)=S
andd finance
fi
Kt. Time
Ti T-value
T l ST-K
Kter(T-t)
ST-K
KT .
Slide 22--22
Fin 501: Asset Pricing
P
Properties
ti off option
ti prices
i
(cont.)
• Different strike prices (K1 < K2 < K3), for both
European and American options
‰A call with a low strike pprice is at least as valuable as
an otherwise identical call with higher strike price:
C(K1) > C(K2)
‰A put with a high strike price is at least as valuable as
an otherwise identical call with low strike price:
P(K2) > P(K1)
‰The premium difference between otherwise identical
calls with different strike prices cannot be greater than
the difference in strike prices:
C(K1) – C(K2) < K2 – K1
K2 – K1
S
Slide 22--23
Fin 501: Asset Pricing
P
Properties
ti off option
ti prices
i
(cont.)
• Different strike prices (K1 < K2 < K3), for both
European and American options
‰The premium difference between otherwise
identical puts with different strike prices cannot be
greater than the difference in strike prices:
P(K1) – P(K2) < K2 – K1
‰Premiums decline at a decreasing rate for calls with
progressively higher strike prices. (Convexity of
option
ti price
i with
ith respectt to
t strike
t ik price):
i )
C(K1) – C(K2) > C(K2) – C(K3)
K2 – K1
K3 – K2
Slide 22--24
Fin 501: Asset Pricing
P
Properties
ti off option
ti prices
i
(cont.)
Slide 22--25
Fin 501: Asset Pricing
P
Properties
ti off option
ti prices
i
(cont.)
Slide 22--26
Fin 501: Asset Pricing
S
Summary
off parity
it relationships
l ti hi
Slide 22--27
Fin 501: Asset Pricing
O
Overview:
i
Pricing
P i i - one period
i d model
d l
1.
2.
3.
4.
5.
6.
7.
LOOP
LOOP,, No arbitrage
Forwards
Parity relationship between options
No arbitrage and existence of state prices
Market completeness and uniqueness of state prices
Pricing kernel q*
Four pricing formulas (state prices, SDF, EMM, beta
pricing)
8 Recovering state prices from options
8.
10:59 Lecture 02
One Period Model
Slide 22--28
Fin 501: Asset Pricing
… back
b k tto the
th big
bi picture
i t
•
•
•
•
•
State space (evolution of states)
(Risk) preferences
Aggregation over different agents
Security structure – prices of traded securities
Problem:
• Difficult to observe risk preferences
• What can we say about existence of state prices
without assuming specific utility
functions/constraints for all agents in the economy
10:59 Lecture 02
One Period Model
Slide 22--29
Fin 501: Asset Pricing
V t N
Vector
Notation
t ti
• Notation:
N t ti
y,x ∈ Rn
‰ y ≥ x ⇔ yi ≥ xi for each i=1,…,n.
‰ y > x ⇔ y ≥ x and y ≠ x.
x
‰ y >> x ⇔ yi > xi for each i=1,…,n.
• Inner product
‰ y · x = ∑i yx
• Matrix multiplication
10:59 Lecture 02
One Period Model
Slide 22--30
Fin 501: Asset Pricing
Th Forms
Three
F
off N
No-ARBITRAGE
No1. Law of one price (LOOP)
If h’X = k’X then p· h = p· k.
2. No strong arbitrage
There exists no portfolio h which is a strong
arbitrage, that is h’X ≥ 0 and p · h < 0.
3 No arbitrage
3.
There exists no strong arbitrage
nor portfolio k with kk’ X > 0 and p · k ≤ 0.
0
10:59 Lecture 02
One Period Model
Slide 22--31
Fin 501: Asset Pricing
Th Forms
Three
F
off N
No-ARBITRAGE
No• Law of one price is equivalent to
every
y portfolio
p
with zero payoff
p y has zero price.
p
• No arbitrage
⇒ no strong arbitrage
No strong arbitrage ⇒ law of one price
10:59 Lecture 02
One Period Model
Slide 22--32
Fin 501: Asset Pricing
Pi i
Pricing
• Define for each z ∈ <X>,
• If LOOP hholds
ld q(z)
( ) is
i a single-valued
i l
l d andd linear
li
functional. (i.e. if h’ and h’ lead to same z, then price has to be the same)
• Conversely, if q is a linear functional defined in
<X> then the law of one price holds.
