Generalized Recovery

Generalized Recovery Christian Skov Jensen Copenhagen Business School David Lando Copenhagen Business School and CEPR Lasse Heje Pedersen AQR Capital Management, Copenhagen Business School, NYU, CEPR December, 2017

Steve Ross One of the greats Ross APT Cox Ingesoll Ross Cox Ross Rubinstein Ross agency, signalling Recovery theorem American Finance Association, 2017

Recovery: the shock, the fight, the resolution Conventional wisdom: the expected return μ disappears from Black-Scholes formula recovery of μ and, more broadly, natural probabilities impossible

Recovery: the shock, the fight, the resolution Conventional wisdom: the expected return μ disappears from Black-Scholes formula recovery of μ and, more broadly, natural probabilities impossible The shock: the recovery theorem, Ross 2015, AQR Insight Award 2012

Recovery: the shock, the fight, the resolution Conventional wisdom: the expected return μ disappears from Black-Scholes formula recovery of μ and, more broadly, natural probabilities impossible The shock: the recovery theorem, Ross 2015, AQR Insight Award 2012 The fight: misspecified recovery, Borovicka-Hansen-Scheinkman 2015 This paper seeks to resolve these issues simple counting arguments (Sard s Theorem vs. Perron-Frobenius) examples and empirical implementation

Recovery: counting argument for impossibility Standard price formula state price = preferences probabilities π ij = δ uj u i p ij Impossibility of recovery: a counting argument in binomial model up current state down 2 equations 4 unknowns 1 probability (p up ) 3 preference parameters (δ, uup u 0, udown u 0 ) t=0 t=1

Recovery: the counting argument in Ross s framework Ross s recovery theorem: one period two parallel universes current state other state t=0 t=1 4 equations 4 unknowns 2 probabilities 2 preference parameters u (δ, up ) u down

Generalized recovery: counting works for any distribution t=0 t=1 t=2 Ross s recovery theorem 4 equations 4 unknowns 2 probabilities 2 preference parameters u (δ, up ) u down

Generalized recovery: counting works for any distribution t=0 t=1 t=2 Ross s recovery theorem 4 equations 4 unknowns 2 probabilities 2 preference parameters u (δ, up ) u down Our generalized recovery 4 equations 4 unknowns 2 probabilities 2 preference parameters t=0 t=1 t=2

Main results: theory Generalized recovery is generically possible when S T where S is #states, T is #time periods result allows for any probability distribution, incl. time-inhomogeneity Ross s recovery is a special case Strictly more general than Ross recovery We have uniqueness when Ross does Small set of parameters without uniqueness: even in Ross s case Closed-form approximate solution Flat term structure: no recovery Recovery for a large state space i.e., when S >> T, S can even be a continuous state space is possible if the pricing kernel is spanned by N < T parameters

Main results: empirics Recovery in specific models Mehra-Prescott (1985) Black-Scholes-Merton Non-Markovian model (outside the frameworks of Ross and Borovicka, Hansen, and Scheinkman) Empirical tests Does the recovered expected return Et (r t+1 ) predict future returns?

Literature review Recovery vs. misspecified recovery Ross (2015) Hansen and Scheinkman (2009), Borovicka, Hansen, and Scheinkman (2015) Rely on Perron-Frobenius Theorem we use Sard s Theorem We follow the tradition of General Equilibrium theory pioneered by Debreu (1970) (we are grateful to Ross for pointing out the link between our work and GE) Empirical implementation difficulties Bakshi, Chabi-Yo, and Gao (2015) Audrino, Huitema, and Ludwig (2014) Extensions Probability bounds without time-homogeneity, Schneider and Trojani (2015) Beyond bounded or discrete state space: Carr and Yu (2012), Linetsky and Qin (2015), Walden (2013) Term structure results: Martin and Ross (2016)

Overview of this talk Theory: Generalized Recovery and our counting argument Large state space Examples: Mehra-Prescott, Black-Scholes-Merton, and beyond Empirical analysis

The equation set From general asset pricing theory we are given π ts = p ts m ts Noah s ark model: two periods and two states π 11 π 21 π 11 = p 11 m 11 π 12 = (1 p 11 ) m 12 π 21 = p 21 m 21 π 12 π 22 π 22 = (1 p 21 ) m 22 t=0 t=1 t=2 4 equations with 6 unknowns

