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I am reading a book by Donald Saari on mathematical finance. I am trying to understand the intuition behind modelling the rate of change of the stock price $S$ (excluding randomness for the moment) as

$\Delta S = \mu S \Delta t$ (eq.1) ($\mu$ is a constant)

For other physical phenomena, such as modelling the rate of change of temperature, I understand that one could try imagine that the rate of change of temperature depending on some forcing function $G(T)$ and then Taylor expanding as follows

$\Delta T = G(T) \Delta t = \mu (T-T^*)\Delta t$, say to $O(T-T^*)$

I can convince myself intuitively that it is not wise (as a first approximation) to include the variable time to model the rate of change of temperature, i.e

$\Delta T = G(T,t) \Delta t$ (is a bad idea)

Thinking in terms of 'worst case scenario' by fixing time and then fixing temperature separately.

With stock prices however, I have no intuition as to why (eq.1) holds? Why not instead

$\Delta S = G(S,t) \Delta t$

or even worst

$\Delta S = R(S,t,p) \Delta t$

where $p$ is some form of running average. Could someone explain why (eq.1) holds and why it is truly a leading order approximation?

Andy
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2 Answers2

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It is true that it would be more appropriate to use $\Delta S = G(S,t)\Delta t$. However, the real-world function $G(S,t)$ is too complicated. So, we use first term in Taylor series approximation of $G(S,t)$ and obtain $\Delta S = \mu S \Delta t$.

"Is that a correct term? No. But it is a reasonable term to start with" - D. Saari UCI Math of Finance, Lecture 7 around 0:0:40.

Such assumptions along with Efficient Market Hypothesis, i.i.d. returns and others are necessary to make math toolbox applicable to financial series. They are questioned in academic literature.

Prof. Saari emphasises the need to be wary about assumptions a lot in the lecture series. They will be a fine complement to the textbook (considering the textbook is essentially lecture notes and problem sheets combined).

DharmaBum
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In general, you could model the stock price as $\Delta S = G(S,t) \Delta t$. But, in practice, you want to work with a model that is tractable while capturing the main behavior of stock price, i.e. always maintaining positive values.

It turns out that the proportional model $\Delta S = \mu S \Delta t$ suits that objective. It has nice close-form solution while guaranteeing that the stock price does not become negative.

Quanto
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