Mathematical Assumptions Behind a Continuous Asset Price Model

Summary

The document asks what mathematical assumptions underlie a continuous asset price model introduced heuristically in an option valuation text. It proposes that log returns over an interval are normally distributed, with a mean adjusted by half the variance and variance proportional to the interval’s length. It also proposes independence between returns over successive, nonoverlapping intervals.

These statements describe the question’s candidate assumptions rather than a confirmed derivation or answer. The document gives no evidence, resolution, or discussion of further conditions, such as continuity of sample paths or the process formulation needed to connect the increment assumptions to geometric Brownian motion. It is therefore useful as a prompt about how distributional and independence assumptions define a model, but it does not establish that the listed conditions are exhaustive.

Key ideas

• The proposed model assigns normally distributed values to log price changes over time intervals.
• The proposed log return mean and variance both scale with the length of the interval.
• The question suggests that returns over successive nonoverlapping intervals are independent.
• The document raises, but does not resolve, whether these assumptions fully characterize the continuous asset model.

Tags

# What exactly is the 'continuous asset price model'?


# What exactly is the 'continuous asset price model'?












I am reading An Introduction to Financial Option Valuation by Higham. In Chapter 6, the book covers two asset price models, a discrete one and a continuous one. In Section 6.3 (Continuous asset model) he "derives" the model using mostly heuristics.

After having read it, I am not sure what are the assumptions here. I.e., the question is: What is exactly the continuous asset model? I.e., what are the underlying assumptions?

Note: I am not talking about "assumptions" such as 'volatility is constant' -- I mean mathematical assumptions.

Is it true that essentially the only things assumed are that

(1) $\log \left( \frac{S(t_2)}{S(t_1)} \right) \sim \text{Normal} \left( (\mu - \frac12 \sigma^2 )(t_2-t_1) , \sigma^2 (t_2-t_1)\right)$ (for all $t_2 > t_1$), and

(2) $\log \left( \frac{S(t_2)}{S(t_1)} \right) $ and $\log \left( \frac{S(t_3)}{S(t_2)} \right) $ are independent, for all $t_3>t_2>t_2$?

Thank you.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.