LECTURE 3: ONE-PERIOD MODEL PRICING FIN501 Asset Pricing Lecture 03 One Period Model: Pricing (2) Overview: Pricing

FIN501 Asset Pricing Lecture 03 One Period Model: Pricing (1)

Markus K. Brunnermeier LECTURE 3: ONE-PERIOD MODEL PRICING FIN501 Asset Pricing Lecture 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 푞∗ 7. Four pricing formulas: state prices, SDF, EMM, beta pricing [L2,3,5,6] 8. Recovering state prices from options [DD10.6] FIN501 Asset Pricing Lecture 03 One Period Model: Pricing (3) Vector Notation

• Notation: 푦, 푥 ∈ ℝ푛 – 푦 ≥ 푥 ⇔ 푦푖 ≥ 푥푖 for each 푖 = 1, … , 푛 – 푦 > 푥 ⇔ 푦 ≥ 푥, 푦 ≠ 푥 – 푦 ≫ 푥 ⇔ 푦푖 > 푥푖 for each 푖 = 1, … , 푛 • Inner product – 푦 ⋅ 푥 = 푦푥 • Matrix multiplication FIN501 Asset Pricing Lecture 03 One Period Model: Pricing (4) Three Forms of No-ARBITRAGE

1. Law of one Price (LOOP) 푋ℎ = 푋푘 ⇒ 푝 ⋅ ℎ = 푝 ⋅ 푘 2. No strong arbitrage There exists no portfolio ℎ which is a strong arbitrage, that is 푋ℎ ≥ 0 and 푝 ⋅ ℎ < 0 3. No arbitrage There exists no strong arbitrage nor portfolio 푘 with 푋푘 > 0 and 푝 ⋅ 푘 ≤ 0 FIN501 Asset Pricing Lecture 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 Pricing Lecture 03 One Period Model: Pricing (6)

specify observe/specify Preferences & existing Technology Asset Prices

• evolution of states NAC/LOOP NAC/LOOP • risk preferences • aggregation

State Prices q absolute (or stochastic discount relative asset pricing factor/Martingale measure) asset pricing

LOOP

derive derive Asset Prices Price for (new) asset

Only works as long as market completeness doesn’t change FIN501 Asset Pricing Lecture 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 Pricing Lecture 03 One Period Model: Pricing (8) Alternative ways to buy a stock • 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: FIN501 Asset Pricing Lecture 03 One Period Model: Pricing (9) Pricing prepaid forwards

• If we can price the prepaid forward (퐹푃), then we can calculate the price for a forward contract: 퐹 = Future value of 퐹푃 • Pricing by analogy – In the absence of dividends, the timing of delivery is irrelevant – Price of the prepaid forward contract same as current stock price 푃 – 퐹0,푇 = 푆0 (where the asset is bought at t = 0, delivered at t = T) FIN501 Asset Pricing Lecture 03 One Period Model: Pricing (10)

Pricing prepaid forwards (cont.)

• Pricing by arbitrage – If at time 푡 = 0, the prepaid forward price somehow exceeded the 푃 stock price, i.e., 퐹0,푇 > 푆0, an arbitrageur could do the following: FIN501 Asset Pricing Lecture 03 One Period Model: Pricing (11)

Pricing prepaid forwards (cont.)

푃 • What if there are deterministic* dividends? Is 퐹0,푇 = 푆0 still valid? – No, because the holder of the forward will not receive dividends that will be 푃 paid to the holder of the stock ⇒ 퐹0,푇 < 푆0 푃 퐹0,푇 = 푆0– PV(푎푙푙 푑푖푣푖푑푒푛푑푠 푝푎푖푑 푓푟표푚 푡 = 0 푡표 푡 = 푇)

– For discrete dividends 퐷푡푖at times 푡푖, 푖 = 1, … , 푛 푃 푛 • The prepaid forward price: 퐹0,푇 = 푆0 − 푖=1 푃푉0,푖 퐷푡푖 (reinvest the dividend at risk-free rate) – For continuous dividends with an annualized yield 훿 푃 −훿푇 • The prepaid forward price: 퐹0,푇 = 푆0푒 −훿푇 (reinvest the dividend in this index. One has to invest only 푆0푒 initially) 푃 푃 푟푇 – Forward price is the future value of the prepaid forward: 퐹0,푇 = FV 퐹0,푇 = 퐹0,푇 × 푒

