Jumps, Realized Densities, and News Premia

Transcription

1 Jumps, Realized Densities, and News Premia Paul Sangrey University of Pennsylvania February 6, 019

2 Return Dynamics S&P 500 Log-Return 1-Second Daily Apr 16, 10:00 Apr 16, 1:00 Apr 16, 4:00 Mar 01 Jul 01 Nov 01 Red lines are large jumps. How do price jumps affect the investors time-varying risk? 1 / 40

3 What Characterizes Investors Time-Varying Risk? What summarizes the information returns time-varying densities: h(r t F t 1 )? / 40

4 What Characterizes Investors Time-Varying Risk? What summarizes the information returns time-varying densities: h(r t F t 1 )? When prices are continuous, the diffusion volatility σ (t) does. 1. Volatility summarizes distributional dynamics.. We can estimate σ (t) using realized volatility or bipower variation. 3. Expected returns are instantaneous covariances. / 40

5 What Characterizes Investors Time-Varying Risk? What summarizes the information returns time-varying densities: h(r t F t 1 )? When prices are continuous, the diffusion volatility σ (t) does. 1. Volatility summarizes distributional dynamics.. We can estimate σ (t) using realized volatility or bipower variation. 3. Expected returns are instantaneous covariances. If prices jump, we need an entire density A Realized Density. / 40

6 What Characterizes Investors Time-Varying Risk? What summarizes the information returns time-varying densities: h(r t F t 1 )? When prices are continuous, the diffusion volatility σ (t) does. 1. Volatility summarizes distributional dynamics.. We can estimate σ (t) using realized volatility or bipower variation. 3. Expected returns are instantaneous covariances. If prices jump, we need an entire density A Realized Density. 1. Existing jump variation measures don t summarize distributional dynamics.. We can t estimate time-varying tail risk. 3. Expected returns aren t just instantaneous covariances. / 40

7 Talk Structure / Contributions 1. Propose a new measure jump volatility a) that yields a tractable, nonparametric expression for the realized density. b) that measures jumps in investor s information (news) 3 / 40

8 Talk Structure / Contributions 1. Propose a new measure jump volatility a) that yields a tractable, nonparametric expression for the realized density. b) that measures jumps in investor s information (news). a) Derive this expression from no-arbitrage. b) Derive estimators for the return s volatilities and realized density. c) Estimate the volatilities using high-frequency data on the S&P 500 providing new stylized facts. 3 / 40

9 Talk Structure / Contributions 1. Propose a new measure jump volatility a) that yields a tractable, nonparametric expression for the realized density. b) that measures jumps in investor s information (news). a) Derive this expression from no-arbitrage. b) Derive estimators for the return s volatilities and realized density. c) Estimate the volatilities using high-frequency data on the S&P 500 providing new stylized facts. 3. a) Show the volatilities jointly determine risk premia. b) Show the jump volatility premium is less than the diffusive volatility premium. 3 / 40

10 Jumps := Discontinuities in the Price Process. What are Jumps? Price changes caused by discrete (possibly small) releases of information. They may come at unpredictable times. They may be observed by only a few investors. Examples: FOMC Announcements, (Andersen, Bollerslev, Diebold, and Vega 003) and (Engelberg 008; Gürkaynak, Kısacıkoǧlu, and Wright 018, WP). A startup announcing a new product line. Effectively anything in a Bloomberg or Associated Press feed relevant for asset pricing. Private communications between investors. Theorem 4 / 40

11 Literature Review 1. Modeling prices in continuous-time: a) The stochastic volatility diffusion case: Barndorff-Nielsen and Shephard (00), Andersen, Bollerslev, Diebold, and Labys (003), and Mykland and Zhang (009) b) Allowing for jumps (We need infinitely many): Barndorff-Nielsen and Shephard (004), Aït-Sahalia and Jacod (009a, 009b, 01), Bollerslev and Todorov (011), and Gallant and Tauchen (018) 5 / 40

12 Literature Review 1. Modeling prices in continuous-time: a) The stochastic volatility diffusion case: Barndorff-Nielsen and Shephard (00), Andersen, Bollerslev, Diebold, and Labys (003), and Mykland and Zhang (009) b) Allowing for jumps (We need infinitely many): Barndorff-Nielsen and Shephard (004), Aït-Sahalia and Jacod (009a, 009b, 01), Bollerslev and Todorov (011), and Gallant and Tauchen (018). Representing prices in continuous-time: a) Dambis (1965), Dubins and Schwarz (1965), Duffie, Pan, and Singleton (000), and Todorov and Tauchen (014) 5 / 40

