What stochastic processes (and corresponding probability distributions) empirically capture spot/forward commodity prices and forward term structures?
Background
I want to use discounted cash flow analysis to value the equity of a commodity producer which assumes the firm's cash flows vary stochastically according to some sensible process. I want to avoid using deterministic models which can potentially yield negative net present values (NPVs).
My motivation is to find/develop a model which will allow investors and management to price contingent claims on a firm's net present value (NPV). This is supported by the observation that equity values are always positive, even if cash flows are negative.
My assumption is this is based on the structure of contingent claims (i.e., investors cannot lose more than principal) as well as real options (management can enhance firm value by investing in growth opportunities and/or opportunistically shuttering/downsizing unprofitable operations).
My core intuitions are that this model would be extremely helpful as a generalize equilibrium pricing model for unprofitable firms, growth companies, etc.
Problem Statement
While GBM initially seemed like an appropriate model, I ran into a stumbling block since there is no closed-form expression for the distribution of the sum or average of lognormal variables. This a problem when evaluating conditional expectations. I understand this is the same challenge as with valuing arithmetic Asian options.
Many accepted alternatives to GBM for modeling commodities prices use variations of exponential Ornstein-Uhlenbeck processes (Gibson and Schwartz (1990), Schwartz (1997), Smith and Schwartz (July 2000), etc.), meaning they too are lognormal and therefore do not have closed-form expressions for integrated values wrt time.
Given problems with analytical tractability, I would like to explore alternatives which are empirically supported.
One Alternative
- Dassios and Nagaradjasarma found that the of the square root (i.e., Cox-Ingersol-Ross or CIR) process has attractive properties (autoregression to long-term mean) and has a time integral which analytically tractable due to fact that sums of Gamma distributions are still Gamma distributed.
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 6, [15], [30]) and other financial quantities.
I could only find one paper which empirically supports this approach. According to Ribeiro and Hodges (2004):
...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 w/stochastic volatility). However, the paper and data are quite dated, and the time frame used to validate goodness-of-fit is quite short.
Specific questions concerning the literature include:
- Is a two-factor (spot + conv yield) mean-reverting CIR process well adapted for capturing the dynamics of commodity prices?
- Is a third term recommended to capture variations in the term structure?
- What are the advantages and disadvantages between standard Gaussian and square root processes?
Wish List
Ideal commodity price dynamics (stochastic process + distributions), from high to low priority, include:
- Fits empirical distributions of commodities prices
- Has a CDF which can be expressed using elementary (differentiable) functions; Has closed form expression for quantile function (inverse CDF)
- Can support negative values (i.e., doesn't necessarily need real support over [0, inf))
- Supports drift (inflationary) parameter
- Supports mean reversion (auto-regression) parameter
- Supports min of 2 stochastic parameters for spot price volatility and conv yield
- May also support additional stochastic parameters for pricing (Poisson) shocks and term structure shape
