Fair games, and the Martingale (or "Random walk") model of stock prices


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1 Economics 236 Spring 2000 Professor Craine Problem Se 2: Fair games, and he Maringale (or "Random walk") model of sock prices Sephen F LeRoy, 989. Efficien Capial Markes and Maringales, J of Economic Lieraure,27, Definiions From saisics: Maringale: E[ x Ω ] Ex = x, + + A sochasic process x is a Maringale wih respec o he sequence of informaion ses, Ω, if, he expecaion of x + (in fac x +, =,2,..) condiional on all currenly available informaion, Ω, equals he curren value.. This says ha x is he opimal predicor of all fuure values of x. Fair Game: E[ y Ω ] E y = 0, + + A sochasic process y is a fair game wih respec o he sequence of informaion ses, Ω, if he condiional expecaion of y + is zero. A Maringale difference sequence, y + = x +  x, is a fair game. From Economics: Fundamenal Value: P = E λ+ i d = i= where, Ω + P is he asse's fundamenal value a ime Technical commen a Maringale has incremens ha are unpredicable, i.e., he expeced value of he incremen is unpredicable. A random walk has independen incremens none of he momens are predicable. The popular "ARCH" class models in finance violae random walks because heir variance is predicable, bu mos of he models are sill Maringales.
2 2 λ + (0,) is he "discoun facor" d + is he asse's flow payoff a + The fundamenal value of an asse is he expeced discouned value of he asse's (flow) payoffs condiional on he available informaion. An operaional definiion of fundamenal value requires a specificaion for he discoun facor. LeRoy uses he popular consan discoun facor model, λ + = λ (0,). Presen Expeced Value: λ + (0) = P = Ed where, eg, P = curren sock price d = nonnegaive random dividend wih a finie variance When he discoun facor is consan, or deerminisic, hen he presen expeced value of he sock's dividend payoffs is he fundamenal value. Seve LeRoy (and Paul Samuelson before him and you and me) recognized ha he Maringale propery mahemaically capures he economic noion of an informaionally efficien marke. The curren value reflecs all he available informaion. Samuelson waned o show ha when sock prices equal he presen expeced value of heir payoffs (heir fundamenal value given a deerminisic discoun facor) hey are unpredicable. The curren price is he bes predicor of fuures values. Samuelson, and LeRoy in his survey, esablish he Maringale propery of "sock prices" using he consan discoun facor specificaion. Your Problem I. Algebra Assume he fundamenal value of he sock equal he presen expeced value, equaion (0). Esablish he Maringale propery of sock prices. Your advisor, Elmo he mah ca, says le's do his in seps o make i easier. (You follow Elmo's advice and you show me he seps.) Sep : The Equilibrium Pricing Equaion Elmo says, "Le's solve he equilibrium pricing condiionha he curren sock price equals he
3 3 discouned expeced dividend payoff plus he sock price nex period, P = λe [ P + + d +]; λ (0,) () as a forward difference equaion and impose he erminal condiion, lim( ) λ E P + 0 o show ha i gives he fundamenal value (presen expeced value) equaion2. (Elmo noes ha you us did somehing really cool. He knows ha bubbles are deviaions from he fundamenal value. The equilibrium pricing condiion rules ou irraional bubbles. The expeced reurn facor equals /λ, bu he price could conain a raional bubble. You us showed ha he equilibrium pricing condiion and he end poin condiion rule ou raional bubbles. Elmo hinks mahemaically his is sraighforward, bu he economics are suble. The equilibrium pricing condiion is he differenial and he fundamenal value is he inegral so he equilibrium pricing condiion is consisen wih any endpoin. ) Sep 2: Fair Games Elmo recognizes ha equaion () (or equaion (0)) is rouble for showing sock prices are a Maringale. He says, Sock prices aren' a Maringale unless dividends are zero. Bu if dividends are zero, hen he fundamenal value of he sock is zero. A sequence of zeros is a Maringale, bu an infinie sequence of zeros is a company ha wen ou of business or doesn' exis. LeRoy claims he fair game model has he same efficien marke implicaions as he Maringale. Afer all, a Maringale difference sequence is a fair game. So le's ransform sock prices ino an obec ha's a fair game. Le's show ha he sochasic process of reurns (reurn facors) is a consan plus a residual, and he residual is a fair game. y R ER R P + d + P, (2) Elmo advises you o noice ha ER + is he uncondiional expecaion of he reurn, so he residual is he deviaion from he uncondiional mean. Elmo hinks ha I guess his is why he finance guys call he consan expeced reurns model he unpredicable reurns model. Sep 3: Maringales 2 Economerofreaks can pick a limiing concep, like meansquared error. The res of us can assume he expecaion is finie.
