Asset Pricing

Lecture 3: OneOne--periodperiod Model Pricing

Prof. Markus K. Brunnermeier

Slide 0303--11 Fin 501: Asset Pricing OiPiiOverview: 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 comppqpleteness and uniqueness of state prices 6. Pricing kernel q* 7. Four pricing formulas: state prices , SDF, EMM, beta pricing [L2L2356],3,5,6] 8. Recovering state prices from options [DD10.6]

Slide 0303--22 Fin 501: Asset Pricing VtNttiVector Notation

• Notation: yxy,x ∈ Rn –y ≥ x ⇔ yi ≥ xi for each i=1,…,n. – y>xy > x ⇔ y ≥ x and y ≠ x. – y >> x ⇔ yi > xi for each i=1,…,n. • Inner product

–y ≤ x = ∑i yx • Matrix multiplication

Slide 0303--33 Fin 501: Asset Pricing

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 AtPiAsset Prices PiPrice for (new ) asse t

Only works as long asSlide market 0303--44 completeness doesn’t change Fin 501: Asset Pricing Three Forms of N oo--ARBITRAGE

1. 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 There exists no strong arbitrage nor portfolio k with k’k X > 0 and 0 ≥ p ≤ k

Slide 0303--55 Fin 501: Asset Pricing Three Forms of N oo--ARBITR AGE

• Law of one price is equivalent to everyyp portfolio with zero pa yoff has zero price. • No arbitrage => no strong arbitrage No strong arbitrage => law of one price

Slide 0303--66 Fin 501: Asset Pricing OiPiiOverview: Pricing

1. LOOP, No arbitrage 2. Forwards 3. Options: Parity relationship 4. No arbitrage and existence of state prices 5. Market comppqpleteness and uniqueness of state prices 6. Pricing kernel q* 7. Four pricing formulas: state prices , SDF , EMM , beta pricing 8. Recovering state prices from options

Slide 0303--77 Fin 501: Asset Pricing Alternati ve ways t o b uy a st ock • 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 0303--88 Fin 501: Asset Pricing PiPric ing prepa idfid forwar ds

• If we can price the prepaid forward (FP), then we can calculate the price for a forward contract: F = FtFuture val ue of FP • 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 P – F 0, T = S0 (where the asset is bought at t = 0, delivered at t = T)

Slide 0303--99 Fin 501: Asset Pricing

PiPric ing prepa idfid forwar ds (cont.) • Pricinggy by arbitra ge – If at time t=0, the prepaid forward price somehow exceeded P the stock price, i.e., F 0, T > S0 , an arbitrageur could do the following:

Slide 0303--1010 Fin 501: Asset Pricing

PiPric ing prepa idfid forwar ds (cont.)

P • What if there are deterministic* dividends? Is F 0, T = S0 still valid? – No, because the holder of the forward will not receive dividends that will be P paid to the holder of the stock Î F 0, T < S0

P F 0, T = S0 – PV(all dividends paid from t=0 to t=T)

– For discrete dividends Dti at times ti, i = 1,…., n P n • The prepaid forward price: F 0, T = S0 – Σ i=1PV0, ti (Dti) (reinvest the dividend at risk-free rate) – For continuous dividends with an annualized yield δ P −δT • The prepaid forward price: F 0, T = S0 e −δT (reinvest the dividend in this index. One has to invest only S0 e initially)

* NB 1: if dividends are stochastic, we cannot apply the one period model

Slide 0303--1111 Fin 501: Asset Pricing

PiPric ing prepa idfid forwar ds (cont.)

• Example 5.1 – XYZ stock costs $100 today and will pay a quarterly dividend of $1.25 . If the risk-free rate is 10% compounded continuously, how much does a 1-year prepaid forward cost?

