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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Simulating Stock's close, high and low prices

I am testing a model in which I need to simulate closing, high and low prices (i.e. 3 dimensions of prices) of any given stock. Using the simple Geometric Brownion Motion equation I can easily ...
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180 views

Geometric Brownian Motion and Energy-Efficiency Investments

Suppose the payoff $X$ on an investment follows a Geometric Brownian Motion: $$ dX/X = \mu dt + \sigma dz\ , $$ for $dz$ an increment of a Wiener process. I wish to compute the expected present value ...
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53 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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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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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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751 views

Properties of Geometric Brownian Motion Integrated w.r.t. Time (i.e., distribution of a Yor Process)

Let $S_t$ be a process which follows a Geometric Brownian Motion: $\frac{dS_\tau}{S_\tau} = \mu \,d\tau + \sigma \,dW_\tau$ By Ito's lemma, we have: $S_T = S_t e^{(\mu-{\sigma^2 \over 2})(T-t) + \...
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1k views

Standard definition of multidimensional Brownian Motion with correlations

I was wondering was the standard definition of a multi-dimensional Brownian motion is. For one-dimension, I consider the following the standard definiton. Brownian motion (or a Wiener process) is a ...
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42 views

Polynomial interpolation of corrected lognormal distribution

Can anyone provide a formula for a polynomial interpolation of the corrected lognormal distribution used to model returns traditionally resulting from the wrong Brownian motion generated model? ...
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2answers
2k views

How to discretize a GBM under P- and Q-measures?

Under the P-measure, a geometric Brownian motion can be specified using the following SDE: $$dS_t=\mu S_tdt+\sigma S_tdW_t^P$$ and its Euler discretization is $$S_{t+\Delta t}=S_t + \mu S_t \Delta ...
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490 views

Is Geometric Brownian Model suitable for long term price forecast?

I was thinking of using Geometric Brownian Motion to forecast future prices of timber (say one variable, the stumpage price of sawtimber). I tested the time series with Augmented Dickey-Fuller test ...
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752 views

Is the Brownian motion multiplication rule a definition or is it a theorem?

Is the Brownian motion multiplication rule a definition or is it a theorem? Refer to the highlight part of http://i.stack.imgur.com/doQuT.png where $dw_1(t)dw_1(t)=dt$
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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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148 views

Itos Lemma Derivation notation

So in Hull (2012) the main point is that $\Delta x^2 = b^2 \epsilon ^2 \Delta t + $higher order terms$ $ has a term of order $\Delta t$ and can not be ignored as the Brownian motion exhibits the ...
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525 views

Ito Integral of functions of Brownian motion

How does one show that: $$ \mathbb{E}\left[ \int f(W_s)dWs \right] = 0 $$ For all $f()$ that are powers of $W(s)$?? I assume that one would have to go via the definition of Ito integral and express ...
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232 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')\,...
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275 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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868 views

The Distribution of Future Stock Price

In Hull, we are presented that $$\frac{\Delta S}{S_{0}}=\mu \Delta t+\sigma\sqrt{\Delta t}\cdot \varepsilon.$$ Following some algebra, $$ \begin{align*} \frac{\Delta S}{S_{0}} &=\mu \Delta t+\...
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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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244 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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140 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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885 views

Why is rate of return on the stock normally distributed under GBM?

Let us assume the geometric Brownian motion, and we have $$dS_t= uS_tdt+\sigma S_tdz,$$ and $S_t$ follows a log-normal distribution, but why is $r_t$, the continuously compounded rate of return, ...
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1answer
453 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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273 views

Is this a poorly written example, or could volatility in fact be negative?

I'm self-studying and I encountered the following example. It seems to suggest that volatility is negative in this example. I was under the impression that volatility can never be negative, both from ...
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752 views

Unique risk neutral measure for Brownian Motion

For a standard geometric Brownian motion model of stock prices: $$ dS = a S dt + \sigma S dZ$$ we can transform the process to be under risk neutral measure: $$ dS = r S dt + \sigma S d \tilde{Z}$$ ...
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417 views

Exchange rate model and Martingales

In exchange rate model explanation, "...If under the domestic risk neutral measure $Q_d$, the process $X(t)$ satisfies $\displaystyle \frac{dX(t)}{X(t)}=\sigma dZ_d(t)$ Since $Z_d(t)$ is $Q_d$-...
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166 views

Expectation and variance of $\int_0^t (W_s)^n ds$ for any positive integer $n$?

It is well known that the integral $$\int_0^t W_s ds,$$ where $(W_s)_s$ is a Brownian motion, can be derived using Ito's Lemma. More precisely, Ito's lemma on $d(tW_t)$ implies that $$d(tW_t) = ...
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132 views

How to find the transition distribution functions of these two processes?

