A Recovery That We Can Trust? Deducing and Testing the Restrictions of the Recovery Theorem

Transcription

1 A Recovery That We Can Trust? Deducing and Testing the Restrictions ohe Recovery Theorem Gurdip Bakshi Fousseni Chabi-Yo Xiaohui Gao University of Houston December 4, 2015 Bakshi & Chabi-Yo & Gao Test restrictions ohe Recovery theorem December 4, / 25

2 The Ross Recovery Theorem (Ross, JF 2015) Asserts that one can separate and identify the natural probability distribution from risk aversion and discounting (the stochastic discount factor) Enormous practical implications, with overtones that OPTIONS markets are omniscient. q density {}}{ q[z t+1 ] = SDF p density {}}{{}}{ E t (ξ t+1 z t+1 ) p[z t+1 ] E, t (ξ t+1 z t+1 ) p[z t+1 ]dz t+1 Bakshi & Chabi-Yo & Gao Test restrictions ohe Recovery theorem December 4, / 25

3 Borovicka, Hansen, and Scheinkman (JF, forthcoming) Show that the treatment in Ross implies that the martingale component ohe stochastic discount factor is an invariant constant equal to unity. But when the martingale component is not trivial, the recovered probabilities MAY not the same as those used by investors. Bakshi & Chabi-Yo & Gao Test restrictions ohe Recovery theorem December 4, / 25

4 Debate rages on... Partial list ohe papers in the conversation Carr and Yu (2012) Martin and Ross (2013) Dubynskiy and Goldstein (2013) Walden (2014) Audrino, Huitema, and Ludwig (2015) Backwell (2015) Qin and Linetsky (2015) Tran and Xia (2015) Schneider and Trojani (2015) Bakshi & Chabi-Yo & Gao Test restrictions ohe Recovery theorem December 4, / 25

5 What do we do? While recovery may not be possible in general terms as counter-examples can refute, our approach is to examine the reliability ohe recovery theorem. Our motivation is that finance theory has derived much of its analytical power and appeal from a few simplifying assumptions, and the defence of a proposed theory eventually resides in its empirical validity. We use the setting ohe long-term bond market (30-year Treasury), as testing may be more difficult in the context ohe market index and individual stocks (can be onerous to extract the transitory component and parameterize the finite-state Markov chain). Bakshi & Chabi-Yo & Gao Test restrictions ohe Recovery theorem December 4, / 25

6 Hypothesis development in the long-term bond market Let mt P be the martingale component and mt T be the transitory component With the treatment mt+1 P = 1, the SDF in the long-term bond market can be expressed as: m t+1 = mt+1m P t+1, T (1) = mt+1 T = 1, (from Alvarez and Jermann (2005)) R t+1, (2) 1 =, (spot-futures no arbitrage) (3) R t+1,rf +1 ( ) 1 ft+1 = e zt+1, defining z t+1 log (,+ ). (4) R t+1,rf This is the form ohe SDF under the treatment of Ross in the long-term bond market. Bakshi & Chabi-Yo & Gao Test restrictions ohe Recovery theorem December 4, / 25

7 Hypothesis development (continued) We consider claims written on the futures ohe long-term discount bond, and let p[z t+1 ] be the physical density of z t+1 log( ft+1 ). The associated risk-neutral density is: q[z t+1 ] = e zt+1 p[z t+1 ] e zt+1 p[z t+1 ]dz t+1 = e zt+1 p[z t+1 ], sincee P t(e zt+1 ) = 1, (5) which can be interpreted as the Esscher transform ohe physical density p[z t+1 ]. For an arbitrary parameter n: E Q t (e (n+1)zt+1 ) = = + + e (n+1)zt+1 q[z t+1 ]dz t+1, (6) e (n+1)zt+1 e zt+1 p[z t+1 ]dz t+1, (7) = E P t (e nzt+1 ). (8) Bakshi & Chabi-Yo & Gao Test restrictions ohe Recovery theorem December 4, / 25

8 Hypothesis development The key takeaway is that the Ross recovery theorem implies a restriction on the moment generating function ohe physical and the risk-neutral distributions: E P t ({ ft+1 } n ) ( ) = E Q t e (n+1)z t+1 }{{} recovered from state prices The function on the right-hand side can be spanned and valued using out-of-the-money options on the 30-year Treasury bond futures. All return moments under P can be recovered via Binomial formula. (9) Bakshi & Chabi-Yo & Gao Test restrictions ohe Recovery theorem December 4, / 25

9 Hypothesis development ( ) One can synthesize E Q t e (n+1)z t+1 from out-of-the-money option prices on the Treasury bond futures as: ( ) E Q t e (n+1)z t+1 = 1 ( + n(n+1)r ( ) n 1 ( ) n 1 t+1,rf K K C t[k]dk + P t[k]dk), ft 2 where C t[k] (P t[k]) is the price ohe one-period European call (put) option with strike price K. Thus, one central testable restriction is that, for n 1, ( ) e nz t+1 where E P t x (n+1) t x (n+1) t E Q t K> 1 K< = 0, (10) ( ) e (n+1)z t+1. (11) Bakshi & Chabi-Yo & Gao Test restrictions ohe Recovery theorem December 4, / 25

