(Mathématiques financières)

Ethanol crush spreads are used to model the value of a facility which produces ethanol from corn. A real options PDE is derived to investigate the effects of model parameters on management's decision to operate the facility through optimal switching and the firm's decision to enter into the project given its expected real options NPV. We present evidence of increased correlation between corn and ethanol prices, perhaps as a result of government policies which have induced more players to enter the market. Numerical solutions of the real options PDE allow us to investigate the resulting negative effects on firms. We also discuss the impact of an abrupt change in government policy, as happened in January 2012, on a firm's decision to enter the corn ethanol business.

Axiomatically based risk measures have been the object of numerous studies and generalizations in recent years. In the literature we find two main schools: coherent risk measures (Artzner et al. [1]) and insurance risk measures (Wang et al. [3]). In this note, we set to study yet another extension motivated by a third axiomatically based risk measure that has been recently introduced. In Heyde et al. [2], the concept of * natural risk statistics* is discussed as a data-based risk measure, i.e. as an axiomatic risk measure defined in the space $\mathbb R^n$. One drawback of these kind of risk measures is their dependence on the space dimension $n$. In order to circumvent this issue, we propose a way to define a family $\{\rho_n\}_{n=1,2,\dots}$ of *natural risk statistics* whose members are defined on $\mathbb{R}^n$ and related in an appropriate way. This construction requires the generalization of *natural risk statistics* to the space of infinite sequences $l^\infty$. We also look at the problem of extending this construction to a bi-variate sequence. Both of these problems are relevant in finance because they give a theoretical framework to study estimators of risk measures in the univariate and bi-variate setting. In fact, these axiomatic risk measures can be seen as defined over the set of univariate and bivariate data sets. In a risk management application this is important because typically one would have a data base of a financial position without any *a-priori* knowledge of the model itself.

**References**

[1] Artzner, Philippe; Delbaen, Freddy; Eber, Jean-Marc; Heath, David (2002) Coherent measures of risk. Risk management: value at risk and beyond (Cambridge, 1998), 145–175, Cambridge Univ. Press, Cambridge.

[2] Heyde, C. C.; Kou, S.G.; Peng, X. H. (2007). What Is a Good Risk Measure: Bridging the Gaps between Data, Coherent Risk Measures, and Insurance Risk Measures. Working Paper, Columbia University.

[3] Wang, S. S., V. R. Young, and H. H. Panjer (1997). Axiomatic characterization of insurance prices. Insurance: Mathematics and Economics, 21, 173-183.

We propose a new method for the numerical solution of backward stochastic differential equations (BSDEs) which finds its roots in Fourier analysis. The method consists of an Euler time discretization of the BSDE with certain conditional expectations expressed in terms of Fourier transforms and computed using the fast Fourier transform (FFT). The problem of error control is addressed, we consider the extension of the method to reflected BSDEs, and some numerical examples are considered from finance demonstrating the performance of the method.

While there are a number of academic papers discussing the importance of accounting for risk aversion in real option valuation, none of them are applicable to discrete cash-flow estimates supplied by managers. Recently, we introduced a method where we use any stochastic (Markov) process that can be mapped to managerial cash-flow estimates, to drive the cash-flows. The mapping allows us to link the cash-flow estimates to many theoretical real options frameworks which currently cannot be applied in practice. Through indifference pricing we are able to model the effect of risk aversion. We provide real world examples of the application of the methodology and show, through simulation, the effect of hedging the correlated portion of the cash-flows to a traded asset.