Leverage Effect, Volatility Feedback, and Self-Exciting Market Disruptions Liuren Wu, Baruch College and Graduate Center Joint work with Peter Carr, New York University and Morgan Stanley CUNY Macroeconomics and Finance Colloquium November 4, 2011 Leverage Effect, Volatility Feedback, and Self-Exciting Market Disruptions 11/4/2011 1 / 24
What are the roles of an option pricing model? 1 Interpolation and extrapolation. Broker-dealers: Calibrate the model to actively traded option contracts, use the calibrated model to generate option values for contracts without reliable quotes (for quoting or book marking). 2 Alpha generation. Investors: Hope that market deviations from model fair valuation are mean reverting. 3 Risk management. The estimated dynamics provide insights on how to manage/hedge the risk exposures of the option positions. 4 Answer economic questions with the help of option observations. Macro/corporate policies interact with investor preferences and expectations to determine financial security prices. Reverse engineering: Learn about policies, expectations, and preferences from security prices. You need a model to link them together. Leverage Effect, Volatility Feedback, and Self-Exciting Market Disruptions 11/4/2011 2 / 24
What can we learn from options... that we cannot learn as easily from primary securities? 1 Time variation in risk and risk premium: Options market is the natural market for volatility trading. Carr and Wu (RFS 2009): Document variance risk premium behaviors on stock indexes and single names. Bakshi, Carr, Wu (JFE, 2008): Identify pricing kernels from options on a currency triangle. 2 Time variation in crash risk: Deep out-of-money (DOOM) puts provide natural insurance for economic dooms. Carr and Wu (RFS 2011): Link DOOM puts to credit insurance. Bakshi and Wu (MS 2010): Understand/predict the Nasdaq bubble from QQQQ options. 3 This paper links capital structure decisions to option price behavior. Increasing financial leverage not only increases market risk (beta, equity vol), but also increase default risk (credit spread). Options market can be the information source for both. To do this, we need to go from statistical to structural... Leverage Effect, Volatility Feedback, and Self-Exciting Market Disruptions 11/4/2011 3 / 24
Background: Reduced-form option pricing models Two general, tractable frameworks: 1 Affine models (Duffie, Pan, Singleton (2000)) Returns and variance rates are linear ( affine ) combinations of some hidden statistical factors. Factors can rotate, causing identification issues. Economic meanings of the factors are not clear. 2 Time-changed Lévy processes (Carr and Wu (2004)) Each source of economic shock is captured by one Lévy process. The time-varying intensity of the financial market response to each economic shock is captured by its stochastic time change business clock versus calendar clock. The framework facilitates the design of well-identified models with economically meaningful structures. 3 This paper: Add capital structure decisions on top of reduced-form option pricing models. Leverage Effect, Volatility Feedback, and Self-Exciting Market Disruptions 11/4/2011 4 / 24
Background: Capital structure models Capital structure decisions and their impacts on equity and bond pricing are often analyzed via structural models. The classic example, Merton (1974): Model asset value dynamics as Geometric Brownian motion. The firm only has a zero-coupon bond maturing at a fixed maturity. Equity is a call option on the asset, with strike equal to the debt principal and maturity equal to the debt maturity. Many extensions are proposed for more realistic asset value dynamics, debt structure, default mechanisms... Pros: Link equity volatility to capital structure decisions (financial leverage) and business composition (asset volatility). Cons: Stylized and not tractable for pricing equity options. Cremers, Driessen, Maenhout (RFS 2008): A big calibration effort to link the average credit spread to equity index option prices, resolving the credit spread puzzle... Leverage Effect, Volatility Feedback, and Self-Exciting Market Disruptions 11/4/2011 5 / 24
