Notation: ti y,x R n. y x y i x i for each i=1,,n. y>x y x and y x. y >> x y i > x i for each i=1,,n. y x = i yx

Transcription

1 Lecture 03: One Period Model: Pricing Prof. Markus K. Brunnermeier 10:59 Lecture 02 One Period Model Slide 2-1

2 Overview: Pricing i 1. LOOP, No arbitrage 2. Parity relationship between options 3. No arbitrage and existence of state prices 4. Market completeness and uniqueness of state prices 5. Pricing kernel q * 6. Four pricing formulas (state prices, SDF, EMM, state pricing) 7. Recovering state prices from options 10:59 Lecture 02 One Period Model Slide 2-2

3 Vector Notation ti Notation: ti y,x R n y x y i x i for each i=1,,n. y>x y x and y x. y >> x y i > x i for each i=1,,n. Inner product y x = i yx Matrix multiplication 10:59 Lecture 02 One Period Model Slide 2-3

4 Three Forms of No-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 < No arbitrage There exists no strong arbitrage nor portfolio k with k X >0andp p k 0. 10:59 Lecture 02 One Period Model Slide 2-4

5 Three Forms of No-ARBITRAGE Law of one price is equivalent to every yportfolio with zero payoff has zero price. No arbitrage no strong arbitrage No strong arbitrage law of one price 10:59 Lecture 02 One Period Model Slide 2-5

6 Overview: Pricing i 1. LOOP, No arbitrage 2. Forwards 3. Parity relationship between options 4. No arbitrage and existence of state prices 5. Market completeness and uniqueness of state prices 6. Pricing kernel q * 7. Four pricing formulas (state prices, SDF, EMM, state pricing) 8. Recovering state prices from options 10:59 Lecture 02 One Period Model Slide 2-6

7 Alternative ti 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: Slide 2-7

8 Pii Pricing prepaid idf forwards If we can price the prepaid forward (F P ), then we can calculate the price for a forward contract: F = Future value of fff P 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 F P 0, T = S 0 (where the asset is bought at t = 0, delivered at t = T) Pricing by discounted preset value (α: risk-adjusted discount rate) If expected t=t stock price at t=0ise 0 )thenf P = αt 0 (S T ), F 0, T E 0 (S T ) e Since t=0 expected value of price at t=t is E 0 (S T ) = S 0 e αt Combining the two, F P 0, T = S 0 e αt e αt = S 0 Slide 2-8

9 Pii Pricing prepaid idf forwards (cont.) Pricing by arbitrage If at time t=0, the prepaid forward price somehow exceeded the stock price, i.e., F P 0, T > S 0, an arbitrageur could do the following: Slide 2-9

10 Pii Pricing prepaid idf forwards (cont.) What if there are deterministic* dividends? Is F P 0, T = S 0 still valid? No, because the holder of the forward will not receive dividends id d that t will be paid to the holder of the stock F P 0, T < S 0 F P 0, T = S 0 PV(all dividends paid from t=0 to t=t) For discrete dividends D t i at times t i, i i = 1,., n The prepaid forward price: F P 0, T = S 0 Σ n i=1 PV 0, ti (D ti ) For continuous dividends with an annualized yield δ The prepaid forward price: F P = δt 0, T S 0 e *NB: if dividends are stochatistic, we cannot apply the one period model Slide 2-10

11 Pii Pricing prepaid idf forwards (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? F P 0, 1 = $100 Σ 4 i=1 $1.25e 0.025i = $95.30 Example 5.2 The index is $125 and the dividend yield is 3% continuously compounded. How much does a 1-year prepaid p forward cost? F P 0,1 = $125e 0.03 = $ Slide 2-11

12 Pii Pricing forwards on stock Forward price is the future value of the prepaid forward No dividends: F 0, T = FV(F P 0, T ) = FV(S 0 ) = S 0 e rt n Discrete dividends: F 0, T = S 0 e rt Σ n i=1 er(t-t i) D t i Continuous dividends: F 0, T = S 0 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 Dfiiti Definition: Forward premium = F 0, T / S 0 Annualized forward premium =: π a = (1/T) ln (F 0, T / S 0 ) (from e π T =F 0,T / S 0 ) Slide 2-12

