Markus K. Brunnermeier LECTURE 3: ONE-PERIOD MODEL PRICING · Lecture 03 One Period Model: Pricing (9) Pricing prepaid forwards ⢠If we can price the prepaid forward ( ð), then
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FIN501 Asset PricingLecture 03 One Period Model: Pricing (1)
LECTURE 3: ONE-PERIOD MODELPRICING
Markus K. Brunnermeier
FIN501 Asset PricingLecture 03 One Period Model: Pricing (2)
Overview: Pricing
1. LOOP, No arbitrage [L2,3]
2. Forwards [McD5]
3. Options: Parity relationship [McD6]
4. No arbitrage and existence of state prices [L2,3,5]
5. Market completeness and uniqueness of state prices
6. Unique ðâ
7. Four pricing formulas:state prices, SDF, EMM, beta pricing [L2,3,5,6]
8. Recovering state prices from options [DD10.6]
FIN501 Asset PricingLecture 03 One Period Model: Pricing (3)
Vector Notation
⢠Notation: ðŠ, ð¥ â âð
â ðŠ ⥠ð¥ â ðŠð ⥠ð¥ð for each ð = 1,⊠, ð
â ðŠ > ð¥ â ðŠ ⥠ð¥, ðŠ â ð¥
â ðŠ â« ð¥ â ðŠð > ð¥ð for each ð = 1,⊠, ð
⢠Inner product
â ðŠ â ð¥ = ðŠð¥
⢠Matrix multiplication
FIN501 Asset PricingLecture 03 One Period Model: Pricing (4)
Three Forms of No-ARBITRAGE
1. Law of one Price (LOOP) ðâ = ðð â ð â â = ð â ð
2. No strong arbitrageThere exists no portfolio â which is a strong arbitrage, that is ðâ ⥠0 and ð â â < 0
3. No arbitrage There exists no strong arbitrage nor portfolio ð with ðð > 0 and ð â ð †0
FIN501 Asset PricingLecture 03 One Period Model: Pricing (5)
Three Forms of No-ARBITRAGE
⢠Law of one price is equivalent to every portfolio with zero payoff has zero price.
⢠No arbitrage => no strong arbitrage No strong arbitrage => law of one price
FIN501 Asset PricingLecture 03 One Period Model: Pricing (6)
specifyPreferences &
Technology
observe/specifyexisting
Asset Prices
State Prices q(or stochastic discount
factor/Martingale measure)
derivePrice for (new) asset
⢠evolution of states⢠risk preferences⢠aggregation
absolute asset pricing
relativeasset pricing
NAC/LOOP
LOOP
NAC/LOOP
Only works as long as market completeness doesnât change
deriveAsset Prices
FIN501 Asset PricingLecture 03 One Period Model: Pricing (7)
Overview: Pricing
1. LOOP, No arbitrage
2. Forwards
3. Options: Parity relationship
4. No arbitrage and existence of state prices
5. Market completeness and uniqueness of state prices
6. Unique q*
7. Four pricing formulas:state prices, SDF, EMM, beta pricing
8. Recovering state prices from options
FIN501 Asset PricingLecture 03 One Period Model: Pricing (8)
Alternative ways to buy a stock⢠Four different payment and receipt timing combinations:
â Outright purchase: ordinary transaction
â Fully leveraged purchase: investor borrows the full amount
â Prepaid forward contract: pay today, receive the share later
â Forward contract: agree on price now, pay/receive later
⢠Payments, receipts, and their timing:
FIN501 Asset PricingLecture 03 One Period Model: Pricing (9)
Pricing prepaid forwards
⢠If we can price the prepaid forward (ð¹ð), then we can calculate the price for a forward contract:
ð¹ = Future value of ð¹ð
⢠Pricing by analogyâ In the absence of dividends, the timing of delivery is irrelevant
â Price of the prepaid forward contract same as current stock price
â ð¹0,ðð = ð0 (where the asset is bought at t = 0, delivered at t = T)
FIN501 Asset PricingLecture 03 One Period Model: Pricing (10)
Pricing prepaid forwards (cont.)
