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Fin 501:Asset Pricing I Lecture 08: Multi Multi--period Model Prof. Markus K. Brunnermeier Multi--period Model Multi Slide 08 08--1 Fin 501:Asset Pricing I … the th remaining i i lectures l t • • • • • • Generalizing to multi-period setting Ponzi Schemes, Schemes Bubbles Forward, Futures and Swaps ICAPM and d hhedging d i ddemand d Multiple factor pricing models Market efficiency Multi--period Model Multi Slide 08 08--2 Fin 501:Asset Pricing I I t d ti Introduction • accommodate multiple and even infinitely many periods. • severall issues: i ¾ how to define assets in an multi period model, ¾ how to model intertemporal preferences, preferences ¾ what market completeness means in this environment, ¾ how the infinite horizon may the sensible definition of a budget constraint (Ponzi schemes), ¾ and how the infinite horizon may affect pricing (bubbles). • This section is mostly based on Lengwiler (2004) Multi--period Model Multi Slide 08 08--3 Fin 501:Asset Pricing I O Overview i 1. Generalizing the setting ¾ Trees, modeling information and learning ¾ Time Time--preferences 2 Multi 2. Multi--period SDF and event prices ¾ Static Static--dynamic completeness 3 Dynamic 3. D i completion l ti 4. Ponzi scheme and rational bubbles 5. Martingale process - EMM Mean--Variance Analysis and CAPM Mean Slide 08 08--4 Fin 501:Asset Pricing I 0 1 2 3 many one period i d models d l how to model information? Multi--period Model Multi Slide 08 08--5 Fin 501:Asset Pricing I 0 1 2 3 0 1 2 3 States s Ω∪ ∅ ∪ F1 Multi--period Model Multi F2 Events Ai,t Slide 08 08--6 Fin 501:Asset Pricing I M d li information Modeling i f ti over time ti • Partition • Field/Algebra • Filtration Multi--period Model Multi Slide 08 08--7 Fin 501:Asset Pricing I S Some probability b bilit theory th • Measurability: A random variable y(s) is measure w.r.t. algebra F if ¾ Pre-image of y(s) are events (elements of F) • for each Ai ∈ F, y(s) = y(s’) for each s ∈ Ai and s’ ∈ Ai, • yt(Ai):= ): yt(s), (s) s ∈ Ai. • Stochastic process: A collection of random variables yt(s) for t = 0,…, T. • Stochastic process is adapted to filtration F ={Ft}tT if each yt((s)) is measurable w.r.t. Ft ¾ Cannot see in the future Multi--period Model Multi Slide 08 08--8 Fin 501:Asset Pricing I f from static t ti to t dynamic… d i asset holdings Dynamic strategy (adapted process) asset payoff x Next period’s payoff xt+1+ pt+1 Pa off of a strategy Payoff strateg span of assets Marketed subspace of strategies Market completeness a) Static completeness (Debreu) b) Dynamic completeness (Arrow) No arbitrage w.r.t. holdings No arbitrage w.r.t strategies State prices q(s) Event prices qt(At(s)) Multi--period Model Multi Slide 08 08--9 Fin 501:Asset Pricing I …from f static t ti to t dynamic d i State prices q(s) _ Risk free rate R Risk neutral _prob. π*(s) = q(s) R Pricingg kernel pj = E[kq x_j] 1 = E[kq] R Event prices qt(A(s)) _ Risk free rate rt varies over time Discount factor from t to 0 ρt(s) Risk neutral prob. π*(At(s)) = qt(At(s)) / ρt (At) Pricingg kernel kt ptj_= Et[kt+1(pjt+1+ xjt+1)] kt = Rt+1 Et[kt+1] Multi--period Model Multi Slide 08 08--10 Fin 501:Asset Pricing I … in more detail A3 We recall the event tree that captures the gradual resolution of uncertainty. ψ1(A4) A1 ψ0(A4) ) ψ2(A4) This tree has 7 events (A0 to A6). (Lengwiler uses e0 to e6) A4 A0 3 time periods (0 to 2). A5 If A is some event, we denote the period it belongs to as τ(A). A2 A6 t 0 t=0 t 1 t=1 So for