13 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 δ) Recall the long forward payoff at expiration: = S T - F 0, T Borrow and purchase shares as follows: Note that t the total t payoff at expiration is same as forward payoff Slide 2-13

14 Creating a synthetic forward 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 Tbl Table Slide 2-14

15 Other issues in forward pricing i Does the forward price predict the future price? According the formula F 0, T = S 0 e (r-δ)t the forward price conveys no additional information beyond what S 0, 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 2-15

16 Overview: Pricing i - one period model 1. LOOP, No arbitrage 2. Forwards 3. Parity relationship between options 4. No arbitrage and existence of state prices 5. Market completeness 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 10:59 Lecture 02 One Period Model Slide 2-16

17 Put-Call CllP Parity For European options with the same strike price and time to expiration the parity relationship is: Intuition: Call put = PV (forward price strike price) or C(K, T) P(K, T) = PV 0,T (F 0,T K) = e -rt (F 0,T K) Buying i a call and selling a put with the strike equal to the forward price (F 0,T = K) creates a synthetic forward contract and hence must have a zero price. Slide 2-17

18 Parity for Options on Stocks If underlying asset is a stock and Div is the deterministic* dividend stream, then e -rt F 0,T = S 0 PV 0,T (Div), therefore Rewriting above, C(K, T) = P(K, T) + [S 0 PV 0,T (Div)] e -rt (K) S 0 = C(K, T) P(K, T) + PV 0,T (Div) + e -rt (K) For index options, S = -δt 0 PV 0,T (Div) S 0 e, therefore C(K, T) = P(K, T) + S 0 e -δt PV 0,T (K) *allows to stick with one period setting Slide 2-18

19 Properties of option prices 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 i European option: C Amer (S, K, T) > C Eur (S, K, T) Option price boundaries P Amer (S, K, T) > P Eur (S, K, T) Call price cannot: be negative, exceed stock price, be less than price implied by put-call parity using zero for put price: S > C Amer (S, K, T) > C Eur (S, K, T) > > max [0, PV 0,T (F 0,T ) PV 0,T (K)] Slide 2-19

20 Properties of option prices (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 > C Amer (S, K, T) > C Eur (S, K, T) > max [0, PV 0,T (F 0,T ) PV 0,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 > P Amer (S, K, T) > P Eur (S, K, T) > max [0, PV 0T 0,T (K) PV 0T 0,T (F 0T 0,T )] Slide

21 Properties of option prices (cont.) Early exercise of American options A non-dividend paying American call option should not be exercised early, because: C Amer > C Eur > S t K + P Eur +K(1-e -r(t-t) ) > S t 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

22 Properties of option prices (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. European call options on dividend-paying stock and European puts may be less valuable than an otherwise identical option with less time to expiration. AE European call option on a non-dividend id d paying stock will be more valuable than an otherwise identical option with less 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 P(T) < P(t) for T>t, then arbitrage. Buy P(T) and sell P(t). At t if S t >K t, P(t)=0, if S t <K t, payoff S t K t. Keep stock and dfinance K t. Time T-value S T -K t e r(t-t) =S T -K T. Slide

23 Properties of option prices (cont.) Different strike prices (K 1 < K 2 < K 3 ), 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: C(K 1 ) > C(K 2 ) A put with a high strike price is at least as valuable as an otherwise identical call with low strike price: P(K 2 ) > P(K 1 ) The premium difference between otherwise identical calls with different strike prices cannot be greater than the difference in strike prices: C(K 1 ) C(K 2 ) < K 2 K 1 Slide K 2 K 1 S

24 Properties of option prices (cont.) Different strike prices (K 1 < K 2 < K 3 ), 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(K 1 ) P(K 2 ) < K 2 K 1 Premiums decline at a decreasing rate for calls with progressively higher strike prices. (Convexity of option price with respect to strike price): C(K 1 ) C(K 2 ) > C(K 2 ) C(K 3 ) K 2 K 1 K 3 K 2 Slide