⢠Pricing by arbitrageâ If at time ð¡ = 0, the prepaid forward price somehow exceeded the
stock price, i.e., ð¹0,ðð > ð0, an arbitrageur could do the following:
FIN501 Asset PricingLecture 03 One Period Model: Pricing (11)
Pricing prepaid forwards (cont.)
⢠What if there are deterministic* dividends? Is ð¹0,ðð = ð0 still valid?
â No, because the holder of the forward will not receive dividends that will be
paid to the holder of the stock â ð¹0,ðð < ð0
ð¹0,ðð = ð0â PV(ððð ððð£ðððððð ðððð ðððð ð¡ = 0 ð¡ð ð¡ = ð)
â For discrete dividends ð·ð¡ðat times ð¡ð , ð = 1,⊠, ð
⢠The prepaid forward price: ð¹0,ðð = ð0 â ð=1
ð ðð0,ð ð·ð¡ð
(reinvest the dividend at risk-free rate)
â For continuous dividends with an annualized yield ð¿
⢠The prepaid forward price: ð¹0,ðð = ð0ð
âð¿ð
(reinvest the dividend in this index. One has to invest only ð0ðâð¿ð initially)
â Forward price is the future value of the prepaid forward: ð¹0,ð = FV ð¹0,ðð = ð¹0,ð
ð Ã ððð
NB: If dividends are stochastic, we cannot apply the one period model
FIN501 Asset PricingLecture 03 One Period Model: Pricing (12)
Creating a synthetic forward⢠One can offset the risk of a forward by creating a synthetic forward to
offset a position in the actual forward contract
⢠How can one do this? (assume continuous dividends at rate ð¿)
â Recall the long forward payoff at expiration ðð â ð¹0,ðâ Borrow and purchase shares as follows:
â Note that the total payoff at expiration is same as forward payoff
â This leads to: Forward = Stock â zero-coupon bond
FIN501 Asset PricingLecture 03 One Period Model: Pricing (13)
Other issues in forward pricing
⢠Does the forward price predict the future price?
â According the formula ð¹0,ð = ð0ððâð¿ ð the forward price conveys no
additional information beyond what ð0, ð, ð¿ provide
â Moreover, if ð < ð¿ the forward price underestimates the future stock price
⢠Forward pricing formula and cost of carryâ Forward price =
Spot price + Interest to carry the asset â asset lease rate
Cost of carry ð â ð¿ ð
FIN501 Asset PricingLecture 03 One Period Model: Pricing (14)
Overview: Pricing
1. LOOP, No arbitrage
2. Forwards
3. Options: Parity relationship
4. No arbitrage and existence of state prices
5. Market completeness and uniqueness of state prices
6. Unique q*
7. Four pricing formulas:state prices, SDF, EMM, beta pricing
8. Recovering state prices from options
FIN501 Asset PricingLecture 03 One Period Model: Pricing (15)
Put-Call Parity
⢠For European options with the same strike price and time to expiration the parity relationship is:
Call â Put = PV Forward Px â Strike Px
ð¶ ðŸ, ð â ð ðŸ, ð = ðð0,ð ð¹0,ð â ðŸ = ðâðð ð¹0,ð â ðŸ
ice (ð¹0,ð = ðŸ) creates a synthetic forward contract and hence must
â creates a synthetic forward contract and hence must have a zero price
â creates a synthetic forward contract and hence must have a zero price
FIN501 Asset PricingLecture 03 One Period Model: Pricing (16)
Parity for Options on Stocks
⢠If underlying asset is a stock and Div is the deterministic* dividend stream, we can plug in ðâððð¹0,ð = ð0 â ðð0,ð Divthus obtaining:
