# Do all assets satisfy the “black scholes type PDE”, or just the stocks?

I am reading Bjork. In it, he says that the martingale measure $Q$ is characterized by the property that all stocks have the short rate as their local rate of return under the $Q$-dynamics.

Is it just stocks, or really all assets that we could imagine pricing in this market?

More generally speaking, say I price an asset with $\pi(t)$. Say this depends on some vector of variables $\textbf{X}$ whose dynamics are known under $Q$ (for example, it could be some stocks). Can I then always proceed by using Ito's formula to compute $$d\pi(t)$$ and then take the drift term and set it equal to $\pi r$? This would give me an equation wich sort of looks like the black scholes pde. Will that PDE always hold if we want to price with no arbitrage?

• It's only a model... – wogsland Mar 6 '17 at 11:38

The martingale (risk-neutral) measure is always defined for some (complete) particular model. This model includes stocks and/or bonds. And by definition of $Q$, discounted prices of all assets in the model are matrtingales under $Q$.
So the answer to your question is yes, you can say that all assets in the model have the short rate as their local rate of return under the $Q$.
• To clarify one thing that might be obvious: The original Black/Scholes PDE is only satisfied by spot assets as well as spot-style settled contingent claims on them (i.e. the premium is paid upfront at $t = 0$). If either the underlying or the contingent claim have futures-style settlement, then you still obtain a PDE that structurally looks very similar to the Black/Scholes one. The risk-neutral portfolio drift is only then equal to $r$ when all assets/contingent claims in it have spot-style settlement. – LocalVolatility Mar 6 '17 at 9:16