Capturing Volatility Smiles with a Perpetual Leverage Model, and its Implications to Fund Overlay Designs
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Structural modeling of leveraged firms treats a firm's equity as a derivative whose underlying instrument is the firm's asset. For example, Merton (1974) modeled the firm's equity as a vanilla call option, and Leland (1994) modeled equity as a perpetual option. Under this assumption, the equity option of a leveraged firm is then a compound option, as demonstrated by Geske (1979) and by Toft and Prucyk (1997). The compound option assumption is a powerful tool to for explaining the volatility smile and skew observed in the equity option market, as demonstrated in Toft and Prucyk (1997), Hull et al. (2004a) and Chen and Kou (2009). However efforts to understand the smile and skew observations through structural modeling have been limited. This thesis further explores the explicit representation of volatility smiles/skews through structural modeling, achieving a better replication of the market with practical modeling and calibration strategies. The equity of a firm is modeled as a perpetual option, but in a more general format, compared to existing publications following Leland. Asymmetry is introduced into asset return distributions through a constant elasticity of variance (CEV) process, so that the model achieves better agreements to skews, smiles, and when the leverage is insignificant. The choice of CEV asset stochastic ensures enough model flexibility to produce various shapes of volatility skew and smile. It also retains the mathematical tractability allowing the calibration of the compound option to the vanilla market to remain practical. Lastly, the model remains moderately parameterized so that the calibration is still meaningful. This calibration produces leverage metrics potentially helpful to fundamental and credit analysis. The equity-asset relation under CEV asset assumption is modeled through a free-boundary differential equation, which reflects the financial aspects of a limited liability firm. The equity value is solved from this free-boundary problem as a closed-form relation with respect to the firm's asset dynamic and its nominal liability. This closed-form representation is then embedded into the equity option pricing model, simplifying the compound option pricing problem as a more approachable barrier option pricing (first seen in Toft and Prucyk (1997)). A practical Monte-Carlo based fitting strategy is proposed, so that the model can be tested on a larger set of candidates within a reasonable amount of computation time. Empirical tests demonstrate the capabilities of this model to produce both volatility smile and skew, to accommodate the volatility skew observed on very low-leverage firms, and to produce credit quality measures that are more consistent to the credit default swap (CDS) market. Distribution analysis on S&P-100 and NASDAQ-100 candidates generates distinct leverage and volatility distributions between the two index pools that are consistent with the component characteristics of each pool. Some desirable features of the perpetual structural model also inspired additional discoveries in retail fund management. A perpetual American put option replication strategy is provided as an investment protection, whose benefits, including extreme loss prevention and path-independency, are also illustrated.