Parametric Inference and Dynamic State Recovery from Option Panels. Torben G. Andersen

Transcription

1 Parametric Inference and Dynamic State Recovery from Option Panels Torben G. Andersen Joint work with Nicola Fusari and Viktor Todorov The Third International Conference High-Frequency Data Analysis in Financial Markets Hiroshima University of Economics November 17, 212

2 Motivation Under realistic assumptions: Derivatives Non-Redundant Assets. Contain important Information about Volatility and Jump risks and their Pricing: Volatility Time-Varying Option Price & + Risk + Observation Error Jumps Premia Derivatives on Equity Indices actively Traded: On Average, 2 plus SPX Option Quotes at Close of Trading, Cover a wide range of Moneyness and Tenor (time-to-maturity). 1

3 Motivation Most Parametric Option-based Estimation Methods of Risk Premia follow Two Steps: (Bates (2), Pan (22), Eraker (24), BCJ (27), Christoffersen at al. (26-8)) 1. Identify Volatility and Jump Risks from underlying Asset Data, 2. Use Price Levels in Option Panel to estimate Risk Premia: typically via restrictive Specification; i.e., small P Q wedge. Bates (2), Pan (22) use aspects of Implied Volatility State Dynamics. Option Price Error ignored or modeled with Normal Distribution. 2

4 .13 IV error kernel regression IVa IVb (IVa+IVb)/ Log Moneyness: log(k/f ) σ τ 3

5 Motivation Goal to develop Estimation Technique that: 1. fully uses the State Dynamics implied by Option Prices; 2. is Robust with respect to Option Price Error Specification; 3. relies on In-Fill Asymptotics (Increasing number of Options each Trading Day); 4. Specifies only Risk-Neutral Dynamics (allows for flexible risk premia). In Sum: Formal Estimation, Inference and Diagnostic Tests for Option Pricing Obtain Path of State Vector Realizations solely from Option Panel Set Stage for Risk Premia Estimation via Semi-Parametric P Estimation 4

6 t 1! Parametric Inference from Option Panels September 212! Underlying price process Maturity! Option'cross'section' t 3! Moneyness! t 2! 5

7 Outline Information in Option Panels Inference in Presence of Noise Semiparametric Tests 6

8 Notation Formally, underlying Price X t has the following P-Dynamics: dx t X t = α t dt + V t dw P t + x> 1 x µ P (dt, dx), W P t is a Brownian motion; Vt is Spot Volatility (under both P and Q); µ P is an Integer-valued Random Measure; µ P = µ P ν P µ P Counts Jumps in X; Jump Compensator is ν P (ds, dx). We assume V t = ξ 1 (S t ), where S t is Latent State Vector (p 1). 7

9 Notation Assumption A. The process X, defined over the fixed interval [, T ], satisfies: (i) For s, t, exists K > : E { V t V s 2 K } K t s. (ii) x> 1 ( x β 1) ν P (dx) <, for some β [, 2). (iii) inf t [,T ] V t > and the processes α t, V t and a t are locally bounded. A(i) satisfied if V t is governed by (multivariate) Stochastic Differential Equation A(ii) restricts so-called Blumenthal-Getoor index of the jumps to be below β A(iii) implies, at each t [, T ], the price process has Non-Vanishing BM Component Assumption A does not involve Integrability or Stationarity Conditions for the Model 8

10 Notation Likewise, X t has Q-Dynamics: dx t X t = ( r t δ t ) dt + V t dw t + x> 1 x µ(dt, dx), ν(dt, dx) = ξ 2 (S t ) ν(dx), where µ = µ ν We denote Options with Log-Moneyness k = log(k/x t ) and Tenor τ by O t,k,τ = E Q t { e t+τ t } (rs δs)ds (X t+τ K) +, We denote associated Black-Scholes Implied Volatility by κ(k, τ, S t ). 9

11 An Empirical Illustration The Double-Jump Model of Duffie, Pan and Singleton (21) has Risk-Neutral Dynamics: dx t X t = (r δ) dt + V t dw t + dl x,t, dv t = κ (v V t ) dt + σ d Vt db t + dl v,t, L x,t and L v,t are Jump Martingales; (L x,t, L v,t ) Jump (simultaneously) with i.i.d. Probability λ j, Jump Size (Z x, Z v ). Z v exp (µ v ); log (Z x + 1) Z v N ( µ x + ρ j Z v, σ 2 x ) ; Cor (dwt, db t ) = ρ d State Vector (Realization): {V t } T t=1 Risk-Neutral Parameters: θ = (ρ d, v, κ, σ d, λ, µ x, σ x, µ v, ρ j ). 1