NB: If dividends are stochastic, we cannot apply the one period model FIN501 Asset Pricing Lecture 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 훿)

– Recall the long forward payoff at expiration 푆푇 − 퐹0,푇 – Borrow and purchase shares as follows:

– Note that the total payoff at expiration is same as forward payoff – This leads to: Forward = Stock – zero-coupon bond FIN501 Asset Pricing Lecture 03 One Period Model: Pricing (13) Other issues in forward pricing

• Does the forward price predict the future price? 푟−훿 푇 – According the formula 퐹0,푇 = 푆0푒 the forward price conveys no additional information beyond what 푆0, 푟, 훿 provide – Moreover, if 푟 < 훿 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 푟 − 훿 푆 FIN501 Asset Pricing Lecture 03 One Period Model: Pricing (14) Overview: Pricing

• For European options with the same strike price and time to expiration the parity relationship is: Call − Put = PV Forward Px − Strike Px −푟푇 퐶 퐾, 푇 − 푃 퐾, 푇 = 푃푉0,푇 퐹0,푇 − 퐾 = 푒 퐹0,푇 − 퐾

ice (퐹0,푇 = 퐾) creates a synthetic forward contract and hence must – creates a synthetic forward contract and hence must have a zero price – creates a synthetic forward contract and hence must have a zero price FIN501 Asset Pricing Lecture 03 One Period Model: Pricing (16) Parity for Options on Stocks

• If underlying asset is a stock and Div is the deterministic* −푟푇 dividend stream, we can plug in 푒 퐹0,푇 = 푆0 − 푃푉0,푇 Div thus obtaining:

−푟푇 퐶(퐾, 푇) = 푃(퐾, 푇) + 푆0 − 푃푉0,푇 Div – 푒 퐾

−훿푇 • For index options, 푆0 − 푃푉0,푇 Div = 푆0푒 , therefore

−훿푇 −푟푇 퐶 퐾, 푇 = 푃 퐾, 푇 + 푆0푒 − 푒 퐾

* allows us to stay in one period setting FIN501 Asset Pricing Lecture 03 One Period Model: Pricing (17) Option price boundaries

• American vs. European – 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: 퐶퐴 푆, 퐾, 푇 ≥ 퐶퐸 푆, 퐾, 푇 푃퐴 푆, 퐾, 푇 ≥ 푃퐸 푆, 퐾, 푇

• Option price boundaries – Call price cannot: be negative, exceed stock price, be less than price implied by put-call parity using zero for put price: + 푆 > 퐶퐴 푆, 퐾, 푇 ≥ 퐶퐸 푆, 퐾, 푇 > 푃푉0,푇 퐹0,푇 − 푃푉0,푇 퐾 – Put price cannot: be negative, exceed strike price, be less than price implied by put-call parity using zero for call price: + 퐾 > 푃퐴 푆, 퐾, 푇 ≥ 푃퐸 푆, 퐾, 푇 > 푃푉0,푇 퐾 − 푃푉0,푇 퐹0,푇 FIN501 Asset Pricing Lecture 03 One Period Model: Pricing (18) Early exercise of American call

• Early exercise of American options – A non-dividend paying American call option should not be exercised early, because: −푟 푇−푡 퐶퐴 ≥ 퐶퐸 = 푆푡 − 퐾 + 푃퐸 + 퐾 1 − 푒 > 푆푡 − 퐾

– That means, one would lose money be exercising early instead of selling the option

• Caveats – 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 FIN501 Asset Pricing Lecture 03 One Period Model: Pricing (19) Options: Time to expiration

• 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. – European call options on dividend-paying stock may be less valuable than an otherwise identical option with less time to expiration. FIN501 Asset Pricing Lecture 03 One Period Model: Pricing (20) Options: Time to expiration