ð¶(ðŸ, ð) = ð(ðŸ, ð) + ð0 â ðð0,ð Div â ðâðððŸ
⢠For index options, ð0 â ðð0,ð Div = ð0ðâð¿ð, therefore
ð¶ ðŸ, ð = ð ðŸ, ð + ð0ðâð¿ð â ðâðððŸ
* allows us to stay in one period setting
FIN501 Asset PricingLecture 03 One Period Model: Pricing (17)
Option price boundaries
⢠American vs. Europeanâ 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 as an otherwise identical European option:
ð¶ðŽ ð, ðŸ, ð ⥠ð¶ðž ð, ðŸ, ðððŽ ð, ðŸ, ð ⥠ððž ð, ðŸ, ð
⢠Option price boundariesâ Call price cannot: be negative, exceed stock price, be less than price
implied by put-call parity using zero for put price:ð > ð¶ðŽ ð, ðŸ, ð ⥠ð¶ðž ð, ðŸ, ð > ðð0,ð ð¹0,ð â ðð0,ð ðŸ
+
â Put price cannot: be negative, exceed strike price, be less than price implied by put-call parity using zero for call price:
ðŸ > ððŽ ð, ðŸ, ð ⥠ððž ð, ðŸ, ð > ðð0,ð ðŸ â ðð0,ð ð¹0,ð+
FIN501 Asset PricingLecture 03 One Period Model: Pricing (18)
Early exercise of American call
⢠Early exercise of American optionsâ A non-dividend paying American call option should not be
exercised early, because:ð¶ðŽ ⥠ð¶ðž = ðð¡ â ðŸ + ððž + ðŸ 1 â ðâð ðâð¡ > ðð¡ â ðŸ
â That means, one would lose money be exercising early instead of selling the option
⢠Caveatsâ If there are dividends, it may be optimal to exercise early
â It may be optimal to exercise a non-dividend paying put option early if the underlying stock price is sufficiently low
FIN501 Asset PricingLecture 03 One Period Model: Pricing (19)
Options: Time to expiration
⢠Time to expiration
â 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.
â European call options on dividend-paying stock may be less valuable than an otherwise identical option with less time to expiration.
FIN501 Asset PricingLecture 03 One Period Model: Pricing (20)
Options: Time to expiration⢠Time to expiration
â When the strike price grows at the rate of interest, European call and put prices on a non-dividend paying stock increases with time.
⢠Suppose to the contrary ð ð < ð(ð¡) for ð > ð¡, then arbitrage.
â Buy ð(ð) and sell ð(ð¡) initially.
â ðð¡ < ðŸð¡, keep stock and finance ðŸð¡, Time ð value ðŸð¡ðð ðâð¡ = ðŸð
0 t T
ðð¡ < ðŸð¡ ðð¡ > ðŸð¡ ðð¡ < ðŸð¡ ðð¡ > ðŸð¡
+ð ð¡ ðð¡ â ðŸð¡ 0
âðð¡ +ðð
+ðŸð¡ âðŸð
âð(ð) max{ðŸð â ðð , 0}
-------------- -------------- -------------- -------------- --------------
> 0 0 0 ⥠0 ⥠0
FIN501 Asset PricingLecture 03 One Period Model: Pricing (21)
Options: Strike price
⢠Different strike prices (ðŸ1 < ðŸ2 < ðŸ3), for both European and American optionsâ A call with a low strike price is at least as valuable as an otherwise
identical call with higher strike price:ð¶ ðŸ1 ⥠ð¶(ðŸ2)
â A put with a high strike price is at least as valuable as an otherwise identical put with low strike price:
ð ðŸ2 ⥠ð ðŸ1
â The premium difference between otherwise identical calls with different strike prices cannot be greater than the difference in strike prices:
ð¶ ðŸ1 â ð¶ ðŸ2 †ðŸ2 â ðŸ1⢠Price of a collar is not greater than its maximum payoff
S
K2 â K1
FIN501 Asset PricingLecture 03 One Period Model: Pricing (22)
Options: Strike price (cont.)