25 Properties of option prices (cont.) Slide

26 Properties of option prices (cont.) Slide

27 Summary of parity relationships Slide

28 Overview: Pricing i - one period model 1. LOOP, No arbitrage 2. Forwards 3. Parity relationship between options 4. No arbitrage and existence of state prices 5. Market completeness 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 10:59 Lecture 02 One Period Model Slide

29 back kt 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 10:59 Lecture 02 One Period Model Slide

30 Vector Notation ti Notation: ti y,x R n y x y i x i for each i=1,,n. y>x y x and y x. y >> x y i > x i for each i=1,,n. Inner product y x = i yx Matrix multiplication 10:59 Lecture 02 One Period Model Slide 2-30

31 Three Forms of No-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 < No arbitrage There exists no strong arbitrage nor portfolio k with k X >0andp p k 0. 10:59 Lecture 02 One Period Model Slide 2-31

32 Three Forms of No-ARBITRAGE Law of one price is equivalent to every yportfolio with zero payoff has zero price. No arbitrage no strong arbitrage No strong arbitrage law of one price 10:59 Lecture 02 One Period Model Slide 2-32

33 Pii Pricing Define for each z <X>, If LOOP holds q(z) ()is a single-valued l and 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 <X> then the law of one price holds. 10:59 Lecture 02 One Period Model Slide 2-33

34 Pii Pricing LOOP q(h X) = p h A linear functional Q in R S is a valuation function if Q(z) = q(z) for each z <X>. Q(z) = q z for some q R S, where q s = Q(e s ), and e s is the vector with e ss = 1 and e si = 0 if i s e s is an Arrow-Debreu security q is a vector of state prices 10:59 Lecture 02 One Period Model Slide 2-34

35 State prices q q is a vector of state prices if p = X q, that is p j =x j q for each j=1 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 hxloop 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 2-35

36 State prices q p(1,1) = q 1 + q 2 c 2 p(2,1) = 2q 1 + q 2 Value of portfolio (1,2) 3p(1,1) p(2,1) = 3q 1 +3q 2-2q 1 -q 2 = q 1 + 2q 2 q 2 c q 1 c 1 10:59 Lecture 02 One Period Model Slide 2-36

37 The Fundamental Theorem of Finance 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 R J. There is no h in R J satisfying h p 0, h X 0 and at least one strict inequality if, and only if, there exists a vector q R S with q >> 0 and p = X q. No arbitrage positive state prices 10:59 Lecture 02 One Period Model Slide 2-37

38 Overview: Pricing i - one period model 1. LOOP, No arbitrage 2. Forwards 3. Parity relationship between options 4. No arbitrage and existence of state prices 5. Market completeness and uniqueness of state prices 6. Pricing kernel q * 7. Four pricing formulas (state prices, SDF, EMM) 8. Recovering state prices from options 10:59 Lecture 02 One Period Model Slide 2-38

39 Multiple State Prices q & Incomplete Markets Fin 501: Asset Pricing bond (1,1) only What state prices are consistent with p(1,1)? c 2 p(1,1) 1) = q 1 +q 2 Payoff space <X> q 2 p(1,1) One equation two unknowns q 1, q 2 There are (infinitely) many. e.g. if p(1,1)=.9 q 1 =.45,,q 2 =.45 or q 1 =.35, q 2 =.55 q 1 c 1 10:59 Lecture 02 One Period Model Slide 2-39

40 Q(x) x 2 complete markets q <X> x 1 10:59 Lecture 02 One Period Model Slide 2-40

41 Q(x) p=xq x 2 <X> incomplete markets q x 1 10:59 Lecture 02 One Period Model Slide 2-41

42 Q(x) p=xq o x 2 <X> incomplete markets q o x 1 10:59 Lecture 02 One Period Model Slide 2-42

43 Multiple l q in incomplete markets c 2 <X> q* p=x q v q o q c 1 10:59 Lecture 02 Many possible state price vectors s.t. p=x q q. One is special: q* - it can be replicated as a portfolio. One Period Model Slide 2-43

44 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 v R S with 0 = Xv. Suppose there is no arbitrage and let q >> 0 be a vector of state t prices. Then q + α v >> 0 provided dα is small enough, and p = X (q + α v). Hence, there are an infinite number of strictly positive state prices. 10:59 Lecture 02 One Period Model Slide 2-44

