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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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2
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2answers
176 views

Geometric Brownian Motion - Price Probabilities

I am modeling a stock price that follows Geometric Brownian Motion and have the following: $E(X)$ = .16 (16%) $\sigma$ = .24 (24%) $X_0$ = 95 $T$ = 1 (12 months) I am trying to find the ...
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2answers
261 views

Stochastic Integral Graph

As we can represent the integration of $f(x)$ on $[a,b]$ with the graph below, I was wondering how to represent the following integral with $X(t)$ a Brownian motion, $f(t)$ any function and $t_j = ...
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0answers
67 views

Integrals with respect to Brownian motion

I have two questions: Let $(B_t)$ be a Brownian motion. Find all constants $a$ and $b$ such that $X_t =\int_a^t (a +b\frac{u}{t}) \mathrm{d}B_u$ is also a Brownian motion. Find all constants $a$, $b$...
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2answers
816 views

probability question about brownian motion

Assume $W_{t}$ is a standard Brownian Motion, calculate the the probability that $W_{t}*W_{2t}$ is negative, i.e., $P(W_{t}*W_{2t}<0)$. I find it tricky to calculate the probability.Thank you.
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0answers
28 views

Swap rate in the annuity measure and Martingale Representation Theorem

As we know, swap rate evolves as a martingale in the appropriate annuity measure. Martingale representation theorem says if I can find a Brownian motion in the annuity measure and the swap rate is ...
5
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2answers
187 views

More questions about integral of Brownian Motion w.r.t time

A similar question have been posted earlier but one part has remained unanswered. Let us define: $$X_t = \int_0^t W_s ds,$$ where $W_t$ is a standard Brownian Motion. Is $X_t$ an Itô process or a ...
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2answers
139 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)...
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3answers
139 views

How to calculate standard deviation of continuously compounded four-year stock returns?

Currently I am preparing for quant interview and I encounter the following question in Heard on the street. Question: If the standard deviation of continuously compounded annual stock returns is $...
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4answers
19k views

Integral of Brownian motion w.r.t. time

Let $$X_t = \int_0^t W_s \,\mathrm d s$$ where $W_s$ is our usual Brownian motion. My questions are the following: Expectation? Variance? Is it a martingale? Is it an Ito process or a Riemann ...
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1answer
115 views

Invariance Scaling of Brownian Motion

Prove $\frac{1}{\sqrt{t}}\log\left(\int_0^t \exp(B_s)\mathrm{d}s\right)$ converges to $\sup\limits_{t\in [0,1]}B_t$ in distribution as $t\to\infty$. I have a sense to use scaling invariance, but no ...
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1answer
72 views

Mathematical proof of $g = \mu - \frac{\sigma^2}{2}$ relationship between CAGR and average returns

I found in a paper the relation between the CAGR and the arithmetic average of returns to be $$g \sim \mu - \frac{\sigma^2}{2}$$ where g is the geometric average, $\mu$ the arithmetic average and $ ...
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1answer
74 views

Integrating Brownian Motion [closed]

I just wonder how to integrate standard Brownian motion on time interval $(t, T)$. Let $Z$ be a standard Brownian motion with mean $0$ and standard deviation $1$, with $dZ^2 = dt$. How to derive the ...
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1answer
37 views

Moments of discrete Asset Price Model

Say if B is standard Brownian motion then: $S(t) = S0e^{((𝜇- σ^2)/2)t+σB(t)}$ The mean of this SDE would be $𝐄[𝑆(𝑡)]=𝑆_0𝑒^{𝜇𝑡}$ I know to do this you use the density function and ...
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2answers
137 views

Understanding $N(d_1)$ and $N(d_2)$

Firstly, if the solution to geometric Brownian motion is $S_t = S_0 \exp((r-\sigma^2)t + \sigma W_t$ then if I have a payment that is not necessarily a full call option e.g. if the exercise price $K$ ...
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0answers
30 views

Risk neutral measure in the binomial approximation of geometric Brownian motion

Suppose an asset is described by geometric Brownian motion with a drift, i.e. $$dS_t = S_t\mu dt + S_t \sigma dW_t$$ for a Wiener process $W_t$ and $S_0=1$. By some consequence of Girsanov's theorem (...
5
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1answer
84 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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0answers
39 views
2
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1answer
86 views

What is the annualized realized volatility of simulated Brownian motion paths?

