Questions tagged [brownian-motion]

In mathematics, Brownian motion is described by the Wiener process; a continuous-time stochastic process named in honor of Norbert Wiener.

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4answers
126 views

Price of Call Option with or without jumps

Suppose two assets in Black Scholes world have the same volatility, but one has downward jumps at random times. How does this affect the option prices? I would have thought that downward jumps would ...
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4answers
327 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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66 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 ...
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How to price a down-and-out leveraged barrier call option using Brownian motion?

I am trying to price a type of leveraged down-and-out (LDAO) barrier call option, using geometric Brownian motion. My python script is below. I am not sure how to correctly model the increasing ...
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How to simulate stock prices with a Geometric Brownian Motion?

I want to simulate stock price paths with different stochastic processes. I started with the famous geometric brownian motion. I simulated the values with the following formula: $$R_i=\frac{S_{i+1}-...
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1answer
123 views

Simulate stock prices with Geometric Brownian Motion motion with mu and signa based on 'normal' or continuous compounding?

I have written a simple script for modelling stock prices using Geometric Brownian Motion. The time series I am downloading are daily adjusted closing prices. My aim is to be able to change the ...
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1answer
80 views

Convert option inputs to standard Brownian motion

I want to know the probability that the strike price of an option is touched. My input values are: P = price S = strike v = vol t = time to expiration According ...
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0answers
33 views

mixing fractional Brownian motions

Given two Brownian motions $W_t^1, W_t^2$, we can have them correlated by $$dW_t^1 = \rho d W_t^2+\sqrt{1-\rho^2}dZ_t$$ where $W_t^{2}$ and $Z_t$ are independent of each other. My question then: is ...
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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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0answers
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How to expand lognormal approximation of Brownian motion

How can we expand this sum? $\sum_{i=1}^n (e^{rt_i-\frac{1}{2}\sigma^2t_i+\sigma w_{t_i}})^2$ where: $w_{t_i}$ is a standard Brownian motion. If we let $m_t=e^{-\frac{1}{2}\sigma^2t_i+\sigma w_{t_i}}$...
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1answer
60 views

Ito's lemma for a Forward

I'm trying to understand the derivation of Ito's process with respect to a Forward $F$ on a stock $S$ that pays a constant dividend yield, say $y$. Stock follows brownian motion $\\$ $dS_{t} = S_{t}(\...
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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 ...
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1answer
70 views

Integration of a deterministic function w.r.t. a Brownian motion

Help me solve this problem: Let $W_t$ be a Brownian motion and suppose $X_t = \int_{0}^{t}\delta _{s}dW_{s}$ where $\delta _{s}$ is a deterministic function. Then show that $X_t$ is a Gaussian ...
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1answer
65 views

Properties of integrated GBM

(I asked this question on MSE but I think it might have more success here) Good day, I was going over some exercises and I stumbled upon a question that, for its solution, requires me to find/...
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1answer
227 views

Correlation between stock prices given correlation between returns

assume I have two stocks with known volatilities and a known correlation coefficient of returns - does anyone know how to determine the correlation between the prices and NOT THE RETURNS
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How to find the derivative for a multi-factor geometric brownian motion model

Does anyone know how to find the derivative for a multi-factor geometric brownian motion model $ \frac { dS_{i}}{S_{i}} $. I have seen solutions for the standard GBM model however I suspect that the ...
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1answer
67 views

Brownian motion and Stochastic Integration

I have two questions relating stochastic integration which perhaps could be answered together. First question: First of all, I don't really understand why we can't use Riemann-Stieltjes integration ...
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1answer
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Prove that $d\hat{W}_t = dW_t - \frac{1}{N_t} \cdot dN_t\cdot dW_t$ gives a Brownian motion under forward measure

Let $N_t$ be a numeraire and $(W_t)$ be the standard Brownian motion under the risk-neutral probability measure $P$. Recall that forward measure $\hat{P}$ is defined as the Radon-Nikodym derivative: $...
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Minimal bounds to enclose most sample paths of a GBM (Geometric Brownian Motion)

For a (generalized) Brownian motion $Y = F(t,W)$, starting at $InitialValue$ and running for a total of $T$ time, if I want to "enclose" (in a visual way) "most" of the possible sample paths, I could ...
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1answer
56 views

