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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235 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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150 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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96 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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135 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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126 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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1answer
250 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
102 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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1answer
4k views

Geometric Brownian Motion - Why Sqrt(dt)? [closed]

I was going to simulate a geometric brownian motion in matlab, when I recognized that I didnt fully understand the underlying Wiener process. Following the instuctions here I am starting from the ...
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1answer
315 views

Why Variations of order higher than two vanish for Brownian motion?

Let $W_{t}$ be a Brownian Motion. Verify that variations of Brownian Motion of higher order, say, of order three, vanishes. I try to prove that $\lim_{n\rightarrow\infty}\sum^{n}_{i=1}(W_{t_{i}}-W_{...
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Are my estimates of parameters of geometric brownian motion correct?

I wrote a simulation of a geometric Brownian motion which works like this: ${ t }_{ i }-{ t }_{ i-1 } \sim Exp(\lambda )$ ${ Z }_{ i }\sim N(0,1)$ ${ Y }_{ i }\sim { e }^{ \sigma \sqrt { { t }_{ i }-{...
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Relationship between risk and return for GBM and riskless bond

Suppose we have $S$, a stock following geometric Brownian motion ($dS_t = S_t (\mu dt + \sigma dZ_t)$ for $Z =$ Brownian motion) and $B$, a zero coupon bond with rate $r$, i.e. $dB_t = rB_t dt$. In ...
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95 views

Estimating constant and local volatility based on passage times

Consider a Brownian motion B_t with constant instantaneous volatility σ and zero drift where ...
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66 views

Theoretical Expected Maximum Drawdown vs Empirical Maximum Drawdown

I have been looking at the approach for calculating the expected maximum drawdown of a Brownian Motion [1] and the corresponding function maxddStats in the fBasics package in R [2]. I do not ...
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77 views

law of absolute of max of brownian motion

What is the law of $\max\left(|B_t|\right)$ for $t$ in $[0,T]$ and $B_t$ is a Brownian motion? Any references for properties of this process?
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70 views

Brownian function and Clark's formula

I was reading a paper (link) from Richard Bass about Brownian functionals, and came across the following passage : Let $X_t$ be a Brownian motion, $F_t$ its filtration and $g$ a real-valued function. ...
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51 views

How to calculate the multiple integrals where the integral domain is based on the sum of normal distribution random variables?

The integral is shown below: And how to use python to calculate pi (better if we don't need to code for each pi)?
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66 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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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 $d\ln{X}$. ...
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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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263 views

What is a stochastic processes which reasonably captures commodity price dynamics?

I ran into a stumbling block earlier when I tried to price stochastic annuities (see Asian options). This is actually technically an acturial problem, but is well adapted to the techniques of quant ...
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1answer
82 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
834 views

Instantaneous Volatility Estimator

Suppose a Stock follows an Itô process with instantaneous volatility $\sigma(S(t),t)$. Precisely $$dS(t)=\mu S(t)dt+\sigma(S(t),t)S(t)dW(t)$$ I have a historical data for the values of $S(t)$.How ...
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222 views

2 Ito processes - $d(X_{t} + X^{'}_{t})^2 = (Y_t + Y^{'}_{t})^2 dt$ why it is true?

Having two Ito processes $dX_{t} =z_{1} dt + Y_{t} dB_t $ $dX^{'}_{t} =z^{'}_{1} dt + Y^{'}_{t} dB_t $ I am analyzing a proof of the product rule $d(X_t X_t^{'})=X_t dX_t^{'}+ X_t^{'} dX_t + Y_t ...
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2answers
483 views

How to compute $\mathbb{E} \left[ (W_s + W_t - 2W_0)^2 \right]$?

The solution to the SDE $$dx_t= -kx_t dt + cx_t dW_t$$ is $$x_t = x_0 e^{\left(c - \frac{k^2}{2} \right)t}e^{-k W_t}$$ with mean $$\mathbb{E} \left[ x_t \right] = x_0 e^{\left(c - \frac{k^2}{2}...
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2answers
92 views

Proof of Feller condition for CIR square root process. Any reference?

