Questions tagged [geometric-brownian]

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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? ...
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59 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 ...
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29 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 ...
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1answer
92 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, $d(lnS)=(r-0.5*σ^2)dt+σdW(t)$, ...
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59 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 ...
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2answers
103 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 ...
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1answer
39 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\...
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1answer
60 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 ...
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1answer
89 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{...
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102 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 ...
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2answers
145 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)...