10:59 Lecture 02
One Period Model
Slide 22--33
Fin 501: Asset Pricing
Pi i
Pricing
• LOOP ⇒ q(h’X)
(h’X) = p · h
• A linear functional Q in RS is a valuation
function if Q(z) = q(z) for each z ∈ <X>.
• Q(z) = q · z for some q ∈ RS, where qs = Q(es),
and es is the vector with ess = 1 and esi = 0 if i ≠ s
‰es is an Arrow-Debreu security
• q is a vector of state p
prices
10:59 Lecture 02
One Period Model
Slide 22--34
Fin 501: Asset Pricing
State prices q
• q is a vector of state prices if p = X q,
that is pj = xj · q for each j = 1,…,J
1 J
• If Q(z) = q · z is a valuation functional then q is a vector
of state prices
• Suppose q is a vector of state prices and LOOP holds.
Then if z = hh’X
X LOOP implies that
• Q(z) = q · z is a valuation functional
⇔ q is a vector of state prices and LOOP holds
10:59 Lecture 02
One Period Model
Slide 22--35
Fin 501: Asset Pricing
State prices q
p(1,1) = q1 + q2
p(2,1) = 2q1 + q2
Value of portfolio (1,2)
(1 2)
3p(1,1) – p(2,1) = 3q1 +3q2-2q1-q2
= q1 + 2q2
c2
q2
c1
q1
10:59 Lecture 02
One Period Model
Slide 22--36
Fin 501: Asset Pricing
Th Fundamental
The
F d
t l Theorem
Th
off Finance
Fi
• Proposition 1. Security prices exclude
arbitrage if and only if there exists a valuation
functional with q >> 0.
• Proposition 1’. Let X be an J × S matrix, and
p ∈ RJ. There is no h in RJ satisfying h · p ≤ 0,
h’ X ≥ 0 and at least one strict inequality if,
and
d only
l if,
if there
th exists
i t a vector
t q ∈ RS with
ith
q >> 0 and p = X q.
No arbitrage ⇔ positive state prices
10:59 Lecture 02
One Period Model
Slide 22--37
Fin 501: Asset Pricing
O
Overview:
i
Pricing
P i i - one period
i d model
d l
1.
2.
3.
4.
5.
6.
7.
8.
LOOP
LOOP,, No arbitrage
Forwards
Parity relationship between options
No arbitrage and existence of state prices
Market completeness and uniqueness of state prices
Pricing kernel q*
Four pricing formulas (state prices, SDF, EMM)
Recovering state prices from options
10:59 Lecture 02
One Period Model
Slide 22--38
Fin 501: Asset Pricing
Multiple State Prices q
& Incomplete Markets
What state prices are consistent
with p(1,1)?
p(1 1) = q1 + q2
p(1,1)
Payoff space <X>
One equation – two unknowns q1, q2
There are (infinitely) many.
e.g. if p(1,1)=.9
p(1,1)
q1 =.45,, q2 =.45
or q1 =.35, q2 =.55
bond (1,1) only
c2
q2
c1
q1
10:59 Lecture 02
One Period Model
Slide 22--39
Fin 501: Asset Pricing
Q(x)
x2
complete markets
<X>
q
x1
10:59 Lecture 02
One Period Model
Slide 22--40
Fin 501: Asset Pricing
p=Xq
Q(x)
x2
<X> incomplete markets
q
x1
10:59 Lecture 02
One Period Model
Slide 22--41
Fin 501: Asset Pricing
p=Xqo
Q(x)
x2
<X> incomplete markets
qo
x1
10:59 Lecture 02
One Period Model
Slide 22--42
Fin 501: Asset Pricing
M lti l q in
Multiple
i incomplete
i
l t markets
k t
c2
<X>
p=X’q
q*
qo
v
q
c1
Many possible state price vectors s.t.
s t p=X
p=X’qq.