45 Overview: Pricing i - one period model 1. LOOP, No arbitrage 2. Forwards 3. Parity relationship between options 4. No arbitrage and existence of state prices 5. Market completeness and uniqueness of state prices 6. Pricing kernel q * 7. Four pricing formulas (state prices, SDF, EMM) 8. Recovering state prices from options 10:59 Lecture 02 One Period Model Slide 2-45

46 Four Asset Pricing i Formulas 1. State prices p j = j s q s x s 2. Stochastic discount factor p j = E[mx j ] 3. Martingale measure p j = 1/(1+r f ) E π ^ [x j ] (reflect risk aversion by over(under)weighing the bad(good) states!) 4. State-price beta model E[R j ] - R f = β j E[R * - R f ] (in returns R j := x j /p j ) 10:59 Lecture 02 One Period Model m 1 m 2 m 3 x j 1 x j 2 x j 3 Slide 2-46

47 1. 1Stt State Price Pi Model Mdl so far price in terms of Arrow-Debreu (state) prices p j p j = q s s x j s 10:59 Lecture 02 One Period Model Slide 2-47

48 2St 2. Stochastic ti Discount tf Factor That is, stochastic discount factor m s = q s /π s for all s. 10:59 Lecture 02 One Period Model Slide 2-48

49 2. Stochastic Discount Factor shrink axes by factor <X> m* m c 1 10:59 Lecture 02 One Period Model Slide 2-49

50 Risk-adjustment t in payoffs p = E[mx j ] = E[m]E[x] + Cov[m,x] Since 1=E[mR], the risk free rate is R f = 1/E[m] p = E[x]/R f + Cov[m,x] Remarks: (i) If risk-free rate does not exist, R f is the shadow risk free rate (ii) In general Cov[m,x] < 0, which lowers price and increases return 10:59 Lecture 05 State-price Beta Model Slide 2-50

51 3. Equivalent Martingale Measure Price of any asset Price of a bond 10:59 Lecture 02 One Period Model Slide 2-51

52 in Rt Returns: R j =x j /p j E[mR j ]=1 R f E[m]=1 E[m(R j -R f )]=0 E[m]{E[R j ]-R f } + Cov[m,R j ]=0 E[R j ] R f = - Cov[m,R j ]/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. 10:59 Lecture 05 State-price Beta Model Slide 2-52

53 4. State-price BETA Model c 2 shrink axes by factor Fin 501: Asset Pricing <X> m* R* R * =α m * let underlying asset be x=(1.2,1) m c 1 p=1 (priced with m * ) 10:59 Lecture 05 State-price Beta Model Slide 2-53

54 4Stt 4. State-price BETA Model E[R j ] R f = - Cov[m,R j ]/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[R h ] R f = - Cov[R *,R h ]/E[R * ] (2 ) Define β h := Cov[R *,R h ]/ Var[R * ] for any portfolio h 10:59 Lecture 05 State-price Beta Model Slide 2-54

55 4Stt 4. State-price BETA Model (2) for R * : E[R * ]-R f =-Cov[R *,R * ]/E[R * ] =-Var[R * ]/E[R * ] (2) for R h : E[R h ]-R f =-Cov[R *,R h ]/E[R * ] = - β h Var[R * ]/E[R * ] E[R h ] -R f = β h E[R * -R f ] where β h := Cov[R *,R h ]/Var[R * ] very general but what is R * in reality? Regression R h s = α h + β h (R * ) s + ε s with Cov[R *,ε]=e[ε]=0 10:59 Lecture 05 State-price Beta Model Slide 2-55

56 Four Asset Pricing i Formulas 1. State prices 1 = j s q s R s 2. Stochastic discount factor 1 = E[mR j ] 3. Martingale measure 1 = 1/(1+r f ) E π ^ [R j ] (reflect risk aversion by over(under)weighing the bad(good) states!) 4. State-price beta model E[R j ] - R f = β j E[R * - R f ] (in returns R j := x j /p j ) 10:59 Lecture 02 One Period Model m 1 m 2 m 3 x j 1 x j 2 x j 3 Slide 2-56

