# 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

## 2 Answers

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

No stock satisfies the requirements of Black-Scholes although some single period bonds probably do. The assumption of Black-Scholes is that all parameter values are known perfectly. This, of course, is not the case. There is a 1958 proof showing that these types of problems lack a Frequentist solution. This does not mean that there is no solution, merely that there is no solution using maximum likelihood or Frequentist solutions. The Bayesian solution does not match the Black-Scholes solution so the Black-Scholes solution is not an admissible solution.

• I think you might have misunderstood the question. It is not whether the Black/Scholes assumptions are satisfied by real-world assets. Instead it is asking if, within the model, all assets satisfy the fundamental valuation PDE. – LocalVolatility Mar 6 '17 at 8:18