Publications

You will find here some of my publications and working papers.


Explicit constructions of martingales calibrated to given implied volatility smiles (with P. Carr)


SIAM Journal on Financial Mathematics, 3, 2012, p. 182-214. [ Link ]
The working paper can be found on SSRN or here.

The construction of martingales with given marginal distributions at given times is a recurrent problem in financial mathematics. From a theoretical point of view, this problem is well-known as necessary and sufficient conditions for the existence of such martingales have been described. Moreover several explicit constructions can even be derived from solutions to the Skorokhod embedding problem. However these solutions have not been adopted by practitioners, who still prefer to construct the whole implied volatility surface and use the explicit constructions of calibrated (jump-) diffusions, available in the literature, when a continuum of marginal distributions is known.
            In this paper, we describe several new constructions of calibrated martingales, which do not rely on a potentially risky interpolation of the marginal distributions but only on the input marginal distributions. These calibrated martingales are intuitive since the continuous-time versions of our constructions can be interpreted as time-changed (jump-) diffusions. Moreover, we show that the valuation of claims, depending only on the values of the underlying process at maturities where the marginal distributions are known, can be extremely efficient in this setting. For example, path-independent claims of this type can be valued by solving a finite number of ordinary (integro-) differential equations. Finally, an example of calibration to the S&P 500 market is provided.


Pricing CDOs with State Dependent Stochastic Recovery Rates (with S. Amraoui, S. Hitier and J-P. Laurent)


Quantitative finance, 12 (8), 2012, p. 1219-1240. [ Link ]
The working paper can be found on SSRN or here.


Up to the 2007 crisis, research within bottom‐up CDO models mainly concentrated on the dependence between defaults. However, due to the substantial increase in the market price of systemic credit risk protection, more attention has been paid to recovery rate assumptions. In this paper, we focus first on deterministic recovery rates in a factor copula framework. We use stochastic orders theory to assess the impact of a recovery markdown on CDOs and show that it leads to an increase of the expected loss on senior tranches, even though the expected loss on the portfolio is kept fixed. This result applies to a wide range of latent factor models.
            We then suggest introducing stochastic recovery rates in such a way that the conditional on the factor expected loss (or equivalently the large portfolio approximation) is the same as in the recovery markdown case. However, granular portfolios behave differently. We show that a markdown is associated with riskier portfolios that when using the stochastic recovery rate framework. As a consequence, the expected loss on a senior tranche is larger in the former case, whatever the attachment point.
We also deal with implementation and numerical issues related to the pricing of CDOs within the stochastic recovery rate framework. Due to differences across names regarding the conditional (on the factor) losses given default, the standard recursion approach becomes problematic. We suggest approximating the conditional on the factor loss distributions, through expansions around some base distribution.
            Finally, we show that the independence and comonotonic cases provide some easy to compute bounds on expected losses of senior or equity tranches.


A PDE approach to jump-diffusions (with P. Carr)


Quantitative Finance, 1 (11), 2011, p. 33-52. [ Link ]
The working paper can be found on SSRN or here.

In this paper, we show that the calibration to an implied volatility surface and the pricing of contingent claims can be as simple in a jump-diffusion framework as in a diffusion one. Indeed, after defining the jump densities as those of diffusions sampled at independent and exponentially distributed random times, we show that the forward and backward Kolmogorov equations can be transformed into partial differential equations. It enables us to (i) derive Dupire-like equations (see Dupire (1994)) for coefficients characterizing these jump-diffusions; (ii) describe sufficient conditions for the processes they induce to be calibrated martingales; and (iii) price path-independent claims using backward partial diff.erential equations. This paper also contains an example of calibration to the S&P 500 market.



Conditions on option prices for absence of arbitrage and exact calibration


Journal of Banking and Finance, 31 (11), 2007, p. 3377-3397. [ Link ]
Presented at the Risk Management and Quantitative Finance Conference, Gainesville, FL, 2005. [ Slides ]
The working paper can be found on SSRN or here.


Under the assumption of absence of arbitrage, European option quotes on a given asset must satisfy well-known inequalities, which have been described in the landmark paper Merton (1973). If we further assume that there is no interest rate volatility and that the underlying asset continuously pays deterministic dividends, cross-maturity inequalities must also be satisfied by the bid and ask option prices.
            In this paper, we show that there exists an arbitrage-free model, which is consistent with the option quotes, if these inequalities are satisfied. One implication is that all static arbitrage strategies are linear combinations, with positive weights, of those described here. We also characterize admissible default probabilities for models which are consistent with given option quotes.



The stochastic intensity SSRD model implied volatility patterns for credit default swap options and the impact of correlation (with D. Brigo)


International Journal of Theoretical and Applied Finance, 9 (3), 2006, p. 315-339. [ Link ]
Presented at the Third Bachelier Conference in Mathematical Finance, Chicago, 2004. [ Slides ]
The working paper can be found on SSRN or here.

In this paper we investigate implied volatility patterns in the Shifted Square Root Diffusion (SSRD) model as functions of the model parameters. We begin by recalling the Credit Default Swap (CDS) options market model that is consistent with a market Black-like formula, thus introducing a notion of implied volatility for CDS options. We examine implied volatilies coming from SSRD prices and characterize the qualitative behavior of implied volatilities as functions of the SSRD model parameters. We introduce an analytical approximation for the SSRD implied volatility that follows the same patterns in the model parameters and that can be used to have a first rough estimate of the implied volatility following a calibration. We compute numerically the CDS-rate volatility smile for the adopted SSRD model. We find a decreasing pattern of SSRD implied volatilities in the interest-rate/intensity correlation. We check whether it is possible to assume zero correlation after the option maturity in computing the option price.