P 4 −0.025i – F 0, 1 = $100 – Σ i=1$1.25e = $95.30 • Example 5.2 – The index is $125 and the dividend yield is 3% continuously compounded. How much does a 1-yyppear prepaid forward cost? P −0.03 – F 0,1 = $125e = $121.31

Slide 0303--1212 Fin 501: Asset Pricing PPiric ing forwar ds on s toc k

• Forward price is the future value of the prepaid forward P rT – No dividends: F0, T = FV(F 0, T ) = FV(S0) = S0 e n rT r(T-ti) – Discrete dividends: F0, T = S0 e – Σ i=1 e Dti (r-δ)T – Continuous dividends: F0, T = S0 e • Forward premium – The difference between current forward price and stock price – Can be used to infer the current stock price from forward price – Definition:

• Forward premium = F0, T / S0

a π T • Annualized forward premium =: π = (1/T) ln (F0, T / S0) (from e =F0,T / S0 )

Slide 0303--1313 Fin 501: Asset Pricing CtiCreating a syntheti c ffdorward • One can offset the risk of a forward by creating a synthetic fdffiiihlfdforward 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, T – BdhhfllBorrow and purchase shares as follows:

– Note that the total payoff at expiration is same as forward payoff

Slide 0303--1414 Fin 501: Asset Pricing Creating a synthetic forward (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

Tbl56Table 5.6

Slide 0303--1515 Fin 501: Asset Pricing OthiOther issues ifin forwar diid pricing

• Does the forward price predict the future price? (r-δ)T – According the formula F0, T = S0 e the forward price conveys no additional information beyond what S0 , 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 0303--1616 Fin 501: Asset Pricing OiPiiOverview: Pricing

Slide 0303--1717 Fin 501: Asset Pricing PtPut-CllPitCall Parity

• For European options with the same strike price and time to expiration the parity relationship is: Call – put = PV (forward price – strike price) or

-rT C(K, T) – P(K, T) = PV0,T (F0,T – K) = e (F0,T – K) • Intuition: – Buying a call and selling a put with the strike equal to the forward

price (F0,T = K) creates a synthetic forward contract and hence must have a zero price.

Slide 0303--1818 Fin 501: Asset Pricing PitfParity for O Otiptions on St Stkocks

• If underlying asset is a stock and Div is the -rT deterministic* dividend stream, then e F0,T = S0 – PV0,T (Div), therefore

-rT C(K, T) = P(K, T) + [S0 – PV0,T (Div)] – e (K) • Rewriting above,

-rT S0 = C(K, T) – P(K, T) + PV0,T (DiDiv) + e (K)

-δT • For index options, S0 – PV0,T (Div) = S0e , therefore

-δT C(K, T)=) = P(K, T)+) + S0e – PV0,T (K)

*a llows t o sti ck with one peri od setti ng

Slide 0303--1919 Fin 501: Asset Pricing OtiOption pri ce boun dar ies • 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 Europppean option:

CAmer(S, K, T) > CEur(S, K, T)

PAmer(S, K, T) > PEur(S, K, T) • 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:

S>S > CAmer(S, K, T) > CEur(S, K, T)>) > > max [0, PV0,T(F0,T) – PV0,T(K)]

Slide 0303--2020 Fin 501: Asset Pricing

OtiOption pri ce boun der ies (cont.)

• 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:

S > CAmer(S, K, T) > CEur(S, K, T) > max [0, PV0,T(F0,T) – PV0,T(K)]

– Put price 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(S, K, T) > max [0, PV0,T (K) – PV0,T(F0,T)]

Slide 0303--2121 Fin 501: Asset Pricing EElarly exerc ise o fAf Amer ican ca ll

• Early exercise of American options – A non-dividend paying American call option should not be exercised early, because: -r(T-t) CAmer > CEur > St – K + PEur+K(1-e ) > St – K – That 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 0303--2222 Fin 501: Asset Pricing OtiOptions: Ti me t o expi rati on

• 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 and European puts may be less valuable than an otherwise identical option with less time to expiration. – A European call option on a non-dividend paying stock will be more valuable than an otherwise identical option with less time to expiration. • Strike price does not grow at the interest rate. – 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 P(T) < P(t) for T>t, then arbitrage. Buy P(T) and sell P(t) initially. At t

–if St>Kt, P(t)=0. – if S t

• Different strik e pri ces ( K1 < K2 < K3)f), for b ot hEh European an d American options – A call with a low strike price is at least as valuable as an otherwise identical call with higgpher 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(K ) – C(K ) < K – K 1 2 2 1 K2 –K1 Price of a collar < maximum payoff of a collar S

Note for K3-K2 more pronounced, hence (next slide) …!