13 Literature Review 1. Modeling prices in continuous-time: a) The stochastic volatility diffusion case: Barndorff-Nielsen and Shephard (00), Andersen, Bollerslev, Diebold, and Labys (003), and Mykland and Zhang (009) b) Allowing for jumps (We need infinitely many): Barndorff-Nielsen and Shephard (004), Aït-Sahalia and Jacod (009a, 009b, 01), Bollerslev and Todorov (011), and Gallant and Tauchen (018). Representing prices in continuous-time: a) Dambis (1965), Dubins and Schwarz (1965), Duffie, Pan, and Singleton (000), and Todorov and Tauchen (014) 3. Modeling and measuring volatility and news premia: a) Modeling the continuous case: Sharpe (1964), Merton (1973), and Bollerslev, Engle, and Wooldridge (1988) b) Measuring volatility and news premia: Ghysels, Santa-Clara, and Valkanov (005), Lettau and Ludvigson (010), Lucca and Moench (015), and Weller (018) c) Allowing for jumps (in information): Drechsler and Yaron (011), Borovička and Hansen (014), Ai and Bansal (018), and Wachter and Zhu (018, WP) 5 / 40

14 Data Generating Process Price Process (Diffusion) Wiener Process dp(t) = σ dw(t) Diffusion (dp D ) 6 / 40

15 Data Generating Process Price Process (Stochastic Volatility Diffusion) Diffusion Volatility Wiener Process dp(t) = σ(t) dw(t) Diffusion (dp D ) 6 / 40

16 Data Generating Process Price Process (Stochastic Volatility Diffusion with Poisson Jumps) Diffusion Volatility Wiener Process Poisson Process dp(t) = σ(t) dw(t) Diffusion (dp D ) + dn(t) Jumps (dp J ) 6 / 40

17 Data Generating Process Price Process (Stochastic Volatility Jump Diffusion) Diffusion Volatility Wiener Process dp(t) = σ(t) dw(t) Diffusion (dp D ) Jump Magnitudes Poisson Random Measure + δ(t, x)(n ν)(dt, dx) R Jumps (dp J ) 6 / 40

18 Data Generating Process Price Process (Stochastic Volatility Jump Diffusion) Diffusion Volatility Wiener Process dp(t) = σ(t) dw(t) Diffusion (dp D ) Jump Magnitudes Poisson Random Measure + δ(t, x)(n ν)(dt, dx) R Jumps (dp J ) Returns r t := t t 1 dp(s), r t F t 1 h (r t F t 1 ) 6 / 40

19 The Literature Focuses on Models of the Form: r t F t 1 h (r t F t 1 ) = f (r t x t ) dg (x t F t 1 ) x t Sufficient statistic (factor) for the dynamics How should we model h(r t F t 1 )? What should we use for x t? What should we use for f? What should we use for G? Example: Simple Stochastic Volatility Model r t σ t N(0, 1) log σ t = ρ log σ t 1 + σ σ N(0, 1) x t is the volatility σ t. f ( r t σ t ) is N(0, σ t ). G is AR(1). 7 / 40

20 Previous Work (Diffusion) Volatility Merton (1973) σ 1 (t) := lim 0 E t+ t dp D (s) F t 8 / 40

21 Previous Work (Diffusion) Volatility Merton (1973) σ 1 (t) := lim 0 E t+ t dp D (s) F t Realized Density Barndorff-Nielsen and Shephard (00), Andersen, Bollerslev, Diebold, and Labys (003) RD t := f ( r t σt ) ( t ) xt=[ = f t t 1 σ (s) ds] = N 0, σ (s) ds t 1 8 / 40

22 This Paper (Jump Diffusion) Diffusion Volatility [ σ 1 (t) := lim 0 E t+ t Two Kinds of Volatility: dp D ] (s) F t Details 9 / 40

23 This Paper (Jump Diffusion) Two Kinds of Volatility: Diffusion Volatility [ σ 1 (t) := lim 0 E t+ t dp D ] (s) F t Jump Volatility [ γ 1 (t) := lim 0 E t+ t dp J ] (s) F t Details 9 / 40