instance, instance τ(A2)=1, )=1 τ(A4)=2. )=2 We denote a path with ψ as follows… t 2 t=2 Multi--period Model Multi Slide 08 08--11 Fin 501:Asset Pricing I M lti l period Multiple i d uncertainty t i t A3 A1 A4 A0 Last period events have prob., π 3 - π 6. The earlier events also have probabilities. To be consistent consistent, the probability of an event is equal to the sum of the probabilities of its successor events. So for fo instance, instance π1 = π3+π + 4. A5 A2 A6 t=0 t=1 t=2 Multi--period Model Multi Slide 08 08--12 Fin 501:Asset Pricing I M lti l period Multiple i d assets t A typical multiple period asset is a coupon bond: ⎧coupon if 0 < τ ( A) < t * , ⎪ rA := ⎨1 + coupon if τ ( A) = t * , * ⎪0 if ( ) > . τ A t ⎩ The coupon bond pays the coupon in each period & pays the coupon plus the principal at maturity t*. A consol is a coupon bond with t* = ∞; it pays a coupon forever. A discount bond (or zero-coupon bond) finite maturity bond with no coupon. coupon It just pays 1 at expiration, and nothing otherwise. Multi--period Model Multi Slide 08 08--13 Fin 501:Asset Pricing I M lti l period Multiple i d assets t • create STRIPS by extracting only those payments that occur in a particular period. ¾ STRIPS are the same as discount bonds. • More generally, arbitrary assets (not just bonds) could ld be b striped. i d Multi--period Model Multi Slide 08 08--14 Fin 501:Asset Pricing I M lti l period Multiple i d assets t • Sh Shares are claims l i to the h future f dividends di id d off a firm. fi • Derivatives: A call option on a share, is an asset that pays either 0 or p - k, k where p is the (random) price of the share and k is the strike (exercise) price. p of this kind will be helpful p when thinkingg • Options about how to make a market system complete. Multi--period Model Multi Slide 08 08--15 Fin 501:Asset Pricing I O Overview i 1. Generalizing the setting ¾ Trees, modeling information and learning ¾ Time Time--preferences 2 Multi 2. Multi--period SDF and event prices ¾ Static Static--dynamic completeness 3 Dynamic 3. D i completion l ti 4. Ponzi scheme and rational bubbles 5. Martingale process - EMM Mean--Variance Analysis and CAPM Mean Slide 08 08--16 Fin 501:Asset Pricing I Ti preference Time f u(c0) + ∑t δ(t) E{u(ct)} • Discount factor δ(t) number between 0 and 1 • Assume δ(t)>δ(t+1) for all t. • Suppose S you are iin period i d 0 andd you make k a plan l off your present and future consumption: c0, c1, c2, … • The relation between consecutive consumption will depend on the interpersonal rate of substitution, which is δ(t). • Time consistency δ(t) = δ t (exponential discounting) Multi--period Model Multi Slide 08 08--17 Fin 501:Asset Pricing I Ti preference Time f • Canonical framework (exponential discounting) U(c) = E[ ∑ δt u(ct)] ¾ prefers earlier uncertainty resolution if it affect action ¾ indifferent, if it does not affect action • Time-inconsistent (hyperbolic discounting) Special case: β−δ formulation U(c) = E[u(c0) + β ∑ δt u(ct)] • Preference for the timing of uncertainty resolution recursive utility formulation (Kreps-Porteus (Kreps Porteus 1978) Slide 08 08--18 Fin 501:Asset Pricing I Preference for the timing of uncertainty resolution Digression: g π 0 $100 $150 $100 $ 25 $150 $100 Kreps-Porteus π $ 25 Early (late) resolution if W(P1,…) is convex (concave) Example: do you want to know whether you will get cancer at the age of 55? Slide 08 08--19 Fin 501:Asset Pricing I Digression: MultiM lti-period Multi i d P tf li Choice Portfolio Ch i