2. Discussion of Structural GARCH: The Volatility-Leverage Connection by Rob Engle and Emil Siriwardane © 2013 by Andrew W. Lo All Rights Reserved Andrew W. Lo MFM Conference October 11, 2013 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAA A A A A AAA A A A

3. 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)

4. MFM © 2013 by Andrew W. Lo All Rights Reserved 11 October 2013 Slide 3 The Basic Framework AssetsLiabilities ATAT E T = Max [ A T X, 0 ] D T = Min [ A T, X ] ATAT ATAT Company XYZ Equity is a call option on the firms assets Option-pricing model provides valuation formulas

5. MFM © 2013 by Andrew W. Lo All Rights Reserved 11 October 2013 Slide 4 The Basic Framework Specific case of geometric Brownian motion for A t

6. MFM © 2013 by Andrew W. Lo All Rights Reserved 11 October 2013 Slide 5 The Basic Framework Itôs formula yields the exact dynamics of E t = LM(A t /X,1,,r, ) is the option elasticity or gearing

7. MFM © 2013 by Andrew W. Lo All Rights Reserved 11 October 2013 Slide 6 The Basic Framework AtAt Call Option Elasticity

8. MFM © 2013 by Andrew W. Lo All Rights Reserved 11 October 2013 Slide 7 The Basic Framework Now assume a more general asset-price process

9. MFM Structural 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

10. MFM Structural 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., VIX 2.Use nonparametric option-pricing model (Ait-Sahalia and Lo, 1998) to estimate SPD from options, which yields LM 3.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

11. MFM Structural 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

12. MFM Leverage 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

13. MFM Leverage 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!

14. MFM Systemic Risk Measurement During periods of financial distress, BSM option- pricing model for firms equity may be inadequate Complex capital structure is particularly relevant, e.g., AIGs credit support agreements A structural alternative is Mertons 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

15. MFM Systemic Risk Measurement Another measure of systemic risk might be the options gamma: © 2013 by Andrew W. Lo All Rights Reserved 11 October 2013 Slide 14 d1d1