# 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?