---
description: "Commodity futures pricing strategy that fits a mean-reverting stochastic model to the term structure and trades futures identified as rich or cheap relative to model-implied fair value."
tags: [commodities, futures, stochastic-model, ornstein-uhlenbeck, pricing, term-structure]
---

# Trading with Pricing Models

**Section**: 9.6 | **Asset Class**: Commodities | **Type**: Relative Value / Model-Based

## Overview
Commodity futures term structures are non-trivial and can be modelled via stochastic processes. Fitting a parametric model (e.g., the Ornstein-Uhlenbeck mean-reverting process) to historical data allows the identification of futures that are rich (sell signal) or cheap (buy signal) relative to the model's predicted fair value. The approach acknowledges that structural mean reversion is a reasonable property for commodity prices.

## Construction / Mechanics
Let S(t) be the spot price and X(t) = ln(S(t)). Model X(t) as a mean-reverting Brownian motion (Ornstein-Uhlenbeck):

```
dX(t) = κ[a - X(t)] dt + σ dW(t)                             (459)
```

Parameters:
- κ: mean-reversion speed
- a: long-run mean of ln(S)
- σ: log-volatility
- W(t): Q-Brownian motion under risk-free measure Q

Under the standard pricing argument, the futures price F(t,T) is:

```
F(t,T) = E_t(S(T))                                           (460)
ln(F(t,T)) = E_t(X(T)) + (1/2) V_t(X(T))                    (461)
```

This gives the closed-form futures price:

```
ln(F(t,T)) = exp(-κ(T-t)) X(t) + a[1 - exp(-κ(T-t))]
             + (σ²/4κ)[1 - exp(-2κ(T-t))]                    (462)
```

**Calibration and trading:**
1. Fit κ, a, σ to historical data (e.g., nonlinear least squares on observed futures prices).
2. Compute the model-implied futures price for each contract.
3. Compare market price to model price:
   - Market price > model price: **sell signal** (futures is rich)
   - Market price < model price: **buy signal** (futures is cheap)

Note: as κ → 0, a → ∞ with κa fixed, this model reduces to the Black-Scholes model.

## Return Profile
Profits when market prices revert toward the model-implied fair values. Returns are driven by mean-reversion in the spread between market and model prices. In-sample fit may be strong but out-of-sample predictive power is model-dependent.

## Key Parameters / Signals
| Parameter | Description |
|-----------|-------------|
| κ | Mean-reversion speed; higher κ → faster reversion |
| a | Long-run mean of log-spot price |
| σ | Log-volatility of the spot price |
| F(t,T) model vs. market | Rich/cheap signal: sell if market > model, buy if market < model |

## Variations
- **Multifactor models**: add stochastic convenience yield or stochastic volatility for richer term structure fitting.
- **Black-box / ML models**: fit any model with desirable qualitative properties (e.g., mean reversion) using machine learning, without explicit stochastic dynamics; valid as long as out-of-sample predictive power is demonstrated.
- Combine with roll-yield (Section 9.1) as a complementary signal.

## Notes
- In-sample fit can be excellent even for models with poor predictive power; out-of-sample backtesting is essential (see Paschke and Prokopczuk, 2012).
- Model mis-specification risk: the true dynamics may not be OU; using a flexible model without theoretical grounding is equally valid if it works out-of-sample.
- Parameter instability: κ, a, σ estimated on historical data may shift during structural changes (supply shocks, geopolitical events).
- "Fancy does not equal better" — complex models do not necessarily outperform simple ones out-of-sample.