# Abstracts

**Estimating Option Implied Risk Neutral Densities: A Novel Parametric Approach**

*Greg Orosi (American University of Sharjah, UAE)*

Tuesday June 3, 11:00-11:30 | session 1.5 | Options, Futures | room G

Extracting risk-neutral density (RND) functions from cross-sections of observed standard option prices is highly important for both academics and practitioners. For instance, Ross (2013), Figlewski (2009), and Figlewski and Birru (2012) employ the RND to obtain information about investorsâ€™ risk preferences and expectations. In the field of option pricing, the implied RND allows to price illiquid exotic derivatives with arbitrary payoffs and are also applicable to the copula-based pricing of multi-asset products. For example, Monteiro et al. (2011) show that the implied RND can be used to accurately price European-style binary options and Cherubini and Luciano (2002) use the implied RND to price bivariate equity options.

In this work, we propose a novel parametric approach to extract the implied risk neutral density function from a cross-section of call option prices. The method is based on the framework proposed by Orosi (2011), who presents a multi-parameter extension of the models of Figlewski (2002) and Henderson, Hobson, and Kluge (2007). By choosing a proper functional form, we show that well-behaved risk neutral densities can be generated by imposing restrictions on the parameters of the model. The results of our numerical experiments indicate that the method is capable of extracting risk neutral densities with complex characteristics. Moreover, we demonstrate the pricing performance of our method by generating arbitrage-free call option prices that can be used to produce well-behaved densities from S&P 500 Index options.

Additionally, further extensions of the model are straightforward. For example, we demonstrate that the model is capable of retrieving the risk neutral probability density function from call options written on defaultable assets.

**Forward Stopping Rule Within HJM Framework**

*Mingxin Xu (University of North Carolina at Charlotte, USA)*

Tuesday June 3, 11:30-12:00 | session 1.5 | Options, Futures | room G

We revisit the optimal stopping problem using Heath-Jarrow-Morton (HJM) approach. The HJM method was originally introduced to model the fixed-income market by Heath et al. (1992). More recently, it was implemented in equity market models by Schweizer and Wissel (2008), and Carmona and Nadtochiy (2008, 2009). Prior work has mainly focused on European derivative pricing, while in this paper we apply the HJM philosophy to American derivative pricing with a focus toward solving optimal stopping problems in general. As a counterpart to forward rate for the fixed-income market and forward volatility for the equity market, we introduce forward drift for the optimal stopping problem. The standard results for HJM-type models are confirmed for the forward drift dynamics: the drift condition and the spot consistency condition. More interestingly, we discover a forward stopping rule that is fundamentally different from the classical stopping rule based on backward induction. Although simple, the binomial model enables us to clearly illustrate the calculation difference between the two approaches in finding the optimal stopping times and show that the binomial tree only needs to be built up to the optimal stopping time in the forward approach, i.e., the decision to stop does not depend on the future evolution of the tree. In the classical backward approach, it is usually much easier to obtain solutions for infinite-horizon problems than finite-horizon problems. With the Black-Scholes model we highlight another novelty for the forward approach: there is fundamentally no difference in terms of level of difficulty for the finite or infinite-horizon case. In addition to the usual minimal optimal stopping time, we characterize the maximal optimal stopping time in the forward approach.

**A Continuous Mixed-Laplace Jump Diffusion model for option pricing, with and without mean reversion.**

*Donatien Hainaut (Rennes Business School, France)*

Tuesday June 3, 12:00-12:30 | session 1.5 | Options, Futures | room G

This paper proposes a jump diffusion model with and without mean reversion, for stocks and commodities prices in which jump sizes are continuous mixtures of Laplace random variables. The jump component is like an infinite sum of jumps whith Laplace distributed amplitudes. The jump times and the jump amplitudes are respectively interpreted as dates of information disclosures and as their impact on asset prices. Given that low frequency information has usually a bigger impact on asset prices than frequent information, the frequencies of jumps decrease and are inversely proportional to the average size of jumps. In this framework, we infer analytical expressions of distribution of jumps size, characteristic functions and moments. Simple series developments of characteristic functions are also proposed and options prices and densities are retrieved by discrete Fourier transforms. To motivate this research from an econometric point of view, we provide some empirical evidence is presented about the ability of the Continuous Mixed-Laplace Jump Diffusion (CMLJD) to represent daily returns of four major stocks indices (MS World, FTSE, S&P and CAC 40), over a period of 10 years. So as to illustrate its utility, the mean reverting CMLJD is fitted to four time series of commodity prices that exhibit this feature (Copper, Soy Beans, Crude Oil WTI and Wheat, observed on four years). Finally, examples of implied volatility surfaces for European Call options are presented. And the sensitivity of this surface to each parameters is analyzed.