A new, structural, reduced-form model We propose a reduced-form model with many economic structures. The model allows three distinct economic channels of interaction between equity return and volatility: 1 The leverage effect: With business risk fixed, an increase in financial leverage level leads to an increase in equity volatility. A financial leverage increase can come from stock price decline while the debt level is fixed Black (76) s classic leverage story. It can also come from active capital structure management. 2 The volatility feedback effect on asset valuation: A positive shock to business risk increases the discounting of future cash flows, and reduces the asset value, regardless of the level of financial leverage. 3 The self-exciting behavior of market disruptions: A downside jump in the index leads to an upside spike in the chances of having more of the same. We explore what structural questions can be answered from observations on stock index options. Leverage Effect, Volatility Feedback, and Self-Exciting Market Disruptions 11/4/2011 6 / 24
Separate the financial leverage variation from the asset value dynamics Decompose the forward value of the equity index F t into a product of the asset value A t and the equity-to-asset ratio (EAR) X t, F t = A t X t. This is just a tautology, (1) Model leverage X t as a stand-alone CEV process: dx t /X t = δx p t dw t, p > 0. (2) Leverage effect: A decline in X increases leverage [by definition], reduces equity value [via (1)], and raises equity volatility [via X p ]. Structural link: When A is fixed, the equity return volatility becomes a pure power function of the leverage ratio. A similar (more restricted) volatility structure can be obtained from the structural model of Leland (94). Firms can (and do) actively manage their leverages (Adrian & Shin (2008)). Leverage Effect, Volatility Feedback, and Self-Exciting Market Disruptions 11/4/2011 7 / 24
The asset value dynamics Model the asset value A t separately, in addition to leverage variation, da t A t = vt Z dz t + 0 (ex 1) ( µ(dx, dt) π J (x)dxvt J dt ) ( ), dvt Z = κ Z θz vt Z dt + σz vt dzt v, E [dzt v dz t ] = ρdt, ( ) dvt J = κ J θj vt J 0 dt σj x ( µ(dx, dt) π J (x)dxvt J dt ). Volatility feedback ρ < 0. Self-exciting crashes σ J > 0. Negative jumps in asset return are associated with positive jumps in the jump arrival rate v J t. Jump specification: Variance gamma high-frequency jump: π J (x) = e x /v J x 1. When X is fixed, the equity dynamics follow the asset dynamics. ρ < 0 is often referred to as the leverage effect, without distinguishing whether the negative correlation is really driven by financial leverage change or just volatility feedback on the business valuation. Leverage Effect, Volatility Feedback, and Self-Exciting Market Disruptions 11/4/2011 8 / 24
A different notation for the asset value dynamics We can write the log asset return ln A T /A t as a summation of two time-changed Lévy processes, [ ln A T /A t = Z T Z t,t 1 ] [ ] 2 T t,t Z + J T J t,t k J (1)Tt,T J, Two types of movements for asset return: (i) normal-time market fluctuations (Brownian motion Z) and (ii) large market disruptions (jump J). Apply separate time changes (Tt,T Z, T t,t J ), defined via their respective activity rates, T T Tt,T Z vs Z ds, Tt,T J vs J ds, t t Think of t as calendar time, and T as the business clock on which each business activity runs. The intensities of the two types of movements can vary separately. Leverage Effect, Volatility Feedback, and Self-Exciting Market Disruptions 11/4/2011 9 / 24