Theorem 4.10 ((Merton, 1971): ) Consider the above canonical multi-period consumption-saving-portfolio allocation problem. Suppose U() displays CRRA, rf is constant and {r} is i.i.d. Then a/st is time invariant. invariant Slide 08 08--20 Fin 501:Asset Pricing I O Overview i 1. Generalizing the setting ¾ Trees, modeling information and learning ¾ Time Time--preferences 2 Multi 2. Multi--period SDF and event prices ¾ Pricing in a static dynamic model 3 Dynamic 3. D i completion l ti 4. Martingale process - EMM Mean--Variance Analysis and CAPM Mean Slide 08 08--21 Fin 501:Asset Pricing I A static i dynamic d i model d l • We consider pricing in a model that contains many periods (possibly infinitely many)… • …and and we assume that information is gradually revealed (this is the dynamic part)… • …but we also assume that all assets are only traded "at the beginning of time" (this is the static part). • There is dynamics y in the model because there is time,, but the decision making is completely static. Multi--period Model Multi Slide 08 08--22 Fin 501:Asset Pricing I M i i ti over many periods Maximization i d • vNM exponential utility representative agent max{∑t=0 δt E{u(ct)} | y-w ∈ B(p)} • If all Arrow securities (conditional on each event) are traded, we can express the first-order conditions as, u'(c0) = λ, δτ(A)πAu'(wA) = λqA. Multi--period Model Multi Slide 08 08--23 Fin 501:Asset Pricing I MultiM lti-period Multi i d SDF • The equilibrium SDF is computed in the same fashion as i the in h static i model d Al we saw before b f qA πA = δ τ ( A) u' (w ) u ' ( w0 ) ⎛ u ' ( wψ 1 ( A) ) ⎞ ⎛ u ' ( wψ 2 ( A) ) ⎞ ⎛ u' (w A ) ⎞ ⎟ ⎟⎟ ⋅ ⎜⎜ δ ⎟L⎜ δ = ⎜⎜ δ ψ ( A ) ψ 1 ( A) ⎟ 0 ⎜ u ' ( w τ ( A )−1 ) ⎟ u ' ( w ) u ' ( w ) ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ mψ 1 ( A) • × mψ 2 ( A) ×L× mψ τ ( A )−1 ( A) =: M A We call mA the "one-period ahead" SDF and MA the multi-period SDF (“state-price density”). Multi--period Model Multi Slide 08 08--24 Fin 501:Asset Pricing I Th fundamental The f d t l pricing i i formula f l • To price an arbitrary asset x, portfolio of STRIPed cash flows, x j = x1j +x2j +L+x∞j, where xt j denotes the cash-flows in period t. • The price of asset x j is simply the sum of the prices of its STRIPed payoffs, so p j = ∑ E{M t xt } j t • This is the fundamental pricing formula. • Note that Mt = δt if the repr agent is risk neutral. The fundamental pricing formula then just reduces to the present value of expected dividends, pj = ∑ δt E{xtj}. Multi--period Model Multi Slide 08 08--25 Fin 501:Asset Pricing I O Overview i 1. Generalizing the setting ¾ Trees, modeling information and learning ¾ Time Time--preferences 2 Multi 2. Multi--period SDF and event prices ¾ Pricing in a static dynamic model 3 Dynamic 3. D i completion l ti 4. Ponzi scheme and rational bubbles 5. Martingale process - EMM Slide 08 08--26 Fin 501:Asset Pricing I D Dynamic i ttrading di • In the "static static dynamic dynamic" model we assumed that there were many periods and information was gradually revealed (this is the dynamic part)… • …but b all ll assets are traded d d "at " the h bbeginning i i off time" i " (this is the static part). • Now consequences of re re-opening opening financial markets. Assets can be traded at each instant. • This has deep implications. ¾ allows ll us to reduce d the h number b off assets available il bl at eachh instant through dynamic completion. ¾ It opens up some nasty possibilities (Ponzi schemes and b bbl ) bubbles), Multi--period