One is special: q* - it can be replicated as a portfolio.
10:59 Lecture 02
One Period Model
Slide 22--43
Fin 501: Asset Pricing
U i
Uniqueness
andd C
Completeness
l t
• Proposition 2. If markets are complete, under no
arbitrage there exists a unique valuation functional.
• If markets are not complete, then there exists
v ∈ RS with 0 = Xv.
Suppose there is no arbitrage and let q >> 0 be a vector
off state
t t prices.
i
Then
Th q + α v >> 0 provided
id d α is
i small
ll
enough, and p = X (q + α v). Hence, there are an infinite
number of strictly positive state prices.
prices
10:59 Lecture 02
One Period Model
Slide 22--44
Fin 501: Asset Pricing
O
Overview:
i
Pricing
P i i - one period
i d model
d l
1.
2.
3.
4.
5.
6.
7.
8.
LOOP
LOOP,, No arbitrage
Forwards
Parity relationship between options
No arbitrage and existence of state prices
Market completeness and uniqueness of state prices
Pricing kernel q*
Four pricing formulas (state prices, SDF, EMM)
Recovering state prices from options
10:59 Lecture 02
One Period Model
Slide 22--45
Fin 501: Asset Pricing
F
Four
A
Assett P
Pricing
i i F
Formulas
l
1. State prices
2. Stochastic discount factor
pj = ∑s qs xsj
pj = E[mxj]
m1
m2
m3
xj1
xj2
xj3
pj = 1/(1+rf) Eπ^ [xj]
3. Martingale measure
((reflect
fl risk
i k aversion
i by
b
over(under)weighing the “bad(good)” states!)
4 State
4.
State--price beta model
(in returns Rj := xj /pj)
10:59 Lecture 02
E[Rj] - Rf = βj E[R*- Rf]
One Period Model
Slide 22--46
Fin 501: Asset Pricing
1.
1 State
St t Price
P i Model
M d l
• … so far price in terms of Arrow-Debreu (state)
p
prices
j
p =
10:59 Lecture 02
j
q
x
s
s
s
One Period Model
Slide 22--47
Fin 501: Asset Pricing
2 St
2.
Stochastic
h ti Discount
Di
t Factor
F t
• That is, stochastic discount factor ms = qs/πs for all s.
10:59 Lecture 02
One Period Model
Slide 22--48
Fin 501: Asset Pricing
2 Stochastic Discount Factor
2.
shrink axes by factor
<X>
m*
m
10:59 Lecture 02
c1
One Period Model
Slide 22--49
Fin 501: Asset Pricing
Ri
Risk
Riskk-adjustment
dj t
t in
i payoffs
ff
p = E[mxj] = E[m]E[x] + Cov[m,x]
Since 1=E[mR], the risk free rate is Rf = 1/E[m]
p = E[x]/Rf + Cov[m,x]
Remarks:
(i) If risk-free rate does not exist, Rf is the shadow risk free
rate
(ii) In general Cov[m,x]
Cov[m x] < 0,
0 which lowers price and
increases return
10:59 Lecture 05
StateState-price Beta Model
Slide 22--50
Fin 501: Asset Pricing
3.
3 Equivalent Martingale Measure
• Price of any asset
• Price of a bond
10:59 Lecture 02
One Period Model
Slide 22--51
Fin 501: Asset Pricing
… in
i Returns:
R t
Rj=xj/pj
E[mRj]=1
Rf E[m]=1
⇒ E[m(Rj-Rf)]=0
E[m]{E[Rj]-Rf} + Cov[m,Rj]=0
E[Rj] – Rf = - Cov[m,Rj]/E[m]
also holds for portfolios h
(2)
Note:
• risk correction depends only on Cov of payoff/return
with discount factor.
• Only compensated for taking on systematic risk not
idiosyncratic risk.
10:59 Lecture 05
StateState-price Beta Model
Slide 22--52
Fin 501: Asset Pricing
4 State4.
State-price BETA Model
shrink axes by factor
c2
<X>
let underlying asset
be x=(1.2,1)
R*
R*=α m*
m*
m
c1
p=1
(priced with m*)
10:59 Lecture 05
StateState-price Beta Model
Slide 22--53
Fin 501: Asset Pricing
4 St
4.
