Risk-Neutral Pricing with Arithmetic Brownian Asset Dynamics

Summary

The document asks how to price derivatives when the underlying follows an arithmetic Brownian motion rather than the geometric Brownian motion used in the Black–Scholes model. It starts with an additive diffusion under the physical measure, then proposes changing measure so the discounted asset price has zero drift under the risk-neutral measure. The central question is how to solve the resulting stochastic differential equation and obtain the terminal asset value.

The post raises a possible connection to the pricing PDE and the Feynman–Kac representation, but it does not resolve the derivation or give a pricing example. Its proposed manipulation of the discounted process into a terminal-value formula is explicitly treated as suspect by the author. The discussion is therefore useful as a prompt to distinguish the dynamics of a discounted martingale from the dynamics and solution of the original asset process. It also leaves model assumptions, admissibility of the measure change, and boundary conditions unspecified.

Key ideas

• The document contrasts arithmetic Brownian asset dynamics with the geometric Brownian motion used in Black–Scholes.
• It proposes risk-neutral dynamics by requiring the discounted asset price to be a martingale.
• The author questions whether integrating discounted martingale dynamics directly gives the terminal asset value.
• The relationship between the resulting stochastic differential equation, its pricing PDE, and Feynman–Kac is raised but not answered.

Tags

# What happens trying to price derivatives starting from a non-geometric brownian motion?


# What happens trying to price derivatives starting from a non-geometric brownian motion?












To get a better understanding, I tried going through BSM-model starting from a non-geometric brownian motion. However, during the derivation I got stuck, which led me to a specific question.

The set-up:

$dS_t = \mu dt + \sigma dW_t^P$

We want $e^{-rt}S_t $ to be a martingale. Applying Ito, Girsanov and enforcing the drift to 0, we get:

$dS_t = rS_t dt + \sigma dW_t^Q \qquad (1) $

Differently from BSM, I can't use the logarithm to solve this SDE, and here lies my question.

May I use the dynamics of the martingale process to solve for $S_T$? Meaning, from $(1)$:

$de^{-rt}S_t = e^{-rt}\sigma dW_t^Q \iff S_T = \sigma dW_T^Q - e^{rt}S_0$

This clearly seems not the case, but I never thought about it since solutions would usually play out more nicely. I checked this in BSM and Vasicek and again the martingale dynamics don't seem to output back the process we start with.

I tried finding a reasonable and intuitive explanation for this but couldn't. The only thing I could think of is that maybe since I am using a different process now (even though is the related martingale process), the implied PDE looks different, and in such a way so that Feynman-Kac formula does not hold anymore and a solution is not achievable via the conditional expectation (or more likely it is beyond my understanding).

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