57 What do we know about 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 l factor is due to time-variation i (wait for multi-period model) Uniqueness if markets are complete 10:59 Lecture 02 One Period Model Slide 2-57

58 Different Asset Pricing i Models p t = E[m t+1 x t+1 ] where m t+1 =f(,, ), f( ) = asset pricing model E[R h ] -R f = β h E[R * -R f ] where β h := Cov[R *,R h ]/Var[R * ] General Equilibrium f( ) = MRS / π Factor Pricing Model a+b 1 f 1,t+1 + b 2 f 2,t+1 CAPM CAPM a+b = M R * =Rf (a+b RM )/(a+b Rf 1 f 1,t+1 a+b 1 R 1 )( 1 ) 10:59 Lecture 05 State-price Beta Model where R M = return of market portfolio Is b 1 < 0? Slide 2-58

59 Different Asset Pricing i Models Theory All economics and modeling is determined by m t+1 = a + b f t+1 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) 10:59 Lecture 05 State-price Beta Model Slide 2-59

60 specify Preferences & Technology evolution of states risk preferences aggregation NAC/LOOP NAC/LOOP observe/specify existing Asset Prices absolute asset pricing State Prices q (or stochastic discount factor/martingale measure) LOOP relative asset pricing derive Asset tprices 10:59 Lecture 02 One Period Model derive Pi Price for (new) asset Only works as long as market Slide 2-60 completeness doesn t change

61 Overview: Pricing i - one period model 1. LOOP, No arbitrage 2. Forwards 3. Parity relationship between options 4. No arbitrage and existence of state prices 5. Market completeness 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 10:59 Lecture 02 One Period Model Slide 2-61

62 Recovering State Prices from Option Prices Suppose that S T, 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 t the following portfolio: for some small positive number ε>0, Buy one call with Sell one call with Sell one call with Buy one call with E. δ E = Ŝ T 2 ˆ δ E = S T 2 δ E = + Ŝ T 2 δ = + Ŝ T 2 ε + ε 10:59 Lecture 02 One Period Model Slide 2-62

63 Recovering State t Prices (ctd.) Payoff C( T S T, E = S ^ T δ 2 ε) C(, E = S ^ T + δ 2 + ε) T S T ε ^ δ2 ε ^ δ 2 S T S T ^ ^ +δ 2 ^ + δ2+ ε S T S T S T S T C( T S T,E = S ^ T δ 2) C(, E = S ^ T + δ 2) T S T 10:59 Lecture 02 Figure 8-2 Payoff Diagram: Portfolio of Options One Period Model Slide 2-63

64 Recovering State t Prices (ctd.) Let us thus consider buying 1 / ε units of the portfolio. The δ δ 1 total payment, py when Ŝ T 2 S T Ŝ T + 2, is ε 1, for ε any choice of ε. Wewanttolet ε a 0,soastoeliminate δ δ the payments in the ranges and S T is : (Ŝ (S T δ +,SŜ 2 T δ + + ε ) 2 S T (Ŝ T ε,ŝ 2.The value of 1 / ε units of this portfolio T ) 2 { [ ]} 1 ( = ˆ ε ) ( = ˆ ) ( = ˆ + ) ( = ˆ + + ε ) δ δ δ δ C S, E Ŝ 2 C S, E ŜT 2 C S, E ŜT 2 C S, E ŜT 2 ε T 10:59 Lecture 02 One Period Model Slide 2-64

65 Taking the limit ε 0 Fin 501: Asset Pricing { ( ˆ ) ( ˆ ) [ ( ˆ ) ( ˆ )] δ δ δ δ C S, E = Ŝ ε C S, E = Ŝ C S, E = Ŝ + C S, E = Ŝ + + ε } 1 lim ε T 2 T 2 T 2 T 2 ε a 0 ( ) ( ) ( ) ( ) δ δ δ δ CS,E = Ŝ ε = = + +ε = + T 2 CS,E ŜT 2 CS,E ŜT 2 CS,E ŜT 2 = lim a0 ε + lim εa0 εa0 a0 0 0 ε Payoff 1 10:59 Lecture 02 S$S δ T 2 One Period Model S $S $S + δ T S T S 2 T Divide by δ and let δ 0 to obtain state price density as 2 C/ E 2. Slide 2-65

