Log-Price Limits in a Discrete Asset Model
Summary
The document examines a concern in deriving a continuous-time asset model from discrete steps. Each step multiplies the current price by a random factor involving drift and a normally distributed shock. Taking logarithms turns the product of step factors into a sum, which can then be expanded to study the limiting return distribution. The question is whether a step factor can be nonpositive, making its logarithm undefined, and whether a Taylor expansion with random arguments is justified.
The response says that as the time step shrinks in a diffusion limit, the chance of a jump to zero is zero, while a finite discrete step can produce such a value as a numerical possibility when variance is sufficiently large. It distinguishes this setting from jump-diffusion models that include a nonzero bankruptcy probability, where zero may be absorbing. The answer offers intuition rather than a rigorous probability bound or justification of the random Taylor expansion, so those mathematical details remain unresolved in the document.
Key ideas
- The discrete model represents price changes as random multiplicative factors over small time steps.
- Taking logarithms converts a product of factors into a sum of log increments.
- A finite step can produce a nonpositive factor, even if the limiting diffusion avoids jumps to zero.
- A model with a bankruptcy jump may treat zero as an absorbing price state.
- The response does not provide a rigorous treatment of Taylor expansion with random inputs.
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# Continuous Time Asset Model in Higham
# Continuous Time Asset Model in Higham
I read Higham's derivation of the Black-Scholes equation in "An Introduction to Financial Option Valuation". The issue I am having is that it relies on some assumptions related to a continuous time asset model, but these assumptions do not seem completely justified. I am hoping that someone has seen this derivation (but this may not be necessary to answer my question).
The idea is to start with a discrete time asset model that models the price $S(t)$ of an asset. The model is \begin{equation} S(t_{i+1}) = (1+\mu \delta t + \sigma \sqrt{\delta t} Y_i)S(t_i) \end{equation} where we are working on a time interval $[0,t]$ with $n$ subintervals $[t_i, t_{i+1}]$ of length $\delta t$. Here, $\mu$ and $\sigma$ are constants and $Y_i$ are I.I.D. random variables with distribution N(0,1). Now, when moving to the continuous time model (which corresponds to the limit $\delta t \to 0$), Higham writes \begin{equation} \text{log}\left(\frac{S(t)}{S(0)} \right) = \sum_{i=1}^n \text{log} \left(1 + \mu \delta t + \sigma \sqrt{\delta t} Y_i \right) \end{equation} and then expands the right hand side as a Taylor series to compute the expected value and variance of the left hand side. Appealing to the central limit theorem then allows one to write down a continuous time model for $S(t)$. The issue I have is with the $\text{log}$ on the right hand side. It seems that the argument can be negative and so this shouldn't be defined. How can one get around this issue? Higham also claims that the Taylor series expansion is valid even though the argument is a random variable. Is there also a way to make this statement rigorous?
## Answer by Ezy (score 0, accepted)
https://quant.stackexchange.com/a/43319
$\delta t \rightarrow 0$ as $n\rightarrow \infty$. In that limit jumping to 0 is probability 0 with a continuous diffusion model.
In the discrete case it is a numerical possibility but more of the artifact kind if the variance at each step is not too big.
On the other hand, if you simulate a jump diffusion model where for instance a bankrupcy probability is not 0 then you should take this possibility into account since $S_t$ is an absorbing state for the SDE.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.