Model Multi Slide 08 08--27 Fin 501:Asset Pricing I C Completion l ti with ith shortshort h t-lived li d assets t • If the horizon is infinite, the number of events is p y that we need an also infinite. Does that imply infinite number of assets to make the market p complete? • Do we need assets with all possible times to maturity and events to have a complete market? • No. Dynamic completion. Arrow (1953) and Guesnerie and Jaffray (1974) Multi--period Model Multi Slide 08 08--28 Fin 501:Asset Pricing I C Completion l ti with ith shortshort h t-lived li d assets t • Asset is `short lived´ if it pays out only in the period immediately after the asset is issued. • Suppose for each event A and each successor event A' there is an asset that pays in A' and nothing otherwise. • It is possible to achieve arbitrary transfers between all events in the event tree by trading only these short-lived assets. • This is straightforward if there is no uncertainty. Multi--period Model Multi Slide 08 08--29 Fin 501:Asset Pricing I C Completion l ti with ith shortshort h t-lived li d assets t • Without uncertainty, and T periods (T can be infinite), there are T one period assets, from period 0 to period 1, from period 1 to 2, etc. • Let pt be the price of the bond that begins in period t-1 andd matures iin period i d t. • For the market to be complete we need to be able to t transfer f wealth lth between b t any two t periods, i d nott just j t between consecutive periods. • This can be achie achieved ed with ith a trading t di strategy. t t Multi--period Model Multi Slide 08 08--30 Fin 501:Asset Pricing I C Completion l i with i h shortshort h -lived li d assets • Example: Suppose we want to transfer wealth from period 1 to period 3. • In period 1 we cannot buy a bond that matures in period 3 because 3, b suchh a bbond d iis nott traded t d d then. th • Instead buy a bond that matures in period 2, for price p2. • In period 2, 2 use the payoff of the period-2 period 2 bond to buy period-3 bonds. • In period 3, collect the payoff. • The result is a transfer of wealth from period 1 to period 3. The price, as of period 1, for one unit of purchasing power in period 3, is p2p3. Multi--period Model Multi Slide 08 08--31 Fin 501:Asset Pricing I C Completion l i with i h shortshort h -lived li d assets With uncertainty the process is only slightly more complicated. It is easily understood with a graph. event 1 event 3 event 4 event 0 event 5 event 2 event 6 Let pA be the price of the asset that h pays one unit i in i event A. A This asset is traded only in the event immediately preceding A. We want to transfer wealth from event 0 to event 4. Go backwards: in event 1, buy one event 4 asset for a price p4. In event 0, buy p4 event 1 assets. The cost of this today is p1p4. The payoff ff iis one unit i in i event 4 and d nothing otherwise. Multi--period Model Multi Slide 08 08--32 Fin 501:Asset Pricing I C Completion l i with i h longlong l -lived li d assets • dynamic completion with long-lived assets, Kreps (1982) • T-period model without uncertainty (T < ∞). • assume there is a single asset: ¾a discount bond maturing in T. ¾bond can be purchased and sold in each period, period for price pt, t=1,…,T. Multi--period Model Multi Slide 08 08--33 Fin 501:Asset Pricing I Completion C l ti with ith a long l maturity t it bond b d • So there are T prices (not simultaneously, but sequentially). • Purchasing power can be transferred from period t to period t' > t by purchasing the bond in period t and