When the Difference of Two Assets Is an Ornstein–Uhlenbeck Process

Summary

The document asks what dynamics for two assets can produce a spread that follows an Ornstein–Uhlenbeck process. The answer gives one sufficient construction: model both component assets as OU processes with the same mean-reversion speed and correlated Brownian shocks. Their difference then has OU dynamics, with a long-run level equal to the difference between the component means and a volatility determined by both volatilities and their correlation.

This construction does not uniquely determine the two asset processes from the spread alone. Many pairs of processes can share the same difference process, and the example's equal-speed condition is important for the stated derivation. The response also does not address how to ensure nonnegative asset values, despite that being mentioned in the question. It is a concise structural example, not a complete model for positive asset prices or a trading strategy.

Key ideas

• An OU spread can be formed as the difference of two OU processes with equal mean-reversion speeds.
• The spread's long-run level is the difference between the component long-run means.
• The spread variance depends on both component volatilities and the correlation of their shocks.
• Knowing the spread dynamics does not uniquely identify the dynamics of either asset.
• The construction does not by itself guarantee that the modeled assets remain nonnegative.

Tags

# If the spread between two assets is an OU process, what processes do the two assets follow?


# If the spread between two assets is an OU process, what processes do the two assets follow?












Let $(\Omega,\mathcal{F}, \mathbb{P}, (\mathcal{F}_{t})_{t\geq0})$ be a filtered probability space. Furthemore, let $(S_{t}^{1},S_{t}^{2})_{t\geq0}$ be two assets (adapted to filtration, etc). Define $X_{t}=S^{1}_{t}-S^{2}_{t}$. If $X_{t}$ satisfies the SDE: $dX_{t}=\xi(\zeta-X_{t})dt+\sigma dW_{t}$ ($W_{t}$ is a $\mathbb{P}$ Brownian motion) then what process does $(S_{t}^{1},S^{2}_{t})$ follow (assuming reasonable conditions like nonnegativity)?

## Answer by Kermittfrog (score 10)

https://quant.stackexchange.com/a/71018

If we allow the mean reversion speeds to be identical, we could assume OU processes for the two components:

Let

$$ \begin{align} dx_1&=\kappa_1(\theta_1-x_1)dt+\sigma_1dW_1\\ dx_2&=\kappa_2(\theta_2-x_2)dt+\sigma_2dW_2 \end{align} $$ with $E(dW_1dW_2)=\rho dt$. Now let $z=x_1-x_2$. Then, if $\kappa_1=\kappa_2=\kappa$,

$$ \begin{align} dz&=dx_1-dx_2\\ &=\kappa_1(\theta_1-x_1)dt+\sigma_1dW_1-\kappa_2(\theta_2-x_2)dt-\sigma_2dW_2\\ &=\kappa(\theta_1-x_1)dt+\sigma_1dW_1-\kappa(\theta_2-x_2)dt-\sigma_2dW_2\\ &=\kappa((\theta_1-\theta_2)-(x_1-x_2))dt+\sigma_1dW_1-\sigma_2dW_2\\ &\equiv\kappa(\theta_z-z)dt+\sigma_zdW_z \end{align} $$

where $\theta_z=\theta_1-\theta_2$ and $\sigma_z^2=\sigma_1^2+\sigma_2^2-2\rho\sigma_1\sigma_2$.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.