Steven Heston: Recovering the Variance Premium. Discussion by Jaroslav Borovička November 2017

Transcription

1 Steven Heston: Recovering the Variance Premium Discussion by Jaroslav Borovička November 2017

2 WHAT IS THE RECOVERY PROBLEM? Using observed cross-section(s) of prices (of Arrow Debreu securities), infer preference parameters investors beliefs imposing as little structure as possible. Only: Markovianity time invariance minimal restrictions on preferences This is an identification problem. 2/16

3 A FINITE STATE SPACE FRAMEWORK Physical environment X a discrete-time stationary and ergodic Markov chain with n states Investor beliefs and preferences P = [p ij ] transition matrix (subjective) beliefs p ij = P (X t+1 = j X t = i) M = [m ij ] stochastic discount factor m ij state-specific discount rate between states i and j Asset prices Q = [q ij ] matrix of prices of one-period Arrow securities q ij price in state i of one unit of state-j cash flow next period 3/16

4 AN IDENTIFICATION PROBLEM Arrow prices encode both beliefs and preferences: q ij = p ij m ij Suppose we observe asset prices [q ij ]. Identification problem: Can we separately identify [p ij ] and [m ij ]? q ij }{{} n n equations = p ij }{{} n (n 1) unknowns m ij }{{} n n unknowns Underidentification!!! 4/16

5 PERRON FROBENIUS THEORY Let Q be a matrix with strictly positive entries. Then there exists a unique strictly positive eigenvector e associated with the largest eigenvalue exp (η): Qe = exp (η) e Hence, given asset prices Q, we can back out e and exp(η). What can we do with them? 5/16

6 A LONG-RUN PRICING PROBABILITY MEASURE Use the results from the Perron Frobenius problem to construct p ij = exp ( η) q ij e j e i P = [ p ij ] is a transition matrix (rows sum up to one) 6/16

7 A LONG-RUN PRICING PROBABILITY MEASURE Use the results from the Perron Frobenius problem to construct p ij = exp ( η) q ij e j e i P = [ p ij ] is a transition matrix (rows sum up to one) Invert to obtain a decomposition q ij = exp (η) e i e j }{{} m ij p ij 6/16

8 A LONG-RUN PRICING PROBABILITY MEASURE Use the results from the Perron Frobenius problem to construct p ij = exp ( η) q ij e j e i P = [ p ij ] is a transition matrix (rows sum up to one) Invert to obtain a decomposition q ij = exp (η) e i e j }{{} m ij p ij There is no claim that M = [ m ij ] is the true stochastic discount factor or that P = [ p ij ] represents investors beliefs. 6/16

9 AN IDENTIFICATION PROBLEM Definition The pair (M, P) explains asset prices Q if q ij = p ij m ij for every i, j. Take any random variable H = [h ij ] with mean one: n j=1 h ijp ij = 1. Define P H = M H = [ ] p H ij with p H ij = h ij p ij [ ] m H ij with m H ij = m ij h ij Then P H is a valid transition matrix and ( M H, P H) also explains asset prices Q. 7/16

10 DECOMPOSITION OF THE STOCHASTIC DISCOUNT FACTOR How are S and S related? q ij = exp (η) e i e j }{{} s ij p ij 8/16

11 DECOMPOSITION OF THE STOCHASTIC DISCOUNT FACTOR How are S and S related? {}}{ q ij = exp (η) e i p ij = exp (η) e i p ij e j e j p ij }{{}}{{} s ij s ij p ij 8/16

12 DECOMPOSITION OF THE STOCHASTIC DISCOUNT FACTOR How are S and S related? h ij {}}{ p ij q ij = exp (η) e i p ij = exp (η) e i e j e j p ij }{{}}{{} s ij s ij p ij 8/16

13 DECOMPOSITION OF THE STOCHASTIC DISCOUNT FACTOR How are S and S related? h ij {}}{ p ij q ij = exp (η) e i p ij = exp (η) e i e j e j p ij }{{}}{{} s ij s ij ] [ hij has conditional expectation equal to one multiplicative martingale increment (change of measure). p ij 8/16

14 DECOMPOSITION OF THE STOCHASTIC DISCOUNT FACTOR How are S and S related? h ij {}}{ p ij q ij = exp (η) e i p ij = exp (η) e i e j e j p ij }{{}}{{} s ij s ij ] [ hij has conditional expectation equal to one multiplicative martingale increment (change of measure). p ij A unique decomposition: Every stochastic discount factor in this environment has to have this form. A deterministic drift exp (η). A stationary component e i /e j. A martingale component h ij. 8/16

15 WHAT IS IDENTIFIED? q ij = exp (η) e i e j }{{} s ij p ij Given Q, we identified the eigenfunction-eigenvalue pair (e, η) = pair ( S, P) 9/16

