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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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60 views

Solving for roots of a stochastic pay-off function

I have a pay-off function for a derivative which is defined by the Heaviside difference between $G$ and $B$ shifted by $-F$. To find the value of $V_{t=0}$, I need to find $\tau$ when $\frac{dV}{dt} = ...
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1answer
440 views

Modelling returns in the real world measure with or without drift

What I would like to discuss is the following. I don't think that this is a pure duplicate, so I would be happy about comments: On one hand it is reasonable to model log-returns as Gaussian: $$ \log(...
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1answer
178 views

Perform scipy Kolmogorov-Smirnov Test for lognormal distribution in GBM

I am simulating asset prices for n days using GMB with Euler scheme, calculate returns and then perform Kolmogorov-Smirnov test on simulated returns. Code for simulating GBM : ...
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3answers
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Simulate correlated Geometric Brownian Motion in the R programming language

In response to this question: How to simulate correlated Geometric brownian motion for n assets? One of the responses provides an implementation in MATLAB: http://www.goddardconsulting.ca/matlab-...
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1answer
14k views

How to simulate correlated Geometric brownian motion for n assets?

So I'm trying to simulate currency movements for several currencies with a given correlation matrix. I have the initial price, drift and volatility for each of the separate currencies, and I want to ...
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1answer
59 views

Why changing measure is necessary? [closed]

I want to understand the logic for why this is: We have our model for the stock price behaviour: $$d{S_t} = \mu {S_t}dt + \sigma {S_t}d{\tilde W_t}$$ It describes the development of a stock price ...
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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
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On the application of Itos lemma to Geometric Brownian motion [closed]

I recently read this from a book: The canonical SDE in financial math, the geometric Brownian motion, ${{d{S_t}} \over {{S_t}}} = \mu dt + \sigma d{W_t}$ has solution $${S_t} = {S_0}{e^{(\mu -...
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1answer
104 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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1answer
236 views

Will volatility smoothing effects exist for returns driven by geometric brownian motion?

Say I randomly simulate a one-year pathway of 252 prices, where the underlying price model is driven by geometric brownian motion. where $t = (1 / 252)$, $mu = 5$% and annual $st.dev = 10%$%. My ...
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2answers
199 views

Stochastic process and brownian motion

I just read the following and i am having some difficulty to interpret it: We begin our analysis in the standard Black-Scholes world consisting of a bank account process of price denoted by $B_t$, ...
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1answer
72 views

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

This question was asked by another user, but was deleted. As it may be useful for others, I re-post it here. What are the transition distribution (or density) functions of two processes defined by \...
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1answer
125 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
592 views

What is the probability that a Brownian Bridge hits an upper barrier $U$ before a lower barrier $L$?

The probability that an arithmetic Brownian motion process $dt = \mu dt + \sigma dW$ hits an upper Barrier $U$ before it hits a lower barrier $L$ is given by $$ \mathbb{P}(\tau_U\leq \tau_L) = \frac{\...
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1answer
1k views

Geometric Brownian Motion: percentage returns vs log-returns

In classical calculus, we know that the limit of percentage return (ie $dS/S$) equals that of the log return (ie. $dln(S)$ ). With uncertainty, we rely on Ito Lemma to draw a relationship between the ...
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1answer
2k views

Given $S$ is a Geometric Brownian Motion, how to show that $S^n$ is also a Geometric Brownian Motion?

Suppose that a stock price $S$ follows Geometric Brownian Motion with expected return $\mu$ and volatility $\sigma:$ $$dS = \mu S dt +\sigma S dz$$ How to find out the process followed by variable $...
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4answers
250 views

Correlation of Asynchronous Brownian Motion

I am trying to use the closing prices of the S&P 500 and the Nikkei Index to see how they are correlated (assuming they are exactly 12 hours apart). In order to test my method, I have generated ...
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6answers
3k 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
108 views

Pricing Secured Barrier Call

A European barrier call with barrier $B = 50$, expiration $T = 31$, and strike $K = 33$ costs $12$. The investor is interested in a product that, unlike this barrier call, offers some protection for ...
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0answers
72 views

Pricing defaultable asset with finite maturity

Assume a stochastic process $X_0 = 0$ and $X_t = \nu t + \sigma W_t$ where $W_t$ is standard Brownian motion and $\nu$ is a drift (can have $\nu \leq 0$ if necessary, but prefer it to be general), ...
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1answer
3k views

Can I always use quadratic variation to calculate variance?