12 Monte Carlo Scenario Inspired by Calibration/Empirical Estimates from BCJ (27); ρ j =. Table 1: Parameter Setting for the Numerical Experiments Under P Under Q Parameter Value Parameter Value Parameter Value Parameter Value ρ d.46 λ j 1.8 ρ d.46 λ j 1.8 v.144 µ x.284 v.144 µ x.51 κ d 4.32 σ x.49 κ d 4.32 σ x.751 σ d.2 µ v.315 σ d.2 µ v.93 Observation Errors on Option Prices: ɛ t,k,τ = σ t,k,τ Z t,k,τ, Z t,k,τ N (, 1). σ t,k,τ = 1 2 ψ k/q.995, ψ k = Bid-Ask Spread at k, Q p = p th Quantile of N (, 1). 11

13 Information in Panels of Options What can we Identify from Options? Different Parts of Volatility Surface Load differently on distinct Risks and their Pricing: Short-Term OTM Options determined largely by Pricing of Jump Risks Role of Volatility Risks more prominent for ATM Options Different Maturities separate Persistent from Transient State Variables Persistence of Smirk Identifies Sources of Leverage type Effects... Can we identify the model? 12

14 Option Sensitivity to Parameters; Double-Jump Model ρ d v κ d σ d σ x Moneyness λ j µ v Moneyness µ x ρ j Moneyness 13

15 Information in Panels of Options... = A Large Cross-Section of Option Prices observed without Error can Identify Risk-Neutral Parameters and the Current Value of the State Vector, Once Risk-Neutral Parameters are Known, Options are Known Transformations of the State Variables = Contain Same Information as observing directly the State Vector, = Options alone Contain Information to Estimate the Risk Premia! 14

16 Information in Panels of Options Assumption A1. Fix T >. For each Date t = 1,.., T and Moneyness τ, # options N τ t with N τ t /N t π τ t and N t/ T t=1 N t ς t, where π τ t, ς t >. Let k(t, τ), k(t, τ) denote Min, Max Log-Moneyness on Day t, Maturity τ. Sequence of Grid Nested: k(t, τ) = k t,τ () < k t,τ (1)... < k t,τ (N τ t ) = k(t, τ). N t (k t,τ (i) k t,τ (i 1)) ψ t,τ (k) Uniformly on (k(t, τ), k(t, τ)). Assumption A2. For every ɛ > and T > finite, we have a.s. inf t=1,..,t : Z t S t >ɛ θ θ >ɛ T k(t,τ) t=1 τ k(t,τ) (κ(k, τ, S t, θ ) κ(k, τ, Z t, θ)) 2 dk >, where θ is the risk-neutral parameter vector. 15

17 Inference in the Presence of Noise Options are Observed with Error, i.e., we observe κ t,k,τ for κ t,k,τ = κ t,k,τ + ɛ t,k,τ, where the errors, ɛ t,k,τ, are defined on an extension of the original probability space. We assume the Error can be averaged out by Pooling Options across Moneyness: Assumption A3. For every ɛ > and T > finite, we have sup t=1,..,t : Z t S t >ɛ θ θ >ɛ Tt=1 1 N t Nt j=1 Tt=1 1 N t Nt j=1 when min t=1,...,t N t for all θ Θ. ( ) κ(k j, τ j, S t, θ ) κ(k j, τ j, Z t, θ) ɛ t,k,τ P ( ) 2 κ(k j, τ j, S t, θ ) κ(k j, τ j, Z t, θ), 16

18 Estimation We define our estimator of risk-neutral parameters and state variables as ( {Ŝn t } t=1,...,t, θ n) = argmin {Z t } t=1,...,t, θ Θ T { 1 t=1 N t N t j=1 ( κ t,k,τ κ(k j, τ j, Z t, θ)) 2 ( ) 2 } n + λ n V t ξ 1 (Z t ), λ n, V n t is Nonparametric Estimator of Volatility from High-Frequency Data. 17

19 t 1! Parametric Inference from Option Panels September 212! Underlying price process Maturity! Option'cross'section' t 3! Moneyness! t 2! 18