⢠Different strike prices (ðŸ1 < ðŸ2 < ðŸ3), for both European and American optionsâ The premium difference between otherwise identical puts with
different strike prices cannot be greater than the difference in strike prices:
ð ðŸ2 â ð ðŸ1 †ðŸ2 â ðŸ1
â Premiums decline at a decreasing rate for calls with progressively higher strike prices. (Convexity of option price with respect to strike price):
ð¶ ðŸ1 â ð¶ ðŸ2
ðŸ1 â ðŸ2<
ð¶ ðŸ2 â ð¶ ðŸ3
ðŸ2 â ðŸ3
FIN501 Asset PricingLecture 03 One Period Model: Pricing (23)
Options: Strike price
⢠Proof: suppose to the contraryð¶ ðŸ1 â ð¶ ðŸ2
ðŸ2 â ðŸ1â€
ð¶ ðŸ2 â ð¶ ðŸ3
ðŸ3 â ðŸ2
⢠(Asymmetric) Butterfly spreadâ Price †0:
1
ðŸ2âðŸ1ð¶ ðŸ1 â
1
ðŸ2âðŸ1+
1
ðŸ3âðŸ2ð¶ ðŸ2 +
1
ðŸ3âðŸ2ð¶ ðŸ3 †0
â Payoff > 0: (at least in some states of the world)
â â arbitrage ðŸ1 ðŸ2 ðŸ3
FIN501 Asset PricingLecture 03 One Period Model: Pricing (24)
Overview: Pricing - one period model
1. LOOP, No arbitrage
2. Forwards
3. Options: Parity relationship
4. No arbitrage and existence of state prices
5. Market completeness and uniqueness of state prices
6. Unique q*
7. Four pricing formulas:state prices, SDF, EMM, beta pricing
8. Recovering state prices from options
FIN501 Asset PricingLecture 03 One Period Model: Pricing (25)
⊠back to the big picture
⢠State space (evolution of states)
⢠(Risk) preferences
⢠Aggregation over different agents
⢠Security structure â prices of traded securities
⢠Problem:
â Difficult to observe risk preferences
â What can we say about existence of state prices without assuming specific utility functions/constraints for all agents in the economy
FIN501 Asset PricingLecture 03 One Period Model: Pricing (26)
specifyPreferences &
Technology
observe/specifyexisting
Asset Prices
State Prices q(or stochastic discount
factor/Martingale measure)
derivePrice for (new) asset
⢠evolution of states⢠risk preferences⢠aggregation
absolute asset pricing
relativeasset pricing
NAC/LOOP
LOOP
NAC/LOOP
Only works as long as market completeness doesnât change
deriveAsset Prices
FIN501 Asset PricingLecture 03 One Period Model: Pricing (27)
Three Forms of No-ARBITRAGE
1. Law of one Price (LOOP) ðâ = ðð â ð â â = ð â ð
2. No strong arbitrageThere exists no portfolio â which is a strong arbitrage, that is ðâ ⥠0 and ð â â < 0
3. No arbitrage There exists no strong arbitrage nor portfolio ð with ðð > 0 and ð â ð †0
FIN501 Asset PricingLecture 03 One Period Model: Pricing (28)
Pricing
⢠Define for each ð§ â ðð£ ð§ â ð â â: ð§ = ðâ
⢠If LOOP holds ð£ ð§ is a linear functionalâ Single-valued, because if hâ and hâ lead to same z, then price
has to be the same
â Linear on ð
â ð£ 0 = 0
⢠Conversely, if ð£ is a linear functional defined in ð then the law of one price holds.