10 Implications for Return and Volatility Forecasts We will examine return and volatility forecasts, relying on the recovery theorem. Framework is more tractable than the finite-state Markov chain setting. Via Binomial formula, our expressions match those from finite-state Markov chain. E t (R t+1, ) R t+1,rf = 1 R t+1,rf Var Q t [R t+1, ]. (12) Thus, the above result corresponds to the one in Martin and Ross (2013, Result 7), and an equivalence can also be established for n > 1. The linchpin ohe Ross recovery theorem is mt+1 P = 1, and our work is aimed at empirically understanding the nuances underlying mt+1 P = 1. Bakshi & Chabi-Yo & Gao Test restrictions ohe Recovery theorem December 4, / 25

11 Data We use daily data of futures on the 30-year Treasury bond, and the daily data of out-of-the-money options traded on the 30-year Treasury bond futures from 1985:01 to 2013:12. We build one set of option prices at the end ohe month, and two time series of option returns: Nearest maturity options at the monthly frequency: These options usually expire on the last Friday, at least two business days from the last business day of the next month. The options so constructed have an average maturity of 27 days. Returns of a straddle: The gross return of a straddle at the end of each month t is constructed as: R t+1,straddle max(+1 K,0)+max(K +1,0), where C[K] and P[K] C[K]+P[K] are respectively the prices of calls and puts, with moneyness closest to zero. Returns of a 3% out-of-the-money put: At the end of each month, we search for a put that is closest to 3% out-of-the-money. The grosss return is constructed as: R t+1,otm put max(k +1,0) P[K], where K e Bakshi & Chabi-Yo & Gao Test restrictions ohe Recovery theorem December 4, / 25

12 Convex Minimization Problem to Extract the Martingale Component Does Data Support m P t = 1, as under the Treatment of Ross? We pose the following convex optimization problem: inf 2 E((mP ) 2 ) (13) ( ) subject to E m P Z = 0, E(m P ) = 1, m P 0. (14) m P 1 We also look at an alternative: inf m P E(m P log(m P )). Note R t+1 (R t+1,,r t+1,j) and Z t+1 R t+1 R t+1,rf 1 R t+1,, (15) where 1 is a conformable vector of ones. Moreover, R t+1,j is a K 1 vector of gross returns that excludes the long-term bond return, and Z t+1 is a vector of excess returns (over the riskfree return) divided by the gross return ohe long-term bond. Bakshi & Chabi-Yo & Gao Test restrictions ohe Recovery theorem December 4, / 25

13 Case 1 Solution We consider properties that incorporates nonnegativity of m P and the optimal solution is: m P t = max ( ν +λ Z t,0 ), and min (λ,ν ) ν E( [(ν +Z λ) 1 {ν+z λ 0}] 2). Bakshi & Chabi-Yo & Gao Test restrictions ohe Recovery theorem December 4, / 25

14 Case 2 Solution Consider ψ[m P ] = m P log(m P ) for m P > 0. The optimal solution can be characterized as: mt P = exp ( 1+ν +λ ) Z t, (16) where (λ,ν ) solve inf (λ,ν) ν + E ( exp ( 1+ν +λ Z )). (17) The exponential form ohe solution ensures m P > 0. Since E(ψ[m P ]) ψ[e(m P )] = E( 1 }{{} 2 (mp 1) (mp 1) (mp 1) 4 +O((m P 1) 5 )), =0 the objective weights higher-order moments of m P. Bakshi & Chabi-Yo & Gao Test restrictions ohe Recovery theorem December 4, / 25

15 Test Assets With the understanding that R t+1, = R t+1,rf +1, we consider the following Z: Z t+1 = Also for robustness or Z t+1 = Z t+1 = 1 R t+1,rf +1 1 R t+1,f +1 1 R t+1,f +1 f R t+1 t+1,rf R t+1,rf R t+1,3% otm put R t+1,rf R t+1,1% otm put R t+1,rf R t+1,1% otm call R t+1,rf. R t+1,3% otm call R t+1,rf ( Rt+1,f +1 R t+1,f R t+1,straddle R t+1,f ( Rt+1,f +1 R t+1,f R t+1,otm put R t+1,f ) ). Bakshi & Chabi-Yo & Gao Test restrictions ohe Recovery theorem December 4, / 25

16 Table 1: Importance of martingale component, bond market Panel A: Panel B: The convex function ψ[m P ] is 1 2 (mp ) 2 The convex function ψ[m P ] is m P log(m P ) Martingale component Martingale component Variance Skewness Kurtosis ρp,t Variance Skewness Kurtosis ρp,t Solution Block bootstrap Mean Std th th th th th p-val., ρ P,T = 0 {0.064} {0.069} Bakshi & Chabi-Yo & Gao Test restrictions ohe Recovery theorem December 4, / 25