State
Statet -price
i BETA Model
M d l
E[Rj] – Rf = - Cov[m,Rj]/E[m]
(2)
also holds for all portfolios h and
we can replace m with m*
Suppose (i) Var[m*] > 0 and (ii) R* = α m* with α > 0
E[Rh] – Rf = - Cov[R*,Rh]/E[R*]
(2’)
Define βh := Cov[R*,Rh]/ Var[R*] for any portfolio h
10:59 Lecture 05
StateState-price Beta Model
Slide 22--54
Fin 501: Asset Pricing
4 St
4.
State
Statet -price
i BETA Model
M d l
(2) for
f R*: E[R*]-R
] Rf=-Cov[R
C [R*,R
R*]/E[R*]
=-Var[R*]/E[R*]
(2) for Rh: E[Rh]-R
] Rf=-Cov[R
= Cov[R*,R
Rh]/E[R*]
= - βh Var[R*]/E[R*]
E[R
E[
Rh] - Rf = βh E[R*- Rf]
where βh ::= Cov[R*,Rh]/Var[R*]
very general – but what is R* in reality?
Regression Rhs = αh + βh (R*)s + εs with Cov[R*,ε]=E[ε]=0
10:59 Lecture 05
StateState-price Beta Model
Slide 22--55
Fin 501: Asset Pricing
F
Four
A
Assett P
Pricing
i i F
Formulas
l
1. State prices
2. Stochastic discount factor
1 = ∑s qs Rsj
1 = E[mRj]
m1
m2
m3
xj1
xj2
xj3
1 = 1/(1+rf) Eπ^ [Rj]
3. Martingale measure
((reflect
fl risk
i k aversion
i by
b
over(under)weighing the “bad(good)” states!)
4 State
4.
State--price beta model
(in returns Rj := xj /pj)
10:59 Lecture 02
E[Rj] - Rf = βj E[R*- Rf]
One Period Model
Slide 22--56
Fin 501: Asset Pricing
Wh t do
What
d we know
k
about
b t q, m, π, R*?
^
• Main results so far
‰Existence iff no arbitrage
• Hence, single factor only
• but doesn’t famos Fama-French factor model has 3 factors?
• multiple
lti l factor
f t is
i due
d to
t time-variation
ti
i ti
(wait for multi-period model)
‰Uniqueness if markets are complete
10:59 Lecture 02
One Period Model
Slide 22--57
Fin 501: Asset Pricing
Diff
Different
t Asset
A tP
Pricing
i i Models
M d l
pt = E[mt+1 xt+1]
where
⇒
mt+1=f(·,…,·)
(, ,)
f(·) = asset pricing model
General Equilibrium
f(·) = MRS / π
Factor Pricing Model
a+b1 f1,t+1 + b2 f2,t+1
CAPM
a+b1 f1,t+1 = a+b1 RM
10:59 Lecture 05
E[Rh] - Rf = βh E[R*- Rf]
where βh := Cov[R*,Rh]/Var[R*]
CAPM
f
R*=Rf ((a+b1RM))/(a+b
(
1R )
where RM = return of market portfolio
Is b1 < 0?
StateState-price Beta Model
Slide 22--58
Fin 501: Asset Pricing
Diff
Different
t Asset
A tP
Pricing
i i Models
M d l
• Theory
‰All economics and modeling is determined by
mt+1= a + b’ f
‰Entire content of model lies in restriction of SDF
• Empirics
‰m* (which is a portfolio payoff) prices as well as m
(which is e.g. a function of income, investment etc.)
‰ measurement error of m* is smaller than for any m
‰Run regression on returns (portfolio payoffs)!
(
(e.g.