FIN501 Asset PricingLecture 03 One Period Model: Pricing (29)
Pricing
⢠LOOP â ð£ ðâ = ð â â
⢠A linear functional ð â âð is a valuation function if
ð ð§ = ð£ ð§ for each ð§ â ð
⢠ð ð§ = ð â ð§ for some ð â âð, where ðð = ð ðð , and ðð is the vector with ðð
ð = 1 and ðð ð = 0 if ð â ð
â ðð is an Arrow-Debreu security
⢠ð is a vector of state prices
⢠ð extends ð£ on âð
FIN501 Asset PricingLecture 03 One Period Model: Pricing (30)
State prices q
⢠ð is a vector of state prices if ð = ðâ²ð, that is ðð = ð¥ð â ð for each ð = 1,⊠, ðœ
⢠If ð ð§ = ð â ð§ is a valuation functional then ð is a vector of state prices
⢠Suppose ð is a vector of state prices and LOOP holds. Then if ð§ = ðâ LOOP implies that
ð£ ð§ =
ð
âððð
=
ð
ð
ð¥ð ððð âð =
ð
ð
ð¥ð ðâð ðð = ð â ð§
⢠ð ð§ = ð â ð§ is a valuation functional âð is a vector of state prices and LOOP holds
FIN501 Asset PricingLecture 03 One Period Model: Pricing (31)
ð 1,1 = ð1 + ð2ð 2,1 = 2ð1 + ð2
Value of portfolio (1,2)3ð 1,1 â ð 2,1 = ð1 + 2ð2
State prices q
ð¥1
ð¥2
21
12
FIN501 Asset PricingLecture 03 One Period Model: Pricing (32)
The Fundamental Theorem of Finance
⢠Proposition 1. Security prices exclude arbitrage if and only if there exists a valuation functional with ð â« 0
⢠Proposition 1â. Let ð be a S à ðœ matrix, and ð â âðœ. There is no â in âðœ satisfying â â ð †0, ðâ ⥠0 and at least one strict inequality â there exists a vector ð â âð with ð â« 0 and ð = ðâ²ð
No arbitrage , positive state prices
FIN501 Asset PricingLecture 03 One Period Model: Pricing (33)
Overview: Pricing
1. LOOP, No arbitrage
2. Forwards
3. Options: Parity relationship
4. No arbitrage and existence of state prices
5. Market completeness and uniqueness of state prices
6. Unique ðâ
7. Four pricing formulas:state prices, SDF, EMM, beta pricing
8. Recovering state prices from options
FIN501 Asset PricingLecture 03 One Period Model: Pricing (34)
Multiple State Prices ð& Incomplete Markets
ð1
ð2
ð¥1
ð¥2
ð 1,1
Payoff space âšðâ©
bond (1,1) only
What state prices are consistent with ð 1,1 ?ð 1,1 = ð1 + ð2
One equation â two unknowns ð1, ð2There are (infinitely) many.
e.g. if ð 1,1 = .9ð1 = .45, ð2 = .45,
or ð1 = .35, ð2 = .55
FIN501 Asset PricingLecture 03 One Period Model: Pricing (35)
âšðâ©
ð
complete markets
ð¥1
ð¥2
ð(ð¥)
FIN501 Asset PricingLecture 03 One Period Model: Pricing (36)
ð(ð¥)
âšðâ©
ð
ð = ðâ²ð
incomplete markets
ð¥1
ð¥2
FIN501 Asset PricingLecture 03 One Period Model: Pricing (37)
âšðâ©
ðâ
ð = ðâ²ðâ
incomplete markets
ð¥1
ð¥2
ð(ð¥)
FIN501 Asset PricingLecture 03 One Period Model: Pricing (38)
Multiple q in incomplete marketsâšðâ©
ðv
ðâ
ðâ
ð = ðâ²ð
Many possible state price vectors s.t. ð = ðâ²ð.One is special: ðâ - it can be replicated as a portfolio.
ð¥2
ð¥1
FIN501 Asset PricingLecture 03 One Period Model: Pricing (39)
Uniqueness and Completeness
⢠Proposition 2. If markets are complete, under no arbitrage there exists a unique valuation functional.
⢠If markets are not complete, then there exists ð£ â âð with 0 = ðð£
⢠Suppose there is no arbitrage and let ð â« 0 be a vector of state prices. Then ð + ðŒð£ â« 0 provided ðŒ is small enough, and ð = ð ð + ðŒð£ . Hence, there are an infinite number of strictly positive state prices.