selling it in period tt'. Multi--period Model Multi Slide 08 08--34 Fin 501:Asset Pricing I A simple i l information i f ti tree t 1 0 event 1 q1,1 q2,1 q1,0 q2,0 20 0 1 1 0 event 0 q1,2 q2,2 event 2 0 asset 1 1 This information tree has three non trivial events plus four final non-trivial states, so seven events altogether. It seems as if we would need six Arrow securities (for events 1 and 2 and for the four final states) to have a complete market. Yet we have only two assets. So the market cannot be complete, right? Wrong! Dynamic trading provides a way to fully insure each event separately. Note that there are six prices because each asset is traded in three events. asset 2 Slide 08 08--35 Fin 501:Asset Pricing I O -period OneOne i d holding h ldi • Call “tradingg strategy gy [j,A]” [j ] the cash flow of asset j that is purchased in event A and is sold one period later. • How many such assets exist? What are their cash flows? p1,1 p2,1 p1,0 p2,0 1 0 0 1 1 0 p1,2 p2,2 0 1 There are six such strategies: [1,0], [1,1], [1,2], [2,0], [2,1], [2,2]. (Note that (N h this hi iis potentially i ll sufficient ffi i to span the complete space.) “Strategy [1,1]" costs p1,1 and pays outt 1 iin th the fi firstt final fi l state t t and d zero in all other events. “Strategy [1,0]" costs p1,0 and pays o t p1,1 in e out event ent 1, 1 p1,2 in event e ent 2, 2 and zero in all the final states. Multi--period Model Multi Slide 08 08--36 Fin 501:Asset Pricing I Th extended The t d d payoff ff matrix ti The trading strategies [1,0] … [2,2] give rise to a new 6x6 payoff matrix. matrix asset event 0 event 1 event 2 state 1 state 3 state 3 state 4 [1,0] –p1,0 10 p1,1 p1,2 0 0 0 0 [2,0] –p2,0 20 p2,1 p2,2 0 0 0 0 [1,1] 0 –p1,1 0 1 0 0 0 [2,1] 0 –p2,1 0 0 1 0 0 [1,2] 0 0 –p1,2 0 0 1 0 [2,2] 0 0 –p2,2 0 0 0 1 This matrix is regular (and hence the market complete) if the grey submatrix is regular (= of rank 2). Multi--period Model Multi Slide 08 08--37 Fin 501:Asset Pricing I Th extended The t d d payoff ff matrix ti • Is the gray submatrix regular? • Components of submatrix are prices of the two assets, conditional on period 1 events. • There Th are cases in i which hi h (p ( 11, p21) and d ((p12, p22) are collinear in equilibrium. 2 in • If per capita endowment is the same in event 1 and 2, state 1 and 3, and in state 2 and 4, respectively, and if the probability of reaching state 1 after event 1 is the same as the h probability b bili off reaching hi state 3 after f event 2 Æ submatrix is singular (only of rank 1). • But then events 1 and 2 are effectively identical identical, and we may collapse them into a single event. Multi--period Model Multi Slide 08 08--38 Fin 501:Asset Pricing I Th extended The t d d payoff ff matrix ti • A random square matrix is regular. regular So outside of special cases, the gray submatrix is regular (“almost surely”). • The 2x2 submatrix may still be singular by accident accident. • In that case it can be made regular again by applying a small perturbation of the returns of the long long-lived lived assets, by perturbing aggregate endowment, the probabilities, or the utility function. • Generically, the market is dynamically complete. Multi--period Model Multi Slide 08 08--39 Fin 501:Asset Pricing I H many assets to complete How l market? k ? • b branching hi number b = The Th maximum i number b off branches b h fanning out from any event in the uncertainty tree. • This is also the number of assets necessary to achieve dynamic completion. • Generalization