Fama-Frenchh three
h ffactor model)
d l)
10:59 Lecture 05
StateState-price Beta Model
Slide 22--59
Fin 501: Asset Pricing
observe/specify
existing
Asset Prices
specify
Preferences &
Technology
•evolution of states
•risk preferences
•aggregation
absolute
asset pricing
NAC/LOOP
NAC/LOOP
State Prices q
(or stochastic discount
factor/Martingale measure)
relative
asset pricing
LOOP
derive
A t Prices
Asset
Pi
10:59 Lecture 02
derive
P i for
Price
f (new)
(
) assett
One Period Model
Only works as long as market
Slide 22--60
completeness doesn’t change
Fin 501: Asset Pricing
O
Overview:
i
Pricing
P i i - one period
i d model
d l
1.
2.
3.
4.
5.
6.
7.
LOOP
LOOP,, No arbitrage
Forwards
Parity relationship between options
No arbitrage and existence of state prices
Market completeness and uniqueness of state prices
Pricing kernel q*
Four pricing formulas (state prices, SDF, EMM, betabetapricing)
8 Recovering state prices from options
8.
10:59 Lecture 02
One Period Model
Slide 22--61
Fin 501: Asset Pricing
Recovering
ecove g S
Statee Prices
ces from
o
Option Prices
• Suppose that ST, the price of the underlying portfolio (we
may think of it as a proxy for price of “market portfolio”),
assumes a "continuum"
continuum of possible values.
• Suppose there are a “continuum” of call options with
different strike/exercise prices ⇒ markets are complete
• Let
L t us construct
t t the
th following
f ll i portfolio:
tf li
for some small positive number ε>0,
‰ Buy one call with
‰ Sell one call with
‰ Sell one call with
‰ Buy one call with
10:59 Lecture 02
E = ŜT − 2δ − ε
E = Sˆ T − δ
2
E = ŜT + 2δ
E. = ŜT + 2δ + ε
One Period Model
Slide 22--62
Fin 501: Asset Pricing
Recovering
R
i State
St t Prices
Pi
… (ctd.)
( td )
Payoff
^− δ 2 − ε)
CT( ST , E = S
T
^ + δ 2 + ε)
CT( ST, E = S
T
ε
S^T − δ 2− ε S^T −δ 2
S^T
S^T +δ 2
CT( ST , E = S^T− δ 2)
S^T +δ 2+ε
ST
CT( ST , E = S^T+ δ 2)
Figure 8-2 Payoff Diagram: Portfolio of Options
10:59 Lecture 02
One Period Model
Slide 22--63
Fin 501: Asset Pricing
Recovering
R
i State
St t Prices
Pi
… (ctd.)
( td )
• Let us thus consider buying 1/ε units of the portfolio. The
1
ŜT − 2δ ≤ ST ≤ ŜT + 2δ , is ε ⋅ ≡ 1 , for
total ppayment,
y
, when
ε
any choice of ε. We want to let ε a 0 , so as to eliminate
the payments in the ranges ST ∈ (ŜT − δ − ε, ŜT − δ ) and
2
2
δ
δ
1
ST ∈ (S
ŜT + , S
ŜT + + ε) .The
Th value
l off /ε units
it off this
thi portfolio
tf li
2
2
is :
{(
) (
) [(
) (
1
C S, E = Ŝˆ − 2δ − ε − C S, E = Ŝˆ T − 2δ − C S, E = Ŝˆ T + 2δ − C S, E = Ŝˆ T + 2δ + ε
T
ε
10:59 Lecture 02
One Period Model
Slide 22--64
)]}
Fin 501: Asset Pricing
Taking the limit ε→ 0
{(
) (
) [(
) (
lim 1ε C S, E = Ŝˆ T − 2δ − ε − C S, E = Ŝˆ T − 2δ − C S, E = Ŝˆ T + 2δ − C S, E = Ŝˆ T + 2δ + ε
εa0
(
) (
)
(
) (
)]}
)
⎧C S, E = ŜT − 2δ − ε − C S, E = ŜT − 2δ ⎫
⎧C S, E = ŜT + 2δ + ε − C S, E = ŜT + 2δ ⎫
= − lim⎨
⎬ + lim
⎨
⎬
εa0
ε
a
0
−ε
ε
⎩
⎭
⎩
⎭
≤0
≤0
Payoff
1
$T −δ
S
2
S$ T
S$ T + δ 2
ST
Divide by δ and let δ → 0 to obtain state price density as ∂2C/∂ E2.
