I mentioned the other day that I'd like to look into the Black Scholes equation in terms of information equilibrium. John Cochrane's

*Asset Pricing*(2005) derives it using the asset pricing equation and a bit of stochastic calculus. I already looked at the basic asset pricing equation here, so I'm just going to set up the price in the information transfer framework as a stochastic process.$$

p \equiv \frac{dD}{dS} = \frac{1}{\kappa} \; \frac{D}{S}

$$

So that

$$

dp = \frac{1}{\kappa} \left( \frac{dD}{S} - \frac{D}{S^{2}} dS \right)

$$

$$

\frac{dp}{p} = \frac{dD}{D} - \frac{dS}{S}

$$

Using the information equilibrium condition again, we have:

$$

\frac{dp}{p} = \frac{dD}{D} - \kappa \frac{dD}{D}

$$

$$

\frac{dp}{p} = (1 - \kappa) \frac{dD}{D}

$$

Let's say $D$ has a deterministic part and a stochastic part ...

$$

\frac{dp}{p} = (1 - \kappa) \left( \frac{1}{D} \frac{\partial D}{\partial t} dt + \frac{1}{D} \frac{\partial D}{\partial x} dx \right)

$$

Now if we identify

$$

\mu \equiv (1 - \kappa) \frac{1}{D} \frac{\partial D}{\partial t}

$$

$$

\sigma \equiv (1 - \kappa) \frac{1}{D} \frac{\partial D}{\partial x}

$$

We end up with

$$

\frac{dp}{p} = \mu dt + \sigma dx

$$

Which is a geometric stochastic process that would become the price $p$ of the underlying stock in the Black-Scholes option pricing model.

Solid work

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