Commodity Derivative Pricing with Spot Dynamics and Futures Measures

# Use of Non-Risk-Neutral Measure for Pricing Derivatives


# Use of Non-Risk-Neutral Measure for Pricing Derivatives












While trying to understand the risk-neutral pricing of derivatives when the underlying is the spot price of a commodity, I encountered the situation that the measure used for pricing derivatives is not actually the risk-neutral measure. See, for example, the model proposed by Schwartz [1] where the spot price dynamic is given by \begin{align*} dS_t = \kappa S_t (\mu - \ln(S_t))dt + \sigma S_t dB_t \end{align*} where $\kappa,\sigma>0,\mu \in \mathbb{R}$ and $(B_t)_{t\geq 0}$ is a standard Brownian Motion under the real-world measure $\mathbb{P}$. Schwartz states that under the assumption of a constant market price of risk, the logarithm of the spot price $X_t = \ln(S_t)$ evolves according to $$ dX_t = \kappa(\mu-\frac{1}{2\kappa}\sigma^2-\lambda-X_t)dt+\sigma dB^{\mathbb{Q}_1}_t,$$ with $dB^{\mathbb{Q}_1}_t=dB_t+\frac{\kappa \lambda}{\sigma} dt$ being a Brownian Motion under the measure $\mathbb{Q}_1$ given by Girsanov's Theorem. The risk-neutralized spot price dynamic is given by $$ dS_t = \kappa S_t (\mu-\lambda-\ln(S_t))dt+\sigma S_t dB^{\mathbb{Q}_1}_t.$$

However, the discounted spot price $(\exp(-rt)S_t)_{t\geq 0}$ is clearly not a martingale under $\mathbb{Q}_1$, which leads me to question how $\mathbb{Q}_1$ can be used for pricing derivatives?

In contrast, my naive approach would be to set $\theta(S_t,t) = (\kappa \mu-\kappa \ln(S_t) -r)/\sigma$ as the market price of risk. Applying Girsanov's Theorem, I obtain a measure $\mathbb{Q}_2$ under which $dB^{\mathbb{Q}_2}_t=dB_t+\theta(S_t,t)dt$ is a Brownian Motion and $$dS_t =r S_t dt + \sigma S_t dB^{\mathbb{Q}_2}_t.$$ The logarithm of the spot price follows $dX_t= (r-\frac{1}{2}\sigma^2)dt + \sigma dB^{\mathbb{Q}_2}_t$. Here, the discounted spot price is a martingale under the measure $\mathbb{Q}_2$ and would yield different results in pricing derivatives than the approach made by Schwartz[1].

A different way to pose the question would be: Why is the risk-neutralized version of a process $(Y_t)_{t\geq0}$ with $dY_t = \mu(Y_t,t)dt+\sigma(Y_t,t)dB_t$ not always of the form $dY_t = rY_tdt+\sigma(Y_t,t)dB^{*}_t$?

[1] Schwartz, Eduardo S. "The stochastic behavior of commodity prices: Implications for valuation and hedging." The Journal of finance 52.3 (1997): 923-973.

## Answer by Tim (score 2, accepted)

https://quant.stackexchange.com/a/80362

After more research and considering Kurt's comment, I believe I have found a reasonable explanation for why it is appropriate to use the measure $\mathbb{Q}_1$ to price derivatives. First, since we are dealing with commodities, the storability of these "assets" is not guaranteed. As a result, it is not possible to hedge a future position by buying the commodity in the spot market today. Therefore, the standard approach of using the measure $\mathbb{Q}_2$ does not make sense, as commodities purchased in the spot market cannot be used to create a portfolio that replicates the payoff of an option.

In contrast, futures contracts on the commodity behave more like traditional assets and can be used as the underlying for options. This approach is also used in another paper by Schwartz (see [2]). In this context, we use a spot-market process $(S_t)_{t\geq 0}$ that captures the behavior of the spot market and a market price of risk (or an equivalent martingale measure $\mathbb{Q}_1$) so that $\mathbb{E}^{\mathbb{Q}_1}[S_T]$ captures the behavior of the observed futures price.

We use a (conditional) expected value to describe the relationship between the spot price and the futures price because this guarantees that the price of future contract behaves like a martingale under $\mathbb{Q}_1$: $$F(T,s)=\mathbb{E}^{\mathbb{Q}_1}[S_T|\mathcal{F}_s]=\mathbb{E}^{\mathbb{Q}_1}[\mathbb{E}^{\mathbb{Q}_1}[S_T|\mathcal{F}_t]|\mathcal{F}_s]=\mathbb{E}^{\mathbb{Q}_1}[F(T,t)|\mathcal{F}_s]$$ where $s<t<T,$ and $F(T,s)$ is the price of a futures contract at time $s$ with maturity $T$. We now can use the future contract as the underlying of an option.

[2] Schwartz, E., & Smith, J. E. (1998). Short term-variations and long-term dynamics in commodity prices: Incorporating a stochastic growth rate.