Could you please give me some reference for the proof of the so-called Feller condition as to a stochastic differential equation of the form: $$dr_t=a(b-r_t)dt+\sigma\sqrt{r_t}dB_t\tag{1}$$ with $\...
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2answers
270 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
965 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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1answer
253 views

Measure of a Brownian motion = normal distribution?

Consider some model where the process increments are normally distributed, e.g. Vasicek: $$dr(t) = \left(\theta - ar(t)\right)dt + \sigma dW(t).$$ We usually say that $W(t)$ is a Brownian motion ...
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1answer
434 views

Stochastic differential equation of a Brownian Motion

I have two questions about Ito's Lemma with respect to calculating SDEs. The examples are simple enough, but I haven't found an answer yet. Take $W_t$ as a standard Brownian motion and $g(s)$ as some ...
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2answers
311 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
90 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
97 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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1answer
261 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
1k views

Simulation of the geometric Brownian motion under risk-neutral measure

I hope you can help me again. It is clear how to simulate the GBM: $S_{t_{k}}=S_{t_{k}}exp[(\mu-\frac{\sigma^2}{2})\Delta t_{k+1}+\sigma\sqrt{\Delta t_{k+1}}Z]$, where Z is a stand. norm. dis. RV. ...
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1answer
493 views

Given Brownian motion $B_t,B_s$ and $t>s$, how to calculate $P(B_t>0,B_s<0)$?

As stated, this is an interview question. Given Brownian motion $B_t,B_s$ and $t>s$, how to calculate $P(B_t>0,B_s<0)$?
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2answers
500 views

How to fix my Ornstein-Uhlenbeck parameter MLE in Python?

I am trying to fit time-series data into an Ornstein-Uhlenbeck process. Here is my code so far: ...
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1answer
89 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
123 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
698 views

How to calculate mean and volatility parameters for Geometric Brownian motion?

Say I have a time series $S_K$ for monthly asset prices for the last 30 years. I want to run a monte carlo simulation using geometric brownian motion $$S_t = S_0\exp\left(\left(\mu - \frac{\sigma^2}{...
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1answer
177 views

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
78 views

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
2k views

Partial derivative of an integral

Suppose I have a model for the short rate $r$ as ($W(t)$ is standard Brownian motion) $r(t) = c+ \int_0^t \sigma (s) ^2 (t-s) ds+ \int_0^t \sigma (s) dW(s)$ I then want to find the dynamics of $r$, ...
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2answers
3k views

Confidence Intervals of Stock Following a Geometric Brownian Motion

In preparation for my Options, Future's and Risk Management examination next week, I have been presented with a series of questions and their answers. Unfortunately, my lecturer, one of the less ...
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2answers
479 views

For the Dothan model $E^Q[B(t)]=\infty$?

How can I show that for the Dothan short rate model We have $E^Q[B(t)]=\infty$ ? Where Dothan short rate model is " $dr_t=ar_tdt+\sigma r_tdW_t$ ". I appreciate any help. Thanks.
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1answer
142 views

Differenced Brownian Motion covariance

I am having some difficult showing what the following equals, where $x$ and $y$, $x>y$, distinct times: $\mathbb{E}[\Delta W_x \Delta W_y]$ where each $\Delta W_t = W_t - W_{t-1}$. I have ...
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2answers
152 views

Integral of the square of Brownian motion using definition of variance

Let $B = \{ B(t); t \ge 0\}$ and let $Z = \{ Z(t); t \ge 0 \}$ where $$Z(t) = \int_0^t B^2(s) ds.$$ How do we find $E[Z(t)]$ and $E[Z^2 (t)]$ in order to get the variance $Var [Z^2(t)] = E[Z^2 (t) ] -...
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1answer
301 views

If $W_t$ is standard Brownian motion, what is $\int_0^T W_t \ln(W_t) dW_t$?

If $W_t$ is standard Brownian motion, what is meant by $\int_0^T W_t dW_t$ in finance? Furthermore, what then is the meaning of $\int_0^T W_t \ln(W_t) dW_t$?
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1answer
69 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
63 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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1answer
107 views

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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