Risk-Neutral Pricing and the Black–Scholes-Type PDE
Summary
The discussion asks whether risk-neutral pricing applies only to stocks or to every asset, and whether applying Itô’s formula to an asset’s value gives a Black–Scholes-type partial differential equation. One answer says the risk-neutral measure is defined relative to a model containing specified assets, such as stocks and bonds. Under that measure, discounted prices of modeled assets are martingales, so their expected local return reflects the short rate. This is the core condition behind deriving pricing equations for claims whose values depend on modeled state variables.
A second answer disputes the idealized assumptions of Black–Scholes, emphasizing that its parameters are not known perfectly in practice and raising a Bayesian-versus-Frequentist objection. The exchange does not develop that objection or derive a PDE, and it does not establish that every imaginable asset belongs to a given model. The practical takeaway is conditional: risk-neutral pricing applies to assets represented within the model, while a PDE also depends on the modeled dynamics and pricing assumptions.
Key ideas
- A risk-neutral measure is defined relative to a particular model and its included assets.
- Under that measure, discounted prices of modeled assets are martingales.
- Pricing equations can be derived from a claim’s dependence on modeled state variables and their risk-neutral dynamics.
- The discussion challenges the practical assumptions of Black–Scholes but does not develop that critique.
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Full text
# Do all assets satisfy the "black scholes type PDE", or just the stocks?
# 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?
## Answer by zer0hedge (score 2)
https://quant.stackexchange.com/a/32840
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 $Q$.
## Answer by Dave Harris (score -3)
https://quant.stackexchange.com/a/32838
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
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