Timeline for Solution to geometric Brownian motion with time dependent volatility and drift?
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
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| Sep 6 at 21:34 | comment | added | Freek | @Kevin how can I prove the resulting process is still log-normal? | |
| May 6, 2021 at 18:46 | comment | added | user53249 | Now, imagine that we aim to make a prediction for the future, then after fitting the model and checking for the accuracy of the model, we need to simulate paths for the future to see how well our model works. In this case, we need to consider a dynamic for the expected value (like a mean-reverting model) to replace the unknown expression with a specific expression characterized by additional parameters. That is why I am asking how we can distinguish the form of diffusion and drift term function. | |
| May 6, 2021 at 18:46 | comment | added | user53249 | For example, I want to model cash inflow from revenue using a time-dependent Geometric Brownian motion. In this case, drift term and diffusion term are deterministic functions of time but unknown. If we find the analytical solution of the process, we will end up with an integral expression for the drift term and diffusion term (Applying Ito lemma). | |
| May 6, 2021 at 13:39 | comment | added | Kevin | @user53249 depends on what you mean with "suitable"? Subject to some constraints (to ensure the SDE has a strong solution), you can choose any arbitrary function of time. | |
| May 6, 2021 at 13:37 | comment | added | user53249 | Could you please let me know how we can specify a suitable diffusion term and drift term when they are time-dependent? | |
| Mar 23, 2021 at 20:03 | comment | added | PlatinumMaths | yeah that probably a good idea! Cheers! | |
| Mar 23, 2021 at 19:37 | comment | added | Kevin | Hey, you can make $\mu$ and $\sigma$ random, yes. Look for example for stochastic volatility model. You may first want to revise some stochastic calculus though. A time dependent mean could for example model seasonal patterns... | |
| Mar 23, 2021 at 19:36 | comment | added | PlatinumMaths | hey what is the idea behind simulating stocks with time dependent mu and sigma? Would it be the case of using a random mu and sigma at each time step? | |
| Mar 23, 2021 at 19:23 | vote | accept | PlatinumMaths | ||
| Mar 23, 2021 at 19:21 | comment | added | PlatinumMaths | yeah makes sense, Thank you so much! God bless! | |
| Mar 23, 2021 at 19:20 | comment | added | Kevin | @PlatinumMaths I hope you enjoy your modelling module :) The $s$ is just an integral dummy variable. You can use $u$ or $\tau$ instead (or whatever really). The point is that $S_t$ contains integrals that go from $0$ to $t$ (depend on the entire history of the path up to time $t$). Thus, the integrals of the drift and volatility need to have another variable, other than $t$. By default, I picked $s$. It's like writing $t^2 = 2\int_0^t s\text{d}s$. Makes sense? | |
| Mar 23, 2021 at 19:18 | comment | added | PlatinumMaths | Oh sorry, I just realised that 's' must be the values of mu and sigma that change at everytime step. Thanks though! | |
| Mar 23, 2021 at 19:15 | comment | added | PlatinumMaths | Thanks @Kevin, that is actually what I want to do, I am doing this for project in mathematical modelling module I study at my final year bachelor's studies. I notice that $\mu(t)$ changes to $\mu(s)$, could you clarify what does 's' represent? Is it same as t (time) but different notation? | |
| Mar 23, 2021 at 19:08 | history | answered | Kevin | CC BY-SA 4.0 |