FIN501 Asset PricingLecture 03 One Period Model: Pricing (40)
Overview: Pricing - one period model
1. LOOP, No arbitrage
2. Forwards
3. Options: Parity relationship
4. No arbitrage and existence of state prices
5. Market completeness and uniqueness of state prices
6. Unique q*
7. Four pricing formulas:state prices, SDF, EMM, beta pricing
8. Recovering state prices from options
FIN501 Asset PricingLecture 03 One Period Model: Pricing (41)
Four Asset Pricing Formulas
1. State prices ðð = ð ðð ð¥ð ð
2. Stochastic discount factor ðð = ðž ðð¥ð
3. Martingale measure ðð =1
1+ðððž ð ð¥ð
(reflect risk aversion by over(under)weighing the âbad(good)â states!)
4. State-price beta model ðž ð ð â ð ð¹ = ðœððž ð â â ð ð
(in returns ð ð âð¥ð
ðð)
ð1
ð2
ð3
ð¥1ð
ð¥2ð
ð¥3ð
FIN501 Asset PricingLecture 03 One Period Model: Pricing (42)
1. State Price Model
⢠⊠so far price in terms of Arrow-Debreu (state) prices
ðð =
ð
ðð ð¥ð ð
FIN501 Asset PricingLecture 03 One Period Model: Pricing (43)
2. Stochastic Discount Factor
ðð =
ð
ðð ð¥ð ð=
ð
ðð
ðð ðð
ð¥ð ð
⢠That is, stochastic discount factor ðð âðð
ðð
ðð = ðž ðð¥ð
Now, probability inner product between ð and ð¥ð
FIN501 Asset PricingLecture 03 One Period Model: Pricing (44)
âšðâ©
2. Stochastic Discount Factor
shrink axes by factor ðð
ð
ðâ
ð2 ð2
ð1 ð1
With m: Probability inner product = 0 (âprobability orthogonalâ)
FIN501 Asset PricingLecture 03 One Period Model: Pricing (45)
Risk-adjustment in payoffs
ð = ðž ðð¥ = ðž ð ðž ð¥ + cov ð, ð¥
Since ðbond = ðž ð Ã 1 , the risk free rate 1
1+ðð=
1
ð ð = ðž ð .
ð =ð¬ ð
ð¹ð+ cov ð, ð
Remarks:
(i) If risk-free rate does not exist, ð ð is the shadow risk free rate
(ii) Typically cov ð, ð¥ < 0, which lowers price and increases return
FIN501 Asset PricingLecture 03 One Period Model: Pricing (46)
3. Equivalent Martingale Measure
⢠Price of any asset ðð = ð ðð ð¥ð ð
⢠Price of a bond ðbond = ð ðð =1
1+ðð
ðð =1
1 + ðð
ð
ðð ð â² ðð â²
ð¥ð ð=
1
1 + ðððž ð ð¥ð
where ðð âðð
ð â²
ðð â²
FIN501 Asset PricingLecture 03 One Period Model: Pricing (47)
⊠in Returns: ð ð =ð¥ð
ðð
ðž ðð ð = 1, ð ððž ð = 1 â ðž ð ð ð â ð ð = 0
ðž ð ðž ð ð â ð ð + cov ð, ð ð = 0
â ðž ð ð â ð ð = âcov ð, ð ð
ðž ð(also holds for portfolios â)
Note:
⢠risk correction depends only on Cov of payoff/return with discount factor.
⢠Only compensated for taking on systematic risk not idiosyncratic risk.