by Duffie and Huang (1985): continuous time Æ continuity of events Æ but a small number of assets is sufficient. • The large power of the event space is matched by continuouslyy tradingg few assets, thereby y ggenerating ga continuity of trading strategies and of prices. Multi--period Model Multi Slide 08 08--40 Fin 501:Asset Pricing I E Example: l BlackBlack Bl k-Scholes S h l formula f l • Cox, Ross, Rubinstein binominal tree model of B-S • Stock price goes up or down (follows binominal tree) i interest rate is i constant • Market is dynamically complete with 2 assets ¾ Stock St k ¾ Risk-free asset (bond) • Replicate payoff of a call option with (dynamic Δ-hedging) • (later more) Multi--period Model Multi Slide 08 08--41 Fin 501:Asset Pricing I O Overview i 1. Generalizing the setting ¾ Trees, modeling information and learning ¾ Time Time--preferences 2 Multi 2. Multi--period SDF and event prices ¾ Pricing in a static dynamic model 3 Dynamic 3. D i completion l ti 4. Ponzi schemes and rational bubbles 5. Martingale - EMM Slide 08 08--42 Fin 501:Asset Pricing I P i schemes: Ponzi h infinite horizon max. problem • IInfinite fi it horizon h i allows ll agents t to t borrow b an arbitrarily bit il large amount without effectively ever repaying, by rolling over debt forever forever. ¾ Ponzi scheme - allows infinite consumption. • Consider an infinite horizon model, no uncertainty, and a complete set of short-lived bonds. [zt is the amount of bonds maturing in period t in the portfolio, βt is the price of this bond as of period t-1] ⎧⎪ ∞ t ⎫⎪ c0 − w0 ≤ − β1 z1 max ⎨∑ δ u (ct ) ⎬. c1 − w1 ≤ zt − β t +1 zt +1 for t > 0⎪⎭ ⎪⎩ t =0 Multi--period Model Multi Slide 08 08--43 Fin 501:Asset Pricing I P i schemes: Ponzi h rolling lli over debt d b forever f • The followingg consumption p path p is possible: p ct = wt+1 for all t. • Note that agent consumes more than his endowment in each period, forever. • This can be financed with ever increasing debt: z1=-1/β1, z2=(-1+z1)/β2, z3=(-1+z2)/β3 … • Ponzi schemes can never be part of an equilibrium. In fact, such a scheme even destroys the existence of a utility maximum because the choice set of an agent is unbounded b d d above. b W We needd an additional ddi i l constraint. i Multi--period Model Multi Slide 08 08--44 Fin 501:Asset Pricing I P i schemes: Ponzi h transversality t lit • The constraint that is typically imposed on top of the budget constraint is the transversality condition, limt→∞ βt zt ≥ 0. • This constraint implies that the value of debt cannot diverge to infinity. ¾More precisely, it requires that all debt must be redeemed eventuallyy (i.e. ( in the limit). ) Multi--period Model Multi Slide 08 08--45 Fin 501:Asset Pricing I R ti l Bubbles: Rational B bbl The Th price i off a consoll • In an infinite dynamic model, model in which assets are traded repeatedly, there are additional solutions besides the "fundamental fundamental pricing formula formula." ¾New solutions have “bubble component”. ¾No market clearing at infinity. infinity • Consider ¾model without uncertainty ¾Consol bond delivering $1 in each period forever Multi--period Model Multi Slide 08 08--46 Fin 501:Asset Pricing I Th price The i off a consoll • Our formula pt = Et[Mt+1 (pt+1+ dt+1)] • in the case of certainty with d=1 pt = Mt+1 pt+1 + Mt+1 • solve forward – (many solutions) ⇒ p0 = M ∑ 1 424 3 ∞ t =1 t fundamental value + lim T →∞ M T pT . 