I saw this following question in an exam. Take a Brownian motion simulation with drift 5% and annualized volatility of 20% for a period of 1 year. Then the annualized realized volatility of the ...
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0answers
32 views

Negative drift when calibrating GBM parameters

Setup for question: Consider a basket of $N$ stocks $\{S^1, S^2, \dots, S^N\}$. For fixed strike $K$, each stock in the basket, $S^i$, follows the SDE $$dS_t^i = \mu^i(t) S_t^i dt + \sigma^i(K, t) ...
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2answers
841 views

Risk neutrality correction for Monte Carlo Bootstrapping according to PRIIP regulation for products of category III

The PRIIP (packaged products) regulation prescribes Monte Carlo bootstrapping simulation for calculation of VaR for products of category III (non-linearly leveraged products). The idea is based on ...
2
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1answer
65 views

Valuation of Cash-Or-Nothing option

Studying options pricing, I'm stuck with the following problem: The price of a stock is described by the dynamic: $$dS_t = \mu\, dt + \sigma\,dW_t$$ Compute the fair price of a Cash or Nothing ...
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2answers
280 views

Probability that the price of stock following a brownian motion goes under a certain value

The price of the stock XYZ follows a brownian motion pattern with starting price = 10, μ = 0 and σ = 20 (on annual basis). What's the probability that in 6 months the price is less or equal to 8? ...
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1answer
43 views

In search of double barrier out option on a BM

We have a BM $X_t$ with $dX_t=\sigma dB_t$ ($X_0$ not necessarily zero!) under the risk neutral measure $\Bbb Q$. Given upper barrier $U$, lower barrier $L$, "strike" $K$ such that $L<X_0<U, L&...
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1answer
85 views

Proof standard Brownian Motion under change of measure

Let's split the usual time horizon $[0,T]$ like $0=T_{0}<T_{1}<\dots<T_{n}=T$ and consider the bond price $P(t,T_{i})$ for $i=1,...,n$. We assume $$\frac{dP(t,T_{i})}{P(t,_{i})}=r_{t}dt+\xi_{...
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2answers
259 views

What are some examples of non-solvable SDE where Monte Carlo discretization is necessary

Reading Glasserman - "Monte Carlo Methods in Finance" it says in the introduction to Chapter 6 - Discretization Methods, that moste models arising in derivatives pricing can be simulated only ...
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74 views

How to solve these SDE Problems

Quuestion1. I make a solution $r(t)$ used by Ito's lemma $r(t)=e^{-a t}r(0)+\int _{0}^{t}e^{a (s-t)}\theta (s)ds+\sigma e^{-a t}\int _{0}^{t}e^{a u}\,dB^{1}(u)$ Is this right? and I try to make ...
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1answer
148 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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1answer
110 views

Brownian Motions theorems

I know that if $W$ and $W′$ are two independent brownian motions, then $dWt \ dWt′$ = 0. How can I prove/demonstrate this theorem? Additionaly, how can we prove that if $W$ and $W′$ are dependent, ...
4
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1answer
459 views

Ito`s Lemma problem

Can someone help me with calculus for this problem. I have these 3 equations and with Ito`s Lemma I have to find $dXt$. \begin{cases} dY= μYdt+σYdB \\ X=\frac{1}{2}cY\\ dc =-aαcdt\end{cases}
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0answers
92 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 ...
1
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0answers
45 views

Brownian motion from price-series, what is the time step?

If I assume a given empirical price-series is a brownian motion, I can estimate the drift and standard deviation as long as I know what the time step was when the process was 'generated'. But since ...
7
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1answer
167 views

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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0answers
86 views

The conditional expectation of a geometric brownian motion

In this question it states that $$\mathbb{E}[e^{\sigma(W_t-W_s)}|\mathcal{F}_s] = \mathbb{E}[e^{\sigma(W_t-W_s)}],$$ and I assume that $0 \leq s \leq t$. The accepted answer states that this step is ...
3
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2answers
129 views

Find the brownian motion associated to a linear combination of dependant brownian motions

I have $N$ correlated standard one-dimensional Brownian motions $W_1,\ldots,W_N$ with correlation matrix $\rho$ and I consider the process $Z_t \equiv \sum_{i=1}^N \mu_i (t) W_t$ where the $\mu_i$ are ...
4
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1answer
116 views

Geometric Brownian Motion unable to model / predict jumps

In my finance course, we were talking about the flaws of modelling Stock Prices with the geometric Brownian Motion. According to my professor: "Since the geometric Brownian Motion has continous time ...
2
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0answers
56 views

