Why Jumps and Multiple Random Drivers Exceed Generalized Black–Scholes Dynamics

Summary

The document examines whether every stock process consistent with a risk-neutral probability measure can be written in the generalized Black–Scholes form, where price changes are driven by drift and diffusion terms against Brownian motion. It explains that the martingale representation theorem does not imply this for arbitrary market dynamics: the stated representation applies to random variables measurable with respect to the filtration generated by a Brownian motion.

The answer gives jumps at random times, jumps of random sizes, and additional sources of randomness as examples of features that can fall outside a single-Brownian-driver model. It also notes that extra sources may be useful, though it does not develop a particular model. The discussion is conceptual and does not specify conditions under which a given process can be represented or provide empirical comparisons. Its central caveat is that the representation result depends on the information structure and driving randomness; existence of a risk-neutral measure alone does not guarantee the stated diffusion form.

Key ideas

• The generalized Black–Scholes setup describes a price process with drift and Brownian diffusion terms.
• The martingale representation theorem discussed applies to randomness generated by the Brownian motion in its filtration.
• Random-time or random-size jumps can produce dynamics outside that single-driver diffusion form.
• Additional independent sources of randomness can also prevent the proposed representation.
• A risk-neutral measure by itself does not establish that all stock dynamics fit the generalized Black–Scholes model.

Tags

# Are there stocks dynamic that cannot be represented by Generalized Black Scholes model?


# Are there stocks dynamic that cannot be represented by Generalized Black Scholes model?












The generalized Black Scholes Model refers to a stock dynamic that satisfy

$$ dS(t)=S(t)(\mu_t dt+ \sigma_t dW(t)) $$

By martingale representation theorem, it seems that if there is a risk neutral probability measure, then all stock dynamic is enclosed by the GBS.

Are there exceptions?

## Answer by Bob Jansen (score 3)

https://quant.stackexchange.com/a/55868

KeSchn and I pointed out in the comments that this it is not possible to represent all stock dynamics using the Generalized Black Scholes model. For example, there can be jumps at random moments and not just at random moments but also jumps of random size. These jumps can affect either $\mu_t$ or $\sigma_t$. Models with too many sources of randomness are not considered useful but at least 2 extra source can be useful.

What does this say about the Martingale Representation Theorom. Doesn't that claim that stock dynamics can be captured using an Itô process? Unfortunately, the theorem is a bit more narrow (Wikipedia):

> The martingale representation theorem states that a random variable that is measurable with respect to the filtration generated by a Brownian motion can be written in terms of an Itô integral with respect to this Brownian motion.

Emphasis mine. As long if there is one Brownian motion driving the randomness, all is good. The theorem doesn't hold with more sources than one.

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.