Sample path simulation using two random variables

I was wondering if there is a way of generating a sample path of a Geometric Brownian Motion using two independent standard normal random variables instead of just one. The exact scheme that uses ...
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Compo/Quanto Adjustment & Multivariate Ito

Related to the issue that I have raised here, I am facing another question. As the rule here is 1 question / 1 post, I take the opportunity to ask it below: By exploring StackExchange, I noticed the ...
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1answer
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Difference between $W_t$ and $X_t= \sqrt{t}Z$

$W_t$ is a brownian motion and $X_t= \sqrt{t}Z$, where: $Z\sim N(0,1)$. How to show that for a bounded continuous $f$ process, $$U_t = \int_0^t (f(W_s))ds$$ and $$V_t = \int_0^t (f(X_s))ds$$ have the ...
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1answer
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Conditional distribution of $X_t = \int_0^t W_s \mathrm{d}s$

What is the conditional distribution of $$X_t = \int_0^t W_s \mathrm{d}s$$with respect to $W_t = x$?
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1answer
132 views

Fractional Brownian motion references

Does anyone know any good references to understand the fractional Brownian motion and its numerical simulation, preferably applied to finance.Thank you.
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1answer
47 views

Exact solution stock price with Vasicek interest rate model

Define two correlated stock price- and interest rate (Vasicek) processes, governed by the Wiener processes $W^{S}(t)$ and $W^{r}(t)$ $$dS(t)=r(t)S(t)dt+\sigma S(t)dW^{S}(t)$$ $$dr(t)=\kappa(\theta-r(...
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0answers
37 views

Simulate correlated Brownian motions conditioned on future state(s)

Consider a model defined by 2 geometric Brownian motions $$dY_{1}(t) = \sigma_{2} Y_{1}(t)dW_{1}(t)$$ $$dY_{2}(t) = \sigma_{2} Y_{2}(t)dW_{2}(t)$$ with $Y_{1}(0) = y_{1}$, $Y_{2}=y_{2}$ and $dW_{1}(...
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0answers
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Distribution of the random variable $X (t) = ∫ s*B_sds$ [duplicate]

What is the distribution of the random variable $$X (t) = ∫ s*B_sds ?$$ The integral is taken over [0,t]. $$B_s$$ is a brownian motion.
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2answers
180 views

Why is $S(t) = e^{\alpha + \beta t + \sigma W(t)}$ used as a model for prices?

Why is the Geometric Brownian Motion defined as $S(t) = e^{\alpha + \beta t + \sigma W(t)}$ used as a model for stock prices? $S(t)$ has a lognormal distribution which is right skewed. Another problem ...
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4answers
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Is it really possible to create a robust algorithmic trading strategy for intraday trading?

I'm an engineer doing academic research for my master thesis in the area of quantitative finance, basically the purpose is to study the possibility to create an intraday-trading algorithm. I've tried ...
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1answer
64 views

Intuition behind prices modeled by Geometric Brownian Motion

Suppose that we model a price $P_t$ to evolve per $$\frac{dP_t}{P_t}=\mu dt+\sigma dW_t$$ for $\mu\in\mathbb{R}$ and $\sigma>0$. The unique strong solution to this diffusion is $$P_t=P_0e^{(\mu-\...
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1answer
25 views

Find Arithmetic Brownian Motion's transition density

Consider the following stochastic differential equation, an Arithmetic Brownian Motion: 𝑑𝑆(𝑡) = 𝑟 𝑑𝑡 + 𝜎 𝑑𝑊(𝑡) . Find its solution, integrating from t to T, then find its transition density. ...
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1answer
99 views

Calculation of a process's drift

Let $X_t:=e^{W_t}$ where $W_t$ follows the Wiener process. Calculate the drift. The answer is given as $X_t/2$. My attempt at a solution (which I'm afraid is poor from a mathematical standpoint): I ...
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1answer
175 views

Predicting time series using Jump Diffusion model and Neural Networks

I am trying to understand the difference between using Jump diffusion model and Neural Networks or more precisely LSTM to predict time series data regardless what that data contains for example a ...
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0answers
50 views

Two- (multi) dimensional geometric Brownian Motion

I am trying to calculate the value of a Basket Option with two stocks and the following information: S1 = 100, S2 = 120, r = 0.06 L = Volatilitymatrix = ((0.3, 0.1), (0.0, 0.2)), weight of Stock 1 = 1/...
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1answer
93 views

Advantages of pathwise calculus over stochastic calculus in continuous self-financing trading models

I am new to stochastic calculus but the statement below confuses me: Beside the issue of the impossible consensus on a probability measure, the representation of the gain from trading lacks a ...
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1answer
306 views

What is a Brownian motion “under the risk-neutral” measure?

