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1
vote
0answers
35 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}}$] = ...
2
votes
3answers
111 views

Calibration of a GBM - what should dt be?

I have a time series of daily data that I want to calibrate GBM parameters $\mu$ and $\sigma$ to. Using the discretized solution $$ S_{t_{i+1}} = S_{t_i}\exp\left(\left(\mu - ...
3
votes
3answers
169 views

Show that $E[B_t|\mathscr{F}_s] = B_s$

Given prob space $(\Omega, \mathscr{F}, P)$ and a Wiener process $(W_t)_{t \geq 0}$, define filtration $\mathscr{F}_t = \sigma(W_u : u \leq t)$ Let $(B_t)_{t \geq 0}$ where $B_t = W_t^3 - 3tW_t$. ...
3
votes
5answers
310 views

Why should we expect geometric Brownian motion to model asset prices?

Disclaimer: I am a complete ignoramus about finance, so this may be an inappropriate forum for me to ask a question in. I am a mathematician who knows nothing about finance. I heard from a popular ...
2
votes
1answer
68 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 ...
1
vote
1answer
95 views

Cholesky Decomposition on Correlation Matrix for Correlated Asset Paths

I found a matlab example for modelling correlated asset paths: http://www.goddardconsulting.ca/matlab-monte-carlo-assetpaths-corr.html In this model the author uses the matlab code chol() in order to ...
4
votes
1answer
64 views

Discounted risky asset stochastic process problem

$S_t$ is the random variable representing the risky asset price at time $t$. M_t is the riskless asset. They are governed by the equations $\frac{dS_t}{dt}=\mu dt + \sigma dZ_t$ and $dM_t = rM_t ...
3
votes
0answers
65 views

Particular Conditional Expectation of Geometric Brownian Motion

If we have the density function $$f_{Y}(y,t)=\frac{1}{y \sqrt {2\pi\sigma^2t}}exp(-\frac{(ln \ y - \mu t)^2}{2\sigma^2t})$$ Then the mean of $Y(t)=e^{X(t)}$ conditional on $Y(0)=y_0$ is found to be ...
1
vote
2answers
206 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 ...
3
votes
1answer
93 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; ...
0
votes
2answers
165 views

Getting the next price of a GBM with reversion

Here is the "twin" question of Getting the next price of a GBM (Geometric Brownian Motion) but for GBM with reversion As in that case, I'd like to write a formula for the next price, as function of: ...
5
votes
1answer
85 views

Estimating the Hurst exponent in short terms in developed markets

In the Proceedings of the Estonian Academy of Sciences, Physics and Mathematics (2003), I saw the following sentence: Surprisingly, in the case of developed markets, short-term $H$ results showed ...
3
votes
2answers
116 views

Modelling driftless stock price with geometric Brownian motion

I wish to understand some basic fact about the (primitive) simulation of stock prices with geometric Brownian motion. If $S(t)$ is the stock price at time $t$, and the stock price follows geometric ...
1
vote
0answers
84 views

Distribution of Brownian Bridge

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 ...
1
vote
1answer
110 views

FX Rate dynamics

Let's suppose USD/EUR price in USD follows a GBM with $$ dS_t = rS_tdt + \sigma S_tdW_t $$ What process does EUR/USD follow in EUR?
5
votes
3answers
179 views

Usage of Brownian Bridge?

I was recommended to read something about Brownian Bridge. Could someone familiar with BB give some recommendation? It was mentioned that BB benefits in 2 places BB could reduce the simulation ...
1
vote
1answer
86 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 ...
1
vote
2answers
92 views

Simple question about expected value of brownian motion

I would appreciate some help with the math in this paper : High Frequency Trading in a Limit Order Book Specifically, I would like to understand how the authors calculated the expected value of price ...
5
votes
3answers
265 views

Geometric Brownian motion - Volatility Interpretation (in the drift term)

A Geometric Brownian motion satisfying the SDE $dS_t = rS_t dt+\sigma S_t dW_t$ has the analytic solution $$S_t = S_0\exp\left\{\left(r-\frac{\sigma^2}{2}\right)t\right\}\exp\{\sigma W_t\}$$ Recently ...
4
votes
1answer
191 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) = ...
4
votes
1answer
65 views

Linear-Boundary Crossing Problem for Brownian Motion

This is a question I came across while reading: $W = (W_t)_{t\geq{0}}$ is a standard BM. Let $\mu\in \mathbb{R}$, and let $\tau_{a}^{\mu}$ = $\inf(t>0;W_t = a + \mu{t})$ be the first passage time ...
9
votes
3answers
2k views

Is there an intuitive explanation for the Feynman-Kac-Theorem?