• Time to expiration – 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 푃 푇 < 푃(푡) for 푇 > 푡, then arbitrage. – Buy 푃(푇) and sell 푃(푡) initially. 푟 푇−푡 – 푆푡 < 퐾푡, keep stock and finance 퐾푡, Time 푇 value 퐾푡푒 = 퐾푇 0 t T

푆푡 < 퐾푡 푆푡 > 퐾푡 푆푡 < 퐾푡 푆푡 > 퐾푡

+푃 푡 푆푡 − 퐾푡 0

−푆푡 +푆푇

+퐾푡 −퐾푇

−푃(푇) max{퐾푇 − 푆푇, 0} ------> 0 0 0 ≥ 0 ≥ 0 FIN501 Asset Pricing Lecture 03 One Period Model: Pricing (21) Options: Strike price

• Different strike prices (퐾1 < 퐾2 < 퐾3), for both European and American options – A call with a low strike price is at least as valuable as an otherwise identical call with higher strike price: 퐶 퐾1 ≥ 퐶(퐾2) – A put with a high strike price is at least as valuable as an otherwise identical put with low strike price: 푃 퐾2 ≥ 푃 퐾1 – The premium difference between otherwise identical calls with different strike prices cannot be greater than the difference in strike prices: 퐶 퐾1 − 퐶 퐾2 ≤ 퐾2 − 퐾1 • Price of a collar is not greater than its maximum payoff

K2 – K1 S FIN501 Asset Pricing Lecture 03 One Period Model: Pricing (22)

Options: Strike price (cont.)

• Different strike prices (퐾1 < 퐾2 < 퐾3), 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: 푃 퐾2 − 푃 퐾1 ≤ 퐾2 − 퐾1

– Premiums decline at a decreasing rate for calls with progressively higher strike prices. (Convexity of option price with respect to strike price): 퐶 퐾 − 퐶 퐾 퐶 퐾 − 퐶 퐾 1 2 < 2 3 퐾1 − 퐾2 퐾2 − 퐾3 FIN501 Asset Pricing Lecture 03 One Period Model: Pricing (23) Options: Strike price

• Proof: suppose to the contrary 퐶 퐾 − 퐶 퐾 퐶 퐾 − 퐶 퐾 1 2 ≤ 2 3 퐾2 − 퐾1 퐾3 − 퐾2 • (Asymmetric) Butterfly spread – Price ≤ 0: 1 1 1 1 퐶 퐾1 − + 퐶 퐾2 + 퐶 퐾3 ≤ 0 퐾2−퐾1 퐾2−퐾1 퐾3−퐾2 퐾3−퐾2

– Payoff > 0: (at least in some states of the world)

– ⇒ arbitrage 퐾1 퐾2 퐾3 FIN501 Asset Pricing Lecture 03 One Period Model: Pricing (24) Overview: Pricing - one period model

• 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 FIN501 Asset Pricing Lecture 03 One Period Model: Pricing (26)

specify observe/specify Preferences & existing Technology Asset Prices

• evolution of states NAC/LOOP NAC/LOOP • risk preferences • aggregation

State Prices q absolute (or stochastic discount relative asset pricing factor/Martingale measure) asset pricing

LOOP

derive derive Asset Prices Price for (new) asset

Only works as long as market completeness doesn’t change FIN501 Asset Pricing Lecture 03 One Period Model: Pricing (27) Three Forms of No-ARBITRAGE

• Define for each 푧 ∈ 푋 푣 푧 ≔ 푝 ⋅ ℎ: 푧 = 푋ℎ • If LOOP holds 푣 푧 is a linear functional – Single-valued, because if h’ and h’ lead to same z, then price has to be the same – Linear on 푋 – 푣 0 = 0 • Conversely, if 푣 is a linear functional defined in 푋 then the law of one price holds. FIN501 Asset Pricing Lecture 03 One Period Model: Pricing (29) Pricing

