discussion of structural garch: the volatility-leverage connection by rob engle and emil siriwardane...
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Discussion of “Structural GARCH: The Volatility-Leverage Connection”
by Rob Engle and Emil Siriwardane
© 2013 by Andrew W. LoAll Rights Reserved
Andrew W. LoMFM Conference
October 11, 2013
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MFM
© 2013 by Andrew W. Lo All Rights Reserved
11 October 2013
Modigliani-Miller Proposition II:
The Basic Framework
Slide 2
Equity volatility increases linearly with leverage If debt is risky, the relationship is more complex Use option-pricing model (Merton, 1974)
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MFM
11 October 2013 © 2013 by Andrew W. Lo All Rights Reserved
Slide 3
The Basic Framework
Assets Liabilities
AT
ET = Max [ AT X , 0 ]
DT = Min [ AT , X ]
AT AT
Company XYZ
Equity is a call option on the firm’s assets Option-pricing model provides valuation formulas
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MFM
11 October 2013 © 2013 by Andrew W. Lo All Rights Reserved
Slide 4
The Basic Framework
Specific case of geometric Brownian motion for At
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MFM
11 October 2013 © 2013 by Andrew W. Lo All Rights Reserved
Slide 5
The Basic Framework
Itô’s formula yields the exact dynamics of Et
l = LM(At/X,1,s,r,t) is the option elasticity or “gearing”
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MFM
11 October 2013 © 2013 by Andrew W. Lo All Rights Reserved
Slide 6
The Basic Framework
At
Call Option Elasticity
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MFM
11 October 2013 © 2013 by Andrew W. Lo All Rights Reserved
Slide 7
The Basic Framework Now assume a more general asset-price process
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MFMStructural Interpretation From a formal perspective, not a fully “structural”
specification– e.g., suppose asset prices followed a CEV process
© 2013 by Andrew W. Lo All Rights Reserved
11 October 2013 Slide 8
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MFMStructural Interpretation The benefit of BSM is specificity, not flexibility Consistency of BSM functions with GARCH? If flexibility is the goal, then:
1. Use implied volatilities, e.g., VIX2. Use nonparametric option-pricing model (Ait-Sahalia and
Lo, 1998) to estimate SPD from options, which yields LM3. Use M&M II and multiple factors
– Estimate GARCH models with D/E, CDS, etc.
© 2013 by Andrew W. Lo All Rights Reserved
11 October 2013 Slide 9
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MFMStructural Interpretation Complex capital structure makes this more complex
– Coupon bonds are compound options (Geske, 1977) – Bond indenture provisions (Black and Cox, 1976)– Particularly problematic for financial firms
Potential issue with option-pricing approach– Pricing model hinges on dynamic completeness, i.e.,
dynamic replication of option payoff via trading strategy– Market for corporate assets are relatively illiquid– Delta-hedging strategy may be very expensive, i.e., option-
pricing formula may be a poor approximation (important for systemic risk measurement)
© 2013 by Andrew W. Lo All Rights Reserved
11 October 2013 Slide 10
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MFMLeverage Effect
Black (1972), Christie (1982), Duffie (1995)
Hasanhodzic and Lo (2013):
January 2, 1973 to December 31, 2010 (241 firms)
© 2013 by Andrew W. Lo All Rights Reserved
11 October 2013 Slide 11
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MFMLeverage Effect
© 2013 by Andrew W. Lo All Rights Reserved
11 October 2013 Slide 12
But leverage effect is the same for all-equity-financed companies!
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MFMSystemic Risk Measurement
During periods of financial distress, BSM option-pricing model for firm’s equity may be inadequate
Complex capital structure is particularly relevant , e.g., AIG’s credit support agreements
A structural alternative is Merton’s jump-diffusion model:
– For lognormal Z, LM can be derived in closed-form; compare and contrast to the pure diffusion case
A reduced-form alternative is regime-switching
© 2013 by Andrew W. Lo All Rights Reserved
11 October 2013 Slide 13
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MFMSystemic Risk Measurement
Another measure of systemic risk might be the option’s gamma:
© 2013 by Andrew W. Lo All Rights Reserved
11 October 2013 Slide 14
d1