24 This Paper (Jump Diffusion) Two Kinds of Volatility: Diffusion Volatility [ σ 1 (t) := lim 0 E t+ t dp D ] (s) F t Jump Volatility [ γ 1 (t) := lim 0 E t+ t dp J ] (s) F t Realized Density: (Derived Below) RD t := f ( r t σ t, γt ) = f t x t= t t 1 σ (s) ds t 1 γ (s) ds Details 9 / 40

25 This Paper (Jump Diffusion) Two Kinds of Volatility: Diffusion Volatility [ σ 1 (t) := lim 0 E t+ t dp D ] (s) F t Jump Volatility [ γ 1 (t) := lim 0 E t+ t dp J ] (s) F t Realized Density: (Derived Below) RD t := f ( r t σ t, γt ) = f t x t= t t 1 σ (s) ds t 1 γ (s) ds ( t ) = N 0, σ (s) ds t 1 Laplace Distribution Convolution ( t ) L 0, γ (s) ds t 1 Details 9 / 40

26 How Should We Model Daily Returns? Recall: r t F t 1 h (r t F t 1 ) = f (r t x t ) dg (x t F t 1 ) x t Sufficient statistic (factor) for the dynamics 10 / 40

27 How Should We Model Daily Returns? Recall: r t F t 1 h (r t F t 1 ) = f (r t x t ) dg (x t F t 1 ) x t Sufficient statistic (factor) for the dynamics Two Factors: x t = σ t and γ t 10 / 40

28 How Should We Model Daily Returns? Recall: r t F t 1 h (r t F t 1 ) = f (r t x t ) dg (x t F t 1 ) x t Sufficient statistic (factor) for the dynamics Two Factors: Model: x t = σ t and γ t r t = σ t N(0, 1) + γ t L(0, 1) ( ) σ t G (F t 1 ) γ t 10 / 40

29 Talk Structure / Contributions (Again) 1. Proposed a new measure of jump volatility: γ t. a) that yields r t N ( ) ( ) 0, σt L 0, γ t b) that measures jumps in investor s information (news). a) Derive r t N ( ) ( ) 0, σt L 0, γ t from no-arbitrage. b) Derive estimators for σ t, γ t, and RD t. c) Apply these estimators to high-frequency data on the S&P 500 providing multiple new stylized facts. 3. a) Show σ t and γ t jointly determine risk premia. b) Show the γ t premium is less than the σ t premium. 11 / 40

30 Derive r t N ( 0, σ t ) L ( 0, γ t ) from no-arbitrage. Why does dp J (t) = δ(t, x)(n ν) dx = R t t 1 ( t ) dp J (t) L 0, γ (s) ds? t 1

31 Properties of Laplace Distributions PDF: 1 exp ( x ) S exp(1), Z N(0, 1) = S Z L(0, 1) Laplace Density Gaussian Density Standardized random sums with a geometrically-distributed number of terms L geometric-stable, (Klebanov, Maniya, and Melamed 1985; Mittnik and Svetlozar 1993) Laplace is a geometric-stable distribution with finite variance. 1 / 40

32 Standard Variance-Gamma Process A variance-gamma process with zero mean and all scale parameters equal to one. The variance-gamma process is used to represent processes with frequent jumps. - Option prices easily computed. Its increments are i.i.d. Laplace random variables. = It is a Lévy process. Compound Poisson arrival-rate, jumps in every interval, finite variance, Gaussian magnitudes = standard variance-gamma 13 / 40

33 The First Time-Changed Laplace Theorem Assumptions: 1. p(t) is a semimartingale. (Delbaen and Schachermayer (1994) showed no-arbitrage is a sufficient condition.). p J (t) has infinite-activity jumps. (p J (t) jumps in every interval.) 3. p J (t) is locally-square integrable. (r t has finite variance.) 4. p J (t) has no predictable jumps. Theorem p J (t) time-changed by its predictable quadratic variation is a standard variance-gamma process. Details 14 / 40

34 Proof Intuition A jump process has two kind of variation: jump magnitudes and jump locations. Partition the y-axis. 1. These locations are conditionally independent across strips.. They form time-changed Poisson processes, (Time between jumps is distributed exp(1).) Condition on the jump times in each strip. 1. This is a standardized sum of infinitely-many i.i.d. random variables.. The is a time-changed Wiener process. (C.L.T.) Magntidues Locations Integrating out the locations and taking limits implies the original process is a time-changed variance-gamma process. 15 / 40

35 What is the Appropriate Time Change? We need the composition of the previous time-changes. Changing the magnitude or the intensity affects the variance in the same way. = We need the predictable quadratic variation of the original process p J (t). γ (t) is the time-derivative of p J (t). Details 16 / 40