Slide 0303--2424 Fin 501: Asset Pricing

OtiOptions: St Stikrike pri ce (cont.)

• Different str ike pr ices (K1 < K2 < K3)f), for b ot h European and American options – The premium difference between otherwise identical puts with diff erent st rik e pr ices canno t be grea ter than the difference in strike prices:

P(K1) – P(K2) < K2 – K1 – Premiums decline at a decreasing rate for calls w ith progressively higher strike prices. (Convexity of option price with respect to strike price):

C(K1) – C(K2) C(K2) – C(K3)

K1 – K2 K2 – K3

Slide 0303--2525 Fin 501: Asset Pricing OtiOptions: St Stikrike pri ce

Slide 0303--2626 Fin 501: Asset Pricing

PtiftiiProperties of option prices (cont.)

Slide 0303--2727 Fin 501: Asset Pricing Summ ar y of parit y relationships

Slide 0303--2828 Fin 501: Asset Pricing OOevervie w: Pri cin g - on e peri od model 1. LOOP, No arbitrage 2. Forwards 3. Options: Parity relationship 4. No arbitrage and existence of state prices 5. Market comppqpleteness and uniqueness of state prices 6. Pricing kernel q* 7. Four pricing formulas: state prices , SDF , EMM , beta pricing 8. Recovering state prices from options

Slide 0303--2929 Fin 501: Asset Pricing … bktthbiitback to the big picture

• 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

Slide 0303--3030 Fin 501: Asset Pricing

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 AtPiAsset Prices PiPrice for (new ) asse t

Only works as longSlide as market 0303--3131 completeness doesn’t change Fin 501: Asset Pricing Three Forms of N oo--ARBITRAGE

Slide 0303--3232 Fin 501: Asset Pricing PiPric ing

• Define for each z ∈ ,

• If LOOP h old s q()(z) iilis a single-valdlued an d linear 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 then the law of one price holds.

Slide 0303--3333 Fin 501: Asset Pricing PiPric ing

• LOOP q(hhX’X)=) = php .h ⇒ • A linear functional Q in RS is a valuation function if Q(z) = q(z) for each z . ∈ · • Q(z) = q .z for some q RS, where qs = Q(e ), and e ∈ s s s i is the vector with es = 1 and es = 0 if i ≠ s

– es is an Arrow-Debreu security • q is a vector of state prices

Slide 0303--3434 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 jj,, = 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 = h’X LOOP implies that

• Q(z) = q ≤ z is a valuation functional iff q is a vector of state prices and LOOP holds

Slide 0303--3535 Fin 501: Asset Pricing State prices q

p(1,1) = q1 + q2 p(2,1) = 2q1 + q2 Value of portfolio (1 ,2) c2 3p(1,1) – p(2,1) = 3q1 +3q2-2q1-q2 = q1 + 2q2 q2

c1 q1

Slide 0303--3636 Fin 501: Asset Pricing Theeudaetaeoeo Fundamental Theorem of Finance • Proposition 1. Security prices exclude arbitrage if and only if there exists a valuation functional with q>>0q >> 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 only if, there exists a vector q RS with ∈ q >> 0 and p = X q. No arbitrage ⇔ positive state prices

Slide 0303--3737 Fin 501: Asset Pricing OiPiiOverview: Pricing

Slide 0303--3838 Fin 501: Asset Pricing Multiple State Prices q & Incomplete Markets bond (1,1) only What state prices are consistent with p(1,1)? c2 p(1, 1) = q 1 +q+ q2 Payoff space

One equation – two unknowns q1, q2 There are (infinitely) many. e.g. if p(1,1)=.9 p(1,1) q1 =.45,,q q2 =.45 q2 or q1 =.35, q2 =.55

c1 q1

Slide 0303--3939 Fin 501: Asset Pricing

Q(x)

x2 complete markets

Slide 0303--4040 Fin 501: Asset Pricing

p=Xq Q(x)

x2 incomplete markets

Slide 0303--4141 Fin 501: Asset Pricing

p=Xqo Q(x)

x2 incomplete markets

Slide 0303--4242 Fin 501: Asset Pricing Muupeqltiple q in in copeecomplete markets c2

p=X’q q* qo v q

Many possible state price vectors s .t . p=X’q . One is special: q* - it can be replicated as a portfolio. Slide 0303--4343 Fin 501: Asset Pricing Uni quen ess an d Completeness

• Proposition 2. If markets are complete, under no arbitrage there exists a unique valuation functional.

