1 Implications of No Arbitrage, State Prices, and SDF (Notes 4) 1.1 Case 1: Finite State Space Prop 1: Complete markets = ' (state prices) unique ) Prop 2: Incomplete markets = in…nitely-many ' (state prices) ) 9 s Theo 1: No arbitrage strictly positive state price vector' R++ s.t. P = X' () 9 2 s Corr 1: No arbitrage strictly positive SDF m R++ s.t. P = E (mX) () 9 2 s Corr 2: No arbitrage in a complete market unique and strictly positive state price vector ' R++ (and SDF s () 9 2 m R++) 2 s Corr 3: No arbitrage in an incomplete market = in…nitely-many ' (state prices) and SDF m R++ (Not everyone has to be str +, but at least one is str + under no) arbitrage)9 2
1.2 Case 2: General State Space Prop 1: LOOP Linear Price Functional [p (ax + by) = ap(x) + bp(y)] () Prop 2: LOOP basis vector of random payo¤s x is nondegereate (i.e. non-redundant) () ! 1 Prop 3: LOOP = E x x T (i.e. E x x T invertible) ) 9 ! ! ! ! Prop 4: No arbitrage LOOP (But LOOP ; No arbitrage) ) Theo 1: Under frictionless portfolio formation and additional technical assumptions, there is NO ARIBTRAGE a strictly positive m L2 s.t. x X; p(x) = E (mX) () 9 2 8 2 Theo 2: Under frictionless portfolio formation, LOOP holds a SDF x X and it is unique. (so that x X, () 9 2 8 2 p(x) = E (xx))
2 Theo 3: Under LOOP (so that x exists), m L is a SDF (m X) = x 2 () j 2 Mean Variance Frontier and CAPM (Notes 5)
Prop 1: For a mean portfolio return of , the MV portfolio has variance and portfolio weights given by: p A2 2B + C 2 = and ! = g + h p p n 1 p D 1 1 1 1 C !1 B !E A !E B !1 where gn 1 = ; hn 1 = ; D D T 1 T 1 T 1 2 A = !1 !1 ;B = !1 !E;C = !E !E;D = AC B
Prop 2: MV frontier form a convex set. (i.e. if !1;!2 frontier portfolios with 1; 2. Then, R;! !1+(1 ) !2 also frontier.) 8 2 Prop 3: Mean, variance, covariance of the MVP (minimum variance portfolio) is given by: B 2 1 1 2 = ; = ; Cov (Ri;Rmvp) = = i on frontier mvp A mvp A A mvp 8
Prop 4: If pT is the tangency portfolio, then all securities / portfolios can be priced according to the following beta- representation: T Cov Ri;R E (R ) R = E RT R i i f V ar(RT ) f 8 Prop 5: If R.F. asset, then ANY portfolio p on the MV frontier (other than the r.f. asset) can be used to price other assets/portfolios9 according to the following beta-representation:
Cov (Ri;Rp) E (Ri) Rf = [E (Rp) Rf ] i V ar(Rp) 8
1 Prop 6: If no R.F. asset, then s.t. ANY portfolio p on the MV frontier (other than the MV portfolio) can be used to price other assets/portfolios9 according to the following beta-representation:
Cov (Ri;Rp) E (Ri) = [E (Rp) ] i V ar(Rp) 8
and = E (Rzcp) where zcp is the unique portfolio on the MV frontier that has 0 correlation with p
Cov(Ri;Rp) Prop 7: If a portfolio p and a s.t. E (Ri) = [E (Rp) ] i, p must be on the MV frontier. 9 V ar(Rp) 8 )
Cov(Ri;Rp) Prop 7 + 8: A portfolio p on MV frontier E (Ri) = [E (Rp) ] i () V ar(Rp) 8 3 CAPM (The beta-rep model with market portfolio as factor. Or, the model with SDF m = a + bRm) (Notes 6) Theo 1: (CAPM with RF) Suppose a R.F. asset, and suppose CAPM assumptions hold (agents optimize, have full information, and no trading frictions)9 in equilibrium the market portfolio is the tangency portfolio. ) Cov (Ri;Rm) So, E (Ri) Rf = [E (Rm) Rf ] i V ar(Rm) 8