36 Continuous-Time Model Recall: dp(t) = σ(t) dw(t) + δ(t, x)(n ν)(dt, dx) R Details 17 / 40

37 Continuous-Time Model Recall: Allow for drift µ(t): dp(t) = σ(t) dw(t) + δ(t, x)(n ν)(dt, dx) R dp(t) = µ(t) dt + σ(t) dw(t) + δ(t, x)(µ ν)(dt, dx) R Details 17 / 40

38 Continuous-Time Model Recall: Allow for drift µ(t): Simplify jump representation: dp(t) = σ(t) dw(t) + δ(t, x)(n ν)(dt, dx) R dp(t) = µ(t) dt + σ(t) dw(t) + δ(t, x)(µ ν)(dt, dx) R dp(t) = µ(t) dt + σ(t) dw(t) + δ(t, x)(n ν)(dt, dx) R γ(t) dl(t) Details 17 / 40

39 How Should We Model Continuous-Time Returns? Proposed (Conditionally) Exponentially-Affine (Duffie, Pan, and Singleton 000; Calvet, Fearnley, Fisher, and Leippold 015) x(t) := ( σ (t) γ (t) ) x(t) := ( σ (t) M(t) ) 18 / 40

40 How Should We Model Continuous-Time Returns? Proposed (Conditionally) Exponentially-Affine (Duffie, Pan, and Singleton 000; Calvet, Fearnley, Fisher, and Leippold 015) x(t) := ( σ (t) γ (t) ) x(t) := ( σ (t) M(t) ) ( ) σ (t) dp(t) = µ 1 γ dt+σ(t) dw 1 (t)+ γ(t) dl(t) (t) d ( ) ( ) σ (t) σ (t) γ = µ (t) γ dt (t) ( ) σ +Σ 1/ (t) γ dw (t) (t) dp(t) = µ 1 (σ (t), M(t)) dt + σ(t) dw 1 (t) + dj(t) J M(t) dt dσ (t) = µ (σ (t)) dt + c σ(t) dw (t) J(t) = s t J M(t) N M(t) (t), J M(t) (t) := E [J(t) F t ] M(t) follows a Markov-switching process. 18 / 40

41 How Should We Model Continuous-Time Returns? Proposed (Conditionally) Exponentially-Affine (Duffie, Pan, and Singleton 000; Calvet, Fearnley, Fisher, and Leippold 015) x(t) := ( σ (t) γ (t) ) x(t) := ( σ (t) M(t) ) ( ) σ (t) dp(t) = µ 1 γ dt+σ(t) dw 1 (t)+ γ(t) dl(t) (t) d ( ) ( ) σ (t) σ (t) γ = µ (t) γ dt (t) ( ) σ +Σ 1/ (t) γ dw (t) (t) dp(t) = µ 1 (σ (t), M(t)) dt + σ(t) dw 1 (t) + dj(t) J M(t) dt dσ (t) = µ (σ (t)) dt + c σ(t) dw (t) J(t) = s t J M(t) N M(t) (t), J M(t) (t) := E [J(t) F t ] M(t) follows a Markov-switching process. No latent processes and easy to simulate. Adapt variance-gamma methods to price options. Requires particle filtering. Affine process price American options in closed-form. 18 / 40

42 Discrete-Time: r t = t t 1 dp(s) Additional Assumptions: 1. Innovations to prices and volatilities are independent. The drift and volatilities may be correlated. Theorem ( t t ) ( t ) RD t = N µ(s), σ (s) L 0, γ (s) t 1 t 1 t 1 and h (r t F t 1 ) = f (r t t t ) ( t t t µ(s), σ (s), γ (s) dg µ(s), σ (s), t 1 t 1 t 1 t 1 t 1 ) γ (s) F t 1 19 / 40

43 Estimators

44 Estimator for σ (τ ) Adapted from Jacod (008) and Aït-Sahalia and Jacod (009a). Assumptions: 1. k n.. k n n p(t) jumps in every interval. 4. v n 1 tightly bounds the diffusive deviations. 5. Standard technical conditions. Theorem σ n(k n, τ, p) := 1 k n n k n 1 n in p 1{ n in p v n 1 } P σ (τ ) m=0 0 / 40

