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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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What is the Brownian motion in the model for the return of a stock price trying to capture?

I have read that in the derivation of the Black-Scholes PDE, we assume that the return of a stock $S$ is given by $$\frac{dS}{S}=\mu dt+\sigma dB$$ where $\mu$ is the average growth of $S$, $\sigma$ ...
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Reference request for research on the maximum drawdown **ratio** (NOT value)

Let's suppose the asset price process follows a Geometric Brownian motion $S_t \sim GBM(\mu, \sigma),\,t\ge 0$, and define the two process: $$ \begin{align} \text{MSF}_t &:= \max_{\tau\in[0,t]} S_\...
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What are the underlying events that the random variables map to the real line in the derivation of the Black-Scholes PDE?

When we first try and set up a model for the evolution of S, the value of the underlying stock, I have seen in a lot of textbooks that they model the evolution by the formula $$\frac{dS_t}{S_t}=\mu dt+...
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Euler scheme to simulate trajectories + Python Code

Question: Euler scheme to simulate trajectories of S + Python code to get independent trajectories of S? For solving a problem I have the following assumptions: stochastic basis (Ω,FT,(Ft)t≥0,P) ...
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59 views

Distribution of time integral of Brownian motion squared (where the Brownian motion occurs in square root time)?

Let $I_t = \int_0^t W_{\sqrt{u}}^2du$. What is the distribution of $I$? If I recall correctly, if the Brownian motion were instead $W_u$, then it would be $I_t \sim N\left(\frac{t^2}{2},\frac{t^4}{3}\...
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49 views

Discounted asset price is martingale in BS model

I want to verify that the discounted stock price process $\mathrm{e}^{-r(T-t)}V(S_t,t)$ is a martingale in the BS-model. Using Ito's formula and the BS-PDE I get that $$ \mathrm{d}\mathrm{e}^{-r(T-t)}...
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Random Walk with normal increments and n time periods why is the increment $\sqrt{(t/n)}$?

Question is basically in the title. I have found several sources stating that $R_i = \sqrt{\frac{t}{n}}$, but I couldn't find the intuition behind taking the square root. And it seems to be crucial ...
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78 views

Integral of Wiener process over time

This should hopefully be an easy question to answer, but I am new to Stochastic Calculus and am gapping as to why the following is true, for a brownian motion $W_t$: $$d(\int W_t dt ) = W_t dt$$ I ...
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1answer
56 views

Autocovariance of increments of a semimartingale

Say that $X_t$ is an Itō process with \begin{equation} dX_t = \mu_t dt + \sigma_t dW_t \end{equation} where $\mu_t$ and $\sigma_t$ are adapted processes. Is it always true that \begin{equation} E[...
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57 views

Expectation of the product of two Brownian motions [closed]

Could you please let me know the steps to follow to get to the solution?
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1answer
47 views

Does GBM stock price model have E[S(t)] unaffected by volatility?

Many an author claims that, if you model stock prices through GBM, $E[S(t)]=e^{\mu t}$, and the expectation is thus not related to volatility. I keep running around in circles on this one. First ...
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120 views

How to calculate the covariance between two stochastic integrals?

How to calculate the covariance between the integral of a Brownian motion at different times: $$\text{Cov}\left(\int^{t_1}_0\sigma(t)dW_t,\int^{t_2}_0\sigma(t)dW_t\right)\ ?$$ I know the answer is: $$\...
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Model of asset substitution/risk shifting in continuous time

Consider a firm with cash flows $X_t$, which under a risk-neutral probability measure, follows a geometric brownian motion: $$dX_t = X_t[(r-\beta)dt + \sigma dZ_t]$$ where $r>0$ is the risk-free ...
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92 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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63 views

Differential product Correlated processes

I am trying to derive the differential of the product of two processes, but I got stuck. This is what I have until now: We have the following two stochastic processes: $dX_t= \mu_t dt +\sigma_t dW_t$...
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63 views

Fourth moment of a itos integral

$I(t)=\int_0^t \sqrt sdW_s$ What is $E(I(t)^4)$
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Theoretical distribution of (geometric) Brownian motion (with drift)

I am working on a simulation study which focuses on both the Brownian motion with drift (1) and the geometric Brownian motion (2). I denote them by $X_t$. What are the theoretical distributions of ...
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88 views

Quadratic variation of an integral of a function of a Brownian motion

I'm asked to find the quadratic variation of the integral $\int_{0}^{t} W_s^2 ds$.
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48 views