10:59 Lecture 02
One Period Model
Slide 22--65
Fin 501: Asset Pricing
Recovering
R
i State
St t Prices
Pi
… (ctd.)
( td )
Evaluating following cash flow
[
[
⎧
0 if ST ∉ ŜT − 2δ , ŜT + 2δ
CFT = ⎨
δ ˆ
δ
ˆ
⎩50000 if ST ∈ ŜT − 2 , ŜT + 2
~
]⎫⎬.
]⎭
The value today of this cash flow is :
10:59 Lecture 02
One Period Model
Slide 22--66
Fin 501: Asset Pricing
Table 8.1 Pricing an Arrow-Debreu State Claim
E
7
C(S,E)
Cost of
position
7
8
Payoff if ST =
9
10
11
12
13
ΔC
Δ(ΔC) qs
Δ(ΔC)=
3.354
-0.895
8
2 459
2.459
0 106
0.106
-0.789
9
1.670
+1.670
0
0
0
1
2
3
4
0.164
-0.625
10
1.045
-2.090
2.090
0
0
0
0
-2
2
-4
4
-6
6
0.184
-0.441
11
0.604
+0.604
0
0
0
0
0
1
2
0.162
-0.279
12
0.325
0.118
-0.161
0 161
13
0.164
0.184
10:59 Lecture 02
0
0
0
1
0
One Period Model
0
0
Slide 22--67
Fin 501: Asset Pricing
observe/specify
existing
Asset Prices
specify
Preferences &
Technology
•evolution of states
•risk preferences
•aggregation
absolute
asset pricing
NAC/LOOP
NAC/LOOP
State Prices q
(or stochastic discount
factor/Martingale measure)
relative
asset pricing
LOOP
derive
A t Prices
Asset
Pi
10:59 Lecture 02
derive
P i for
Price
f (new)
(
) assett
One Period Model
Only works as long as market
Slide 22--68
completeness doesn’t change
Fin 501: Asset Pricing
The following is for later lecture
10:59 Lecture 02
One Period Model
Slide 22--69
Fin 501: Asset Pricing
F t
Futures
contracts
t t
• Exchange-traded “forward contracts”
• Typical features of futures contracts
‰ Standardized,
Standardized with specified delivery dates,
dates locations,
locations procedures
‰ A clearinghouse
• Matches buy and sell orders
• Keeps track of members’
members obligations and payments
• After matching the trades, becomes counterparty
• Differences from forward contracts
‰ Settled daily
dail through
thro gh the mark-to-market
mark to market process Î low
lo credit risk
‰ Highly liquid Î easier to offset an existing position
‰ Highly standardized structure Î harder to customize
Slide 22--70
Fin 501: Asset Pricing
E
Example:
l S&P 500 Futures
F t
•
WSJ listing:
•
Contract specifications:
Slide 22--71
Fin 501: Asset Pricing
E
Example:
l S&P 500 Futures
F t
(cont.)
•
•
•
•
Notional
N
ti l value:
l $250 x Index
I d
Cash-settled contract
Open interest: total number of buy/sell pairs
M i and
Margin
d mark-to-market
k
k
‰
‰
‰
‰
•
Initial margin
Maintenance margin (70-80% of initial margin)
Margin call
Daily mark-to-market
Futures prices vs. forward prices
‰ The difference negligible especially for short-lived contracts
‰ Can be significant for long-lived contracts and/or when interest rates are
correlated with the price of the underlying asset
Slide 22--72
Fin 501: Asset Pricing
E
Example:
l S&P 500 Futures
F t
(cont.)
•
Mark-to-market
Mark
to market proceeds and margin balance for 8 long futures:
Slide 22--73
Fin 501: Asset Pricing
E
Example:
l S&P 500 Futures
F t
(cont.)