144244 3 bubble component Multi--period Model Multi Slide 08 08--47 Fin 501:Asset Pricing I Th price The i off a consoll • Consider case without uncertainty. (event tree is line) • According to the static-dynamic static dynamic model (the fundamental pricing formula), the price of the consol is p0 = ∑t Mt. Mt := m1 × m2 × L × mt • • • • Consider re-opening markets now. The price at time 0 is just the sum of all Arrow prices, so p0 = ∑t=1∞ Mt. qt is the marginal rate of substitution between consumption in period t and in period 0, qt = δt ((u'(w ( t))/u'(w ( 0)) )), Multi--period Model Multi so p0 = ∑t =1 δ ∞ t u '( w t ) u '( w 0 ) Slide 08 08--48 . Fin 501:Asset Pricing I Th price The i off a consoll att t=1 1 • At time 1, 1 the price of the consol is the sum of the remaining Arrow securities, ∞ p1 = ∑t = 2 δ t −1 uu ''(( ww )) . t 1 • This can be reformulated, p1 = δ • −1 u '( w0 ) u '( w1 ) ∑ ∞ δ t =2 t u '( w t ) u '( w 0 ) . The second part (the sum from 2 to ∞) is almost equal to p0, ∑ ∞ t =2 δ t u '( w t ) u '( w 0 ) = (∑ ∞ t =1 δ t u '( w t ) u '( w 0 ) )− δ Multi--period Model Multi u '( w1 ) u '( w 0 ) = p0 − δ u '( w1 ) u '(w ( w0 ) . Slide 08 08--49 Fin 501:Asset Pricing I Consol: Solving forward • More generally, we can express the price at time t+1 as a function of the price at time t, ( pt +1 = δ −1 uu'('(wwt +1)) qt − δ t pt = δ • u '( wt +1 ) u '( w t ) +δ ), u '( wt +1 ) u '( wt ) u '( wt +1 ) u '( w t ) thus pt +1 = mt +1 + mt +1 pt +1. We can solve this forward by substituting the t+1 version of this equation into the t version, ad infinitum,, p0 = m1 + m1 p1 = m1 + m1[m2 + m2 p2 ] = L ⇒ p0 = M ∑ 1 424 3 ∞ t =1 t fundamental value + limT →∞ MT pT . 14 4244 3 Multi--period Model Multi bubble component Slide 08 08--50 Fin 501:Asset Pricing I M Money as a bbubble bbl p0 = M ∑ 1 424 3 ∞ t =1 t fundamental value + lim T →∞ M T pT . 144244 3 bubble component • The fundamental value = p price in the static-dynamic y model. • Repeated trading gives rise to the possibility of a bubble. • Fiat money can be understood as an asset with no dividends. In the static-dynamic model, such an asset would have no value (the present value of zero is zero). But if there is a bubble on the price of fiat money, then it can have positive value (Bewley, 1980). • In asset pricing theory, we often rule out bubbles simply by i imposing i lim li T→∞ MT pT = 0. 0 Multi--period Model Multi Slide 08 08--51 Fin 501:Asset Pricing I O Overview i 1. Generalizing the setting ¾ Trees, modeling information and learning ¾ Time Time--preferences 2 Multi 2. Multi--period SDF and event prices ¾ Pricing in a static dynamic model 3 Dynamic 3. D i completion l ti 4. Ponzi schemes and rational bubbles 5. Martingale - EMM Mean--Variance Analysis and CAPM Mean Slide 08 08--52 Fin 501:Asset Pricing I M ti l Martingales • Let X1 be a random variable and let x1 be the realization of this random variable. • Let X2 be another random variable and assume that the distribution of X2 depends on x1. • Let X3 be a third random variable and assume that the di ib i off X3 depends distribution d d on x1, x2. • Such a sequence of random variables, (X1,X2,X3,…), is called ll d a stochastic t h ti process. • A stochastic process is a martingale if E[ t+1 | xt , … , x1] = xt . E[x Multi--period Model Multi Slide 08 08--53 Fin 501:Asset Pricing I Hi t History off the th wordd martingale ti l • • • • “martingale” originally refers to a sort of pants worn by “Martigaux” people living in Martigues located in Provence in the south of France. By analogy, it is used to refer to a strap in equestrian. This