Theo 2: (Zero-Beta CAPM) If no R.F. asset, under CAPM assumptions in equilibrium the market portfolio is on the MV frontier and ) Cov (Ri;Rm) E (Ri) Rzc = [E (Rm) Rzc] i V ar(Rm) 8
4 SDF, Beta Representation, and MV Frontier (Notes 7) 4.1 Beta Representation of Linear Factor Pricing Models T T (E(Ri) = + k where : Ri = ai + fk + "i)
Prop 1: If a factor fk is a gross return on a portfolio (e.g. Rm in CAPM) and R.F. asset k should always be a 9 ) risk-premium on that portfolio (i.e. k = E (Rm) Rf ) Prop 2: If a factor fk is an excess return (e.g. RH RL in FF3) k is always the expected excess return (i.e. ) k = E (fk))
4.2 SDF and Beta Representation m = a + bT f Beta Representation with f as factor () Theo 1: a R & b Rk s.t. m = a + bT f~ is a SDF R & Rk s.t. securities/portfolios i, 9 2T 2 () 9 2 2 8 E (Ri) = + i i 8 1 where ~ Cov f; Ri is the reg coe¤ of Ri on f and a constant i f Prop 3: m = a + bT f~ in beta representation, = E f~ E mf~ where = 1 ) E(m) 4.3 Single-Factor Special Case: CAPM (CAP M m = a + bRm) () Prop 4: CAPM m = am + bmRm for some m; bm R () 2 Moreover,
1 1 E (Rm)(ERm Rf ) 1 ERm Rf if R.F.asset, am = + and bm = 9 Rf Rf V ar (Rm) Rf V ar (Rm)
1 1 E (Rm)(ERm Rzc) 1 ERm Rzc if no R.F.asset, am = + and bm = Rzc Rzc V ar (Rm) Rzc V ar (Rm) Corr 1: (Complete Mkt + CAPM don’tmix!) Suppose market is complete. If CAPM holds and market portfolio return distribution is normal (or has large enough support) arbitrage opportunities. ) 9
2 4.4 Single-Factor Special Case: SDF as Factor Prop 5: Any valid SDF m yields a beta representation with m as factor. 1 V ar (m) Cov (m; Ri) i:e: E (R ) = + i E (m) E (m) V ar (m)
| {z } | {z }| {z } Prop 6: Given any valid SDF m and under LOOP (so that x exists). We can use m-mimicking payo¤ x (m X) as factor to yield a beta representation. j
1 V ar (x) Cov (x;Ri) i:e: E (Ri) = + E (x) E (x) V ar (x) | {z } | {z }| {z } x x Prop 7: Given SDF m and under LOOP (so that x exists). Then, beta representation with R = 2 as 9 p(x) E(x ) factor. 2 E R V ar (R) Cov (R;Ri) i:e: E (R ) = + i E (R ) E (R ) V ar (R ) | {z } | {z }| {z } 4.5 MV and an Orthogonal Characterization of the MV Frontier (Under LOOP so that x* exists)
x x Prop 8: (R*on MV frontier) R = 2 is on the MV frontier bc E (Ri) = i;R [ER ] p(x) E(x ) Prop 9: (Any frontier portf can be a factor) a; b R and a return Rp s.t. m = a + bRp is a SDF Rp is on the MV frontier 9 2 ()
e e Prop10: (Representation of R (1 R )) j
e R a) If Rf ; then R = 1 9 Rf
E (R) 1 e T T b) If no Rf ; then R = (1 X) 2 R where (1 X) E ~x E ~x~x ~x j E (R ) j Prop11: Every Ri can be expressed as e Ri = R + !iR + i e e where ! R; i an excess return with E (i) = 0; and E (RR ) = E (Ri) = E (R i) = 0 2 mv mv e Corr 2: R is on the MV frontier R = R + !R for some ! () x x Prop12: Under LOOP (so that x exists), R = 2 is the MV frontier portfolio with the smallest second p(x) E(x ) moment.