The Feynman-Kac theorem states that for an Ito-process of the form $$dX_t = \mu(t, X_t)dt + \sigma(t, X_t)dW_t$$ there is a measurable function $g$ such that $$g_t(t,x) + g_x(t, x) \mu(t,x) + ...
2
votes
1answer
163 views

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 ...
3
votes
2answers
94 views

What is the difference between these two equations for GBMs?

The two equations commonly found online for GBM are: $\begin{matrix} S_{ t }=S_{ 0 }\exp\left( \left( \mu -\frac { \sigma ^{ 2 } }{ 2 } \right) t+\sigma W_{ t } \right) \\ S_{ t }=S_{ 0 ...
-1
votes
2answers
221 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 - ...
1
vote
1answer
55 views

Scaling Intervals in Diffusion Process

I know this is a very elementary question but... when modeling asset prices through a stochastic process as in $$dS_t=S_t μ dt+S_t σdW_t,$$ where the following is a wiener process ...
1
vote
1answer
156 views

Basics about the scaling property of volatility

It is a usual practice to calculate realized volatility $\sigma$ using the square root of the usual variance estimator $\hat{{\sigma}²}$. This is done using the stock log returns (practitioners ...
2
votes
2answers
198 views

Shortcomings of generalized Brownian motion for asset price modelling

I'm simply interested on hearing some views on which shortcomings arise by using the (multidimensional) SDE $$dS(t)=S(t)\alpha(t,S(t))dt+S(t)\sigma(t,S(t))dW(t)$$ as a model for asset prices. I know ...
7
votes
1answer
401 views

How to compute the Radon-Nikodym derivative?

Suppose $B(t)$ is a standard Brownian motion, and $B_{1}(t)$ is given by $dB_{1}(t)=\mu dt+dB(t)$. Suppose $P$ is the Wiener measure induced by $B(t)$ on the $C[0,\infty)$, and $P_{1}$ is the Law ...
4
votes
5answers
518 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 ...
1
vote
2answers
465 views

Brownian motion - first passage time

Can anyone point me to the expression for the first passage time for a geometric Brownian motion process X(t) as a function of the starting point, threshold, drift and diffusion parameters. I am ...
4
votes
1answer
223 views

Covariance of brownian motion and its time average

It's a question pertaining to the correlation of a log asset process (following BM) and its time average, to put it into form, if $$X(t)=\mu t+\sigma W(t)$$ then $$ ...
5
votes
1answer
3k 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 ...
1
vote
2answers
146 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, ...
5
votes
2answers
402 views

Correlation decay in lognormal distribution

I noticed that if you use two correlated geometric brownian motions, the correlation structure decays in time pretty fast even for really high correlation values. I think that is not replicating ...
5
votes
1answer
620 views

Derivation of Ito's Lemma

My question is rather intuitive than formal and circles around the derivation of Ito's Lemma. I have seen in a variety of textbooks that by applying Ito's Lemma, one can derive the exact solution of a ...
5
votes
2answers
741 views

Simulation of GBM

I have a question regarding the simulation of a GBM. I have found similar questions here but nothing which takes reference to my specific problem: Given a GBM of the form $dS(t) = \mu S(t) dt + ...
3
votes
2answers
208 views

Comparison of Brownian Motion Expected Drawdown and simulated results

Can anyone tell me whether results as predicted by Brownian Motion for a given mean and std, match what you get by measuring actual drawdown from simulated results over a number of iterations?
3
votes
0answers
95 views

Estimating two normal random numbers with one equation

Subtitle: Estimating the correlation of the shocks driving two commodities in two multi-factor models I am fitting two 2-factor models to electricity and gas futures, respectively. In order to ...
6
votes
2answers
750 views

What is the average stock price under the Bachelier model?

Let's say stock price follows following process: $$dS(t) = \sigma dW(t)$$ where $W(t)$ is Standard Brownian motion. The initial level for the stock is $S(0)$. Define the average of stock price ...
8
votes
2answers
327 views

GBM 3d plot with R

I want to plot the density of the GBM in a 3d plot. So I have on one axis the stock price, on the other the time and on the z axis the density. At the end I want to produce this graph. The formula I ...
7
votes
2answers
10k views

How to simulate stock prices with a Geometric Brownian Motion?

I want to simulate stock price paths with different stochastic processes. I started with the famous geometric brownian motion. I simulated the values with the following formula: ...
10
votes
1answer
3k views

Estimation of Geometric Brownian Motion drift

One can find many papers about estimators of the historical volatility of a geometric Brownian motion (GBM). I'm interested in the estimation of the drift of such a process. Any link on this topic ...
5
votes
3answers
573 views

What are the limitations of brownian motion in finance? [duplicate]

What are the limitations of brownian motion in its applications to finance?
4
votes
4answers
529 views

Expected Growth

The model assumption of the Black-Scholes formula has two parameters for the geometric Brownian motion, the volatility $\sigma$ and the expected growth $\mu$ (which disappears in the option formulae). ...