Introduction Optimality and Asset Pricing
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1 Introduction Optimality and Asset Pricing Andrea Buraschi Imperial College Business School October 2010
2 The Euler Equation Take an economy where price is given with respect to the numéraire, which is our consumption good. Utility defined over current and expected future consumption levels [ T U(c) = E β t u(c t ) t=0 Two choices: consume today Vs invest for tomorrow. Rational agents maximise their utility w.r.t financial wealth. W t+1 = (W t + Y t c t ) R t+1 where R t+1 is the stochastic return in period t, Y t is any income from holding our good, and c t is the consumption of that good. Rearranging we can obtain the consumption recursively: c t = c t+1 = [ W t + Y t W t+1 R t+1 [ W t+1 + Y t+1 W t+2 R t+2
3 The Euler Equation Two Period Problem: Agents choose consumption levels such to maximise deterministic utility today plus expected utility tomorrow. max u(c t ) + βe t [u(c t+1 ) W t+1 max u(w t + Y t W t+1 ) + βe t W t+1 R t+1 [ u(w t+1 + Y t+1 W t+2 ) R t+2 Taking first order conditions w.r.t second period wealth we obtain the Euler equation where, u (c t ) = E t [ βu (c t+1 )R t+1 and so, R t+1 = P t+1 + D t+1 P t P t u (c t ) = E t [ βu (c t+1 )(P t+1 + D t+1 )
4 The Euler Equation: The Price of Debt Now consider the special case where the per period interest rate, R t,t+1, is constant (and therefore locally risk free), then: u [ (c t ) = (β R t,t+1 )E t u (c t+1 R t,t+1 = E t [ 1 β P ZCB = E t [λ t+1 u (c t ) u (c t+1 = 1 E t [λ t+1 Log utility: A limiting case of Power utility (homework) u(c t ) = log u(c t ) u (c t ) = 1 c t R t,t+1 = 1 [ β E ct+1 t c t
5 The Euler Equation: The Price of Debt Remarks: Real Interest rates are proportional to consumption growth. We can test empirically since 1/R f is the price of a zero coupon bond from t t + 1. For example, Hall(1988).
6 The Euler Equation: The Price of Equity Starting from the Euler equation. U (C t ) = E t [βu (C t+1 ).R t+1 or, [ 1 = E t β U (C t+1 ) U (C t ).P t+1 + D t+1 P t P t = E t [λ t+1.(p t+1 + D t+1 ) Price is stochastic and dependent on a future uncertain state of the world. Note that our stochastic dependence originates from three terms in three expectation!
7 The Euler Equation: The Price of Equity The product of two stochastic variables is given by: therefore E t [XY = E t [X E t [Y + Cov t [X, Y 1 = E t [λ t+1 E t [ Pt+1 + D t+1 P t [ + Cov t λ t+1, P t+1 + D t+1 P t Interpreting E t [λ t+1 = 1/R f and rearranging we obtain the traditional corporate finance pricing function plus a risk correction. P t = E t[p t+1 + D t+1 R f + Cov t [λ t+1, P t+1 + D t+1 Note that the pricing function of both debt and equity contain common terms - how are they related?
8 The stochastic discount factor The price of zero coupon bond is given by: whereas the price of equity is given by: P ZCB = E t [λ t+1 P t = E t[p t+1 + D t+1 R f + Cov t [λ t+1, P t+1 + D t+1 Both debt and equity pricing formulae contain λ t+1 which is the stochastic discount factor or otherwise known as the pricing kernel. Remarks: λ t+1 = β U (C t+1 ) U (C t ) This is a way of testing our model. Is the stochastic discount factor correct for the prices observed in the data? When we talk about an asset pricing model, it is the same thing as talking about the stochastic discount factor implied by the model. How does correlation with the stochastic discount factor affect price? Why?
9 From two periods to multiple periods Price is the discounted expected utility gained from consuming that assets dividend stream. Recursively substituting the Euler equation we obtain: P t = E t [λ t+1 (P t+1 + D t+1 ) P t = E t [λ t+1 D t+1 + E t+1 [λ t+2 D t+2 + E t+2 [λ t+3 D t+3...e T [P T P t = T [ E t λt+j D t+j + Et [λ T P T j=1 Transversality condition (homework): lim E t [λ T P T = 0 j The price of every asset is given by the expected discounted dividend stream of that asset.