• LOOP ⇒ 푣 푋ℎ = 푝 ⋅ ℎ • A linear functional 푉 ∈ ℝ푆 is a valuation function if 푉 푧 = 푣 푧 for each 푧 ∈ 푋 푆 푠 • 푉 푧 = 푞 ⋅ 푧 for some 푞 ∈ ℝ , where 푞 = 푉 푒푠 , 푠 푖 and 푒푠 is the vector with 푒푠 = 1 and 푒푠 = 0 if 푖 ≠ 푠 – 푒푠 is an Arrow-Debreu security • 푞 is a vector of state prices • 푉 extends 푣 on ℝ푆 FIN501 Asset Pricing Lecture 03 One Period Model: Pricing (30) State prices q

• 푞 is a vector of state prices if 푝 = 푋′푞, that is 푝푗 = 푥푗 ⋅ 푞 for each 푗 = 1, … , 퐽 • If 푉 푧 = 푞 ⋅ 푧 is a valuation functional then 푞 is a vector of state prices • Suppose 푞 is a vector of state prices and LOOP holds. Then if 푧 = 푋ℎ LOOP implies that

푣 푧 = ℎ푗푝푗 푗 푗 푗 푗 푗 = 푥푠 푞푠 ℎ = 푥푠 ℎ 푞푠 = 푞 ⋅ 푧 푗 푠 푠 푗 • 푉 푧 = 푞 ⋅ 푧 is a valuation functional ⇔ 푞 is a vector of state prices and LOOP holds FIN501 Asset Pricing Lecture 03 One Period Model: Pricing (31) State prices q

푝 1,1 = 푞1 + 푞2 푝 2,1 = 2푞1 + 푞2

Value of portfolio (1,2) 푥2 1 3푝 1,1 − 푝 2,1 = 푞1 + 2푞2 2

2 1

푥1 FIN501 Asset Pricing Lecture 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 푞 ≫ 0 • Proposition 1’. Let 푋 be a S × 퐽 matrix, and 푝 ∈ ℝ퐽. There is no ℎ in ℝ퐽 satisfying ℎ ⋅ 푝 ≤ 0, 푋ℎ ≥ 0 and at least one strict inequality ⇔ there exists a vector 푞 ∈ ℝ푆 with 푞 ≫ 0 and 푝 = 푋′푞

No arbitrage positive state prices , FIN501 Asset Pricing Lecture 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 푞∗ 7. Four pricing formulas: state prices, SDF, EMM, beta pricing 8. Recovering state prices from options FIN501 Asset Pricing Lecture 03 One Period Model: Pricing (34) Multiple State Prices 푞 & Incomplete Markets bond (1,1) only What state prices are consistent 푥2 Payoff space ⟨푋⟩ with 푝 1,1 ? 푝 1,1 = 푞1 + 푞2 One equation – two unknowns 푞1, 푞2 There are (infinitely) many. e.g. if 푝 1,1 = .9 푞1 = .45, 푞2 = .45, 푝 1,1 or 푞1 = .35, 푞2 = .55 푞2

푞1 푥1 FIN501 Asset Pricing Lecture 03 One Period Model: Pricing (35)

푉(푥)

푥2 complete markets

⟨푋⟩ 푞

푥1 FIN501 Asset Pricing Lecture 03 One Period Model: Pricing (36)

푝 = 푋′푞 푉(푥)

푥2 ⟨푋⟩ incomplete markets

푞

푥1 FIN501 Asset Pricing Lecture 03 One Period Model: Pricing (37)

푝 = 푋′푞∘ 푉(푥)

푥2 ⟨푋⟩ incomplete markets

푞∘

푥1 FIN501 Asset Pricing Lecture 03 One Period Model: Pricing (38) Multiple q in incomplete markets

푥2 ⟨푋⟩

푝 = 푋′푞 푞∗

푞∘ v 푞

푥1

Many possible state price vectors s.t. 푝 = 푋′푞. One is special: 푞∗ - it can be replicated as a portfolio. FIN501 Asset Pricing Lecture 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 푣 ∈ ℝ푆 with 0 = 푋푣 • Suppose there is no arbitrage and let 푞 ≫ 0 be a vector of state prices. Then 푞 + 훼푣 ≫ 0 provided 훼 is small enough, and 푝 = 푋 푞 + 훼푣 . Hence, there are an infinite number of strictly positive state prices. FIN501 Asset Pricing Lecture 03 One Period Model: Pricing (40) Overview: Pricing - one period model