20 Estimation Theorem 1. Suppose Assumptions A1-A3 Hold for some T N fixed, and { V n t } t=1,...,t is Consistent for {V t } t=1,...,t, as n. Then, if min t=1,...,t N t and λ n λ for some finite λ, as n, we have that (Ŝn t, θ ) n t exists with probability approaching 1, and Ŝn t S t P, θ n θ P, t = 1,..., T. 19

21 Estimation To quantify precision of estimation we need slightly stronger assumption on errors: Assumption A4. For the error process, ɛ t,k,τ, we have, ( (i) E ɛ t,k,τ F ()) = (ii) E ( ɛ 2 t,k,τ F ()) = φ t,k,τ, for φ t,k,τ continuous in its second argument (iii) ɛ t,k,τ, ɛ t,k,τ are independent, conditional on F (), for (t, k, τ) (t, k, τ ) (iv) E ( ɛ t,k,τ 4 F ()) <, almost surely where F () is the σ-algebra associated with X. 2

22 Estimation Theorem 2. Assume Assumptions A1-A4 Satisfied for T N Fixed, and κ(t, τ, Z, θ) is twice Continuously-Differentiable in its arguments. Then, if min t=1,...,t N t and λ 2 n min t=1,...,t N t, for n : N1 (Ŝn 1 S 1). NT (Ŝn T S T ) N N T T ( θ n θ ) L s H 1 T (Ω T ) 1/2 E 1. E T E, E 1,...,E T are p 1 vectors, E is q 1 vector, all are i.i.d. Standard Normal and Defined on an Extension of the original Probability Space, H T, Ω T are F () T -adapted Random Matrices, for which Consistent Estimates can be Constructed from Options Data. 21

23 MC Estimation; Double-Jump Model; 1, Replications 2 ρd 2 v κd 2 σd λj 4 μx σx 2 μv

24 Empirical Application We use the following Data Set in the Application CBOE European-style (SPX) Options on the S&P 5 index, The Options have Maturity Ranging from 8 Days to 1 Year, The Data Covers Period for a Total of 3, 5 Days, We Apply Standard Filters; Retain only OTM and ATM Options; Wide Strike Range, For Semi-Parametric Tests: 5-minute S&P 5 Futures, Same Sample Period. 23

25 Model-free vs Option-Implied Volatility We specify and estimate the risk-neutral distribution of the underlying process X and we do not impose any parametric structure for the dynamics under the true statistical measure P. Absence of arbitrage implies that recovered volatility from options should be the same with that observed in the underlying asset X. This is a semiparametric restriction: it is based on a parametric specification for the risk-neutral distribution as well as nonparametric estimate for the stochastic volatility. 24

26 An Empirical Illustration The Double-Jump Model of Duffie, Pan and Singleton (21) has Risk-Neutral Dynamics: dx t X t = (r δ) dt + V t dw t + dl x,t, dv t = κ (v V t ) dt + σ d Vt db t + dl v,t, L x,t and L v,t are Jump Martingales; (L x,t, L v,t ) Jump (simultaneously) with i.i.d. Probability λ j, Jump Size (Z x, Z v ). Z v exp (µ v ); log (Z x + 1) Z v N ( µ x + ρ j Z v, σ 2 x ) ; Cor (dwt, db t ) = ρ d State Vector (Realization): {V t } T t=1 Risk-Neutral Parameters: θ = (ρ d, v, κ, σ d, λ, µ x, σ x, µ v, ρ j ). 1

27 Empirical Application Qualitative Features of the Double-Jump Model Both Stochastic Volatility and Price Jumps Present, Volatility can Move through Jumps, Price and Volatility may be Correlated through Small and Big Moves, Jump Intensity is Constant. (Only) One Volatility Factor. Slightly more General than BCJ State-of-Art 25

28 Empirical Application Table 2: Parameter Estimates of One-Factor Model Parameter Estimate Standard Error Parameter Estimate Standard Error ρ d λ.15.6 v µ x κ σ x σ d µ v ρ j Risk Neutral Mean of Volatility: 24.3% vs Sample RV Estimate: 21.35% Mean Risk-Neutral (Log-Return) Jump: 8% Annual Jump Probability: 1.5% Mean Risk-Neutral Volatility (Level) Jump: 1.62 = 127% 26

29 Diagnostic Tests We design the following Diagnostic Tests of Model Performance: Fit to the Volatility Surface over some Period of Time Parameter Stability across Time Distance between Model-Free and Option-Model Implied Volatility 27

