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?



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