Suppose we have a Brownian Motion $BM(\mu,\sigma)$ defined as $X_t=X_0 + \mu ds + \sigma dW_t$ The quadratic variation of $X_t$ can be calculated as $dX_t dX_t = \sigma^2 dW_tdW_t = \sigma^2 dt$ ...
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1answer
194 views

Determine the conditions for Brownian motion to be a Martingale

Let $W_T$ denote normalised univariate Brownian motion and let $X_t = W_t^2 + \alpha W_t + \beta t + \gamma$ where $\alpha, \beta$ and $\gamma$ are constants. Determine conditions on these constants ...
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1answer
245 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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1answer
604 views

Matlab implementation for modelling stock price process

I am trying to model the stock's price process. Let's assume volatility and risk-free rate is given. I've come up with the code below to try and model the price process with the geometrical Brownian ...
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2answers
1k views

Black Scholes in Practice: Delta Hedging

From the Wikipedia page, we know call option as an example is price through delta hedging. $$\Pi=-V+V_SS$$ and over $[t,t+\triangle t]$ $$\triangle\Pi=-\triangle V+V_S\triangle S$$ My questions ...
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1answer
636 views

Bond price and its process

Suppose that x is the yield to maturity with continuous compounding on a discount bond that pays off $1 at time T. Assume that the x follows the process $dx=a(x_0-x)dt + sxdz$ where $a, x_0$ and $s$ ...
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2answers
168 views

Moment Ito's Process Proof

I have a following stochastic integral - related problem that I have difficulty to solve: Given \begin{equation} dX_t = -\alpha X_tdt+\sigma\sqrt{X_t}dW_t \end{equation} and the second moment of $...
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1answer
82 views

Soft: Interpretation Fractional BM in finance

Suppose we are in the BS framework. If we replace the Brownian Motion with a more general fractional Brownian motion therein, how can it be interpreted? That is what is a financial interpretation of ...
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1answer
266 views

How to prove we have a $\mathbb{Q}$-Brownian motion?

Background Information: This question comes from the book Financial Calculus by Baxter and Rennie. WE start with looking at the marginal of $W_T$ under $\mathbb{Q}$. We need to find the likelihood ...
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1answer
76 views

Do we have a Brownian motion

Background Information: The process $W = (W_t:t\geq 0)$ is a $\mathbb{P}$-Brownian motion if and only if i) $W_t$ is continuous, and $W_0 = 0$ ii) the value of $W_t$ is distributed, under $\mathbb{...
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2answers
1k views

Brownian Bridge's first passage time distribution

Let's say we have a Brownian Bridge $Y_{b,T}(t)$ such that $Y_{b,T}(0)=0$, $Y_{b,T}(T)=b$. Let's say we are interested in the first passage time of $Y_{b,T}(t)$ at level $b$: $\tau_b = \{\min \tau; ...
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1answer
310 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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0answers
112 views

Brownian bridge density and risk neutral density for derivative pricing

The book The Volatility Surface by Gatheral (2006) introduces the Brownian bridge like density $q(x_t,t;x_T,T)$ of $x_t$ conditional on $x_T = log(K)$. Can we use $q(x_t,t;x_T,T)$ as the risk neutral ...
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0answers
109 views

Smooth ornstein uhlenbeck process

I want to simulate paths for a commodity price. I use the historic data in the following way: $X_t$ is the price. $\ln\left(\frac{X_t}{X_{t-1}}\right)$ is the daily return. I calculate the slope of ...
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1answer
1k 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
94 views

How to show the equality of these two events?