45 What should we use for v n 1? Intuition: We need a tight bound for the diffusive variation as 0. The law of the iterated logarithm provides such a bound: lim sup 0 σ(τ )W(τ+ ) W(t) σ(τ ) log(log(1/ )) = 1. Algorithm: 1. Use 1.5 times the Bipower estimator for σ(t).. Calculate v n 1 as implied by the bound above. 3. Then use σ n(k n, τ, p) 4. Iterate to convergence. 1 / 40

46 Estimator for γ (τ ) Assumptions: 1. k n.. k n n p(t) jumps in every interval. 4. σ n (τ ) P σ(τ ). 5. Standard technical conditions. 6. g( ) is a convex weight function. Theorem γ n (k n, τ, p) := argmin γ g 1 k n n k n 1 m=0 n i n+mp E [ n i n p γ, σ ] σ n (τ ) P γ(τ ) / 40

47 Advantages of γ n(k n, τ, p) 1. Shows γ (t) is identified.. Enables extending results that depend on instantaneous volatility to the jump case. Examples: Mykland and Zhang (009), Xiu (010), Barndorff-Nielsen, Hansen, Lunde, and Shephard (011), Bandi and Renò (01), and Li, Todorov, and Tauchen (017) 3. The first consistent estimator for any instantaneous jump variation measure. 3 / 40

48 Estimators for σ t, γ t, and RD t Theorem σ t := 1 #i n [t 1, t] t 1<t n t t σ (k n, t n, p) P σ (s) ds t 1 Theorem γ t := 1 #i n [t 1, t] t 1<t n t t γ (k n, t n, p) P γ (s) ds t 1 Theorem Estimate RD t by plugging σ t and γ t into N ( t ) ( t ) 0, σ (s) ds L 0, γ (s) ds. t 1 t 1 4 / 40

49 Simulation Setup Set parameters to match the data s discrete-time dynamics. Simulate volatilities, γ (t) and σ (t), as square-root (Cox-Ingersoll-Ross) processes and the price using the mixture representation. dσ (t) = a σ (b σ σ (t)) + σ σ σ(t) dw(t) dγ (t) = a γ (b γ γ (t)) + σ γ γ(t) dw(t) dp(t) = σ(t) dw(t) + γ(t) dl(t) 5 / 40

50 Simulation Results σ t γ t Truth Proposed Barndorff-Nielsen and Shepard (004) [BNS] 5 Minute Li, Todorov, and Tauchen (016) [LTT] Aug Sep Oct Nov Dec Jan Aug Sep Oct Nov Dec Jan Obs. per Min. E[( σ t σ t ) ]/E[σ t ] E[( γ t γ t ) ]/E[γ t ] BNS LTT 5 Minute Proposed BNS LTT 5 Minute Proposed Details 6 / 40

51 Empirics

52 Data SPY S&P 500 ETF : January 003 September 017 Tick data from TAQ, sampled once per second days. 4 thousand observations per day. 87 million total observations. Removed extraneous trades and used pre-averaging to filter out the market-microstructure. Only finely sampled data is sufficient to separate the jump and diffusion parts. Pre-averaging is commonly used to handle market-microstructure in this case. 7 / 40

53 Volatilities & Returns 0.10 r t 0.05 t t / 40

54 Log Volatility Distributions 8.0 log(σ t ) log(γt ) log ( ) σt log ( ) γt Mean Standard Deviation Skewness Kurtosis Corr ( log(σ t ), log(γ t ) ) 0.90 Black lines are Gaussian distributions fit to the data. Details 9 / 40

55 ( Does RD t = N 0, ) ( t t 1 σ (s) ds L 0, ) t t 1 γ (s) ds work well? 1. Lower tail measured almost perfectly.. Skewness is not a large problem in practice. QQ Plot Probability Integral Transform (PIT) PIT ACF 1.00 RDt RDt / 40

56 Talk Structure / Contributions (Again) 1. Proposed a new measure of jump volatility: γ t. a) that yields r t N ( ) ( ) 0, σt L 0, γ t b) that measures jumps in investor s information (news). a) Derived r t N ( ) ( ) 0, σt L 0, γ t from no-arbitrage. b) Derived estimators for σ t, γ t, and RD t. c) Applied these estimators to high-frequency data on the S&P 500 providing multiple new stylized facts. 3. a) Show σ t and γ t jointly determine risk premia. b) Show the γ t premium is less than the σ t premium. 31 / 40

