Questions tagged [geometric-brownian]
The geometric-brownian tag has no usage guidance.
24
questions
-1
votes
0answers
44 views
0
votes
1answer
57 views
What is the meaning that Geometric Brownian motion is leptokurtic? [closed]
Does this have any relation to the symmetry of the normal distribution?
0
votes
0answers
65 views
How to do Monte Carlo simulation given the stochastic ODE of a Brownian motion [closed]
I've learn the theoretical basis and lots of Brownian motion in quantitate finance. But i'm wondering how to actually simulate something based on brownian motion and make into something code-able or ...
2
votes
0answers
60 views
What are some alternatives to Geometric Brownian motion that can be used in the Black-Scholes? [closed]
I hear that there are many extensions to the black scholes model to make it more realistic, however, GBM does not account for volatile swings. Is there any sort of alternative approach to use instead?
3
votes
4answers
296 views
+50
Find a formula for the price of a derivative paying $\max(S_T(S_T-K),0)$
Develop a formula for the price of a derivative paying
$$\max(S_T(S_T-K))$$
in the Black Scholes model.
Apparently the trick to this question is to compute the expectation under the stock measure. So,...
3
votes
2answers
157 views
Normality or Log-Normality of Regular Returns
Another old question on this site (How to simulate stock prices with a Geometric Brownian Motion?) inspired me to ask the following question: if we assume that regular returns could be normally ...
0
votes
1answer
72 views
Correctly simulating BEKK series to model asset returns
I am trying to create financial data as close as possible to that of asset returns. Using the R code I can collect some stock data and compute the return:
...
11
votes
1answer
170 views
From VG and NIG processes to GBM
I would like to find out if it is possible to reduce:
the Madan-Seneta Variance Gamma (VG) model;
the Barndorff-Nielsen Normal Inverse Gaussian (NIG) model
to the standard Black-Scholes through a ...
0
votes
1answer
61 views
Sampling from SDE
In the case of the classic Geometric Brownian motion
$$dS_t = \mu S_t dt + \sigma S_tdW_t$$
we solve it as
$$ S_t = S_0 \exp\left[ \left(\mu - \frac{\sigma^2}{2}\right)t + \sigma dW_t\right] $$
and ...
0
votes
0answers
60 views
Geometric brownian motion and probabilities
A stock's price movement is described by the equations $dS_t=0.02S_tdt+0.25S_tdW_t$ and $S_0=100$. An investor buys a call option on said stock with a strike price $K=95$ which expires in $T=2$ years. ...
1
vote
1answer
78 views
true or false: the risk-neutral measure is useless in this situation
Example 2 of this Wiki article on the risk-measure describes how a stock price $S_t$ that is modeled with Geometric Brownian motion with drift $\mu$
$$
dS_t = \mu S_t dt + \sigma S_t dW_t
$$
can be ...
1
vote
0answers
63 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 ItƓ's lemma and formulas $(dt)^2=dtdW_t=dW_tdt=0$ and $(dW_t)^2=...
0
votes
0answers
48 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
$$S_{t+1} = S_{t}e^{[(\mu-\sigma^2/2)]\Delta t + \sigma \sqrt{\...
1
vote
1answer
80 views
Why do I get this difference when simulating geometric Brownian motion?
I tried simulating GBM using both the SDE definition and the closed form solution. The paths I get through these methods are very different. Can someone help me figure my mistake?
...
2
votes
1answer
114 views
Covariance of logarithms of geometric Brownian motion
Suppose I have a Geometric Brownian Motion process,
$$dX_t=\mu X_t dt + \sigma X_t dW_t$$
I'd like to find the covariance of $\log(X_t)$ and $\log(X_s)$ where $s<t$. We can write $\log(X_t)$ in ...
0
votes
0answers
31 views
How to mathematically calculate the probability of GBM generating difference of less than some value
I have a custom index that follows Geometric Brownian Motion (GBM) with volatility v. I started this index at 10k with 4 decimal places i.e the starting price of ...
2
votes
1answer
108 views
How To Understand the Drift of ln(S) if S Follows Geometric Brownian Motion
As we know, if an asset S follows geometric Brownian motion, under risk neutral measure, it can be expressed as $\frac{dS}{S}=rdt+\sigma dW$, by applying Ito's lemma, $d(lnS)=(r-0.5*Ļ^2)dt+ĻdW(t)$, ...
3
votes
0answers
67 views
GBM probability of hitting non constant barrier
I know there is a formula for probability of hitting a constant barrier for GBM/BM (See page 651 in Martinagle Methods in Financial Modelling).
Is there a formula for non-constant barrier?
The ...
1
vote
2answers
126 views
How to Understand Lognormal Distribution in the Following Case
I got a question and corresponding solution, but have some difficulties in understand the lognormal distribution part of it, so I really appreciate your advice:
Question: assume zero interest rate ...
0
votes
1answer
76 views
Drawing values from a lognormal distribution of a GBM
I'm looking at a GBM with parameters
$$
r=0.05 \\
\sigma=0.2 \\
K=130\\
T=0.25\\
S_0 = 100
$$
This is a process that is lognormally distributed with mean and variance given by
$
\mu = S_0e^{r T+0.5\...
3
votes
1answer
69 views
Boundaries for Call Spread
I'm reading an interview book called A Practical Guide to Quantitative Finance Interview and I have some doubts regarding part of its solution and highlighted them in bold:
Question:
What are the ...
1
vote
1answer
581 views
Simulation of Geometric Brownian Motion in R
Using R, I would like to simulate a sample path of a geometric Brownian motion using
\begin{equation*}
S(t) = S(0) \exp\left(\left(\mu - \frac{\sigma^{2}}{2}\right)t + \sigma B_{t}\right),
\end{...
4
votes
1answer
127 views
What the expectation of S^2 is from GBM? [closed]
I was at an interview and was asked to write down the SDE for GBM.
$$
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
171 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)...