30 Diagnostic Test I: Fit to Volatility Surface Corollary 1. Let K ( ) k(t, τ ), k(t, τ ) be a set with positive Lebesgue measure and N K t be the number of options on day t with tenor τ and log-moneyness in K. Then, given our Assumptions, we have, ( j:k j K κ t,kj,τ κ(k j, τ, Ŝn t, θ ) n ) Π Ξ T T ΠT L s N (, 1), where Π T and Ξ T are some F () T -adapted random matrices. 28

31 1 Z score: short maturity DOTM Put options Z score: short maturity OTM Put options Z score: short maturity OTM Call options Year 29

32 1 Z score: long maturity DOTM Put options Z score: long maturity OTM Put options Z score: long maturity OTM Call options Year 3

33 Diagnostic Test II: Parameter Stability Parameters Estimated over Non-Overlapping Periods should, up to Statistical Error, be Identical. Thus, ( θn 1 θ ) ) 1 n 2 (Âvar( θn 1 ) + Âvar( θ n 2 ( θn ) 1 θ ) n L s 2 χ 2 (q), where Âvar( θ n j ) is Consistent Asymptotic Variance Estimate for θ n j. Note: Under Model Misspecification, Parameter Estimates Converge to Pseudo-True Values. However, as State Vector changes over Time = Pseudo-True Values Change as well. 31

34 SPX Options Parameter Stability Test Table 3: Parameter Stability; S&P 5 Options Data Parameter Nominal size of test Parameter Nominal size of test 1% 5% 1% 5% Panel A: One-Factor Model ρ d 62.86% 7.48% λ j 56.19% 67.62% v 71.43% 73.33% µ x 2.% 25.71% κ 91.43% 93.33% σ x 49.52% 61.91% σ d 77.14% 8.95% µ v 31.43% 36.19% ρ j 13.33% 17.14% Panel B: Two-Factor Model ρ d,1 8.57% 16.19% λ j, 37.14% 49.52% v % 12.38% λ j, % 33.33% κ d, % 7.5% µ x.% 4.76% σ d, % 53.33% σ x 9.52% 17.14% ρ d,2 7.62% 16.19% µ v 23.81% 34.29% v % 8.95% ρ j 1.91% 2.86% κ d, % 77.14% σ d,2.95%.95% 32

35 Model-Free vs Option-Implied Volatility Two Nonparametric Estimators for Spot Volatility from High-Frequency Data: V ±,n t = n ( t,n k n i I ±,n i X) 2 ( 1 t,n i X αn ϖ), t,n X = log i ) ( ) (X log X t+ in t+ i 1, n where α >, ϖ (, 1/2), k n is Deterministic sequence, k n /n, and I,n = { k n + 1,..., } and I +,n = {1,..., k n }. V,n t is Estimator for Spot Variance from Left; V +,n t V,n t and V +,n t Estimator from Right. Differ only if Volatility Jumps at t (Probability Zero Event). 33

36 Diagnostic Test III: Model-free vs Option-Implied Volatility Corollary 2. Under the same conditions as in Theorem 3, we have for k n, min t=1,...,t N t and λ 2 n min t=1,...,t N t, ξ 1 (Ŝn t S ξ 1 (Ŝ n t ) χ t S ξ 1 (Ŝ n t ) N t ) V +,n t + +,n 2( V t ) 2 kn t=1,...,t L s where χ t is the part of Ĥ 1 Ω T T (Ĥ 1 T ) corresponding to the variance-covariance of Ŝn t and (Ĕ1,..., ĔT ) is a vector of standard normals independent of each other and of F. Ĕ 1. Ĕ T, 34

37 Nonparametric volatility estimate Option recovered volatility Z score: recovered nonparametric volatility ACF: in level Year ACF: in log Lag Lag 35

38 Empirical Application Two-Factor Model We now extend the Model to Include Two SV Factors: dx t X t = (r δ)dt + V 1,t dw 1,t + V 2,t dw 2,t + dl x,t, dv 1,t = κ 1 (v 1 V 1,t )dt + σ 1,d V1,t db 1,t + dl v,t, dv 2,t = κ 2 (v 2 V 2,t )dt + σ 2,d V2,t db 2,t, Now, Jump Intensity is λ + λ 1 V 1,t : Jumps Self-Exciting, as Jumps Impact Volatility. Note: Extension allows Fears to be Time-Varying. Breaks Tight Link between Pricing of Risk and its Level. 36