57 Volatility and News Premia γ t is a new factor for the dynamics. Is it priced?

58 Preferences (Example: Epstein-Zin) ρ: risk aversion ψ: intertemporal elasticity of substitution (IES) U t = [ C 1 1/ψ t [ + βe U 1 ρ t+ ] 1 1/ψ 1 ρ F t ] 1 1 1/ψ 3 / 40

59 Preferences (Example: Epstein-Zin) ρ: risk aversion ψ: intertemporal elasticity of substitution (IES) U t = [ C 1 1/ψ t Define V t := U 1 1/ψ t. V t = [ + βe U 1 ρ C 1 1/ψ t t+ [ + βe V ] 1 1/ψ 1 ρ F t 1 ρ 1 1/ψ t+ ] 1 1 1/ψ F t ] 1 1/ψ 1 ρ 3 / 40

60 Preferences (Example: Epstein-Zin) ρ: risk aversion ψ: intertemporal elasticity of substitution (IES) U t = [ C 1 1/ψ t Define V t := U 1 1/ψ t. V t = [ + βe U 1 ρ C 1 1/ψ t t+ [ + βe V ] 1 1/ψ 1 ρ F t 1 ρ 1 1/ψ t+ ] 1 1 1/ψ F t ] 1 1/ψ 1 ρ Define ϕ(v) := 1 ρ 1 1/ψ V 1 ρ 1 1/ψ. = [ u(ct ) + βϕ 1 (E [ϕ(v t+ ) F t ]) ] 3 / 40

61 Investor Problem V (Ξ(t ), P(t)) = t+ max C(t), Ξ(t) t u(c(s)) ds + exp( κ )ϕ 1 (E [ϕ (V (Ξ(t), P(t + ))) F t ]) C(t) + i P i (t)ξ i (t) = i P i (t)ξ i (t ) Assumptions: 1. Both u and ϕ are Lipschitz continuous with Lipschitz derivatives.. Both u and ϕ are strictly increasing. 3. Consumption C(t) is an Itô semimartingale. 4. A representative investor prices all assets. 33 / 40

62 Risk Premia Theorem 1. Let the investor face the problem described above as 0.. Assume preferences are such that optimal consumption is strictly positive. Then ( ) log u (W(t)) E[u (W(t))] [ ] Risk Premium on Asset i Cov m(t), p D i (t) F t dt 34 / 40

63 Risk Premia with Recursive Utility Theorem 1. Let the investor face the problem described above as 0.. Assume preferences are such that optimal consumption is strictly positive. Then ( ) log V (W(t)) E[u (W(t))] E [ϕ (V(W(t))) F t ] [ ] Risk Premium on Asset i Cov m(t), p D i (t) F t dt 34 / 40

64 Risk Premia with Recursive Utility and Jumps Theorem 1. Let the investor face the problem described above as 0.. Assume preferences are such that optimal consumption is strictly positive. Then ( ) log V (W(t)) E[u (W(t))] E [ϕ (V(W(t))) F t ] ( log ϕ (V(W(t))) E[ϕ (V(W(t))) F t ] ) [ Risk Premium on Asset i Cov m(t), ] [ ] p D i (t) + p J i (t) F t dt Cov m UP (t), p J i (t) F t dt 34 / 40

65 Takeaway 1. Risk premia are predictable quadratic variations. (i.e., expected returns are covariances even with recursive utility and jumps.). σ (t) and γ (t) jointly determine risk-premia. 3. The theory requires two factors that move high-frequency in general. 35 / 40

66 ( ) γ Jump Proportion t σt + γt γ t σ t +γ t 30 Day Rolling Average Mean σ t Volatility Correlations γ t γ t σ t +γ t rx t 1{FOMC} t σt γt γ t σ t +γ t rx t Details 36 / 40

67 rx t = β 0 + β 1 log ( ) ( ) σt + γt + β log γ t + ϵ σt +γt t 1 1. Adjust for heteroskedasticity using. σt +γ t. Instruments: γ t σ t +γ t Lagged σt + γt and Predetermined variables are independent of innovations. Use the heterogeneous autoregressive (HAR) model lags. Robust to other choices. Details 37 / 40

68 rx t = β 0 + β 1 log ( ) ( ) σt + γt + β log γ t + ϵ σt +γt t 1 1. Adjust for heteroskedasticity using. σt +γ t. Instruments: γ t σ t +γ t Lagged σt + γt and Predetermined variables are independent of innovations. Use the heterogeneous autoregressive (HAR) model lags. Robust to other choices. Weighted-Least Squares with Instruments Regressors Intercept log ( ) σt + γt ( ) γ t log σt +γ t.95 [6.61] 0.4 [5.88] Specifications Details 37 / 40

