Questions tagged [geometric-brownian]
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13
questions
0
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49 views
On Geometric Brownian motion and Itô's formula
Let $S_t$ be a geometric brownian motion such as
$$d S(t) = rS(t)dt +\sigma S(t)dW(t),$$
where $W$ is a standard Brownian motion.
With Itô's lemma and formulas $(dt)^2=dtdW_t=dW_tdt=0$ and $(dW_t)^2=...
0
votes
0answers
46 views
Convert drift and diffusion term in terms of time in the Geometric Brownian Motion framework
Assume that we have daily prices covering the period of 10 years. For calibrating the drift and diffusion parameters of the GBM model
$$S_{t+1} = S_{t}e^{[(\mu-\sigma^2/2)]\Delta t + \sigma \sqrt{\...
1
vote
1answer
74 views
Why do I get this difference when simulating geometric Brownian motion?
I tried simulating GBM using both the SDE definition and the closed form solution. The paths I get through these methods are very different. Can someone help me figure my mistake?
...
2
votes
1answer
80 views
Covariance of logarithms of geometric Brownian motion
Suppose I have a Geometric Brownian Motion process,
$$dX_t=\mu X_t dt + \sigma X_t dW_t$$
I'd like to find the covariance of $\log(X_t)$ and $\log(X_s)$ where $s<t$. We can write $\log(X_t)$ in ...
0
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0answers
30 views
How to mathematically calculate the probability of GBM generating difference of less than some value
I have a custom index that follows Geometric Brownian Motion (GBM) with volatility v. I started this index at 10k with 4 decimal places i.e the starting price of ...
2
votes
1answer
102 views
How To Understand the Drift of ln(S) if S Follows Geometric Brownian Motion
As we know, if an asset S follows geometric Brownian motion, under risk neutral measure, it can be expressed as $\frac{dS}{S}=rdt+\sigma dW$, by applying Ito's lemma, $d(lnS)=(r-0.5*σ^2)dt+σdW(t)$, ...
3
votes
0answers
63 views
GBM probability of hitting non constant barrier
I know there is a formula for probability of hitting a constant barrier for GBM/BM (See page 651 in Martinagle Methods in Financial Modelling).
Is there a formula for non-constant barrier?
The ...
1
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2answers
106 views
How to Understand Lognormal Distribution in the Following Case
I got a question and corresponding solution, but have some difficulties in understand the lognormal distribution part of it, so I really appreciate your advice:
Question: assume zero interest rate ...
0
votes
1answer
44 views
Drawing values from a lognormal distribution of a GBM
I'm looking at a GBM with parameters
$$
r=0.05 \\
\sigma=0.2 \\
K=130\\
T=0.25\\
S_0 = 100
$$
This is a process that is lognormally distributed with mean and variance given by
$
\mu = S_0e^{r T+0.5\...
3
votes
1answer
62 views
Boundaries for Call Spread
I'm reading an interview book called A Practical Guide to Quantitative Finance Interview and I have some doubts regarding part of its solution and highlighted them in bold:
Question:
What are the ...
1
vote
1answer
167 views
Simulation of Geometric Brownian Motion in R
Using R, I would like to simulate a sample path of a geometric Brownian motion using
\begin{equation*}
S(t) = S(0) \exp\left(\left(\mu - \frac{\sigma^{2}}{2}\right)t + \sigma B_{t}\right),
\end{...
4
votes
1answer
112 views
What the expectation of S^2 is from GBM? [closed]
I was at an interview and was asked to write down the SDE for GBM.
$$
dS = S\mu dt + S\sigma dX
$$
Then I was asked how I would compute the expectation of S^2. I didn't know where to start. Any ...
1
vote
2answers
149 views
What is the stock price expectation?
The Hull textbook (and accompanying technical note) says that the expected stock price $\mathbb{E}[S_T]=S_0 \exp(\mu T)$. However, the answers to a British actuarial examination (Q4 for September 2018)...