The stock price dynamics The stock price dynamics: 1 {}}{ 2 3 ( ) p { }}{{ }}{ Ft df t /F t = δ dw t + vt Z dz t + (e x 1) ( µ(dx, dt) π(x)dxvt J dt ) A t Linking/extending 3 strands of literature: 1 The local volatility effect of Dupire (94): Scaling F t by A t (both in dollars) makes the return variance a unitless quantity, and renders the dynamics scale free. Power dependence on leverage is supported by structural models. 2 The stochastic volatility of Heston (93): Used purely for volatility feedback, regardless of leverage. 3 The high-frequency dampened power law jumps (Wu, 2006): Arrival rate varies stochastically, and jumps synchronously with jumps in return. Leverage Effect, Volatility Feedback, and Self-Exciting Market Disruptions 11/4/2011 10 / 24
Alternative representation Let vt X = δ 2 Xt 2p. The stock price dynamics can be written as a three-factor stochastic volatility model: df t /F t = v X t dw t + v Z t dz t + R 0 (e x 1) ( µ(dx, dt) π(x)dxv J t dt ). where dv X t = κ X (v X t ) 2 dt σ X (v X t ) 3/2 dw t, a 3/2-process. (3) with κ X = p(2p + 1) and σ X = 2p. Henceforth, normalize δ = 1. Financial leverage variation is a separate source of stochastic volatility for stock return. The 3/2-vol of vol dependence in (3) has been shown to perform better than square-root dependence, e.g., Bakshi, Ju, Yang (2006). The structurally derived non-stationarity (κ > 0) helps to match long-dated option pricing behaviors. The model can be represented either as a local vol model with level dependence or a pure scale-free stochastic volatility model without level dependence unifying the two strands of literature. Leverage Effect, Volatility Feedback, and Self-Exciting Market Disruptions 11/4/2011 11 / 24
Market prices of risks and Active capital structure decisions The specifications so far are on the risk-neutral dynamics: Simplifying assumptions: Constant market prices (γ v, γ J ) for diffusion variance risk (Z t ) and jump risk (J t ). Capital structure questions: Financial managers make financial leverage decisions based on the current levels of all three types of risks: dx t = Xt 1 p ( ax κ XX X t κ XZ vt Z κ XJ vt J ) dt + X 1 p t dwt P. Market price of W t risk is γ X t = a X κ XX X t κ XZ v Z t κ XJ v J t. κ XX : Mean reversion, leverage level targeting. κ XZ : Response to diffusion (normal-market) business risk. κ XJ : Response to jump (crash) business risk. Leverage Effect, Volatility Feedback, and Self-Exciting Market Disruptions 11/4/2011 12 / 24
Data OTC implied volatility quotes on SPX options, January 1996 to March 2008, 583 weeks. 40 time series on a grid of 5 relative strikes: 80, 90, 100, 110, 120% of spot. 8 fixed time to maturities: 1m, 3m, 6m, 1y, 2y, 3y, 4y, 5y. Listed market focuses on short-term options (within 3 years). OTC market is very active on long-dated options. Why include long-dated options? At one maturity, an implied volatility smile/skew can be generated by many different mechanisms all you can learn is a heavy-tailed distribution. To distinguish the different roles played by the different mechanisms and answer the structural questions, we need to look at how these smiles/skews evolve across a wide range of maturities and over different time periods. Leverage Effect, Volatility Feedback, and Self-Exciting Market Disruptions 11/4/2011 13 / 24
