Market Price of Risk in the One-Factor Schwartz Commodity Model
Summary
The document derives the risk-neutral drift for the one-factor Schwartz commodity model. Starting from a mean-reverting process for the logarithm of spot price, it applies a constant Brownian-motion measure change. The resulting drift retains the mean-reverting form, with its long-run log-price level shifted by a term proportional to volatility, the measure-change parameter, and the inverse mean-reversion speed. The author asks whether this shifted parameter is the model's market price of risk, given that other finance texts often use a drift-minus-risk-free-rate expression divided by volatility.
The central distinction is that a market price of risk depends on how the model's state variable and drift are specified; the familiar ratio applies in a particular asset-pricing setup and should not be transferred mechanically to every parameterization. The document presents an algebraic derivation and points to a related formulation, but it does not provide a full resolution of notation or conventions. Its conclusion should therefore be read as a focused question about reconciling definitions, not as a complete treatment of commodity risk premia.
Key ideas
- Applying a constant change of measure shifts the long-run level in the log-price mean-reverting process.
- The shift depends on volatility and the speed of mean reversion as well as the measure-change parameter.
- Market-price-of-risk formulas depend on the chosen state variable and model parameterization.
- The familiar excess-drift-over-volatility expression should not be assumed to match every model parameter directly.
- The document derives a relationship but leaves the convention and interpretation question unresolved.
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# Why is the market price of risk in the one factor Schwartz model different from the usual one?
# Why is the market price of risk in the one factor Schwartz model different from the usual one?
Assume that the commodity spot price follows the stochastic process (see Schwartz article page 926) $$ dS = \kappa(\mu-\log S)Sdt+\sigma SdW, $$ where $\kappa >0$ measures the degree of mean reversion and $dW$ is the increment to a Brownian motion. Applying Ito lemma to $X = \log S$ we obtain $$ dX = \kappa(\alpha-X)dt+\sigma dW, \quad \text{ with } \alpha = \mu-\frac{\sigma^2}{2\kappa}. $$ We can obtain an arbitrage-free model by applying Girsanov theorem to find a risk neutral measure equivalent to the original measure. Let $dW^* = dW + \nu dt$ for a constant $\nu$, then \begin{align} dX &= \kappa(\alpha-X)dt+\sigma (dW^* - \nu dt) \\ &= \kappa\Big(\alpha-\frac{\sigma\nu}{\kappa}-X\Big)dt+\sigma dW^* \\ &= \kappa(\alpha^*-X)dt+\sigma dW^* \tag1, \end{align} where $\alpha^* = \alpha - \lambda$ and $$\tag2 \lambda = \frac{\sigma\nu}{\kappa} $$ is the market price of risk. I wrote the equation in this way so that (1) is of the same form as (4) in the article. Unfortunately in the article by Schwartz there is no formula for $\lambda$, so I cannot verify that (2) is actually the formula for the market price of risk in the Schwartz model. However, equation (4) in the article is linked with a footnote which says "See for example Bjerksund and Ekern (1995)", a preview of this article can be found here where we see that equation (12.16) coincide with (1), and (12.17) coincide with (2) (in (12.17) there is $\lambda$ instead of $\nu$).
However, in many papers and books, for example Stochastic Calculus for Finance by Shreve, the market price of risk is defined as $$\tag3 \frac{\mu-r}{\sigma}. $$ Does the formula for market price of risk change for different models? Or maybe I made a mistake in the above derivation of $\lambda$ in the Schwartz model?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.