Correlated GBM and OU processes

I want to model two different stochastic processes, such that: $X_t , V_t$ are correlated with coefficient $\rho$. Where: $\frac{dX_t}{X_t}=\mu_1dt+\sigma_1 dW_{1,t}$ and $dV_t=\theta(\mu_2-V_t)dt+\...
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1answer
142 views

Calculate drift of Brownian Motion using Euler method

I am working on a project to approximate numerically the solution $X_t$ of a stochastic differential equation (SDE) using the Euler method. I have do to this for the Brownian motion with drift. I am ...
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218 views

Integral of Brownian Motion w.r.t Time: what is wrong with this solution? [duplicate]

My question is about a stochastic integral of brownian motion w.r.t time. Let $W(t)$ the Wiener process (or brownian motion). I want to calculate this: \begin{eqnarray} X(t)=\int_{0}^t dt' W(t'). \...
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Brownian bridge with time varying volatility

I have a question to ask about the Brownian bridge for a process with deterministic volatility varying over time. In other words, we have this dynamic: $dS_t = \sigma_{t} * dW_t$. We want to know the ...
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1answer
85 views

Asset price simulation under Monte Carlo for option pricing using market data

I am trying to use Monte Carlo to price some exotic options. I have in mind to simulate asset prices under GBM (say S&P prices) using Monte Carlo and price the option accordingly from the payoffs ...
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1answer
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Conditional Probability - Geometric Brownian Motion

Background I am trying to find a way to price a variant of a gap option by using closed-end expressions. What makes this option a bit tricky is that it can be exercised at four predetermined dates (t=...
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2answers
223 views

Does Black Scholes need to assume no arbitrage?

Since Girsanov's theorem guarantees a risk neutral measure for Geometric Brownian motion, by the fundamental theorem of asset pricing there can be no arbitrage. So, why does the model assume no ...
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1answer
79 views

Using Geometric Brownian Motion for Index Options

As far as I understand, in most of the cases we derive the option valuation assuming that the log-return of the asset is partly driven by its own Brownian motion, and we use Geometric Brownian motion (...
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About a Trivariate Density of Brownian motion, its local and occupation times

I need help on a technical detail in the article by Ioannis Karatzas and Steven E. Shreve (1984) titled: " Trivial density of Brownian motion, its local time and occupation,with application to ...
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Combining two returns forecasts of different lengths

Hi Quantitative Finance stack exchange, I'm analyzing a price process. I've made two forecasts, namely $Ret_{\text{10min}}$ = the return after 10 mins and $Ret_{\text{20min}}$ = the return after 20 ...
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About a formula concerning the occupation time of a Brownian motion (The arc-sine formula)

Let the process $Z_t \buildrel\textstyle\over={W_t}+ {\lambda}t $, $t\geq 0$ a brownian Motion with Drift and $A_{T}^{+k, Z}$ his occupation time above the barrier $k$ defined by $$A_{T}^{+k, Z}=\...
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1answer
73 views

Determining the probability of arriving at a price by a time T

A useful calculation for ascertaining the risk of something might be determining the probability of a realization of a set of stock prices $X$ being greater than or equal to some future price $x$. I ...
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Does it make sense to simulate from the multidimensional GBM?

Suppose I have times series data on 3 assets and I do $N$ simulations (GBM) first for each of assets individually and then from a multidimensional GBM since their log-returns are correlated (I use ...
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1answer
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GBM in R giving negative numbers?

I was under the impression that simulations involving geometric brownian motion are not supposed to yield negative numbers. However, I was trying the following Monte Carlo simulation in R for a GBM, ...
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1answer
186 views

Expectation of an Integral of a function of a Brownian Motion

I would really appreciate some guidance on how to calculate the expectation of an integral of a function of a Brownian Motion. Let $B(t)$ be a Brownian motion with drift $\mu$ and standard ...
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3answers
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Does the Ito correction term in GBM result in 'real money', or is it illusory?

There are two ways to think about investment returns and randomness. First is sort of like 'bank interest', with randomness. Suppose we invest 100 units of currency. Suppose each year there is a ...
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1answer
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Basic question on Ito integrals

$Let \space X(t) =\begin{cases} 2, \qquad\text{if} \space 0\le t \le 1 \\ 3, \qquad\text{if} \space 1 < t \le 3 \\ -5, \qquad\text{if}\space 3 < t \le 4 \end{cases} $ or in one forumala $...
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How might I answer this past exam question relating to the value of a European option under the BMS market model?

