# Abstracts

**Optimal Capital Structure in case of Stochastic Volatility**

*Monique Pontier (Université de Toulouse, France)*

Tuesday June 3, 11:00-11:30 | session 1.7 | Stochastic Volatility | room I

We analyze the capital structure of a firm in a infinite time horizon framework following Leland's idea [1] under the more general hypothesis that the firm value process belongs to a fairly large class of stochastic volatility models.

Following Fouque et al. approximating methods [2], in such a scheme, we describe and analyze the effects of stochastic volatility on all variables describing the capital structure. The endogenous failure level is derived in order to exploit the optimal amount of debt chosen by the firm. Exploiting optimal capital structure we find that the stochastic volatility framework seems to be a robust way to improve results in the direction of both higher spreads and lower leverage ratios in a quantitatively significant way.

[1] Leland H.E. (1994), ``Corporate debt value, bond covenant, and optimal capital structure', The Journal of Finance, 49, 1213-1252.

[2] Fouque J.P., Papanicolaou G., Ronnie K.R., Derivatives in Financial Markets with Stochastic Volatility, Cambridge University Press (2000).

**A dynamic view of the Heston model**

*Patrick Roome (Imperial College London, UK)*

Tuesday June 3, 11:30-12:00 | session 1.7 | Stochastic Volatility | room I

The Heston stochastic volatility model was introduced over 20 years ago [1] and is arguably the most widely used stochastic volatility model in the industry.This is in large part due to analytical tractability and the existence of semi-closed formulae for European option pricing. Furthermore, asymptotics of the (spot) implied volatility smile have been thoroughly studied in Heston, giving insight into the behaviour of model-generated spot smiles. However, there are virtually no analytical results on the dynamics of model implied volatility smiles, a key model-risk metric for assessing the suitability of a model for exotic option pricing.

In this talk we will first derive small and large-maturity asymptotics for the Heston forward implied volatility smile ([1],[2],[3]) using the theory of sharp large deviations (and refinements). We will then use these results to gain insight into some core dynamical properties of the model. We will provide a number of cases of degenerate large deviations behaviour and we will show that it is exactly the analysis of these pathological cases that gives the most insight into the dynamical features of the model.

[1] S.Heston. A closed-form solution for options with stochastic volatility with applications to bond and currency options.The Review of Financial Studies, 6 (2): 327-342, 1993.

[2] A. Jacquier and P. Roome. Asymptotics of forward implied volatility. Submitted, http://arxiv.org/abs/1212.0779. 2013.

[3] A. Jacquier and P. Roome. Large-maturity regimes of the Heston forward smile. In progress.

[4] A. Jacquier and P. Roome. The small-maturity Heston forward smile. SIAM J. Finan. Math., 4 (1): 831-856, 2013.

**The dynamics of option pricing and the market price of volatility risk**

*Peter Gruber (Università della Svizzera Italiana, Switzerland)*

Tuesday June 3, 12:00-12:30 | session 1.7 | Stochastic Volatility | room I

We study the joint properties of option pricing together with the market prices of volatility risks, with the help of a parsimonious three-factor model featuring interdependent volatilities and an unspanned skewness component. Using closed-form expressions for option prices and the market price of variance risk, we perform a consistent estimation for a large panel of S&P 500 options and infer on the dynamics and term structure of variance risk premia. We find that our model produces an excellent pricing performance, improving on that of more parameterized three-factor affine models. It also yields variance risk premium dynamics consistent with economic intuition and with the features of unconstrained volatility predictive regressions. Our findings also highlight (i) the inaccuracy of lower-dimensional (e.g., two-factor) models for capturing the S&P 500 volatility dynamics; (ii) a systematic link between volatility risk premia and model-implied unspanned skewness, (iii) a rich dynamics of the term structure of volatility risk premia, which is systematically related to market-wide conditions and sentiment, (iv) the usefulness of model specifications introducing an unspanned skewness dimension, and (v) the necessity of specifying jumps in the underlying price process, at least.

**First-passage times of regime switching models**

*Peter Hieber (TU Munich, Germany)*

Tuesday June 3, 12:30-13:00 | session 1.7 | Stochastic Volatility | room I

This talk studies the first-passage times of regime switching Brownian motion on an upper and/or a lower level. In the 2- and 3-state model, the Laplace transform of the first-passage times is – up to the roots of a polynomial of degree 4 (respectively 6) – derived in closed-form by solving the matrix Wiener-Hopf factorization. This extends the one-sided results in the 2-state model by Guo (Methodol Comput Appl Probab 3(2):135–143, 2001a). If the quotient of drift and variance is constant over all states, we show that the Laplace transform can even be inverted analytically.

Due to their analytical tractability and their ability to well explain many empirical phenomena, regime switching models have recently gained remarkable attention. In contrast to Lévy models, they can capture effects like persistently changing long-term trends – a desirable feature especially for long dated contracts in Finance and Insurance. In Finance, the presented results can be used for the pricing and risk management of, for example, barrier or lookback options.