10 Lucas Tree Economy: Pure Exchange Economy: Supply Demand Market Dt Technology Consumption U(Ct)
11 Lucas Tree Economy: General equilibrium argument: At equilibrium we have the market clearing condition that SUPPLY = DEMAND (C t = D t ). We go back to the Euler equation and use this result. [ 1 = E t β U (C t+1 ) U (C t ). P t+1 + D t+1 P t and substituting in our market clearing condition, [ 1 = E t β U (D t+1 ) U (D t ). P t+1 + D t+1 P t Recursively using the law of iterated expectations we obtain P t = E t P t = E t β j j=1 [ j 1 s=0 β j U (D t+j ) U j=1 (D t ) U (D t+s+1 ) U D (D t + s) t+j D t+j
12 Lucas Tree Economy Example: Price Dividend Ratio Log Utillity U(C ) = ln(c ) U (C ) = 1 C Then, P t = E t β j j=1 and since D t is known at time t 1/D t+j 1/D t D t+j or, P t = D t E t β j β = D t 1 β j=1 D t = β P t 1 β Constant dividend yield! Notice also that with log-preference, investors are MYOPIC.
13 Black-Scholes Economy A stochastic control problem: Filtered probability space (Ω, F, {F t }, P), - natural filtration is generated by a 1-d BM. Investment opportunities given by α units of a money market account and θ units of a stock following a geometric Brownian motion. t B(t) = B(0) + S(t) = S(0) + rb(s)ds 0 t 0 t µs(s)ds + σs(s)dw(s) 0 where r, µ, σ are constants. Consider an investor with CRRA preference, the problem this investor faces is one of stochastic optimal control : max E α θ W T [ 1 γ W T 1 γ s.t α 0 B 0 + θ 0 S 0 = W 0 α T B T + θ T S T = W T
14 Dynamic Programming A general stochastic optimal control problem: [ T max E u(c t, Yt c, t)dt + V (YT c ) c C s.t 0 T Yt c = Y T µ(c s, Ys c, s)ds + σ(c s, Ys c, s)dw(s) 0 where C denotes the set of admissible controls given initial state Y0 c = y. We assume that the primitives (C, u, V, µ, σ) are such that, given any initial state y Y, the utility of any admissible control c is well defined, i.e., the above expectation exists, and there is a unique solution to the SDE. Looking for a feedback (Markovian) control c t = c(y c t, t) since the Markov property of Brownian motion allows us to conclude that the complimentary set of controls is suboptimal. We can thus rewrite our state process as a Markov process T T Y t = Y 0 + µ(y s, s)ds + σ(y s, s)dw(s) 0 0
15 Dynamic Programming Define the value function: J(Y, t) = sup J c (Y, t) J c (Y, t) E c C [ T 0 u(c t, Y c t, t)dt + V (Y c T ) F t where the information filtration is generated by the BM. Proceed by conjecturing that there exists an optimal control such that J(Y, t) = J c (Y, t) that solves the Hamilton-Jacobi-Bellman (HJB) equation: sup {DJ(c, Y, t) + u(c, Y, t)} = 0 c C where D is the Dynkin operator DJ(c, Y, t) = J Y (Y, t)µ(c t, Y c t, t) + J t (Y, t) J Y Y (Y, t)σ(c t, Y c t, t) 2 with the boundary condition J(Y, T ) = V (Y, T )
16 Hamilton-Jacobi-Bellman (HJB) equation The General Case: Using the law of iterated expectations we obtain the following recursive problem by ito s lemma J(Y, t) = sup J c (Y, t) c C = sup c C = sup c C E t [ τ t T u(c, Y, s)ds + u(c, Y, s)ds τ [ τ E t u(c, Y, s)ds + J(Y, τ) F t t dj(y, t) = DJ(Y, t)dt + J Y σ(c, Y, t)dw F t Integration from t to τ, taking expectations and assuming the last term is a martingale we obtain [ [ τ E J(Y, τ) F t = J(Y, t) + E DJ(Y, t)ds F t t