I understand that the risk-neutral measure is one under which the discounted price (acc. to the risk-free rate) of any asset is a martingale. But we also see notation like $\mathbb{W}^Q_t$ to denote a ...
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0answers
48 views

Are the increments of a stochastic process driven by fractional Brownian motion independent?

I'm studying the following equation $$\tag1 dX_t = \mu X_t dt + \sigma X_t dB^H_t $$ where $B^H$ is the fractional Brownian motion (fBm) of Hurst parameter $H\in(0,1)$, that is a continuous ...
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1answer
156 views

Solving Stochastic Differential Equation for Geometric Brownian Motion with time-dependent drift

Given the stochastic differential equation: $$dZ_t = -Z_t \theta_t dB_t, \quad Z_0 = 1.$$ for an adapted process $\theta_t$ and Brownian motion $B_t$, how exactly do I apply Itô's Lemma to obtain: ...
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2answers
171 views

Solution to SDE being Evolution of Price Process

I am trying to explain the concept of a solution to SDE being the model for the evolution of a price process. How would you do this to someone who doesn't have a financial engineering background? ...
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1answer
75 views

Showing BM $W(s)$ is independent of $W(t)-W(s)$ [closed]

Consider $0\leq s<t$ where $t,s$ represent time index. I want to show a Brownian motion $W(s)$ is independent of $W(t)-W(s)$. Specifically, show that $E[W(s)(W(t)-W(s))]=0$ Proof: Writing $W(s)$...
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0answers
38 views

Risk-neutral Simple Return Moment Log-return Moment

I am trying to find a way to link Risk-neutral moment of simple return to risk-neutral moment of log-returns. Specifically, by making the same standard assumptions of the Black-Scholes model with the ...
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1answer
127 views

Process with negative quadratic variation

Today seems to be question day for me, sorry. The complex process $$ dX = i\sigma dW $$ where $i = \sqrt{-1}$ and $dW$ is a standard (real-valued) Brownian motion will have a negative variance ...
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1answer
181 views

Steven Shreve: Stochastic Calculus and Finance

The lecture notes have the following theorem: Let $\theta\in \mathbb{R}$ be given and $B(t)$ stands for the Brownian motion which is a martingale, then $Z(t)=exp\{-\theta B(t)-\dfrac{1}{2}\theta^2t\}$...
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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? ...
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1answer
103 views

Simulation Heston Model, markovianity

I am trying to simulate the instanteneous volatility of a Heston process. My equations are the following : wealth process: $$dX_t = r_t X_t + \theta \sqrt {V_t} u_t dt + u_t dW_{1t}$$ Volatility: $$...
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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 ...
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1answer
91 views

Brownian motion Price and Hedge problem

Let $W_t$ be a Brownian Motion and let $S_t= S_0e^{(rt- \frac{\sigma^2}{3!}t^3 +\int_{0}^{t}\sigma W_s ds )}$ Price and Hedge at time $t=0$ European call with maturity $T$ and strike price $K$, ...
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2answers
331 views

Variance of a time integral with respect to a Brownian Motion function

Let process $$I_t = \int_0^t f(s) W_s \,\mathrm d s $$ where $W_s$ is standard Brownian motion. My question are the following: We know that $\mathbb{E} (I_{t})=0$ for all $t$ and $f$ a integrable ...
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7answers
4k views

Consensus on Cauchy distribution for stock prices

What is the general consensus for using a Cauchy distribution to model stock prices? I can't find much after researching online and wonder if it has been tried and discarded. My motivation is to find ...
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1answer
169 views

Mark Joshi uses forward price to price an option that pays $S_t^2-K$ if $S_t^2>K $ and zero otherwise? Why can we do that?

The following question is taken from Mark Joshi's Concepts and Practice of Mathematical Finance, second edition, Exercise $6.6$ Suppose a stock follows geometric Brownian motion in a Black-Scholes ...

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