푗 푗 1. State prices 푝 = 푠 푞푠푥푠 2. Stochastic discount factor 푝푗 = 퐸 푚푥푗 푚 푗 1 푥1 푚2 푗 푥2 푚3 푥푗 1 3 3. Martingale measure 푝푗 = 퐸휋 푥푗 1+푟푓 (reflect risk aversion by over(under)weighing the “bad(good)” states!) 4. State-price beta model 퐸 푅푗 − 푅퐹 = 훽푗퐸 푅∗ − 푅푓

푥푗 (in returns 푅푗 ≔ ) 푝푗 FIN501 Asset Pricing Lecture 03 One Period Model: Pricing (42) 1. State Price Model

• … so far price in terms of Arrow-Debreu (state) prices

푗 푗 푝 = 푞푠푥푠 푠 FIN501 Asset Pricing Lecture 03 One Period Model: Pricing (43) 2. Stochastic Discount Factor

푗 푗 푞푠 푗 푝 = 푞푠푥푠 = 휋푠 푥푠 휋푠 푠 푠

푞푠 • That is, stochastic discount factor 푚푠 ≔ 휋푠 푝푗 = 퐸 푚푥푗

Now, probability inner product between 푚 and 푥푗 FIN501 Asset Pricing Lecture 03 One Period Model: Pricing (44) 2. Stochastic Discount Factor

푐2 휋2 shrink axes by factor 휋푠

⟨푋⟩

푚∗

푚 푐1 휋1

With m: Probability inner product = 0 (“probability orthogonal”) FIN501 Asset Pricing Lecture 03 One Period Model: Pricing (45) Risk-adjustment in payoffs

푝 = 퐸 푚푥 = 퐸 푚 퐸 푥 + cov 푚, 푥

1 1 Since 푝bond = 퐸 푚 × 1 , the risk free rate = = 퐸 푚 . 1+푟푓 푅푓

푬 풙 풑 = + cov 풎, 풙 푹풇 Remarks: (i) If risk-free rate does not exist, 푅푓 is the shadow risk free rate (ii) Typically cov 푚, 푥 < 0, which lowers price and increases return FIN501 Asset Pricing Lecture 03 One Period Model: Pricing (46) 3. Equivalent Martingale Measure

푗 푗 • Price of any asset 푝 = 푠 푞푠푥푠 1 • Price of a bond 푝bond = 푞 = 푠 푠 1+푟푓

푗 1 푞푠 푗 1 푗 푝 = 푓 푥푠 = 푓 퐸휋 푥 1 + 푟 ′ 푞푠′ 1 + 푟 푠 푠 푞 where 휋 ≔ 푠 푠 푞 푠′ 푠′ FIN501 Asset Pricing Lecture 03 One Period Model: Pricing (47) 푥푗 … in Returns: 푅푗 = 푝푗

퐸 푚푅푗 = 1, 푅푓퐸 푚 = 1 ⇒ 퐸 푚 푅푗 − 푅푓 = 0

퐸 푚 퐸 푅푗 − 푅푓 + cov 푚, 푅푗 = 0 cov 푚, 푅푗 ⇒ 퐸 푅푗 − 푅푓 = − 퐸 푚 (also holds for portfolios ℎ) 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 Pricing Lecture 03 One Period Model: Pricing (48) 4. State-price BETA Model

푐2 휋2 shrink axes by factor 휋푠

⟨푋⟩ let underlying asset be 푥 = 1.2,1 ∗ 푅 푅∗ = 훼푚∗ 푚∗

푚 푐1 휋2

p=1 (priced with m*)

With m: Probability inner product = 0 (“probability orthogonal”) FIN501 Asset Pricing Lecture 03 One Period Model: Pricing (49) 4. State-price BETA Model

cov 푚, 푅푗 퐸 푅푗 − 푅푓 = − 퐸 푚 (also holds for all portfolios ℎ, we can replace m with 푚∗)