69 rx t = β 0 + β 1 log ( ) ( ) σt + γt + β log γ t + ϵ σt +γt t 1 1. Adjust for heteroskedasticity using. σt +γ t. Instruments: γ t σ t +γ t Lagged σt + γt and Predetermined variables are independent of innovations. Use the heterogeneous autoregressive (HAR) model lags. Robust to other choices. Weighted-Least Squares with Instruments Regressors Intercept log ( ) σt + γt ( ) γ t log σt +γ t Specifications [6.61] [ 5.1] 0.4 [5.88] 5.01 [ 5.86] Details 37 / 40

70 rx t = β 0 + β 1 log ( ) ( ) σt + γt + β log γ t + ϵ σt +γt t 1 1. Adjust for heteroskedasticity using. σt +γ t. Instruments: γ t σ t +γ t Lagged σt + γt and Predetermined variables are independent of innovations. Use the heterogeneous autoregressive (HAR) model lags. Robust to other choices. Weighted-Least Squares with Instruments Regressors Intercept log ( ) σt + γt ( ) γ t log σt +γ t Specifications [6.61] [ 5.1] [ 0.58] [5.88] [.68] [ 5.86] [ 4.93] Details 37 / 40

71 Takeaway 1. Changes in σ t + γ t drive large (dozens of percent) changes in risk premia, as in Martin (017).. Changes in γ t σ t +γ t drive changes in risk premia of approximately the same magnitude. 3. The data require two factors that move high-frequency to explain risk premia. 4. Preferences are not time-separable. 38 / 40

72 Recap 1. Proposed a new measure of jump volatility: γ t. a) that yields r t N ( ) ( ) 0, σt L 0, γ t b) that measures jumps in investor s information (news). a) Derived r t N ( ) ( ) 0, σt L 0, γ t from no-arbitrage. b) Derived estimators for σ t, γ t, and RD t. c) Applied these estimators to high-frequency data on the S&P 500 providing multiple new stylized facts. 3. a) Showed σ t and γ t jointly determine risk premia. b) Show γ t premium is less than the σ t premium. 39 / 40

73 Avenues for Future Work 1. Measure tail risk in real-time, Sangrey (018b, WP). a) This paper reduces forecasting the density to forecasting the volatilities. b) I consistently estimated the volatilities and showed they are predictable and log volatility is close to Gaussian. = We can track and forecast standard risk measures such as Value-at-Risk and Expected Shortfall. They are statistics of the daily return density (h (r t F t 1)).. Develop a multivariate version of this paper. Conjecture: tail dependence can be reduced to volatility dependence. 3. Develop the appropriate noise-robust inference theory. γ t σ t +γ t 4. Why does command a statistically and economically significant negative premium? One would think a tail-risk measure would command a positive premium. 40 / 40

74 Jump Times are News Times Theorem Consider a stopping time τ. Let P(t) be a price process satisfying no-arbitrage. Then its natural filtration Ft p contains all of the information in the representative investor s information set relevant for asset pricing, and Fτ p F p τ if and only if P(t) jumps at τ, where F p t is the associated predictable filtration. Return 1 / 14

75 Decomposition of γ t in the Compound Poisson Case [ γt 1 = lim 0 E t 1 = lim 0 E E t+ dp J (s) F t ] s (t,t+ ] (jump magnitude) (s)) F t, s is a jump 1 = lim 0 E [ (jump magnitude) ] E [# of jumps in (t, t + ] Return / 14

76 Representation Theory Theorem (Time-Changing Jump Martingales) Let p J (t) be an infinitely-active, purely discontinuous, square-integrable martingale with no unpredictable jumps that can be represented as H (n ν) where H(t) is a predictable process, n a Poisson random measure, and ν its predictable compensator with Lebesgue base Levy measure. Then p J (t) time-changed by its predictable quadratic variation is a standard variance-gamma process. In other words, p J (t) L = L ( p J (t) ). Return 3 / 14

77 Time-Changed Laplace Corollaries Corollary Let p(t)t be a purely-discontinuous Itô semimartingale that is locally-square integrable and has infinite-activity jumps. Then p(t) = 1 t γ(s) dl(s), where L is a standard variance-gamma process, 0 and γ(s) is predictable function. Corollary Let p(t) be a purely discontinuous, locally-square integrable martingale that can be represented as H (µ ν) where H(t) is a predictable process, µ a Poisson random measure, and ν its predictable compensator with Lebesgue base Levy measure λ. Further assume that it has no predictable jumps. Then p(t) time-changed by its predictable quadratic variation is a mixture of the 0 process δ 0 and the standard variance-gamma process where the mixing weights are determined by the intensity of the jump process. Return 4 / 14

