Questions tagged [geometric-brownian]

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Transition density of geometric Brownian motion with time-dependent drift and volatility

Can you provide a reference to the transition density of the scalar geometric Brownian Motion with time-dependent drift and volatility, i.e. the scalar process $X = (X_t)_{t\geq 0}$ defined by the SDE ...
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40 views

Simulating correlated stock paths to calculate VaR

So I wanted to generate a Monte Carlo simulation for two correlated assets to derive then the VaR as a quantile of the generated distributions. My code is the following, where the input parameters are ...
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1answer
45 views

Can a down-and-out barrier call option be priced using the Black & Scholes formula or should it be approximated?

I am trying to price of a Down-and-Out Barrier call option with leverage. When the price of the underlying asset hits a certain barrier (B), the option becomes worthless. The issuer of these options ...
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2answers
275 views

Market price of risk on two assets

Under the assumptions of the Black--Scholes model, I read that the market price of risk of two assets $S_1$ and $S_2$ are the same, if they both follow Geometric Brownian motion driven by the same ...
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40 views

Applicability of the Ito's lemma [duplicate]

Ito's lemma is used to find the stochastic process of the function of a ...
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0answers
58 views

Arbitrage free pricing of option to trade stocks

Consider Black-Scholes model with constant interest rate r and stocks with prices $S_t^A$ and $S_t^B$ that satisfy the SDE's $dS_t^A = S_t^A(\mu^A dt + \sigma^A dB_t)$ and $dS_t^B = S_t^B(\mu^B dt + \...
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1answer
68 views

What is the meaning that Geometric Brownian motion is leptokurtic? [closed]

Does this have any relation to the symmetry of the normal distribution?
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0answers
70 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?
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4answers
398 views

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,...
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2answers
193 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 ...
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1answer
82 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: ...
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1answer
185 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 ...
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1answer
68 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 ...
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61 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. ...
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1answer
81 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 ...
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0answers
64 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=...
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0answers
53 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{\...
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1answer
84 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? ...
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1answer
118 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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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 ...
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1answer
112 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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0answers
68 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
129 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
89 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
79 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
814 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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1answer
131 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
181 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)...