Questions tagged [brownian-motion]

Question 1

I'm trying to visualize the path transformations coming from the application of the Girsanov Theorem in a Monte Carlo Simulation. Below the (Python) code where I'm trying to "adjust" the ...

Question 2

I want to find the joint distribution of 2 correlated Brownian motions, $X$ and $Y$, at time $T$ after crossing boundaries $a<0$ and $b<0$ respectively with initial values $X_0=Y_0=0$, drifts $\...

Question 3

I am simulating barrier option pricing by implementing Fractional Brownian Motion (Cholesky decomposition approach). Is the adjustment of the Riemann correction of the barrier by the Hurst exponent ...

Question 4

Playing around with the idea of controlling the dynamics between two processes. There are some papers around this building on the SDE $\frac{dS}{S} = \alpha (\mu - \ln S)dt + \sigma dW_{t}$ (Schwartz) ...

Question 5

I've been learning about quantitative finance for the past few months, though I’m still far from an expert. I’ve read about applications of Black-Scholes concepts outside traditional financial options....

Question 6

I am trying to implement a standard stretched Brownian motion 1,2in Python. This requires finding a fixed point of the equation $\mathcal{A}:CDF \to CDF$, $\mathcal{A}F = F_\mu \circ(\phi*(Q_\nu \circ(...

Question 7

Consider two correlated Brownian Motions $W_{1,t}$ and $W_{2,t}$ for which it holds: $$dW_{1,t}\sim N(0, \sqrt{dt})$$ $$dW_{2,t}\sim N(0, \sqrt{dt})$$ $$Cov(dW_{1,t},dW_{2,t}) = E[dW_{1,t}dW_{2,t}] = \...

Question 8

In this problem we assume the interest rate $r=0$. I have a European contingent claim with maturity $T$ and payout $$Y = S_T 1_{\{S_T \leq K\}} - K1_{\{S_T > K\}}$$ I need to show that the initial ...

Question 9

Brownian Bridge, denoted $B_t$ can be described as a Brownian motion pinned at zero and $T$, with the following mathematical definition: $$B_t=W_t-\frac{t}{T}W_T$$ From the above, we can deduce that $\...

Question 10

I want to solve an SDE: $$ dX_t=X^3_t\,dt+X^2_t\,dB_t, $$ with $X_0$=1. In my understanding (I'm currently using J. Michael Steele's textbook "Stochastic calculus and financial applications")...

Question 11

I'm currently working on a stock market prediction project using Geometric Brownian Motion (GBM). My goal is to showcase that GBM, as a stochastic model, does not require a large dataset compared to ...

Question 12

I am trying to derive some properties of geometric brownian motion: $dS_t = \mu S_t dt + \sigma S_t dW_t$ I am interested in analyzing paths that 'survive' a lower boundary $X$ i.e. always stay above $...

Question 13

I typically see people define realized variance as the squared difference in log returns, i.e. $$RVar = \frac{1}{T} \sum_{n=1}^N \log \left( \frac{S_{n}}{S_{n-1}} \right)^2$$ where $t_n - t_{n-1} = \...

Question 14

Show that $$E \left[ \int_s^t W_u \, du \,|\, \mathcal{F}_s \right] = (t - s) W_s$$ where $W_u$ is a standard Brownian motion and $\mathcal{F}_s$ is the filtration up to time $s $.

Question 15

Suppose I have a process that follows an arithmetic brownian motion $dX_t = \sigma dW_t$ How do I calculate, within a certain interval $\Delta t$ , the expected number of times that the process will &...

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Questions tagged [brownian-motion]

Girsanov Theorem: how to implement it with code?

Joint distribution correlated Brownian motions weighted by stopping probabilities

Adjusting Riemann correction by Hurst Exponent for Barrier Option Pricing

Cointegrated log normal processes

Can Black-Scholes-style modeling help with CapEx forecasting? Does it make sense to apply Black-Scholes-related concepts this way?

Implementation of standard stretched Brownian motion in Python

Generation of two correlated Brownian Motions using two different approaches in numpy Python

Initial price of contingent claim in terms of Black-Scholes call option initial price with new volatility parameter

Conceptual questions on Brownian Bridge

Trying to solve an SDE with a power function

Should Geometric Brownian Motion Prediction Use Rolling Window or Train-Test Split?

Trying to derive some properties of geometric brownian motion

Estimate of GBM Return Variance

Show that $E [ \int_s^t W_u \, du \,|\, \mathcal{F}_s ] = (t - s) W_s$

ABM Crossing Times

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