The average implied volatility surface and skew: CLT? 9 8 Implied volatility, % 30 25 20 15 80 90 100 Relative strike, % 110 120 1 2 3 4 5 Maturity, years Mean implied volatility skew, % 7 6 5 4 3 2 1 0 1 2 3 4 5 Maturity, years Implied volatilities show a negatively sloped skew along strike. Return distribution has a down-side heavy tail. The skew slope becomes flatter as maturity increases due to scaling: 80% strike at 5-yr maturity is not nearly as out of money as 80% strike at 1-month maturity. When measured against a standardized moneyness measure d = ln(k/100)/(iv τ), the skew defined as, SK t,t = IV t,t (80%) IV t,t (120%) d t,t (80%) d t,t (120%), does NOT flatten as maturity increases. Central limit theorem does NOT kick in up to 10 years. Leverage Effect, Volatility Feedback, and Self-Exciting Market Disruptions 11/4/2011 14 / 24
Principal component analysis 80 A. Relative contributions B. Loading of the 1st PC Normalized eigenvalue, % 70 60 50 40 30 20 10 0 80 90 100 4 110 3 2 1 2 3 4 5 6 7 8 9 10 1 120 0 Principal component Relative strike, % Maturity, years C. Loading of the 2nd PC D. Loading of the 3rd PC Factor loading on P1 0.6 0.4 0.2 0 0.2 0.4 5 0.6 0.6 Factor loading on P2 0.4 0.2 0 0.2 0.4 Factor loading on P3 0.4 0.2 0 0.2 0.4 80 80 90 100 110 Relative strike, % 120 0 1 4 3 2 Maturity, years 5 90 100 110 Relative strike, % 120 0 1 4 3 2 Maturity, years 5 Leverage Effect, Volatility Feedback, and Self-Exciting Market Disruptions 11/4/2011 15 / 24
Asymmetric interactions 4 Weekly implied volatility level change, % 3 2 1 0 1 2 3 4 Weekly implied volatility skew change, % 1 0.5 0 0.5 1 5 10 5 0 5 10 Weekly index return, % 10 5 0 5 10 Weekly index return, % Self-exciting behavior: Implied volatility and skew respond more to large downside index jumps than upside index jumps. Leverage Effect, Volatility Feedback, and Self-Exciting Market Disruptions 11/4/2011 16 / 24
Option pricing and model estimation Option pricing: Quadrature integration of FFT option prices on asset [ c (F t, K, T ) = E t E t [(X A T K) + ] ] (XT = X ) Xt Estimation: Fix model parameters and use the 3 states (X t, v Z t, v J t ) to capture the variation of the 5 8 implied volatility surface: Let V t [X t, v Z t, v J t ] be the state. State propagation equation: V t = f (V t 1 ; Θ) + Q t 1 ε t. Let the 40 option series be the observation. Measurement equation: y t = h(v t ; Θ) + Re t, (40 1) y: OTM option prices scaled by the BS vega of the option. Assume that the pricing errors on the scaled option series are iid. Estimate 17 parameters over 23,320 options (11 years, 583 weeks, 40 options each day), using (quasi) maximum likelihood method joint with unscented Kalman filter. The model is very parsimonious and well-identified. Option pricing performance is miles better than any models in the literature. Leverage Effect, Volatility Feedback, and Self-Exciting Market Disruptions 11/4/2011 17 / 24
Relative variance and skew contributions Θ Estimates Std Error p 2.9952 0.0086 ρ -0.8388 0.0038 σ J 5.6356 0.0321 v J 0.1926 0.0004 Average instantaneous return variance contributions from (X t, v Z t, v J t ): Source Variance Volatility Leverage: E P [Xt 2p ] = 0.0117 (10.84%) Asset diffusion: E P [vt Z ] = 0.0231 (15.19%) Asset jump: E P [(vj 2 + v 2 + J )v J t ] = 0.0136 (11.67%) All three types of interactions are strong: Leverage effect: p = 2.8427 volatility feedback: ρ = 0.8354 self-excitement: σ J = 5.6355 Leverage Effect, Volatility Feedback, and Self-Exciting Market Disruptions 11/4/2011 18 / 24
Implied volatility responses to financial leverage shocks 0.27 Effects of X t variation 0.45 Effects of X t variation At the money implied volatility, % 0.26 0.25 0.24 0.23 0.22 0.21 0.2 0.19 0.18 One month implied volatility, % 0.4 0.35 0.3 0.25 0.2 0.15 0.17 0 1 2 3 4 5 Maturity, years 0.1 80 85 90 95 100 105 110 115 120 Relative strike, % Leverage v X t : Mean-repelling (drift=p(2p + 1)(v X t ) 2 dt) Responses to shocks become larger at longer maturities. An increase in leverage increase ATM vol more than OTM vol. Leverage Effect, Volatility Feedback, and Self-Exciting Market Disruptions 11/4/2011 19 / 24