5 Arbitrage Pricing Theory (APT) (Notes 8)
Prop 1 (APT): Suppose all N security returns are described by the following statistical factor decomposition T Ri = ai + f~ + "i i (1) i 8
where E ("i) = E "if~k = Cov "i; f~k = 0 i; k (2) 8 8 2 2 and E ("i"j) = if i=j and 0 o.w. (3) i or in a system form... RN 1 = AN 1 + BN K fK 1 + EN 1 (1’) E (E) = E (Ef) = 0 (2’) T N N E EE is diagonal with + and bdd entries (3’)
3 Then, "arbitrage considerations" alone expected returns of securities "approximately" satisfy beta representation with factors f~ ) T i:e:E (Ri) = + i (4) Moreover, if R.F. asset and f~ are returns, then 9 T E (Ri) = Rf + E f~ Rf i (5) i 8 h i T Ex 1: (Exact pricing with "i = 0) When "i = 0, under APT / no arbitrage, E (Ri) Rf = E (f) Rf~1 i T Ex 2: (Approx pricing with "i = 0) When "i = 0, under APT / no arbitrage, E (Ri) Rf = E (f) Rf~1 6 6 i T Rf Cov (m; "i) = E (f) Rf~1 i Prop 2: (Exact beta-rep at the limit) For a …xed SDF m that prices the factors f~, as V ar ("i) 0 for a security/portfolio T ! i E (Ri) Rf + E (f) Rf p f~ ) ! i Corr 1: (Exact beta-rep at the limit for well-div portfolio) For a …xed SDF m that prices the factors f~, consider a T portfolio p with equilibrium weights in n securities. As n , E (Rp) Rf + E (f) Rf p f~ ! 1 ! i Prop 3: (Approx APT Pricing for all but …nite securities) For a …xed SDF m that prices the factors f~, > 0 small, only …nitely many sucurities s.t. 8 9 T E (Ri) Rf E f~ Rf > i h i
6 Option Pricing (Notes 11) 6.1 13 Important Properties of Options We can derive the following with the following 2 no-arbitrage principles: 1) Portfolios with non-neg payo¤ has non-neg cost. 2) If one portfolio’spayo¤ dominates another then so must its cost. 1. c; C; p; P 0 for all parameter values 2. C (S; t; E;T ) S (t) E and P (S; t; E;T ) E S (t) 3. For T2 > T1;C (S; t; E;T2) C (S; t; E;T1) and P (S; t; E;T2) P (S; t; E;T1) 4. C (:) c (:) and P (:) p (:) given same input values 5. For E1 > E2;all else equal, C (:; E1) C (:; E2) c (:; E1) c (:; E2) P (:; E1) P (:; E2) p (:; E1) p (:; E2) 6. S (t) C (S; t; 0; ) C (S; t; E;T ) c (S; t; E;T ) 1 7. C (0; t; E;T ) = c (0; t; 0; ) = 0 1 8. If the stock pays no dividends between t and T, then c (S; t; E;T ) S E B (t; T ) where B(t,T) is the time t PV of receiving $1 at T.
9. European call options are convex functions of E: i.e. c (:; E1) + (1 ) c (:; E2) c (:; E1 + (1 ) E2) 10. "A portfolio of options is more valuable than an option on a portfolio" (i.e. call option on a portfolio of securities w/ an exercise price of X is less valuable than a portfolio of call options on the underlying with equal exercise) 11. Put-Call Parity (0 dividend stock): p (S; t; E;T ) = c (S; t; E;T ) S (t) + E B (t; T ) 12. European put option price is convex in E: i.e. p (:; E1) + (1 ) p (:; E2) p (:; E1 + (1 ) E2) 13. For a stock paying no dividends over (t,T), an American call will never be exercised early. So, C = c when S pays no dividends.
4 7 Binomial Pricing
We can replicate the derivatives by holding a portfolio of the underlying stock and cash bond (N;B) : fu fd N = and B = fnow N Snow Su Sd We can also use risk-neutral pricing 1 f = E f now Q R tomor 1 = qfu + (1 q) fd R f g
RSnow Sd where q Su Sd
5