•
S&P index arbitrage: comparison of formula prices with actual prices:
Slide 22--74
Fin 501: Asset Pricing
U off iindex
Uses
d futures
f t
•
•
•
Whyy buyy an index futures contract instead of synthesizing
y
g it usingg the stocks
in the index? Lower transaction costs
Asset allocation: switching investments among asset classes
Example: Invested in the S&P 500 index and temporarily wish to temporarily
invest in bonds instead of index. What to do?
‰ Alternative #1: Sell all 500 stocks and invest in bonds
‰ Alternative #2: Take a short forward position in S&P 500 index
Slide 22--75
Fin 501: Asset Pricing
U off iindex
Uses
d futures
f t
(cont.)
• $100 million portfolio with β of 1.4 and rf = 6 %
1. Adjust
j for difference in $ amount
• 1 futures contract $250 x 1100 = $275,000
• Number of contracts needed $100mill/$0.275mill = 363.636
2. Adjust for difference in β
363.636 x 1.4 = 509.09 contracts
Slide 22--76
Fin 501: Asset Pricing
U off iindex
Uses
d futures
f t
(cont.)
• Cross-hedging
g g with pperfect correlation
• Cross-hedging with imperfect correlation
• General asset allocation: futures overlay
• Risk management for stock-pickers
Slide 22--77
Fin 501: Asset Pricing
C
Currency
contracts
t t
•
•
Widely used to hedge against changes in
exchange
h
rates
t
WSJ listing:
Slide 22--78
Fin 501: Asset Pricing
C
Currency
contracts:
t t pricing
i i
• Currency prepaid forward
‰ Suppose you want to purchase ¥1 one year from today using $s
‰ FP0, T = x0 e−r T
y
• where x0 is current ($/ ¥) exchange rate, and ry is the yen-denominated
interest rate
• Why? By deferring delivery of the currency one loses interest income
from bonds denominated in that currency
• Currency forward
‰ F0, T = x0 e(r( −r )T
y
• r is the $-denominated domestic interest rate
• F0, T > x0 if r > ry (domestic risk-free rate exceeds foreign risk-free rate)
Slide 22--79
Fin 501: Asset Pricing
C
Currency
contracts:
t t pricing
i i (cont.)
• Example 5.3:
‰ ¥-denominated interest rate is 2% and current ($/ ¥) exchange rate is 0.009.
To have ¥1 in one year one needs to invest today:
• 0.009/¥ x ¥1 x e-0.02 = $0.008822
• Example 5.4:
‰ ¥-denominated interest rate is 2% and $-denominated rate is 6%. The
current ($
($/ ¥)) exchange
g rate is 0.009. The 1-year
y forward rate:
• 0.009e0.06-0.02 = 0.009367
Slide 22--80
Fin 501: Asset Pricing
C
Currency
contracts:
t t pricing
i i (cont.)
•
•
Synthetic currency forward: borrowing in one currency and lending in
another creates the same cash flow as a forward contract
Covered interest arbitrage: offset the synthetic forward position with
an actual forward contract
Table 5.12
Slide 22--81
Fin 501: Asset Pricing
E d ll futures
Eurodollar
f t
•
WSJ listing
•
Contract specifications
Slide 22--82
Fin 501: Asset Pricing
Introduction to Commodity
Forwards
•
Commodity
C
di forward
f
d prices
i
can be
b described
d
ib d by
b the
h same formula
f
l as that
h for
f
financial forward prices:
δ)T
F0,T = S0 e(r(r–δ)T
‰For financial assets, δ is the dividend yield. For
commodities, δ is the commodity lease rate. The lease
rate is the return that makes an investor willing to buy
and then lend a commodity.
y
‰The lease rate for a commodity can typically be
estimated only by observing the forward prices.
prices
Slide 22--83
Fin 501: Asset Pricing
Introduction to Commodity
Forwards
•
The set of prices for different expiration dates for a given commodity
is called the forward curve (or the forward strip) for that date.
•
If on a given date the forward curve is upward-sloping, then the market
is in contango. If the forward curve is downward sloping, the market
is in backwardation.
backwardation
‰Note that forward curves can have portions in
backwardation and portions in contango.
Slide 22--84