What are the transition distribution (or density) functions of processes defined by $dX_t=\mu dt +\sigma dW_t$ and $dX_t= \theta(\mu-X_t) dt +\sigma dW_t,$ where $\theta>0$, $\mu$ is a real ...
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1answer
260 views

How to show that $E\left[ \int_0^t \sigma(s) e^{iuX(s)} dW(s)\right] = 0$?

Let $\sigma(t)$ be a given deterministic function of time and define the process $X_t$ by $$X(t) = \int_0^t \sigma(s)dW(s)$$ I want to show $$E\left[ \int_0^t \sigma(s) e^{iuX(s)} dW(s)\right] = 0$$...
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196 views

example Hamilton-Jacobi-Bellman Equation - clarification of $dX_t$ derivation using $\pi_t$, $\Pi_t$

I have a market with safe rate r and risky asset S $$ \frac{dS_t}{S_t}=(r+Y_t)dt+\sigma dW_t \quad \quad (1)$$ $$ dY_t = - \lambda Y_t +dB_t \quad \quad (2)$$ where W, B are Brownian Motions with ...
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1answer
74 views

Can the differential operator be removed to get the mean/variance of an Ito process?

If $X_t$ is an Ito process, such that: $dX_t = \mu(t, X_t)dt + \sigma(t, Xt)dW_t$ where $W_t$ is a standard brownian motion. Then we can say that: $E(dX_t) = \mu(t, X_t)dt$ and that $Var(dX_t) = \...
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85 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
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 ...
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1answer
114 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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186 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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492 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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1answer
328 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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2answers
185 views

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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1answer
111 views

Explanation on the application of CLT in bionomial tree model

We have a stock price binomial tree model of $n$ steps, with step length $\Delta t=T/n$, stock price volatility $\sigma$ s.t. $u_n=e^{\sigma\Delta t}$ and $d_n=1/u_n$, and the risk neutral probability ...
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2answers
308 views

Ito calculus problem

given $S^1$ satifying the SDE $\quad dS_{t}^{1}=S_{t}^{1}((r+\mu)dt + \sigma dW_t), \quad S_{0}^{1}=1 $ and the safe asset $S_{t}^{0}$ $\quad S_{t}^{0}:=e^{rt} \quad for \quad r\geq 0$ Q1. how ...
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107 views

Meaning of w in SDE

I'm missing meaning of $w$ in typical SDE like $dX_t(w) = f_t(X_t(w)) + \sigma(X_t(w))dW_t$, in context of $w \in F_{xxx}$. Does it mean that both $w$ is one of events that could happen before ...
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2answers
176 views

Statistics of difference between two GBMs

if I have two asset prices modeled separately as geometric brownian motions. How do i go about calculating the expected statistics of their difference? Like given the sigmas and mus of both processes, ...
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1answer
128 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 $...
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1answer
186 views

Dynamical Behavior of Hurst Exponent

I feel that the dynamic of financial market is not really modeled by standard Brownian motion, but fractional Brownian motion or even multifractional Brownian motion. I have read some references on ...
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1answer
84 views

Spot Interest Rate at time $t$

I know that the general model for the dynamics of the spot interest rate is $$dr(t)=\mu(r,t)dt+\sigma(r,t)dB(t)$$ My question is, if $P(t,T)$ is the bond value at time $t$, how would I derive $dP$?
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1answer
233 views

Brownian motion. Solve stoc. integral by using Ito's lemma

I want to show that following statement is true by using Ito's lemma to solve stochastic integrals: I define the functions in Ito's model: a()=0, b()= (2wt-2)^2. f(t)=Integrate[(2wt-2)^2] Then df=(b^...
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146 views

Optional Sampling Theorem Application

Let x, y > 0. Defint eh first passage time of a Brownian motion $W_t$ as $\tau_a$ = min{t $\ge$ 0: $W_t$ = a}. I need to show that E[$e^{-u\tau_x}$$1_{\tau_x < \tau_{-y}}$] = $\frac{sinh(y\sqrt{2u})...
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1answer
77 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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1answer
133 views

Why it's related to stock price

I am reading a paper High-frequency trading in a limit order book by Avellaneda and Stoikov. I verified the formula (6) should be correct. However it doesn't make sense to me when I use it for ...
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1answer
244 views

Modeling the Stock Market [closed]

Hi I was wondering what is the model that best describes the price movement of the stock market? A Brownian motion Process with drift? An Ornstein Uhlenbeck_process? (where the long term mean is ...
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1answer
101 views

investor terminal value of portfolio with two risky assets 1) correlated 2)not correlated $\phi_t^1=S^{2}_{t}, \ \phi_t^2=S^{1}_{t}$

I am analyzing a problem where I have two stocks described by the equations $$ \frac{dS^{1}_{t}}{S^{1}_{t}}=\mu_{1} dt + \sigma_{1} dW^{1}_{t}$$ $$ \frac{dS^{2}_{t}}{S^{2}_{t}}=\mu_{2} dt + \sigma_{2}...

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