Novikov condition for Vasicek process

Suppose that we have a money account $S^{(0)}$ with dynamics \begin{align} dS^{(0)}_{t} = r_{t} S^{(0)}_{t}\, dt, \end{align} where \begin{align} dr_t = a(b-r_t)\, dt + \sigma_{r} \, dW_t^{(0)}. \...
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1answer
128 views

Bitcoin dynamics - C++ Simulation

I would like perform a simulation of Bitcoin future prices given a sample of the 4 past years (2014-2018). My problem is that I do not know what model to use! For common stocks I used the geometric ...
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0answers
80 views

Distribution of portfolio values with constant spending rate

If your portfolio is invested in an asset that follows a geometric Brownian motion, and you withdraw a constant dollar amount at the beginning of each year, is there an approximate analytical ...
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1answer
65 views

Expectation and variance of standard brownian motion

Assuming that the price of the stock follows the model $ S(t) = S(0) exp ( mt − (σ^2/ 2) t + σW(t) ) , $ where W(t) is a standard Brownian motion; σ > 0, S(0) > 0, m are some ...
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0answers
64 views

Brownian motion for modelling future asset values

Assume that an asset price $S$ is given by a Brownian motion. Argue from the definition why it is not possible to predict future values of the asset based on the past values of $S$. I am not sure ...
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0answers
167 views

Geometric Brownian Motion with Dividends

I am working on a problem and had a quick question. I understand that for Geometric Brownian Motion we use the formula: $$X_{t_n} = X_{t_{n-1}} + \mu X_{t_{n-1}} \Delta t + \sigma X_{t_{n-1}} \...
3
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1answer
219 views

Why is it more accurate to simulate ln(S) rather than S?

Let's take a process $S$ that satisfies: \begin{equation} dS = \mu S dt + \sigma S dz \end{equation} with $dz$ a Wiener process, $\sigma$ the volatility of $S$, $\mu$ the expected return of $S$. From ...
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0answers
103 views

For an Ito Process, $d\ln{X} \neq \frac{dX}{X}$ and $(d\ln{X})^2 = (\frac{dX}{X})^2$, but $d\ln{X} \neq \pm \frac{dX}{X}$

In normal calculus we can write $d\ln{x} = \frac{dx}{x}$ since there is no quadratic variation to deal with. This isn't true for stochastic processes, and Ito's Lemma is used to calculate $d\ln{X}$. ...
1
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1answer
37 views

CDF&density of stock price modeled by standard brownian motion

Assume that the price of the stock follows the model $S(t) = S(0) exp ( mt − ((σ^2)/2 ) t + σW(t) )$ , (1) where W(t) is a standard Brownian motion; σ > 0, S(0) > 0, m are some constants. Derive the ...
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2answers
113 views

How to numerically simulate exponential stochastic integral

For example given an integral $$ \int^t_0 \exp(aW(t'))\,dt', t\in\mathbb R_+ $$ where $W(t')$ is a standard Wiener process. I've been very confused about stochastic integrals like $\int^t_0 W(t')\,...
2
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1answer
81 views

If S(t) is geometric Brownian motion, what is the distribution of S(t+h)-S(t)?

Suppose we have a geometric Brownian $S(t)$ which follows a lognormal process. Say $$ \begin{equation} dS_t = \mu S_t dt + \sigma S_tdW_t \end{equation} $$ My question is what is the distribution of $...
8
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2answers
9k views

Two correlated brownian motions

Is it true (see here, footnote 2, p.22 / p.14, without proof) that we can obtain two discretized brownian motions $W_t^1, W_t^2$ with correlation $\rho$ by doing $$d W_t^1 \sim \mathcal N(0,\sqrt{dt}...
1
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1answer
161 views

Correlated stock prices and geometric Brownian motion

I have two uncorrelated stocks which follow geometric Brownian motion, as follows $$\begin{aligned} dS_a &= \mu_aS_adt + \sigma_aS_adW\\ dS_b &= \mu_bS_bdt + \sigma_bS_b dW \end{aligned}$$ ...
4
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1answer
181 views

Ito's Lemma for this problem

I'm attempting to prove a lemma from a paper, in the context of optimal contracts. $r,\rho,\gamma,\alpha,\sigma$ are all known constants. $dR_t = (\alpha + r)dt + \sigma dZ_t$ where $Z_t$ is a ...
10
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4answers
5k views

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 ...