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Right now I am taking a course on financial math and we want to model the stock price.

For continuous compound interest we have the formula $B(t) = b_0 e^{rt}$ where $r$ denotes the interest rate and $b_0$ the initial principal.

Now we want to create a model for the stock price and do so by assuming that the stock price results from a random error $\epsilon$ around a bond price with another interest rate $\tilde{r}$.

This leads to the following plain model for the stock price: $S(t) = s_0 e^{\tilde{r}t + \epsilon}$

This model gets more and more refined over the lecture(especially the properties of our error $\epsilon$) but there are two things a don't get:

  1. Where does this assumption come from that the stock price roughly behaves like a bond price?
  2. Why is it $S(t) = s_0 e^{\tilde{r}t + \epsilon}$ and not for example $S(t) = s_0 e^{\tilde{r}t} + \epsilon$?
Josh.K
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    That is a finance specific question, not mathematics, but the simple assumption is that stock prices might be expected to grow steadily over time, combined with some noise. The noise is in a sense multiplicative (e.g. $\pm 20%$) rather than additive – Henry May 30 '23 at 13:34
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    There's no bond here...bonds have stated maturity and coupons. This (very stripped down model) just says that stocks drift up steadily with some noise. More broadly, you can model a stochastic process anyway you like...the question is whether you can sensibly calibrate your model to the market. Can you parametrize your model in a practical and sensible manner. Here, you'd want to match forward prices and maybe some options. – lulu May 30 '23 at 13:35
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    Note too that $e^{rt+\epsilon}=e^{rt}e^{\epsilon}\approx e^{rt}\times (1+\epsilon)$ if $\epsilon$ is small so the model you propose isn't terribly different. – lulu May 30 '23 at 13:37
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    Another reason to use the exponential form is that we want stock prices to stay non-negative. Nonetheless: a model named after Louis Bachelier where stock prices follow a Brownian motion (not geometric) is of great historical interest. – Kurt G. May 30 '23 at 16:25

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