Leverage Effect, Volatility Feedback, and Self-Exciting MarketAFA, Disruptions 1/7/ / 14

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1 Leverage Effect, Volatility Feedback, and Self-Exciting Market Disruptions Liuren Wu, Baruch College Joint work with Peter Carr, New York University The American Finance Association meetings January 7, 2011 Leverage Effect, Volatility Feedback, and Self-Exciting MarketAFA, Disruptions 1/7/ / 14

2 Background: Structural versus reduced-form models Structural models, such as Merton (1974): Derive equity price dynamics as a function of firm value dynamics. Pros: Link equity volatility to financial leverage and asset volatility. Cons: Stylized, and/or not tractable for pricing equity options. (Cremers, Driessen, Maenhout, Weinbaum (2008)) Reduced models, such as Merton (1976): Directly model equity dynamics (as a function of several reduced-form factors) to price stock options. Pros: Tractable, practical, operational. Cons: Economic meanings (of hidden factors) are unclear. One can specify stochastic volatility for the equity return and allow volatility to be correlated with equity return, but it is unclear: Where the stochastic volatility comes from? Asset volatility variation or leverage variation? What is causing the interactions equity return and volatility? Leverage effect or volatility feedback? Leverage Effect, Volatility Feedback, and Self-Exciting MarketAFA, Disruptions 1/7/ / 14

3 A structural reduced-form model We propose a reduced-form model that can answer structural questions. We propose a model for the equity index that allows three distinct economic channels of interaction between return and volatility: 1 The leverage effect: With business risk fixed, an increase in financial leverage level leads to an increase in equity volatility level. 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 leverage 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 MarketAFA, Disruptions 1/7/ / 14

4 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 MarketAFA, Disruptions 1/7/ / 14

5 The asset value dynamics Model the asset value A t separately, in addition to leverage variation, da t A t = vt Z dz t + (e x 1) ( µ + (dx, dt) π 0 J +(x)dxvt J dt ) + 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, 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. Leverage Effect, Volatility Feedback, and Self-Exciting MarketAFA, Disruptions 1/7/ / 14

6 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 Leland (94). 2 The stochastic volatility of Heston (93): Used purely for volatility feedback, regardless of leverage. 3 The high-frequency jump of Madan,Carr,Chang (98): Arrival rate varies stochastically, and jumps synchronously with VG jump in return. Leverage Effect, Volatility Feedback, and Self-Exciting MarketAFA, Disruptions 1/7/ / 14

7 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 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 MarketAFA, Disruptions 1/7/ / 14

8 Market prices of risks and Active managerial decisions on financial leverage 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 ). Structural 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 business risk. κ XJ : Response to jump business risk. Leverage Effect, Volatility Feedback, and Self-Exciting MarketAFA, Disruptions 1/7/ / 14

9 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 MarketAFA, Disruptions 1/7/ / 14

10 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. Leverage Effect, Volatility Feedback, and Self-Exciting Market AFA, Disruptions 1/7/ / 14

11 Relative variance and skew contributions Θ Estimates Std Error p ρ σ J v J v J Average instantaneous return variance contributions from (X t, v Z t, v J t ): Source Variance Volatility Leverage: E P [Xt 2p ] = (10.91%) Asset diffusion: E P [vt Z ] = (15.19%) Asset jump: E P [(vj 2 + v 2 + J )v J t ] = (10.79%) All three types of interactions are strong: Leverage effect (p = ), volatility feedback (ρ = ), and self-excitement ( σ J = ). Much larger downside jumps than upside jumps (v J v J +). Leverage Effect, Volatility Feedback, and Self-Exciting Market AFA, Disruptions 1/7/ / 14

12 Implied volatility term structure variations 0.27 Effects of X t variation 0.26 Effects of v Z t variation 0.25 Effects of v J t variation At the money implied volatility, % At the money implied volatility, % At the money implied volatility, % Maturity, years Maturity, years Maturity, years 1 Leverage v X t : Mean-repelling (drift=p(2p + 1)(v X t ) 2 dt) Responses to shocks become larger at longer maturities. 2 Diffusion business risk v Z t : Strong mean reversion (κ Z = ) Responses decline quickly as option maturity increases. 3 Jump business risk v J t : Slow mean reversion (κ J = ) Responses do not decline. The impacts are vt Z (volatility feedback) are mainly an short term options. X t (leverage) and vt J (self-exciting jump) extend to long-term options. Long term structure is needed to differentiate the three risk sources. Leverage Effect, Volatility Feedback, and Self-Exciting Market AFA, Disruptions 1/7/ / 14

13 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 κ XX κ XZ κ XJ ) dt + X 1 p t dwt P κ XX = : Capital structure is very persistent. κ XZ = : High diffusion business risk reduces X t and hence increases the financial leverage. κ XJ = : 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 AFA, Disruptions 1/7/ / 14

14 Concluding remarks Many structural models have been proposed to answer structural questions, in stylized manner. Many reduced-form models/mechanisms have been proposed to match implied volatility smiles, without worrying about 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 AFA, Disruptions 1/7/ / 14