Reducing dimensionality: generalized recovery Ross uses Assumptions 1 and 2; we only use 1; 2 is violated in data Assumption 1 (Time-separable utility) There exists δ (0, 1] and u s > 0 s.t., for all times t, the pricing kernel is m ts = δ t us. u 1 Assumption 2 (Time-homogeneous probabilities) For all states i, j and time horizons τ > 0, p i,j t,t+τ does not depend on t. Our method: 4 equations and 4 unknowns (linear for given δ) Let hs = us and notice that h u 1 1 = 1 when the current state is 1 π 11 π 21 π 11 = p 11 δ π 12 = (1 p 11 ) δh 2 π 21 = p 21 δ 2 π 12 π 22 π 22 = (1 p 21 ) δ 2 h 2 t=0 t=1 t=2

Intuition: our general counting argument S unknown preference parameters: S 1 marginal utility ratios 1 discount rate For each of the T time periods: S equations S 1 probabilities 1 extra equation for each time period Hence, we need T S

Solving the generalized recovery problem General equation set with S states T time periods: Π = DPH π 11... π 1S δ... 0 p 11... p 1S 1... 0.. =...... π T 1... π TS 0... δ T p T 1... p TS 0... h S

Solving the generalized recovery problem General equation set with S states T time periods: Π = DPH π 11... π 1S δ... 0 p 11... p 1S 1... 0.. =...... π T 1... π TS 0... δ T p T 1... p TS 0... h S Eliminate P probabilities: ΠH 1 e = De 1 h2 1 δ Π... =. hs 1 δ T Separate Π into block matrices Π = [ [ ] ] Π11 Π Π 1 Π 2 = 12 Π 21 Π 22 [ ] [ δ Π11 Π12 + Π 21 Π 22 ] h 1 2. = hs 1. δ T

Solving the generalized recovery problem Solving the first S 1 equations for (h2 1,..., h 1 S ) h2. 1 hs 1 = Π 1 12 δ. δ S 1 The remaining T S + 1 equations gives us h 1 2 Π 21 + Π... 22 hs 1 = π 11. π S 1,1 δ S+1. δ T We end up with T S + 1 equations, each of which involves a polynomium in δ of degree a most T

Generalized recovery: main result Proposition 1 (Generalized Recovery) The recovery problem has 1. a continuum of solutions if S > T 2. at most S solutions if the matrix Π has full rank and S = T 3. no solution generically in terms of an arbitrary Π matrix of positive elements and S < T 4. a unique solution generically if Π has been generated by the model and S < T

Generalized recovery: main result Proposition 1 (Generalized Recovery) The recovery problem has 1. a continuum of solutions if S > T 2. at most S solutions if the matrix Π has full rank and S = T 3. no solution generically in terms of an arbitrary Π matrix of positive elements and S < T 4. a unique solution generically if Π has been generated by the model and S < T Proofs based on Sard s theorem, which formalizes the counting argument without relying on the probability distribution, stationary etc. (as Perron- Frobenius theorem) Parts 1-2: Clear from calculations above Part 3: Consider the mapping of all possible (δ, h, P) to prices Π. The image has measure zero by Sard because the dimension(π) = ST > Dimension(δ, h, P) = 1 + (S 1) + T (S 1) = TS + S T Part 4: Still Sard, but more involved dimension argument

What About Uniqueness as in Ross? Steve Ross: Why do you have up to S solutions when my method ensures uniqueness?

What About Uniqueness as in Ross? Steve Ross: Why do you have up to S solutions when my method ensures uniqueness? Proposition 2 (Generalized Recovery Works in a Ross Economy) When Ross Recovery Theorem has a solution then our Generalized Recovery generates the same solution For almost all price matrices ˉΠ corresponding to a Ross economy, there is a unique solution to our generalized problem as well Can we get rid of almost? No (counterexamples exist). Same problem is hidden in Ross: Different one-period Ross economies may be observationally equivalent when we do not observe parallel universes (i.e., they generate the same state prices seen from the current state) I.e., if you use Ross and start with state prices seen from current state, non-uniqueness problem happens when trying to find parallel universe prices (which Ross take as given) A verbal version - for exact statement see paper

Strictly More General Proposition 3 (Generalized Recovery is More General) When S = T, there exists a set of parameters for the generalized problem with positive Lebesgue measure for which no solution exists for Ross problem When S < T and parameters are chosen from the generalized recovery set-up, then generically Ross problem has no solution A verbal version - for exact statement see paper