78 Estimating γ (t) Theorem Let p(t) be a locally-square integrable infinite-activity Itô semimartingale whose characteristics have locally-square integrable càdlàg densities. Let k n, k n n 0, and 0 < T < be a deterministic time. Define i n = i k n 1. Let ˆσ T converge in probability to σ T. Let γ(t) > 0 for all t, then we have the following. k 1 n 1 ˆγ := argmin n i γ k n n+mx E N(0, 1) ˆσ T γ ( ) ˆσT P erfcx γ γ T m=0 Return 5 / 14

79 Local Absolute Variation Theorem Let p(t) be a locally-square integrable infinite-activity Itô semimartingale whose characteristics have locally-square integrable càdlàg densities. Let k n and k n n 0, and O < T < be a deterministic time. Define i n = i k n 1. k 1 n 1 ( ) r in, k n P σt E N(0, 1) σ T + erfcx γ T n n γt m=0 Return 6 / 14

80 Simulation Results with Microstructure σ t γ t Truth Estimator Bipower 5 Minute LTT Aug Sep Oct Nov Dec Jan Aug Sep Oct Nov Dec Jan Obs. / Minute E[( σ t σ t ) ]/E[σ t ] E[( γ t γ t ) ]/E[γ t ] Bipower LTT 5 Minute Proposed Bipower LTT 5 Minute Proposed Return 7 / 14

81 Poisson Simulation Results (1 jump per day) σ t γ t Truth Proposed Barndorff-Nielsen and Shepard (004) [BNS] 5 Minute Li, Todorov, and Tauchen (016) [LTT] Aug Sep Oct Nov Dec Jan Aug Sep Oct Nov Dec Jan Obs. per Min. E[( σ t σ t ) ]/E[σ t ] E[( γ t γ t ) ]/E[γ t ] BNS LTT 5 Minute Proposed BNS LTT 5 Minute Proposed Return 8 / 14

82 rx t = β 0 + β 1 log ( ) ( ) σt + γt + β log γ t + β σt +γt 3 log ( ) σt + β4 log ( ) γt + ϵt Regressors Intercept log ( ) σt + γt ( ) γ t log σt +γ t News Premia Estimates Specifications [6.61] [ 5.1] [ 0.58] [7.5] [6.07] [4.15] [6.61] [.68] [ 5.86] [ 4.93] log(σt ) [6.53] [5.18] log ( ) γt [5.40] [ 4.53] Return 9 / 14

83 rx t = β 0 + β 1 log ( ) ( ) σt + γt + β log γ t + ϵ σt +γt t Regressors Intercept log ( ) σt + γt ( ) γ t log σt +γ t log ( ) ( ) σt + γt γ log t σt +γ t OLS Specifications [5.81] [6.48] [ 3.94] [ 0.54] [ 5.58] [ 4.1] [ 0.85] [5.81] [4.06] [1.07] 0.9 [0.80] R.67 % 1.61 % 3.35 % 3.4 % Return 10 / 14

84 Ai and Bansal (018) Use FOMC dates to proxy for γt. σt +γt {FOMC} Dates t tγ t t σ+ t +γ t t Apr-008 Oct-008 Apr-009 Return 11 / 14

85 Is 1{FOMC} t is a good proxy for γ t σ t +γ t? Regressand t = β 0 + β 1 1{FOMC} t + ϵ t ) Regressand log log ( ) σt + γt ( γ t σ t +γ t Intercept [ ] [ ] 1{FOMC} t [7.90] [7.68] R 0.78% 0.58% #1{FOMC} t = / 14

86 Volatility Summary Statistics σ t γ t γ t σ t +γ t log(σt ) log(γt ) log ( ) σt + γt ( ) γ t log σt +γ t Mean Std. Dev Skew Kurt Return 13 / 14

87 Density Forecast Jump-Diffusion Forecast Bollerslev, Kretschmer, Pigorsch, and Tauchen (009) (Slightly Simplified) rt 75% Median 95% 5% 99% 50% 99.9% rt 75% Median 95% 5% 99% 50% 99.9% / 14