I ran into a stumbling block earlier when I tried to price stochastic annuities (see Asian options). This is actually technically an acturial problem, but is well adapted to the techniques of quant finance.

My requirement is to capture the distribution of net present value (i.e., discounted cash flows) of firms with uncertain revenue streams (e.g., firms which are price takers such as commodities producers) in order to ultimately price contingent claims on them. Geometric Brownian Motion seemed like a natural choice to model cash flow volatility due to the breadth of research on this topic (see Black-Scholes). However, the time integral of GBM (i.e., the distribution of the sum of lognormals) has no closed form solution.

On further reflection, this model may be ill-prescribed since commodities prices are widely believed to exhibit different characteristics, such as mean reversion and term structures which reflect carry/convenience yields. However, most canonically accepted stochastic commodities price models (Gibson and Schwartz (1990), Schwartz (1997), Smith and Schwartz (July 2000), etc.) are variations on themes of an exponential Ornstein-Uhlenbeck processes, meaning they too are Gaussian, meaning that I expect the same difficulties as with GBM.

If I was able to get around the Gaussian operator, the problem might be tractable. Indeed, Dassios and Nagaradjasarma found that the distribution of the integral of the square root (i.e., Cox-Ross) process is analytically tractable. The paper claims that:

The popularity of the square-root process in all main branches of financial modeling can be explained by its desirable property of positivity and its richness of behaviour. As a result, it has been used to model equities (Cox-Ross [10] alternative process), interest rates (CIR [9] interest rate model and its time-inhomogeneous [27], multivariate [7] and other derivatives), stochastic volatility ( Heston [22] model and its various extensions 2, [15], [30]) and other financial quantities.

Given the analytical tractability of the integrated CIR process (assuming that this is what paper means by Cox-Ross), I am tempted to switch gears. But is it suitable for modelling commodity price dynamics? Is its use empirically supported?

I could only find one paper on the topic. Ribeiro and Hodges (2004) develop:

...a reduced form two-factor model for commodity spot prices and futures valuation. This model extends the Gibson and Schwartz (1990)-Schwartz (1997) two-factor model by adding two new features. First the Ornstein-Uhlenbeck process for the convenience yield is replaced by a Cox-Ingersoll-Ross (CIR) process. This ensures that our model is arbitrage-free. Second, spot price volatility is proportional to the square root of the convenience yield level. We empirically test both models using weekly crude oil futures data from 17th of March 1999 to the 24th of December 2003.

The paper concludes that the mean reverting CIR process fits the data as good as other models (Gibson and Schwartz (1990) and Schwartz (1997) two factor models and the CIR stochastic volatility) and has the added benefit of being arbitrage free. However, the time frame used is quite short and the model is not able to fully capture the term structure's dynamics.

So, to summarize my questions:

1. Is a mean reverting CIR process well adapted for capturing the dynamics of commodity prices, including their variations and term structures? What other references or arguments (rigorous or heuristic -- I don't care) would support this view?

2. What are the trade-offs in terms intuitions and/or information-content between Gaussian and square root processes?

3. What models are most naturally suited for capturing commodity price dynamics?

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    $\begingroup$ One popular model at some point was the Pilipovic 2 factor model. The main point is that both spot price and convenience yield are stochastic, so modeling requires at a minimum 2 stochastic factors, unless you're only interested in vanilla options (no average) in which case the only thing that needs modeling is the forward. $\endgroup$ – Antoine Conze Mar 21 '18 at 13:09
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    $\begingroup$ Pilipovic, D., 1997, Valuing and Managing Energy Derivatives, New York, McGraw-Hill $\endgroup$ – Alex C Mar 21 '18 at 21:58

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