Closed-form solution The recovery problem is almost linear, except δ τ NB: δ is the simplest parameter we know it is around 0.98 If we fix δ, the problem is linear and we are done Taylor approximation in δ around δ 0 (e.g., t = 2, δ 0 = 0.97) 1.00 δ τ δ τ 0 + τδτ 1 0 (δ δ 0 ) = a τ + b τ δ 0.95 0.90 0.85 a + bδ δ t 0.80 0.90 0.92 0.94 0.96 0.98 1.00 δ The generalized recovery problem is now linear Π 1 + Π 2 h 1 2. h 1 S = a 1 + b 1 δ. a T + b T δ

Closed-form solution Rewrite the T equations in S unknowns as b 1 π 12... π 1S... b T π T 2... π TS In matrix form: δ h 1 2. h 1 S B h δ = a Π 1 a 1 π 11 =. a T π T 1 Closed-form solution { B h δ = 1 (a π 1 ) for S = T (B B) 1 B (a π 1 ) for S < T

Flat Term Structure Proposition 4 (Flat Term Structure) Suppose that the term structure of interest rates is flat, i.e., there exists r > 0 such that 1 = (Πe) (1+r) τ τ for all τ = 1,..., T. Then the recovery problem is solved with equal physical and risk-neutral probabilities, P = Q. This means that either the representative agent is risk neutral or the recovery problem has multiple solutions. Similar to Ross

Allowing for a large state space Assumption 1* (General utility with N parameters) The pricing kernel at time τ in state s (given the initial state 1 at time 0) can be written as m τs = δ τ h s (θ) (1) where δ (0, 1] and h( ) is a smooth function from R N to R S. Proposition 5 (Large State Space) If Assumption 1* holds and N + 1 < T, then the recovery problem generically has 1. no solution for an arbitrary Π 2. a unique solution if Π has been generated by the model.... t=0 t=1 t=2

Large state space: the linear case Suppose S >> T and the inverse pricing kernel is linear in θ h 1 1. h 1 S = a 1 b 11... b 1N. +.. a S b S,1... b S,N θ 1. θ N Combining this equation with the recovery problem gives h 1 1 Π... hs 1 = ΠA + ΠB θ 1... θ N This has the same form as in Proposition 1 = δ. δ T = A + Bθ (2) (3) A Taylor approximation in δ around ˉδ returns us to Proposition 2

Large state space: a non-linear case Assumption 1* also allows marginal utilities be non-linear function of the preference parameters θ For example, consider the CRRA case ( ) θ m sτ = δ τ cs c 1 The large-state-space method also works for continuous state spaces

Recovery in specific economies: Mehra-Prescott (1985) Binomial tree, where the up/down probability changes over time (a Markov process) and preferences are CRRA Generalized recovery not feasible in full generality since S > T is feasible if we know that preferences are CRRA recovering the discount rate δ and risk aversion γ:

Specific economies: Black-Scholes-Merton and iid. growth Binomial tree where up/down probabilities are constant (iid. growth) implies a flat term structure Recovery is not feasible even if we know that preferences are CRRA:

Specific economies: non-markovian Binomial tree where up/down probabilities are random each state outside the frameworks of Ross and Hansen et al. Generalized recovery feasible if we know that preferences are CRRA:

Empirical implementation Data from OptionMetrics S&P 500 index call/put options, January 1996 to June 2014

Expected returns across recovery methods: Correlations Let μ t,i = E t (r t+1 r f t ) recovered using method i. Correlations: Good news: positive and fairly large correlation between excess expected returns obtained using the four methods Bad news: far from perfect correlation indicating that robustness is an issue Slightly stronger association with SVIX than VIX

Recovered expected excess returns

Recovered conditional volatilities

Does the recovered expected return predict future return? r t,t+1 = β 0 + β 1 μ t + β 2 Δμ t+1 + ɛ t,t+1 Δμ t+1 = μ t+1 E t μ t+1 is ex post innovation in expected return Monthly data, 1/1996-12/2015, t-stats in parentheses, 10% significance in bold

Does the recovered expected return predict future return - excluding crisis? r t,t+1 = β 0 + β 1 μ t + β 2 Δμ t+1 + ɛ t,t+1 Monthly data, 1/1996-12/2015, excluding 8/2008-7/2009, t-stats in parentheses, 10% significance in bold

Does the recovered volatility predict future volatility? var(r t,t+1 ) = β 0 + β 1 σ t + ɛ t,t+1 Monthly data, 1/1996-12/2015, t-stats in parentheses, 10% significance in bold

Conclusion Simple characterization of when recovery is feasible vs. not Generalizing Ross s method based on methods also used in GE Practically implementable in closed form Empirical analysis