4 4 Elmo says, Finally, i's ime o show a ransform of sock prices is a Maringale. LeRoy did ha in 989 and we could us copy his resuls. Bu los has happened since hen. A friend of mine ha used o each a he CAL business school, Hua He, (who is a genius and was a Wall S rocke scienis and now is a Yale) showed me a more general represenaion ha capures he essenial idea. LeRoy only looks a a single securiy. Bu, he equilibrium pricing condiion () has o hold for all asses raded in perfec markes. In fac, he basic principle which we us did is no expeced excess reurns. The problem wih showing sock prices are a Maringale, (or random walk) is ha o mee he equilibrium pricing condiion he value of he invesmen mus be expeced o grow over ime so ha he expeced reurn on he invesmen equals he reciprocal of he discoun facor. Socks can do his because eiher he price increases or hey pay dividends3. Turning he sequence ino a Maringale requires derending. So here's Hua He's sraegy () inroduce anoher securiy a risk free bond whose reurn is he required reurn, (2) inves all dividends in risk free bonds, and hen (3) show ha he value of he sock plus he value of he porfolio of accumulaed dividends deflaed by he value of he risk free bond is a Maringale. Le's define a defaul free discoun bond. The bond pays off nex period and he curren price is B = /(+i + ), where i is he reurn. Le v + (B ) denoe he value an invesmen in he bond purchased a in period +. From he equilibrium pricing condiion, v( B) = λe = λ = B = + i v ( B ) =, + (3) he value of he bond when i is purchased equals is (known) discouned payoff which is he curren price. In he following period when he bond pays off is value is he value of he payoff. Now we'll inves he dividends in bonds so ha hey earn he reurn, / λ = +i, (recall LeRoy had o reinves dividends in he muual fund). Le D denoe he curren value of he porfolio of accumulaed dividends invesed in bonds. This period you can buy D bonds, D v B D B D = ( ) = Nex period when he bonds maure he porfolio pays off, v B D D + ( ) = = D ( + i) The porfolio of accumulaed dividends earns he expeced reurn. OK! so each period add he curren dividend o he porfolio and reinves he oal in bonds. Define he value of he porfolio of accumulaed dividends in he recursive form, 3 Noice he fundamenal value of a sock is usually expressed in erms of expeced discouned dividends while he fundamenal value of a firm is he expeced discouned profis. In fac, hese are equivalen. The firm can pay ou profis in dividends, or reinvesmen hem in he firm.
5 5 D D D = d + v( B ) = d + = d + ( + i) D OK! says Elmo enough preliminaries, les form an obec ha's a Maringale. We'll show ha, E P D P + D = v+ ( B) v( B) (4) he raio of he sock price plus "accumulaed dividends" o he bond is a Maringale. Hau He calls he bond is he numeriare securiy. He says here's nohing special abou he sock P. All asses, when you ake care of accumulaed dividends, deflaed by he numeriare securiy follow Maringales if he equilibrium pricing equaion () holds. Elmo muses (wha do you hink cas do when hey preend o sleep 8 hours a day)"i wonder if he Maringale propery rules ou raional bubbles? " II. Empirical: Tes he weakform of he Efficien Markes Hypohesis Maringale Tes he consan expeced reurns model in Sep 2. Use he CRSP daa se. Try all frequencies: daily, monhly, quarerly, and annual. Random Walk The random walk has independen incremens all of he momens of he incremens (error) are consan. Assume you can reec he Maringale (consan expeced reurns) model. How would you es he sronger resricion ha he model is a random walk? For example, how would you es he resricion ha he variance is consan?
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