The goal of the paper is to review the last 35years of continuous-time finance by focusing on two major advances:(i)The powerful elegance of the martingale representation for primitive assets and attainable contingent claims in more and more general settings, thanks to the probabilistic tool of probability change and the economic flexibility in the choice of the numéraire relative to which prices are expressed. This numéraire evolved over time from the money market account to a zero-coupon bond or a stock price, lastly to strictly positive quantities involved in the Libor or swap market models and making the pricing of caps or swaptions quite efficient.(ii)The persistent central role of Brownian motion in finance across the 20th century: even when the underlying asset price is a very general semi-martingale, the no-arbitrage assumption and Monroe theorem [Monroe, I., 1978. Processes that can be embedded in Brownian motion. Annals of Probability 6, 42–56] allow us to write it as Brownian motion as long as we are willing to change the time. The appropriate stochastic clock can be shown empirically to be driven by the cumulative number of trades, hence by market activity. Consequently, starting with a general multidimensional stochastic process S defined on a probability space (Ω,F,Ft,P) and representing the prices of primitive securities, the no-arbitrage assumption allows, for any chosen numéraire, to obtain a martingale representation for S under a probability measure QS equivalent to P. This route will be particularly beneficiary for the pricing of complex contingent claims. Alternatively, changing the clock, i.e., changing the filtration (Ft), we can recover the Brownian motion and normality of returns. In all cases martingales appear as the central representation of asset prices, either through a measure change or through a time change.

The goal of the paper is to show that some types of Lévy processes such as the hyperbolic motion and the CGMY are particularly suitable for asset price modelling and option pricing. We wish to review some fundamental mathematic properties of Lévy distributions, such as the one of infinite divisibility, and how they translate observed features of asset price returns. We explain how these processes are related to Brownian motion, the central process in finance, through stochastic time changes which can in turn be interpreted as a measure of the economic activity. Lastly, we focus on two particular classes of pure jump Lévy processes, the generalized hyperbolic model and the CGMY models, and report on the goodness of fit obtained both on stock prices and option prices.

The aim of the paper is to analyze the diversification effect brought by crude oil Futures contracts, the most liquid commodity Futures, into a portfolio of stocks. The studies that have documented the very low- and essentially negative-correlations between commodities and equities typically rely on normally distributed returns, which is not the case for crude oil Futures and stocks indexes. Moreover, the particular time-to-maturity chosen for the Future contract used as an investment vehicle is an important matter that needs to be addressed, in presence of forward curves switching between backwardation and contango shapes. Our goal in this paper is twofold: (a) we introduce copula functions to have a better representation of the dependence structure of oil Futures with equity indexes; (b) using this copula representation, we are able to analyze in a precise manner the “maturity effect” in the choice of crude oil Future contract with respect to its diversification benefits. Our finding is that, in the case of distant maturities Futures, e.g., 18 months, the negative correlation effect is more pronounced whether stock prices increase or decrease. This property has the merit to avoid the hurdles of a frequent roll over while being quite desirable in the current trendless equity markets. Empirical evidence is exhibited on a database comprising the NYMEX WTI crude oil Futures and S&P 500 index over a 15 year-time period.

We address the problem of managing a storable commodity portfolio, that includes physical assets and positions in spot and forward markets. The vast amount of capital involved in the acquisition of a power plant or storage facility implies that the financing period stretches over a period of several quarters or years. Hence, an intertemporally consistent way of optimizing the portfolio over the planning horizon is required. We demonstrate the temporal inconsistency of static risk objectives based on final wealth and advocate the validity in our setting of a new class of recursive risk measures introduced by Epstein and Zin [Epstein, G., Zin, S., 1989. Substitution, risk aversion, and the temporal behavior of consumption and asset returns: A theoretical framework. Econometrica, 57 (4) 937–969] and Wang [Wang, T., 2000. A class of dynamic risk measures University of British Columbia]. These risk measures provide important insights on the trade-offs between date-specific risks (i.e., losses occurring at a point in time) and time-duration risks represented by the pair (return, risk) over a planning horizon; in a number of situations, they dramatically improve the efficiency of static risk objectives, as exhibited in numerical examples.