17 Table Appendix I: Importance of martingale component, bond market straddles Z contains the long-term bond and the straddle Z contains the long-term bond and the 3% out-of-the-money put option Variance Skewness Kurtosis ρp,t Variance Skewness Kurtosis ρp,t Panel A: The convex function ψ[m P ] is 1 2 (mp ) 2 Solution Block bootstrap Mean Std th th th th th p-val., ρ P,T = 0 {0.073} {0.070} Bakshi & Chabi-Yo & Gao Test restrictions ohe Recovery theorem December 4, / 25

18 Table 2: How Reliable is the Recovery Theorem? Define: y (n) t+1 enz t+1, and ǫ (n) t+1 y(n) t+1 EP t(y (n) t+1 ). (18) Then the empirical restriction via an OLS regression is: y (n) t+1 = α(n) + β (n) x (n+1) t ( e (n+1)z t+1 ) = E P t ( { +1 + ǫ (n) t+1, for n 1, ) } n where x (n+1) t E Q t option prices, as described in equation (10) at the end of date t. and is synthesized using α (n) = 0 and β (n) = 1, for n 1. (19) Reminiscent ohe Fama (1984) and Campbell-Shiller regressions. Bakshi & Chabi-Yo & Gao Test restrictions ohe Recovery theorem December 4, / 25

19 Table 2: Adequacy of return forecasts relying on the recovery theorem and the options on the 30-year Treasury bond futures Dependent variable α β R 2 DW Wald test CORR(y [n] t+1,x[n+1] t ) (%) α = 0, β = 1 Panel A: Using options data to recover the first physical return moment Gross futures return: NW[p] [0.010] [0.000] (0.03) H[p] Panel B: Using options data to recover the physical return variance Variance of futures return: NW[p] [0.000] [0.000] (0.00) H[p] Bakshi & Chabi-Yo & Gao Test restrictions ohe Recovery theorem December 4, / 25

20 Digging deeper into the reliability ohe recovery theorem We compute the deviations: e mean t t+1 y [1] ( e volatility t t+1 t+1 x[2] t, Var t ( ft+1 1 ) ( ) x [3] t {x [2] ft+1 t } )/ 2 Var t 1 which reflects the difference between the realized value and the theoretical counterpart recovered from option prices. The distribution ohe deviations are: Mean Std. 5th 25th 50th 75th 95th t t t t e mean e volatility Bakshi & Chabi-Yo & Gao Test restrictions ohe Recovery theorem December 4, / 25

21 Are deviations from the recovery theorem related to the number of available option strikes? We examine the absolute errors binned across the quintiles ohe number of option strikes: Q1 Q2 Q3 Q4 Q5 Number of puts and calls e t t+1 mean e volatility t t Bakshi & Chabi-Yo & Gao Test restrictions ohe Recovery theorem December 4, / 25

22 Plotted is the time series ohe deviations from the recovery theorem implied quantities, Bakshi & Chabi-Yo & Gao Test restrictions ohe Recovery theorem December 4, / 25 Graphs of deviation Deviation in the mean 0 Deviation in the volatility (%) Feb 85 March 89 May 93 July-97 Sep-01 Nov-05 Jan-10 March-14 time -0.8 Feb 85 March 89 May 93 July-97 Sep-01 Nov-05 Jan-10 March-14 time Figure 1: Deviations ohe recovery theorem implied quantities versus the data

23 Table 3: GMM Restrictions on the Recovered SDF Staying in the same parametric class, we introduce the parameter δ. Under Ross recovery theorem γ = 0. The estimation is for a single equation using Hansen s (1982) GMM: E P( ) u (1) t+1 I t = 0, E P( ) u (2) t+1 I t = 0, where the disturbance terms are defined as u (1) t+1 η e (1 δ1)zt+1 1 1, u (2) x (2) t+1 η 2 t e (2 δ2)zt+1 x (3) t 1, for some parameters (η 1,δ 1 ) and (η 2,δ 2 ). Our interest is again on the first and second moments ohe physical distribution. Bakshi & Chabi-Yo & Gao Test restrictions ohe Recovery theorem December 4, / 25

24 Table 3: Testing the link between the return quantities implied by the recovery theorem and the data: GMM estimation results η 1 δ 1 J T GMM estimates ( ) E u [1] t+1 It = [0.00] [0.00] (0.97) η 2 δ 2 J T GMM estimates ( ) E u [2] t+1 It = [0.00] [0.00] (0.99) Bakshi & Chabi-Yo & Gao Test restrictions ohe Recovery theorem December 4, / 25

25 Conclusions The treatment in Ross (2015) implies that the martingale component of SDF is a unity. Our approach reveals that the martingale component of SDF exhibits pronounced volatility, is positively skewed, and fat-tailed. We deduce testable restrictions based on the moment generating functions under Q and P. Agnostic view ohe recovery theorem Bakshi & Chabi-Yo & Gao Test restrictions ohe Recovery theorem December 4, / 25