• If markets are not complete, then there exists v i\ RS with 0 = X’v. Suppose there is no arbitrage and let q >> 0 be a vector of state prices. Then q + α v >> 0 provided α is small enough, and p =X= X’ (q + α v). Hence , there are an infinite number of strictly positive state prices.

Slide 0303--4444 Fin 501: Asset Pricing OOevervie w: Pri cin g - on e peri od model 1. LOOP, No arbitrage 2. Forwards 3. Options: Parity relationship 4. No arbitrage and existence of state prices 5. Market comppqpleteness and uniqueness of state prices 6. Pricing kernel q* 7. Four pricing formulas: state prices , SDF , EMM , beta pricing 8. Recovering state prices from options

Slide 0303--4545 Fin 501: Asset Pricing FAtPiiFFour Asset Pricing Formul as j j 1. State prices p = ∑s qs xs 2. Stochastic discount factorfactorpj = E[mxj]

j m1 x 1 m j 2 x 2 m 3 j x 3

j f ^ j 3. Martingale measure p = 1/(1+r ) Eπ [x ] (reflect risk aversion by over(under)weighing the “bad(good)” states!) 4. State-price beta model E[Rj] - Rf = βj E[R*- Rf] (in returns Rj := xj /pj) Slide 0303--4646 Fin 501: Asset Pricing 11SttPi. State Price M Mdlodel

• … so far price in terms of Arrow-Debreu (()pstate) prices j j p = s qsxs

Slide 0303--4747 Fin 501: Asset Pricing 2SthtiDi2. Stochastic Discoun tFtt Factor

• That is, stochastic discount factor ms = qs/πs for all s.

Slide 0303--4848 Fin 501: Asset Pricing 2. Stochastic Discount Factor shrink axes by factor

m c1

Slide 0303--4949 Fin 501: Asset Pricing Ris k--aadjus tmen t i n payo ffs

p = E[mxj]E[]E[]+] = E[m]E[x] + Cov[m,x]

Since pbond=E[m1]. Hence, the risk free rate 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) Typically Cov[m,x] < 0, which lowers price and increases return

Slide 0303--5050 Fin 501: Asset Pricing 3. Equivalent Martingale Measure

• Price of any asset • Price of a bond

Slide 0303--5151 Fin 501: Asset Pricing … iRtin Returns: Rjj =xjj /pjj 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] (2) 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.

Slide 0303--5252 Fin 501: Asset Pricing 4. State-price BETA Model shrink axes by factor

let underlying asset be x=(1.2,1)

R* R*=α m* m*

m c1

p=1 (priced with m*)

Slide 0303--5353 Fin 501: Asset Pricing 4Stt4. Statee--prprice BETA M od e 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

Slide 0303--5454 Fin 501: Asset Pricing 4Stt4. Statee--prprice BETA M od e l

(2) for Rh: E[Rh]-Rf=-Cov[R*,Rh]/E[R*] = - βh Var[R*]/E[R*] (2) for R *: E[R*]-Rf=-Cov[R*,R*]/E[R*] =-Var[R*]/E[R*] Hence, E[E[RRh] -R- Rf = βh E[R*--RRf] where βh := Cov[R*,Rh]/Var[R*] very general – but what is R* in reality?

h h h * * Regression R s = α + β (R )s + εs with Cov[R ,ε]=E[ε]=0

Slide 0303--5555 Fin 501: Asset Pricing FAtPiiFFour Asset Pricing Formul as j 1. State prices 1 = ∑s qs Rs 2. Stochastic discount factorfactor1 = E[mRj]

j m1 x 1 m j 2 x 2 m 3 j x 3

f j 3. Martingale measure 1 = 1/(1+r ) E^π [R ] (reflect risk aversion by over(under)weighing the “bad(good)” states!) 4. State-price beta model E[Rj] - Rf = βj E[R*- Rf] (in returns Rj := xj /pj) Slide 0303--5656 Fin 501: Asset Pricing What d o we k now ab ou 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 factor is due to time-variation (wait for multi-period model) – Uniqueness if markets are complete Slide 0303--5757 Fin 501: Asset Pricing Different A sset P ri ci ng M od el s