17 Hamilton-Jacobi-Bellman (HJB) equation The General Case: Using the law of iterated expectations we obtain the following recursive problem J(Y, t) = sup J c (Y, t) c C = sup c C E t [ τ t T u(c, Y, s)ds + u(c, Y, s)ds τ F t by ito s lemma [ τ = sup E t c C t u(c, Y, s)ds + J(Y, τ) F t dj(y, t) = DJ(Y, t)dt + J Y σ(c, Y, t)dw (1) Integration from t to τ, taking expectations and assuming the last term is a martingale we obtain [ [ τ E J(Y, τ) F t = J(Y, t) + E DJ(Y, t)ds F t (2) t
18 Hamilton-Jacobi-Bellman (HJB) equation Substituting?? into?? [ 0 = sup E intt τ DJ(c, Y, s) + u(c, Y, s)ds F t c C Then dividing both sides by τ t and letting τ t we obtain the Hamilton-Jacobi-Bellman (HJB) equation: sup {DJ(c, Y, t) + u(c, Y, t)} = 0 c C where D is the Dynkin operator DJ(c, Y, t) = J Y (Y, t)µ(c t, Y c t, t) + J t (Y, t) J Y Y (Y, t)σ(c t, Y c t, t) 2 with the boundary condition J(Y, T ) = V (Y, T )
19 Hamilton-Jacobi-Bellman (HJB) equation The Discrete Time Case: J(W t, t) = max E t[σ c t,c t+1,..,w t,w t+1,.. j=0 βj U(c t+j ) = max c t,c t+1,..,w t,w t+1,.. U(c t) + E t [Σ j=1 βj U(c t+j ) = max c t,c t+1,..,w t,w t+1,.. U(c t) + βe t [Σ j=0 βj U(c t+1+j ) = max U(c t ) + βe t [ max E t+1[σ c t,w t c t+1,..,w t+1,.. j=0 βj U(c t+1+j ) = max c t,w t U(c t ) + βe t [J(W t+1, t + 1) The objective of the investor is to maximize his expected utility over his infinite wealth and consumption stream. We can split our problem into two periods: a maximization of the first period and then the remaining periods - we are essientially left a two period maximization problem.
20 Hamilton-Jacobi-Bellman (HJB) equation Continuous Time: Utility Maximisation Problem Value function - with a control function u(c s ) (utility of consumption), state vector W t (wealth process - homework), and state variables c t, W t (consumption and wealth): J(W t, t) = [ T sup E t e δ(s t) U(C (s))ds (c s,θ(s)) s 0 t s.tdw t = [W t (θ λ t + r) C t dt + W t θ σdb t We split the problem into a two-period problem - today and tomorrow [ t+ T J(W t, t) = sup E t e δ(s t) u(c s )ds + e δ(s t+ ) u(c s )ds [ t+ J(W t, t) = sup E t e δ(s t) u(c s )ds t t t+ [ [ T + E t e δ E t+ t+ e δ(s t) u(c s )ds
21 Hamilton-Jacobi-Bellman (HJB) equation Continuous Time: Utility Maximisation Note that the state at time s = t +, namely W t+, depends on the state W t prevailing at the time s = t, and the optimal control function chosen over the first time interval. We have an optimal value function with consumption and wealth chosen today such that we can support an optimal consumption and wealth plan tomorrow. Thus, J(W t, t) = sup E t [ t+ Now subtracting J(W t, t) and dividing by we get: 0 = sup E t [ 1 t+ Taking the limit of to zero we obtain t t [ e δ(s t) u(c s )ds + E t e δ J(W t+, t + ) [ e δ(s t) e u(c s )ds + δ J(W E t+, t + ) J(W t, t) t [ 1 t+ [ 0 = sup lim E t e δ(s t) dj(wt, t) u(c s )ds δj(w t+, t + ) + E t 0 t dt δj(w t, t) = sup c,θ u(c t ) + E t [ dj(wt, t) dt
22 Hamilton-Jacobi-Bellman (HJB) equation Remarks: We therefore constructed a two period optimization problem from an infinite period one. Then our control function in the first period has consumption and wealth chosen in such a way to be consistent with optimality over the second period. The consumption plan chosen in the second period is therefore dependent on the prevailing state and the optimal control function chosen in the first period - it is said that we chose consumption and investment today that support optimality over our remaining lifetime.
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