For $X_t$ a brownian motion defined for $t\in[0,T]$, How to show the equality of following events: $$ \{ \displaystyle \max_{[\tau,T]}(X_t)\ge 2u-d, \tau\leq T \}=\{\displaystyle \max_{[0,T]}(X_t)\ge ...
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1answer
50 views

$P(S_T > S_u \mid S_v = s_*)$

Let $u < v < T$ and assume $S_t$ follows a lognormal $((\mu - \sigma^2/2)t, \sigma^2 t)$ process. I'm interested in computing the conditional probability $$ P(S_T > S_u \mid S_v = s_*) $$ ...
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1answer
142 views

Integral with respect of $(dW_s)^n$

I know $$\int _0^t dW_s=W_t-W_0=W_t$$ Since $ dW_s dW_s=ds$ , so $$\int _0^t( dW_s)^2=\int_0^t ds=t-0=t$$ I Want to know why for $n\ge 3$ we have $$\int _0^t (dW_s)^n=0$$ My try $$(dW_s)^2 dW_s (...
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1answer
184 views

Discrete Time to Continuous Time and Summation of Two Geometric Brownian Motions

Could someone please suggest with detailed steps and/or a reference, 1) How to convert the below discrete time summation to continuous time form and write it as an integral? 2) Any methods to ...
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1answer
190 views

On the reflection of a stochastic integral

Let ${(I_t)}_{t\geq 0}$ be a stochastic integral defined by $$ I_t=\int_{0}^{t}\theta_sdW_t, $$ where $W$ is a standard Brownian motion defined on $(\Omega,\mathcal{F},{(\mathcal{F}_t)}_{t\geq 0},\...
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1answer
677 views

What is the distribution of Brownian Bridge over a given time interval?

I know from Karatzas & Shreve (1991) that a Brownian Bridge $B(t)$ from $a$ to $b$ on time interval $[0,T]$ satisfies: $$B(t)=a(1-t/T) + b*t/T + [W(t) - W(T)*t/T]$$ where $W(t)$ is a standard ...
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1answer
938 views

What's the variance of this Ito integral?

I am reading stochastic calculus and I have understood that the process $$X=\int_{0}^{1}\sqrt{\frac{\tan^{-1}t}{t}}dW_t$$ has normal distribution with mean zero. How can I find the variance of $X$?
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1answer
3k views

Correlation coeffitiont between two stochastic processes

I want to find correlation coeffitiont between $W_t$ and $\int_{0}^{t}W_s ds$. I think that these are uncorrelated. But Why? So thanks
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1answer
233 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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1answer
641 views

Given $\mathbb Q$ and $X_t$ is $\mathbb Q$-Brownian, find $\frac{d\mathbb Q}{d\mathbb P}$ / Uniqueness of Brownian or Radon-Nikodym derivative

The problem: Let $T >0$, and let $(\Omega, \mathscr F, \{ \mathscr F_t \}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathscr F_t = \mathscr F_t^W$ where $W = \{W_t\}_{t \...
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1answer
304 views

Brownian motion simulation - scaling issue

I'm trying to simulate some BM for 500 observations. I got correlated increments as I needed and they are not exactly N(0,1), so I standardize them (x-mean(x))/sd(x). But then the resulting Brownian ...
1
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1answer
197 views

Probability that return exceeds a certain level before a certain time (Black-Scholes)

I am self studying for an actuarial exam on financial economics. I encountered the following problem and solution. It seems to me that the author intended to mean what is the probability that the ...
12
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2answers
985 views

Distribution of Geometric Brownian Motion

Please let me know where I have been mistaken! Let the SDE satisfied by the GBM $S(t)$ be $$ \frac{dS(t)}{S(t)} = \mu dt + \sigma dW(t). $$ Then, the underlying BM $X(t)$ will satisfy $$ dX(t) = \...
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1answer
350 views

Probability of Brownian motion particle touching barrier given path starts at $X_0$ and ends at a known $X_t$

I have been reading Su and Rieger's paper on barriers and from there have been able to work out the unconditional probability of the process $dXt = μ dt + σ dWt$ touching a down barrier $α$ to be $\...
3
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1answer
110 views

Drift irrelevance on high frequency data

Let's assume that price of a certain asset follows Brownian Semimartingale process with a drift term and a Brownian-driven continuous part (no jumps for simplicity). In literature it is often stated ...