h f h * f pt = E[mt+1 xt+1] => E[R ] -R- R = β E[R --RR ] where βh := Cov[R*,Rh]/Var[R*] 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 CAPM * f M f a+b f =a+b= a+b RM R =R (a+b1R ))(/(a+b1R ) 1 1t1,t+1 1 where RM = return of market portfolio Is b1 < 0? Slide 0303--5858 Fin 501: Asset Pricing Different A sset P ri ci ng M od el s • 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-French three factor model)

Slide 0303--5959 Fin 501: Asset Pricing OOevervie w: Pri cin g - on e peri od model 1. LOOP, No arbitrage 2. Forwards 3. Options: Parity relationship 4. No arbitrage and existence of state prices 5. Market comppqpleteness and uniqueness of state prices 6. Pricing kernel q* 7. Four pricing formulas: state prices , SDF , EMM , beta pricing 8. Recovering state prices from options

Slide 0303--6060 Fin 501: Asset Pricing

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 AtPiAsset Prices PiPrice for (new ) asse t

Only works as longSlide as market 0303--6161 completeness doesn’t change Fin 501: Asset Pricing Recoecoeverin gSaeg State Pri ces fr om 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" of possible values. • Suppose there are a “continuum” of call options with different strike/exercise prices ⇒ markets are complete • Let us constr uct the follo wing portfolio: for some small positive number ε>0, ˆ δ – Buy one call with E = ST − 2 − ε ˆ δ – Sell one call with E = ST − 2 ˆ δ – Sell one call with E = ST + 2 – Buy one call with ˆ δ E. = ST + 2 + ε

Slide 0303--6262 Fin 501: Asset Pricing Recover ing St at e P ri ces … ( ctd .)

Slide 0303--6363 Fin 501: Asset Pricing Recover ing St at e P ri ces … ( ctd .)

• Let us thus consider buying 1/ units of the portfolio. The ε 1 total payment, when Sˆ − δ ≤ S ≤ S ˆ + δ , is ε ⋅ ≡ 1 , for T 2 T T 2 ε any choice of ε. We want to let ε 0 , so as to eliminate a δ δ the payments in the ranges S ∈[Sˆ − −ε, Sˆ − ) and T T 2 T 2 ˆ δ ˆ δ 1 ST ∈(ST + , ST + + ε ]. The value of / units of this portfolio 2 2 ε is : 1 C(S, K = Sˆ δ/2 ε) C(S, K = Sˆ δ/2) ε { T − − − T − [C(S, K = Sˆ + δ/2) C(S, K = Sˆ + δ/2+ε)] − T − T }

Slide 0303--6464 Fin 501: Asset Pricing Taking the limit ε→ 0 1 lim C(S, K = SˆT δ/2 ε) C(S, K = SˆT δ/2) [C(S, K = SˆT +δ/2) C(S, K = SˆT +δ/2+ε)] ε 0 → ε { − − − − − − } C(S, K = Sˆ δ/2 ε) C(S, K = Sˆ δ/2) C(S, K = Sˆ + δ/2+ε) C(S, K = Sˆ + δ/2) lim T − − − T − +lim T − T − ε 0{ ε } ε 0{ ε } → − → − 0 0 ≤ ≤ | {z } | {z }

Payoff

Divide by δ and let δ → 0 to obtain state price density as ∂2C/∂ K2. 1

S − δ S S + δ S $ T 2 $ T $ T 2 T

Slide 0303--6565 Fin 501: Asset Pricing Recover ing St at e P ri ces … ( ctd .)

Evaluating following cash flow

The value today of this cash flow is :

Slide 0303--6666 Fin 501: Asset Pricing RiSttPiRecovering State Prices (discrete setti ng)

Slide 0303--6767 Fin 501: Asset Pricing

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 ΔC Δ(ΔC)= 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

Slide 0303--6868 Fin 501: Asset Pricing

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 AtPiAsset Prices PiPrice for (new ) asse t

Only works as longSlide as market 0303--6969 completeness doesn’t change