66 Recovering State t Prices (ctd.) Evaluating following cash flow [ ] δ δ ŜT 2,ŜT ˆ ˆ [ ] Ŝ,Ŝ +. δ δ ~ 0 if ST 2 CF = + T if ST T 2 T 2 The value today of this cash flow is : 10:59 Lecture 02 One Period Model Slide 2-66

67 Table 8.1 Pricing an Arrow-Debreu State Claim E C(S,E) Cost of Payoff if S T = position ΔC Δ(ΔC)= q s :59 Lecture 02 One Period Model Slide 2-67

68 specify Preferences & Technology evolution of states risk preferences aggregation NAC/LOOP NAC/LOOP observe/specify existing Asset Prices absolute asset pricing State Prices q (or stochastic discount factor/martingale measure) LOOP relative asset pricing derive Asset tprices 10:59 Lecture 02 One Period Model derive Pi Price for (new) asset Only works as long as market Slide 2-68 completeness doesn t change

69 The following is for later lecture 10:59 Lecture 02 One Period Model Slide 2-69

70 Futures contracts t Exchange-traded forward contracts Typical features of futures contracts Standardized, with specified delivery dates, locations, procedures A clearinghouse Matches buy and sell orders Keeps track of members obligations and payments After matching the trades, becomes counterparty Differences from forward contracts Settled daily through the mark-to-market market process low credit risk Highly liquid easier to offset an existing position Highly standardized structure harder to customize Slide 2-70

71 Example: S&P 500 Futures WSJ listing: Contract specifications: Slide 2-71

72 Example: S&P 500 Futures (cont.) Notional value: $250 x Index Cash-settled contract Open interest: total number of buy/sell pairs Margin and mark-to-market 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 2-72

73 Example: S&P 500 Futures (cont.) Mark-to-market proceeds and margin balance for 8 long futures: Slide 2-73

74 Example: S&P 500 Futures (cont.) S&P index arbitrage: comparison of formula prices with actual prices: Slide 2-74

75 Uses of fi index futures Why buy an index futures contract instead of synthesizing it using 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 2-75

76 Uses of fi index futures (cont.) $100 million portfolio with β of 1.4 and r f = 6 % 1. Adjust for difference in $ amount 1 futures contract $250 x 1100 = $275,000 Number of contracts needed $100mill/$0.275mill = Adjust for difference in β x 1.4 = contracts Slide 2-76

77 Uses of fi index futures (cont.) Cross-hedging gwith perfect correlation Cross-hedging with imperfect correlation General asset allocation: futures overlay Risk management for stock-pickers Slide 2-77

78 Currency contracts t Widely used to hedge against changes in exchange rates WSJ listing: Slide 2-78

79 Currency contracts: t pricing i Currency prepaid forward Suppose you want to purchase 1 one year from today using $s F P 0, T = x r 0 e r T y where x 0 is current ($/ ) exchange rate, and r y 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 F 0, T = x 0 e (r r )T y r is the $-denominated domestic interest rate F 0, T > x 0 if r > r y (domestic risk-free rate exceeds foreign risk-free rate) Slide 2-79

80 Currency contracts: t pricing i (cont.) Example 5.3: -denominated interest rate is 2% and current ($/ ) exchange rate is To have 1 in one year one needs to invest today: 0.009/ x 1 x e = $ Example 5.4: -denominated interest rate is 2% and $-denominated rate is 6%. The current ($/ ) exchange rate is The 1-year forward rate: 0.009e = Slide 2-80

81 Currency contracts: t pricing 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 2-81

82 Eurodollar futures WSJ listing Contract specifications Slide 2-82

83 Introduction to Commodity Forwards Fin 501: Asset Pricing Commodity forward prices can be described d by the same formula as that for financial forward prices: F (r δ)t 0,T = S 0 e 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. The lease rate for a commodity can typically be estimated only by observing the forward prices. Slide 2-83

84 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. Note that forward curves can have portions in backwardation and portions in contango. Slide 2-84