Suppose (i) var 푚∗ > 0 and (ii) 푅∗ = 훼푚∗ with 훼 > 0 cov 푅∗, 푅ℎ 퐸 푅ℎ − 푅푓 = − 퐸 푅∗

cov 푅∗,푅ℎ Define 훽ℎ ≔ for any portfolio ℎ var 푅∗ FIN501 Asset Pricing Lecture 03 One Period Model: Pricing (50) 4. State-price BETA Model

cov 푅∗,푅ℎ var 푅∗ (2) for 푅ℎ: 퐸 푅ℎ − 푅푓 = − = −훽ℎ 퐸 푅∗ 퐸 푅∗

cov 푅∗,푅∗ var 푅∗ (2) for 푅∗: 퐸 푅∗ − 푅푓 = − = − 퐸 푅∗ 퐸 푅∗

Hence, 푬 푹풉 − 푹풇 = 휷풉푬 푹∗ − 푹풇 cov 푹∗,푹풉 where 휷풉 ≔ var 푹∗

ℎ ℎ ℎ ∗ ∗ Regression 푅푠 = 훼 + 훽 푅 푠 + 휀푠 with cov 푅 , 휀 = 퐸 휀 = 0 very general – but what is R* in reality? FIN501 Asset Pricing Lecture 03 One Period Model: Pricing (51) Four Asset Pricing Formulas

푗 푗 1. State prices 푝 = 푠 푞푠푥푠 2. Stochastic discount factor 푝푗 = 퐸 푚푥푗 푚 푗 1 푥1 푚2 푗 푥2 푚3 푥푗 1 3 3. Martingale measure 푝푗 = 퐸휋 [푥푗 ] 1+푟푓 (reflect risk aversion by over(under)weighing the “bad(good)” states!) 4. State-price beta model 퐸 푅푗 − 푅퐹 = 훽푗퐸 푅∗ − 푅푓

푥푗 (in returns 푅푗 ≔ ) 푝푗 FIN501 Asset Pricing Lecture 03 One Period Model: Pricing (52) What do we know about 푞, 푚, 휋 , 푅∗?

• Main results so far – Existence ⇔ no arbitrage • Hence, single factor only

• But doesn’t famous Fama-French factor model have 3 factors?

• Additional factors are due to time-variation (wait for multi-period model) – Uniqueness if markets are complete FIN501 Asset Pricing Lecture 03 One Period Model: Pricing (53) Different Asset Pricing Models

ℎ 푓 ℎ ∗ 푓 푝푡 = 퐸 푚푡+1푥푡+1 ⇒ 퐸 푅 − 푅 = 훽 퐸 푅 − 푅 cov 푅∗,푅ℎ where 푚 = 푓 … and 훽ℎ = 푡+1 var 푅∗ 풇 … = asset pricing model General Equilibrium MRS 푓 … = 휋 Factor Pricing Model 푎 + 푏1푓1,푡+1 + 푏2푓2,푡+1 CAPM CAPM 푀 푀 ∗ 푓 푎+푏1푅 푎 + 푏1푓1,푡+1 = 푎 + 푏1푅 푅 = 푅 푓 푎+푏1푅 where 푅푚 is market return is 푏1 ≷ 0? FIN501 Asset Pricing Lecture 03 One Period Model: Pricing (54) Different Asset Pricing Models

• Theory – All economics and modeling is determined by ′ 푚푡+1 = 푎 + 풃 풇 – Entire content of model lies in restriction of SDF • Empirics – 푚∗ (which is a portfolio payoff) prices as well as m (which is e.g. a function of income, investment etc.) – measurement error of 푚∗ is smaller than for any 푚 – Run regression on returns (portfolio payoffs)! (e.g. Fama-French three factor model) FIN501 Asset Pricing Lecture 03 One Period Model: Pricing (55) Overview: Pricing - one period model

specify observe/specify Preferences & existing Technology Asset Prices

• evolution of states NAC/LOOP NAC/LOOP • risk preferences • aggregation

State Prices q absolute (or stochastic discount relative asset pricing factor/Martingale measure) asset pricing

LOOP

derive derive Asset Prices Price for (new) asset

Only works as long as market completeness doesn’t change FIN501 Asset Pricing Lecture 03 One Period Model: Pricing (57) Recovering State Prices from Option Prices