The following question, which is not homework, was taken from a past paper for a module I will soon be sitting: Consider a Black-Merton-Scholes stochastic market with drift $\mu = 1$, volatility $\...
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1answer
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Which expression of $S_t$ to use under the Black-Scholes model?

I am currently looking at example exam questions relating to the evolution of a stock price under the Black-Scholes model. However, I am confused by seemingly inconsistent expressions used for the ...
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1answer
142 views

Calculating the stochastic integral of $\exp(-rt)S_t$

I am currently reading lecture notes which aim to show that if $$ S_t = S_0 \exp (\mu t + \sigma W_t) $$ then, under the probability measure $\tilde{\mathbb{P}}$ with density $$ \gamma_T = \exp (c W_T ...
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1answer
139 views

Question about the Cameron-Martin-Girsanov (CMG) theorem

Within my lecture notes, the following definition of the CMG theorem is given: Under the probability measure $\mathbb{\tilde{P}}$ with density $\gamma_T = \exp(cW_T - \frac{c^2}{2}T)$, the process $...
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172 views

Applying Ito's formula to complex functions

Within my lecture notes, the following definition is given: We say that the stochastic process $X_t$ has stochastic differential $$ dX_t = b_t dt + \sigma_t dW_t $$ if and only if $$ X_t = ...
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Definition request - Brownian Motion Characterised by Sharpe Ratio

What is the Stochastic Differential Equation for a "Brownian Motion Characterised by Sharpe Ratio"? I saw it in a paper ("Lessons from the Mortician: volatility modulation") and the authors do not ...
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103 views

Prove the Markov property for the stochastic process $Y^x_t$

Prove the Markov property for the stochastic process $Y^x_t=xe^{at+bW_t}$ Given a function $u(t,x)=\mathbb{E}[f(Y^*_t)]$ with $Y^*_0=x$. For $s<t$ we have $\mathbb{E}[f(Y^*_t)]=u(t-s,Y^*_s)$ by ...
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Finding distribution of $\int_0 ^T uW_u du$

I would like to find the distribution of $\int_0 ^T uW_u du$ where $(W_u)_{u\geq0}$ is the Brownian motion. What I have tried: $$\int_0 ^T uW_u du = \int_0 ^T B_udu - \int_0^T \int_0^tB_sdsdt$$ by ...
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1answer
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Calculate $Cov(e^ {B_t} ,e^{B_s})$

Let $(B_t)_{t \geq0} $ be a Brownian Motion. Calculate $Cov(e^ {B_t} ,e^{B_s})$ I would verify the following solution which the result looks a bit weird. My solution: let $0 \leq s \leq t$. $$Cov(e^ {...
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Monte Carlo Pricer for Express Certificate delivers wrong price [Mathematica]

So I wanted to price the following Express Certificate with this specific payout structure: If S1 > S0 -> 105.25 , else -> If S2 > 0.95*S0 -> 110.5 , else -> If S3 > 0.9*S0 -> 115.75 , else -> If ...
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3answers
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Browian motion: $P(B_1<4 | B_2 =1)$

I want to calculate $P(B_1<4 | B_2 =1)$ for the B.M. What I tried: $P(B_1<4 | B_2 =1)=P(B_1 - B_2 < 3- B_2 | B_2 =1)$ but I cant use any independence to calculate further.
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About SDE of Geometric Brownian Motion

It's known that most of the financial assets are subject to Geometric Brownian Motion, which satisfies the following equations: $\frac{dS}{S}=\mu dt + \sigma dX$ (1) $S_t = S_0 e^{(\mu + \frac{1}{2}...
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2answers
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Isn't GBM the equivalent of adding infinitessimally small normally distributed returns?

The classic treatment of GBM for asset pricing leads to a point where eventually one gets a solution that is the same as assuming an underlying arithmetic Brownian motion, $X_t$, which has (over unit ...
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807 views

Integral of Wiener process w.r.t. time

I have a doubt with regards to the calculation of the below integral- $\int_0^t W_sds$ where $W_s$ is the Wiener Process. This has been solved very ably in the following page. It turns out to be a ...
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Continuous returns in BS-market

Let's assume we are in a Black-Scholes market. The price processes in the BS-market are given by $dP_0(t)=P_0(t)rdt, P_0(0)=1$ $dP_i(t)=P_i(t)\cdot(\mu_i dt + \sigma_idW(t))=P_i(t)\cdot \left(\...