FIN501 Asset PricingLecture 03 One Period Model: Pricing (48)
4. State-price BETA Model
ð
ðâ
ð â
p=1(priced with m*)
ð â = ðŒðâ
let underlying asset be ð¥ = 1.2,1
shrink axes by factor ðð
âšðâ©
ð2 ð2
ð1 ð2
With m: Probability inner product = 0 (âprobability orthogonalâ)
FIN501 Asset PricingLecture 03 One Period Model: Pricing (49)
4. State-price BETA Model
ðž ð ð â ð ð = âcov ð, ð ð
ðž ð(also holds for all portfolios â,
we can replace m with ðâ)
Suppose (i) var ðâ > 0 and (ii) ð â = ðŒðâ with ðŒ > 0
ðž ð â â ð ð = âcov ð â, ð â
ðž ð â
Define ðœâ âcov ð â,ð â
var ð â for any portfolio â
FIN501 Asset PricingLecture 03 One Period Model: Pricing (50)
4. State-price BETA Model
(2) for ð â: ðž ð â â ð ð = âcov ð â,ð â
ðž ð â = âðœâ var ð â
ðž ð â
(2) for ð â: ðž ð â â ð ð = âcov ð â,ð â
ðž ð â = âvar ð â
ðž ð â
Hence,ð¬ ð¹ð â ð¹ð = ð·ðð¬ ð¹â â ð¹ð
where ð·ð âcov ð¹â,ð¹ð
var ð¹â
Regression ð ð â = ðŒâ + ðœâ ð â
ð + íð with cov ð â, í = ðž í = 0very general â but what is R* in reality?
FIN501 Asset PricingLecture 03 One Period Model: Pricing (51)
Four Asset Pricing Formulas
1. State prices ðð = ð ðð ð¥ð ð
2. Stochastic discount factor ðð = ðž ðð¥ð
3. Martingale measure ðð =1
1+ðððž ð[ð¥ð ]
(reflect risk aversion by over(under)weighing the âbad(good)â states!)
4. State-price beta model ðž ð ð â ð ð¹ = ðœððž ð â â ð ð
(in returns ð ð âð¥ð
ðð)
ð1
ð2
ð3
ð¥1ð
ð¥2ð
ð¥3ð
FIN501 Asset PricingLecture 03 One Period Model: Pricing (52)
What do we know about ð,ð, ð, ð â?
⢠Main results so far
â Existence â no arbitrage
⢠Hence, single factor only
⢠But doesnât famous Fama-French factor model have 3 factors?
⢠Additional factors are due to time-variation (wait for multi-period model)
â Uniqueness if markets are complete
FIN501 Asset PricingLecture 03 One Period Model: Pricing (53)
Different Asset Pricing Models
ðð¡ = ðž ðð¡+1ð¥ð¡+1 â ðž ð â â ð ð = ðœâðž ð â â ð ð
where ðð¡+1 = ð ⊠and ðœâ =cov ð â,ð â
var ð â
ð ⊠= asset pricing modelGeneral Equilibrium
ð ⊠=MRSð
Factor Pricing Modelð + ð1ð1,ð¡+1 + ð2ð2,ð¡+1CAPM CAPM
ð + ð1ð1,ð¡+1 = ð + ð1ð ð ð â = ð ð ð+ð1ð
ð
ð+ð1ð ð
where ð ð is market returnis ð1 â· 0?
FIN501 Asset PricingLecture 03 One Period Model: Pricing (54)
Different Asset Pricing Models
⢠Theoryâ All economics and modeling is determined by
ðð¡+1 = ð + ðâ²ð
â Entire content of model lies in restriction of SDF
⢠Empiricsâ ðâ (which is a portfolio payoff) prices as well as m (which
is e.g. a function of income, investment etc.)
â measurement error of ðâ is smaller than for any ð
â Run regression on returns (portfolio payoffs)!(e.g. Fama-French three factor model)
FIN501 Asset PricingLecture 03 One Period Model: Pricing (55)
Overview: Pricing - one period model
1. LOOP, No arbitrage
2. Forwards
3. Options: Parity relationship
4. No arbitrage and existence of state prices
5. Market completeness and uniqueness of state prices
6. Unique ðâ
7. Four pricing formulas:state prices, SDF, EMM, beta pricing
8. Recovering state prices from options
FIN501 Asset PricingLecture 03 One Period Model: Pricing (56)
specifyPreferences &
Technology
observe/specifyexisting
Asset Prices
State Prices q(or stochastic discount
factor/Martingale measure)
derivePrice for (new) asset
⢠evolution of states⢠risk preferences⢠aggregation
absolute asset pricing
relativeasset pricing
NAC/LOOP
LOOP
NAC/LOOP
Only works as long as market completeness doesnât change
deriveAsset Prices
FIN501 Asset PricingLecture 03 One Period Model: Pricing (57)
Recovering State Prices from Option Prices
⢠Suppose that ðð, the price of the underlying portfolio (we may think of it as a proxy for price of âmarket portfolioâ), assumes a "continuum" of possible values.