Implied volatility responses to diffusion business shocks 0.26 Effects of v Z variation t 0.5 Effects of v Z variation t At the money implied volatility, % 0.24 0.22 0.2 0.18 0.16 0.14 One month implied volatility, % 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.12 0 1 2 3 4 5 Maturity, years 0.1 80 85 90 95 100 105 110 115 120 Relative strike, % Diffusion business risk v Z t : Strong mean reversion (κ Z = 3.0112) Responses decline quickly as option maturity increases. An increase in diffusion business risk flattens the skew. Leverage Effect, Volatility Feedback, and Self-Exciting Market Disruptions 11/4/2011 20 / 24
Implied volatility responses to crash business shocks 0.24 Effects of v J t variation 0.5 Effects of v J t variation 0.23 0.45 At the money implied volatility, % 0.22 0.21 0.2 0.19 0.18 0.17 One month implied volatility, % 0.4 0.35 0.3 0.25 0.2 0.15 0.16 0 1 2 3 4 5 Maturity, years 0.1 80 85 90 95 100 105 110 115 120 Relative strike, % Jump business risk v J t : Slow mean reversion (κ J = 0.0009) Responses do not decline. An increase in crash business risk increases the skew (OTM put option price). The variation of the implied volatility level and skew at different maturities helps us identify the variations of the three different risk sources. Leverage Effect, Volatility Feedback, and Self-Exciting Market Disruptions 11/4/2011 21 / 24
Sample variation of the three types of risks The S&P 500 Index 1600 1500 1400 1300 1200 1100 1000 900 800 v t X 0.035 0.03 0.025 0.02 0.015 0.01 0.005 700 97 98 99 00 01 02 03 04 05 06 07 08 09 0 97 98 99 00 01 02 03 04 05 06 07 08 09 0.14 0.12 0.1 0.08 4.5 4 3.5 3 2.5 v Z t 0.06 0.04 0.02 v J t 2 1.5 1 0.5 0 97 98 99 00 01 02 03 04 05 06 07 08 09 0 97 98 99 00 01 02 03 04 05 06 07 08 09 Financial leverage (v X t ) reached historical highs before the burst of the Nasdaq bubble. The diffusion variance risk (v Z t ) peaked during the 2003 recession. The crash risk (vt J ) spiked during the LTCM crisis. Leverage Effect, Volatility Feedback, and Self-Exciting Market Disruptions 11/4/2011 22 / 24
The capital structure decision dx t = Xt 1 p ( ax κ XX X t κ XZ vt Z κ XJ vt J Θ Estimates Std Error a X 0.0003 0.0003 κ XX 0.0001 0.0000 κ XZ 17.5341 0.4735 κ XJ -0.0774 0.0000 ) dt + X 1 p t dwt P κ XX = 0.0001: Capital structure is very persistent. κ XZ = 17.534: High diffusion business risk reduces X t and hence increases the financial leverage. κ XJ = 0.0774: High jump business risk increases X t and hence reduces the financial leverage. The key concern of financial leverage decision is default/crash (sustainability), not daily fluctuations Levering up increases your fluctuation, but also increases your return,... if only you can survive. Leverage Effect, Volatility Feedback, and Self-Exciting Market Disruptions 11/4/2011 23 / 24
Concluding remarks Many structural models have been proposed to answer capital structure questions, in stylized manners. Many reduced-form models have been proposed to match implied volatility smiles, without worrying about their structural meanings. We propose a model that is tractable and generates very good option pricing performance. Meanwhile, it can infer structural information from equity options. It is helpful to model the variation of the financial leverage and the business risk separately to bring structures to reduced-form models, to link local volatility models with stochastic volatility models, to generate good pricing performance on equity options over both short and long option maturities. The approach has potentials in analyzing single name stock options. Link the different capital structure management styles to the different behaviors of the option implied volatility surface. Leverage Effect, Volatility Feedback, and Self-Exciting Market Disruptions 11/4/2011 24 / 24