• Suppose that 푆푇, the price of the underlying portfolio (we may think of it as a proxy for price of “market portfolio”), assumes a "continuum" of possible values. • Suppose there are a “continuum” of call options with different strike/exercise prices ⇒ markets are complete • Let us construct the following portfolio: for some small positive number 휀 > 0 훿 – Buy one call with 퐾 = 푆 − − 휀 푇 2 훿 – Sell one call with 퐾 = 푆 − 푇 2 훿 – Sell one call with 퐾 = 푆 + 푇 2 훿 – Buy one call with 퐾 = 푆 + + 휀 푇 2 FIN501 Asset Pricing Lecture 03 One Period Model: Pricing (58) Recovering State Prices … (ctd)

Payoff of the portfolio

휀

훿 훿 훿 훿 푆푇 − − 휀 푆 − 푆푇 + 푆푇 + + 휀 2 푇 2 푆푇 2 2 FIN501 Asset Pricing Lecture 03 One Period Model: Pricing (59) Recovering State Prices … (ctd)

1 • Let us thus consider buying units of the portfolio. 휀 훿 훿 1 • The total payment, when 푆 − ≤ 푆 ≤ 푆 + is 휀 ⋅ = 1, for any 휀 푇 2 푇 푇 2 휀 훿 훿 • Letting 휀 → 0 eliminates payments in the regions 푆 ∈ 푆 − − 휀, 푆 − 푇 푇 2 푇 2 훿 훿 and 푆 ∈ 푆 + , 푆 + + 휀 푇 푇 2 푇 2

1 • The value of units of this portfolio is 휀 1 훿 훿 퐶 푆, 퐾 = 푆 − − 휀 − 퐶 푆, 퐾 = 푆 − 휀 푇 2 푇 2 FIN501 Asset Pricing Lecture 03 One Period Model: Pricing (60) Recovering State Prices … (ctd)

• Taking the limit 휀 → 0 훿 훿 훿 훿 퐶 푆, 퐾 = 푆 − − 퐶 푆, 퐾 = 푆 − − 휀 퐶 푆, 퐾 = 푆 + + 휀 − 퐶 푆, 퐾 = 푆 + 푇 2 푇 2 푇 2 푇 2 = − lim + lim 휀→0 휀 휀→0 휀 훿 훿 휕퐶 푆, 퐾 = 푆 − 휕퐶 푆, 퐾 = 푆 + 푇 2 푇 2 = − + 휕퐾 휕퐾 휕2퐶 as 훿 → 0 we obtain state price density 휕퐾2

푆 푇 − 훿/2 푆푇 푆푇 + 훿/2 FIN501 Asset Pricing Lecture 03 One Period Model: Pricing (61) Recovering State Prices … (ctd.)

• Evaluate the following cash flow

훿 훿 0 푆 ∉ 푆 − , 푆 + 푇 푇 2 푇 2 퐶퐹푇 = 훿 훿 50000 푆 ∈ 푆 − , 푆 + 푇 푇 2 푇 2

• Value of this cash flow today 휕퐶 훿 휕퐶 훿 50000 푆, 퐾 = 푆 + − 푆, 퐾 = 푆 − 휕퐾 푇 2 휕퐾 푇 2 휕퐶 휕퐶 푞 푆1, 푆2 = 푆, 퐾 = 푆1 − 푆, 퐾 = 푆2 푇 푇 휕퐾 푇 휕퐾 푇 FIN501 Asset Pricing Lecture 03 One Period Model: Pricing (62)

Table 8.1 Pricing an Arrow-Debreu State Claim

E C(S,E) Cost of Payoff if ST = position 7 8 9 10 11 12 13 ∆C ∆ (∆C)= 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 Δ퐾 = 1 FIN501 Asset Pricing Lecture 03 One Period Model: Pricing (63)

specify observe/specify Preferences & existing Technology Asset Prices

•evolution of states NAC/LOOP NAC/LOOP •risk preferences •aggregation

State Prices q absolute (or stochastic discount relative asset pricing factor/Martingale measure) asset pricing

LOOP

derive derive Asset Prices Price for (new) asset

Only works as long as market completeness doesn’t change