# Questions tagged [geometric-brownian]

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### Bond price distribution if yield assumed log-normal

Suppose we assume that yields on a zero-coupon bond that matures at time $T$ follow a log-normal process of the type $y(t,T)=y(t_0,T)e^{-0.5\sigma^2t+\sigma W_t}$ under the T-forward measure. Then, I ...
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### Simulating correlated Geometric Brownian Motion in Python

I want to simulate two correlated Geometric Brownian Motion processes in Python. I found an implementation from Matlab (https://www.goddardconsulting.ca/matlab-monte-carlo-assetpaths-corr.html) and ...
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### Why does higher volatility for ATM Call Option lead to a lower risk-neutral probability of expiring ITM?

This is a follow-up question on the discussion in the thread here, from which I borrow the graph below depicting $N(d_2)$ (i.e. the risk neutral probability of a Call option expiring in the money) ...
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### Probability of an Option maturing In-the-money vs. Volatility

How will the probability of an option ending up in the money change if the volatility of the underlying stock increases? Intuitively, I think the answer to this is that if volatility goes up the ...
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### Simulating two correlated time series using GBM [duplicate]

My situation is the following: I have two time series TS1 and TS2, whereas TS1 is a stock price. According to literature, TS2 is positively correlated to TS1. Furthermore, since TS1 is a stock price, ...
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### Is there a relation between the so-called volatility drag and the sigma term in Black-Scholes' model? [duplicate]

The closed-form solution of Black Scholes Dynamics $dS_t=S_t(\mu dt +\sigma dW_t$) is $$S_t=S_0 e^{(\mu -\sigma ^2/2) t+\sigma dW_t}.$$ The $-\sigma^2/2$ term is quite similar to the volatility drag ...
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### VaR and Expected Shortfall for Geometric Brownian Motion

Given that $dS_t=\mu S_tdt+\sigma S_tdW_t$ ,a risk free rate r and defining Value at Risk and Expected Shortfall as $VaR_{t,a}=S_0e^{rt}-x$ where $x$ is the amount such that $P(S_t\leq x)=1-a$ ($a:$...
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### Why am I struggling to replicate the Black-Scholes price of an option stochastically?

I am currently trying to replicate the Black-Scholes price of a call option using stochastic simulations of the price moves of the underlying. My code is as follows: ...
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### Monte Carlo simulations of correlated stocks by Geometric Brownian motion

I am trying to simulate using a Geometric Brownian Motion process three autocorrelated stocks. In particular, I need to simulate three different matrices with 1000 scenarios each using a Monte Carlo ...
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### Simulating artificial asset prices: Random walk vs Brownian motion?

How well can each simulate the real-life behavior of stock prices, and what considerations or (dis-)advantages must we be aware of when deciding to use each: Random walk with drift Random walk ...
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### Source on multivariate correlated geometric Brownian motion returns, not prices

Can anyone provide a source that formulates how to generate multivariate geometric Brownian motion returns using the Cholesky method with target correlation matrix, instead of correlated GBM prices? ...
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### Theoretical Expected Maximum Drawdown vs Empirical Maximum Drawdown

I have been looking at the approach for calculating the expected maximum drawdown of a Brownian Motion  and the corresponding function maxddStats in the fBasics package in R . I do not ...
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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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### 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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### 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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### 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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### 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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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 $dS_t^B = S_t^B(\mu^B dt + \... 1answer 70 views ### What is the meaning that Geometric Brownian motion is leptokurtic? [closed] Does this have any relation to the symmetry of the normal distribution? 0answers 82 views ### What are some alternatives to Geometric Brownian motion that can be used in the Black-Scholes? [closed] I hear that there are many extensions to the black scholes model to make it more realistic, however, GBM does not account for volatile swings. Is there any sort of alternative approach to use instead? 2answers 140 views ### Compute the price of a derivative which pays$\log(S_T)S_T$in the Black Scholes world Compute the price of a derivative which has pays$\log(S_T)S_T$, you can assume that the Black Scholes model is valid. Using the stock measure we can write the expectation as $$D(0) = S_0 \mathbb{E}... 4answers 484 views ### Find a formula for the price of a derivative paying \max(S_T(S_T-K),0) Develop a formula for the price of a derivative paying$$\max(S_T(S_T-K))$$in the Black Scholes model. Apparently the trick to this question is to compute the expectation under the stock measure. So,... 2answers 240 views ### Normality or Log-Normality of Regular Returns Another old question on this site (How to simulate stock prices with a Geometric Brownian Motion?) inspired me to ask the following question: if we assume that regular returns could be normally ... 1answer 83 views ### Correctly simulating BEKK series to model asset returns I am trying to create financial data as close as possible to that of asset returns. Using the R code I can collect some stock data and compute the return: ... 1answer 194 views ### From VG and NIG processes to GBM I would like to find out if it is possible to reduce: the Madan-Seneta Variance Gamma (VG) model; the Barndorff-Nielsen Normal Inverse Gaussian (NIG) model to the standard Black-Scholes through a ... 3answers 177 views ### Probability of a stock price using implied volatility I have attempted to use the fact of having implied volatility, but have not been able to come up with a viable way to calculate the probability, any ideas? Suppose that a stock S_t follows a ... 1answer 72 views ### Sampling from SDE In the case of the classic Geometric Brownian motion$$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] $$and ... 0answers 63 views ### Geometric brownian motion and probabilities A stock's price movement is described by the equations dS_t=0.02S_tdt+0.25S_tdW_t and S_0=100. An investor buys a call option on said stock with a strike price K=95 which expires in T=2 years. ... 1answer 86 views ### true or false: the risk-neutral measure is useless in this situation Example 2 of this Wiki article on the risk-measure describes how a stock price S_t that is modeled with Geometric Brownian motion with drift \mu$$ dS_t = \mu S_t dt + \sigma S_t dW_t $$can be ... 0answers 65 views ### On Geometric Brownian motion and Itô's formula Let S_t be a geometric brownian motion such as$$d S(t) = rS(t)dt +\sigma S(t)dW(t),$$where W is a standard Brownian motion. With Itô's lemma and formulas (dt)^2=dtdW_t=dW_tdt=0 and (dW_t)^2=... 0answers 58 views ### Convert drift and diffusion term in terms of time in the Geometric Brownian Motion framework Assume that we have daily prices covering the period of 10 years. For calibrating the drift and diffusion parameters of the GBM model$$S_{t+1} = S_{t}e^{[(\mu-\sigma^2/2)]\Delta t + \sigma \sqrt{\... 1answer 86 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? ... 1answer 124 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 ... 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 ... 1answer 113 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,$d(lnS)=(r-0.5*σ^2)dt+σdW(t)$, ... 0answers 70 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 ... 2answers 148 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 ... 1answer 111 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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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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### 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{...
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 ...
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)...