Predicting systematic risk: Implications from growth options

In accordance with the well-known financial leverage effect, decreases in stock prices cause an increase in the levered equity beta for a given unlevered beta. However, as growth options are more volatile and have higher risk than assets in place, a price decrease may decrease the unlevered equity beta via an operating leverage effect. This is because price decreases are associated with a proportionately higher loss in growth options than in assets in place. Most of the existing literature focuses on the financial leverage effect: This paper examines both effects. We show, with a simple option pricing model, the opposing effects at work when the firm is a portfolio of assets in place and growth options. Our empirical results show that, contrary to common belief, the operating leverage effect largely dominates the financial leverage effect, even for initially highly levered firms with presumably few growth options. We then link variations in betas to measurable firm characteristics that proxy for the fraction of the firm invested in growth options. We show that these proxies jointly predict a large fraction of future cross-sectional differences in betas. These results have important implications on the predictability of equity betas, hence on empirical asset pricing and on portfolio optimization that controls for systematic risk.     

MCMC maximum likelihood for latent state models

This paper develops a pure simulation-based approach for computing maximum likelihood estimates in latent state variable models using Markov Chain Monte Carlo methods (MCMC). Our MCMC algorithm simultaneously evaluates and optimizes the likelihood function without resorting to gradient methods. The approach relies on data augmentation, with insights similar to simulated annealing and evolutionary Monte Carlo algorithms. We prove a limit theorem in the degree of data augmentation and use this to provide standard errors and convergence diagnostics. The resulting estimator inherits the sampling asymptotic properties of maximum likelihood. We demonstrate the approach on two latent state models central to financial econometrics: a stochastic volatility and a multivariate jump-diffusion models. We find that convergence to the MLE is fast, requiring only a small degree of augmentation.     

Pathwise moderate deviations for option pricing

We provide a unifying treatment of pathwise moderate deviations for models commonly used in financial applications, and for related integrated functionals. Suitable scaling enables us to transfer these results into small‐time, large‐time, and tail asymptotics for diffusions, as well as for option prices and realized variances. In passing, we highlight some intuitive relationships between moderate deviations rate functions and their large deviations counterparts; these turn out to be useful for numerical purposes, as large deviations rate functions are often difficult to compute.     

Large Deviations for the Extended Heston Model: The Large-Time Case

We study here the large-time behaviour of all continuous affine stochastic volatility models [in the sense of Keller-Ressel (Math Finan 21(1):73–98, 2011)] and deduce a closed-form formula for the large-maturity implied volatility smile. We concentrate on (rescaled) strikes around the money, which are the most common in practice, and extend the results in Forde and Jacquier (Finan Stoch 15(4):755–780, 2011) and Gatheral and Jacquier (Quant Finan 11(8):1129–1132, 2011).     

A note on essential smoothness in the Heston model

This note identifies a gap in the proof of Corollary 2.4 in Forde and Jacquier (Finance Stoch., 2011) which arises because the essential smoothness of the family (X _t /t) _t≥1 can fail for the log-spot process X in the Heston model, and it describes how to circumvent the issue by applying a standard argument from large deviation theory.     

Correction note for ��The large-maturity smile for the Heston model��

The papers (Forde and Jacquier in Finance Stoch. 15:755?C780, 2011; Forde et al. in Finance Stoch. 15:781?C784, 2011) study large-time behaviour of the price process in the Heston model. This note corrects typos in Forde and Jacquier (Finance Stoch. 15:755?C780, 2011), Forde et al. (Finance Stoch. 15:781?C784, 2011) and clarifies the proof of Forde et al. (Finance Stoch. 15:781?C784, 2011, Proposition 2.3).     

On VIX futures in the rough Bergomi model

The rough Bergomi model introduced by Bayer et al. [Quant. Finance, 2015, 1–18] has been outperforming conventional Markovian stochastic volatility models by reproducing implied volatility smiles in a very realistic manner, in particular for short maturities. We investigate here the dynamics of the VIX and the forward variance curve generated by this model, and develop efficient pricing algorithms for VIX futures and options. We further analyse the validity of the rough Bergomi model to jointly describe the VIX and the SPX, and present a joint calibration algorithm based on the hybrid scheme by Bennedsen et al. [Finance Stoch., forthcoming].     

Empirical evidence on the dependence of credit default swaps and equity prices

We investigate the common practice of estimating the dependence structure between credit default swap prices on multi‐name credit instruments from the dependence structure of the equity returns of the underlying firms. We find convincing evidence that the practice is inappropriate for high‐yield instruments and that it may even be flawed for instruments containing only firms within a sector. To do this, we model individual credit ratings by univariate continuous time Markov chains, and their joint dynamics by copulas. The use of copulas allows us to incorporate our knowledge of the modeling of univariate processes, into a multivariate framework. However, our test and results are robust to the choice of copula. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 29:695–712, 2009     

The large-maturity smile for the Heston model

Using the Gärtner–Ellis theorem from large deviations theory, we characterise the leading-order behaviour of call option prices under the Heston model, in a new regime where the maturity is large and the log-moneyness is also proportional to the maturity. Using this result, we then derive the implied volatility in the large-time limit in the new regime, and we find that the large-time smile mimics the large-time smile for the Barndorff–Nielsen normal inverse Gaussian model. This makes precise the sense in which the Heston model tends to an exponential Lévy process for large times. We find that the implied volatility smile does not flatten out as the maturity increases, but rather it spreads out, and the large-time, large-moneyness regime is needed to capture this effect. As a special case, we provide a rigorous proof of the well-known result by Lewis (Option Valuation Under Stochastic Volatility, Finance Press, Newport Beach, 2000) for the implied volatility in the usual large-time, fixed-strike regime, at leading order. We find that there are two critical strike values where there is a qualitative change of behaviour for the call option price, and we use a limiting argument to compute the asymptotic implied volatility in these two cases.     

Optimal Estimation of the Risk Premium for the Long Run and Asset Allocation: A Case of Compounded Estimation Risk

It is well known that an unbiased forecast of the terminal valueof a portfolio requires compounding at the arithmetic mean returnover the investment horizon. However, the maximum-likelihoodpractice, common with academics, of compounding at the estimatorof mean return results in upward biased and highly inefficientestimates of long-term expected returns. We derive analyticallyboth an unbiased and a small-sample efficient estimator of long-termexpected returns for a given sample size and horizon. Both estimatorsentail penalties that reduce the annual compounding rate asthe investment horizon increases. The unbiased estimator, whichis far lower than the compounded arithmetic average, is stillvery inefficient, often more so than a simple geometric estimatorknown to practitioners. Our small-sample efficient estimatoris even lower. These results compound the sobering evidencein recent work that the equity risk premium is lower than suggestedby post-1926 data. Our methodology and results are robust toextensions such as predictable returns. We also confirm analyticallythat parameter uncertainty, properly incorporated, producesoptimal asset allocations, in stark contrast to conventionalwisdom. Longer investment horizons require lower, not higher,allocations to risky assets.