⢠Suppose there are a âcontinuumâ of call options with different strike/exercise prices â markets are complete
⢠Let us construct the following portfolio: for some small positive number í > 0
â Buy one call with ðŸ = ðð âð¿
2â í
â Sell one call with ðŸ = ðð âð¿
2
â Sell one call with ðŸ = ðð +ð¿
2
â Buy one call with ðŸ = ðð +ð¿
2+ í
FIN501 Asset PricingLecture 03 One Period Model: Pricing (58)
Recovering State Prices ⊠(ctd)
í
ðð âð¿
2 ðð +
ð¿
2 ðð ðð â
ð¿
2â í ðð +
ð¿
2+ í
Payoff of the portfolio
FIN501 Asset PricingLecture 03 One Period Model: Pricing (59)
⢠Let us thus consider buying 1
units of the portfolio.
⢠The total payment, when ðð âð¿
2†ðð †ðð +
ð¿
2is í â
1= 1, for any í
⢠Letting í â 0 eliminates payments in the regions ðð â ðð âð¿
2â í, ðð â
ð¿
2
and ðð â ðð +ð¿
2, ðð +
ð¿
2+ í
⢠The value of 1
units of this portfolio is1
í ð¶ ð, ðŸ = ðð â
ð¿
2â í â ð¶ ð, ðŸ = ðð â
ð¿
2
Recovering State Prices ⊠(ctd)
FIN501 Asset PricingLecture 03 One Period Model: Pricing (60)
Recovering State Prices ⊠(ctd)
⢠Taking the limit í â 0
= â limâ0
ð¶ ð, ðŸ = ðð âð¿2
â ð¶ ð, ðŸ = ðð âð¿2â í
í+ lim
â0
ð¶ ð, ðŸ = ðð +ð¿2+ í â ð¶ ð, ðŸ = ðð +
ð¿2
í
= âðð¶ ð, ðŸ = ðð â
ð¿2
ððŸ+ðð¶ ð, ðŸ = ðð +
ð¿2
ððŸ
as ð¿ â 0 we obtain state price densityð2ð¶
ððŸ2
1
ðð â ð¿/2 ðð + ð¿/2 ðð
FIN501 Asset PricingLecture 03 One Period Model: Pricing (61)
Recovering State Prices ⊠(ctd.)
⢠Evaluate the following cash flow
ð¶ð¹ð = 0 ðð â ðð â
ð¿
2, ðð +
ð¿
2
50000 ðð â ðð âð¿
2, ðð +
ð¿
2
⢠Value of this cash flow today
50000ðð¶
ððŸð, ðŸ = ðð +
ð¿
2â
ðð¶
ððŸð, ðŸ = ðð â
ð¿
2
ð ðð1 , ðð
2 =ðð¶
ððŸð, ðŸ = ðð
1 âðð¶
ððŸð, ðŸ = ðð
2
FIN501 Asset PricingLecture 03 One Period Model: Pricing (62)
Table 8.1 Pricing an Arrow-Debreu State Claim
E C(S,E) Cost of position
Payoff if ST =
7 8 9 10 11 12 13 âC â (âC)= qs
7 3.354
-0.895
8 2.459 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 0 0 0 0 -2 -4 -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
13 0.164 0.184 0 0 0 1 0 0 0
Note ÎðŸ = 1
FIN501 Asset PricingLecture 03 One Period Model: Pricing (63)
specify
Preferences &
Technology
observe/specify
existing
Asset Prices
State Prices q(or stochastic discount
factor/Martingale measure)
derive
Asset Prices
derive
Price for (new) asset
â¢evolution of states
â¢risk preferences
â¢aggregation
absolute
asset pricing
relative
asset pricing
NAC/LOOP
LOOP
NAC/LOOP
Only works as long as market
completeness doesnât change
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