strap is tied at one end to the girth of the saddle and at the other end to the head of the horse. It has the shape of a fork and divides in two. In comparison to this division, martingale refers to a strategy which consists in playing twice the amount you lost at the previous round. Now, it refers to any strategy used to increase one's probability to win by respecting the rules. The notion of martingale appears in 1718 (The Doctrine of Chance by Abraham de Moivre) referring to a strategy that makes you sure to win in a fair game. See also www.math.harvard.edu/~ctm/sem/martingales.pdf Multi--period Model Multi Slide 08 08--54 Fin 501:Asset Pricing I Pi Prices are martingales… ti l • • Samuelson (1965) has argues that prices have to be martingales in equilibrium. one has to assume that 1. no discounting 2 no dividend payments (intermediate cash flows) 2. 3. representative agent is risk-neutral 1 With 1. Wi h discounting: di i ¾ Discounted price process should follow martingale i l E[δ t+1| pt] = pt . E[δp Multi--period Model Multi Slide 08 08--55 Fin 501:Asset Pricing I R i Reinvesting ti dividends di id d 2 With dividend payments 2. ¾ because pt depends on the dividend of the asset in period t+1, t+1 but pt+1 does not (these are ex-dividend prices). ¾ Consider, Consider value of a fund that keeps reinvesting the dividends follows a martingale LeRoyy ((1989). ) Multi--period Model Multi Slide 08 08--56 Fin 501:Asset Pricing I Reinvesting dividends • Consider a fund owning nothing but one units of asset j. • The value of this fund at time 0 is f0 = p0 = E[∑t=1∞ δt xtj] = δ E[x1j+p1]. (if representative agent is risk-neutral) • After receiving dividends x1j (which are state contingent) it buys more of asset j at the then current price p1, so the f d then fund h owns 1 + x1j/p / 1 units i off the h asset. • The discounted value of the fund is then f1 = δ p1 (1+x1j/p1) = δ (p1+x1j) = p0 = f0, so the discounted value of the fund is indeed a martingale. i l Multi--period Model Multi Slide 08 08--57 Fin 501:Asset Pricing I …and and with risk aversion? • A similar statement is true if the representative agent is risk averse. averse • The difference is that ¾ we must discount with the risk-free interest rate,, not with the discount factor, ¾ we must use the risk-neutral probabilities (also called equivalent martingale measure for obvious reasons) instead of the objective probabilities. • Just as in the 2-period model, we define the riskneutrall probabilities b bili i as π∗A = πA MA / ρτ(A), where ρτ(A) is the discount-factor from event A to 0. (state dependent) Multi--period Model Multi Slide 08 08--58 Fin 501:Asset Pricing I …and with risk aversion? • The initial value of the fund is f0 = p0 = ∑tt=11∞ E[Mt xt j]. • Let us elaborate on this a bit, f0 = E[∑t=1∞ Mt xt j] = E{M1 x1 j + ∑t=2∞ Mt xt j] = E[M1 (x1 j + ∑t=2∞ ∏t'=2t Mt' xt j )] = E[M1 (x1 j + p1)]. • E* = expectations under the risk-neutral distribution π∗, this can be rewritten as f0 = ρ1 E*[x E*[ 1 j + p1] = ρ1 E*[f1]. ] • The properly discounted (ρ instead of δ) and properly expected (π∗ instead of π) value of the fund is indeed a martingale. Multi--period Model Multi Slide 08 08--59 Fin 501:Asset Pricing I O Overview i 1. Generalizing the setting ¾ Trees, modeling information and learning ¾ Time Time--preferences 2 Multi 2. Multi--period SDF and event prices ¾ Static Static--dynamic completeness 3 Dynamic 3. D i completion l ti 4. Ponzi scheme and rational bubbles 5. Martingale process - EMM Slide 08 08--60