39 Empirical Application Table 4: Parameter Estimates of Two-Factor Model Parameter Estimate Standard Error Parameter Estimate Standard Error ρ 1,d λ v λ κ µ x σ 1,d σ x ρ 2,d µ v v ρ j κ σ 2,d Risk Neutral Mean of Volatility: 22.8% vs Sample RV Estimate: 21.35% Mean Risk-Neutral (Log-Return) Jump: 13% Annual Jump Probability: 22.25% Mean Risk-Neutral Volatility (Level) Jump:.151 = 38.7% 37

40 SPX Options Parameter Stability Test Table 3: Parameter Stability; S&P 5 Options Data Parameter Nominal size of test Parameter Nominal size of test 1% 5% 1% 5% Panel A: One-Factor Model ρ d 62.86% 7.48% λ j 56.19% 67.62% v 71.43% 73.33% µ x 2.% 25.71% κ 91.43% 93.33% σ x 49.52% 61.91% σ d 77.14% 8.95% µ v 31.43% 36.19% ρ j 13.33% 17.14% Panel B: Two-Factor Model ρ d,1 8.57% 16.19% λ j, 37.14% 49.52% v % 12.38% λ j, % 33.33% κ d, % 7.5% µ x.% 4.76% σ d, % 53.33% σ x 9.52% 17.14% ρ d,2 7.62% 16.19% µ v 23.81% 34.29% v % 8.95% ρ j 1.91% 2.86% κ d, % 77.14% σ d,2.95%.95% 32

41 Diagnostic Tests I and III Table 5: Tests on S&P 5 options data One-factor Model Two-factor Model Test Nominal size of test Nominal size of test 1% 5% 1% 5% Panel A: Option Fit Tests DOTM, short-maturity puts 18.16% 39.74% 21.84% 45.53% OTM, short-maturity puts 24.87% 53.3% 27.89% 5.39% OTM, short-maturity calls 2.53% 55.% 16.58% 48.55% DOTM, long-maturity puts 41.45% 62.11% 27.5% 41.97% OTM, long-maturity puts 72.63% 8.79% 52.89% 6.53% OTM, long-maturity calls 53.16% 65.53% 79.47% 86.45% Panel B: Distance implied-nonparametric volatility 54.8% 65.66% 49.74% 61.58% Panel C: Root-mean squared error of fit 3.9% 2.32% 38

42 Empirical Application Things to Note for Two-Factor Model: Fit Improves Significantly (RMSE Drops about 25%). Constant Part of Jump Intensity Small = Jump Risk Premia Time-Varying. First Return-Volatility Correlation (remains) Extremely Negative. Second Volatility Factor Smaller, much Less Persistent. Model still Struggles with Short-Maturity OTM Calls, Long-Term OTM Options. Parameters Vary over time, particularly the ones driving Jump Distribution. In Quiet Period Jump Intensity Near Zero Jump Parameters not Identified. Time-Varying Parameters = Missing State Variables? Period very Hard to Fit Reasonably. Model still Misspecified, even on Stretches of One Year. 39

43 1 Z score: short maturity DOTM Put options Z score: short maturity OTM Put options Z score: short maturity OTM Call options Year 4

44 1 Z score: long maturity DOTM Put options Z score: long maturity OTM Put options Z score: long maturity OTM Call options Year 41

45 Nonparametric volatility estimate Option recovered volatility Z score: recovered nonparametric volatility ACF: in level Year ACF: in log Lag Lag 42

46 One-Factor Model Fit to ATM Term Structure of IV ATM IV short maturity ATM IV long maturity ATM IV short maturity ATM IV long maturity Year 43

47 Two-Factor Model Fit to ATM Term Structure of IV ATM IV short maturity ATM IV long maturity ATM IV short maturity ATM IV long maturity Year 44

48 Conclusions We Propose and Derive Asymptotic Properties of Estimation in Large Option Panels with Fixed Time Span and Increasing Cross-Section. Method requires Risk-Neutral Model only, is Nonparametric about Option Pricing Errors, and Allows for Heteroscedasticity in the latter. Battery of Statistics to Detect Sources of Model Misspecification: Testing Model Fit Over Time and different Parts of Volatility Surface, Testing Model Stability, Testing Consistency between Model Option-Implied Volatility and Nonparametric Estimate from High-Frequency Data on underlying Asset. 45