Parametric Inference and Dynamic State Recovery from Option Panels. Nicola Fusari
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1 Parametric Inference and Dynamic State Recovery from Option Panels Nicola Fusari Joint work with Torben G. Andersen and Viktor Todorov July 2012
2 Motivation Under realistic assumptions derivatives are nonredundant assets. They contain important information about Volatility and jump risks and their pricing: Volatility Time-varying Option Price = & + risk + Obs. Erorr Jumps premia Derivatives on stock market indexes are actively traded: on average there are more than 200 reliable option quotes on the S&P 500 index at the close of each trading day, these options cover a wide range of moneyness and time-to-maturity. 1
3 Motivation Most parametric option-based estimation methods of risk-premia (e.g., Bates (2000), Pan (2002), Eraker (2004), BCJ (2007), Christoffersen at al. ( ), etc...) follow two steps: 1. Identify volatility and jump risks from underlying asset data, 2. and then use the information in the price level of a panel of options to estimate these risk premia: this is typically done with a restrictive risk-premia specification; i.e., small P Q wedge. Bates(2000) and Pan (2002) use some of the dynamics of the implied volatility states in the estimation. option price error is either ignored or modeled with normal distributions. 2
4 Motivation Thus the goal is to develop an estimation technique that: 1. fully uses the states dynamics implied by the option prices; 2. is robust with respect to the error specification; 3. relies on in-fill asymptotic (i.e. assumes an ever increasing number of options on each day) 4. requires only the specification of the risk-neutral dynamics (thus allowing a flexible risk premia specification) 3
5 Outline Information in Option Panels Inference in Presence of Noise Semiparametric Tests 4
6 Notation Formally, the underlying price X t has the following dynamics dx t X t = α t dt + V t dw t + x> 1 x µ(dt, dx), where W t is a Brownian motion and µ is a integer-valued random measure counting the jumps in X with compensator is ν(ds, dx). We assume V t = ξ 1 (S t ) and ν Q (ds, dx) = ξ 2 (S t ) ν Q (dx) where S t is a latent state p 1 vector. We denote options with log-moneyness k = log(k/x t ) and time-to-maturity τ with O t,k,τ = E Q t { e t+τ t } (rs δs)ds (X t+τ K) +, and we denote the associated Black-Scholes implied volatility with κ(k, τ, S t ). 5
7 Example: an empirical application We estimate the double-jump model proposed in Duffie, Pan and Singleton (2001) specified by the following 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, The vector of risk-neutral parameters of the model is given by θ = (ρ d, v, κ, σ d, λ, µ x, σ x, µ v, ρ j ). 6
8 Information in Panels of Options What can we identify from options? Different parts of the volatility surface load differently on the different risks and their pricing: short-term out-of-the-money options are determined largely by pricing of jump risks the role of volatility risks is more prominent for at-the-money options different maturities separate persistent from transient state variables persistence of smirk identify sources of leverage type effects... 7
9 Information in Panels of Options... = a large cross-section of option prices observed without error can identify the risk-neutral parameters and the value of the state variables, once risk-neutral parameters are known, options are known transformations of the state variables = they contain the same information as observing directly the state variables, = options alone contain information about the risk premia! 8
10 Inference in 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 that the error can be averaged out when pooling options with different moneyness: 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 ( κt,k τ κ(k j, τ j, Z t, θ) ) 2 ( ) } n 2 + λ n V t ξ 1 (Z t ), where λ n 0 and V n t to be defined later. is a nonparametric estimator of volatility from high-frequency data 9
11 t 1! Parametric Inference from Option Panels June 2012! Underlying price process Maturity! Option'cross'section' t 3! Moneyness! t 2! 10
12 Estimation Theorem 1. Assume A1-A4 are 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 0, for n, we have: N1 (Ŝn 1 S 1). NT (Ŝn T S T ) N N T T ( θ n θ 0 ) L s H 1 T (Ω T ) 1/2 where E 1,...,E T are p 1 vectors and E is q 1 vector all defined on an extension of the original probability space being i.i.d. with standard normal distribution and H T and Ω T are F (0) T -adapted random matrices consistent estimates for which can be constructed from options data. E 1. E T E, 11
13 Diagnostic Tests We design the following tests for model performance: fit to the volatility surface over time parameter stability distance between model-free volatility and the one implied from options 12
14 Fit to Volatility Surface Corollary 1. Let K ( ) k(t, τ ) k(t, τ ) be a set with positive Lebesgue measure and denote by N K t the number of options on day t with time to maturity τ and log-moneyness belonging to the set K. Then, under the assumptions of Theorem 2, we have, ( j:k j K κ t,kj,τ κ(k j, τ, Ŝn t, θ ) n ) Π Ξ T T ΠT where Π T and Ξ T are some F (0) T -adapted random matrices. L s N (0, 1), 13
15 10 Z score: short maturity DOTM Put options Z score: short maturity OTM Put options Z score: short maturity OTM Call options Year 14
16 10 Z score: long maturity DOTM Put options Z score: long maturity OTM Put options Z score: long maturity OTM Call options Year 15
17 Parameter Stability Parameters estimated over different non-overlapping periods of time should be up to statistical error the same. Therefore, ( 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 1 ) and Âvar( θ n 2 ) denote the consistent estimates of the asymptotic variances of θ n 1 and θ n 2. Note: Under model misspecification, the parameter estimates will converge to the pseudo-true values. However, since the state variables change over time = the pseudo-true values will change as well. 16
18 Model-free vs Option-Implied Volatility Corollary 2. Under the same conditions as in Theorem 2, we have for k n, min t=1,...,t N t and λ 2 n min t=1,...,t N t 0, ) V +,n t ξ 1 (Ŝn t S ξ 1 (Ŝ n t ) χ t S ξ 1 (Ŝ n t ) 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 where (Ĕ1,..., ĔT ) is a vector of standard normals independent of each other and of F. Ĕ 1. Ĕ T, 17
19 Nonparametric volatility estimate Option recovered volatility Z score: recovered nonparametric volatility ACF: in level Year ACF: in log Lag Lag 18
20 Conclusions We propose and derive the asymptotic properties of estimation in large panels of options with fixed span and increasing cross-section. The method requires risk-neutral model only, is nonparametric about the option errors, and allows for heteroscedasticity in the latter. We design battery of tests that allows us to detect model misspecification and the reasons for it: to test model fit over time and over different parts of the volatility surface, to test model stability, to test the consistency between option-implied volatility and nonparametric estimate from high-frequency underlying asset data. 19
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