16 WHAT IS IDENTIFIED? q ij = exp (η) e i p ij = exp (η) e i hij e j e j }{{}}{{} s ij s ij p ij Given Q, we identified the eigenfunction-eigenvalue pair (e, η) = pair ( S, P) What remains unidentified? the decomposition p ij = h ij p ij = pair (S, P) i.e., so far we learned nothing about P 9/16

17 HOW TO RECOVER INVESTORS BELIEFS? We must impose economic restrictions on the martingale component H. Ross (2015): H = 1. This implies P = P. Theory: H 1 and volatile recursive preferences consumption with stochastic growth Empirics: H 1 and volatile time series data + imposing rational expectations tests reject H = 1 in broad stock markets and bond markets Alvarez and Jermann (2005), Qin, Linetsky and Nie (2016), Bakshi, Chabi-Yo and Gao (2016), Audrino, Huitema and Ludwig (2016), 10/16

18 CONTINUOUS TIME / CONTINUOUS STATE SPACE Additional mathematical complications The counterpart to the eigenproblem Qe = exp (η) e does not generally have a unique solution. A unique recovered probability measure P that preserves stationarity and ergodicity of X t: S t+j S t = exp (ηj) }{{} deterministic trend e (X t) e (X t+j ) }{{} stationary component H t+j H t }{{} martingale Hansen and Scheinkman (2009), Borovička, Hansen and Scheinkman (2016) This does not address in any way the identification of H and hence of investors beliefs P from cross-sectional asset price data Q. 11/16

19 A SQUARE-ROOT PROCESS EXAMPLE Consider state variable X t following the square-root process 1 dx t = κ (X t µ) dt + σ X tdw t and prices generated by the true stochastic discount factor d log M t = βdt 1 2 α2 X tdt + α X tdw t risk-free rate β price of variance risk α X t risk-neutral price dynamics of X t with κ n = κ σα. dx t = κ n (X t µκ/κ n) + σ X tdw t 1 Borovička, Hansen, Scheinkman (2016), Example 4 12/16

20 RECOVERY In general infinitely many strictly positive eigenfunctions. Consider those of the form e (x) = exp (νx). Two solutions Which one to pick? ν = 0 ν = 2 (κ ασ) σ 2 Borovička, Hansen, Scheinkman (2016): At most one solution is such that the recovered probability measure preserves stationarity and ergodicity. This provides a unique way of decomposing M. But does not in any way address the problem of identifying H from Q. In fact, neither ν above recovers the original dynamics! 13/16

21 HESTON (2017) A standard estimation approach Impose parameterized physical and risk-neutral dynamics Infer stochastic discount factor ( M t = S γ t exp βt + ξv t + η t Dislike the martingale arising from t 0 vsds. = Y t. martingale in S t is fine 0 ) v sds Impose a restriction η = 0 (not needed here!) Given risk-neutral measure (prices), this restricts the physical measure. Estimate parameters γ, β, ξ using time-series information. 14/16

22 CONNECTION TO RECOVERY Original recovery problem Impose transition independence in stationary state variables. Here, v t is the only natural stationary state variable. Solution (given fixed r): γ = η = ξ = 0 = risk-neutrality. 15/16

23 CONNECTION TO RECOVERY Original recovery problem Impose transition independence in stationary state variables. Here, v t is the only natural stationary state variable. Solution (given fixed r): γ = η = ξ = 0 = risk-neutrality. This paper allows transition independence in nonstationary S t, too. New state variable S t = vastly expands the set of solutions. Anything goes : e.g., M t is transition independent in itself. Or use Y t = t vsds as another state variable. 0 See Borovička, Hansen and Scheinkman, Section 7. 15/16

24 CONNECTION TO RECOVERY Original recovery problem Impose transition independence in stationary state variables. Here, v t is the only natural stationary state variable. Solution (given fixed r): γ = η = ξ = 0 = risk-neutrality. This paper allows transition independence in nonstationary S t, too. New state variable S t = vastly expands the set of solutions. Anything goes : e.g., M t is transition independent in itself. Or use Y t = t vsds as another state variable. 0 See Borovička, Hansen and Scheinkman, Section 7. This makes the identification problem much worse. For conceptually different reasons than the misspecified recovery. Still unable to pin down H t from cross-sectional data. 15/16

25 Recovery Restrictions Restrictions on Recovered Parameters 60 Unrestricted Ksi Gamma

26 Restrictions on Risk Premia Restrictions on Risk Premia 10% Monthly variance Premium 0% 10% 20% 30% 40% 50% Unrestricted 60% 4% 3% 2% 1% Monthly Equity Premium 0% 1% 2%

27 TAKEAWAYS Recovery using cross-sectional information can pin down transitory component of SDF cannot pin down investors beliefs P without additional assumptions using non-stationary variables as states makes the identification problem worse What do we need time-series information economically motivated structural restrictions on the form of SDF And this is exactly what this paper aims for. 16/16