61
votes
Integral of Brownian motion w.r.t. time
This type of integral has appeared so many times and in so many places; for example, here, here and here.
Basically, for each sample $\omega$, we can treat $\int_0^t W_s ds$ as a Riemann integral. ...
- 21k
20
votes
Accepted
Finding distribution of $\int_0 ^T uW_u du$
Using the Ito Formula
The general approach that often works for these kinds of question is to search for functions such that their Ito differential contains the terms that we are interested in. In ...
- 5,959
18
votes
Accepted
Processes used in quant finance
Here is a short list (to be edited and improved - community wiki) :
Standard brownian motion (also called Wiener process) for which:
$d\, W_t \sim \mathcal N(0, \sqrt{d t})$
Geometric brownian ...
18
votes
Accepted
Why is Brownian motion useful in finance?
Brownian motion is simply the limit of a scaled (discrete-time) random walk and thus a natural candidate to use. It is very intuitive and arguably one of the simplest and best understood time-...
- 15.2k
14
votes
Integral of Brownian motion w.r.t. time
Just to add to the already nice answers, the result can also be obtained using the (stochastic) Fubini theorem.
\begin{align}
\int_0^t W_s ds &= \int_0^t \int_0^s dW_u\, ds \tag{$W_s=\int_0^s ...
- 14.5k
14
votes
Accepted
Find a formula for the price of a derivative paying $\max(S_T(S_T-K),0)$
I provide a solution in three steps.
The first step carefully outlines how to split up the expectation and what new measures are used. This first step does not require any special model assumption ...
- 15.2k
13
votes
Accepted
Two correlated brownian motions
First you need to correct the formula to:
$$
W_t^2 = \rho W_t^1 + \sqrt{1-\rho^2} Z_t,
$$
where $Z_t$ is a BM independent of $W_t^1$
If you calculate the variance and the covariance, then you see ...
- 13.6k
13
votes
Accepted
Can I always use quadratic variation to calculate variance?
Quadratic variation and variance are two different concepts.
Let $X $ be an Ito process and $t\geq 0$.
Variance of $X_t$ is a deterministic quantity where as quadratic variation at time $t $ that ...
- 2,412
12
votes
Two correlated brownian motions
Here is the general approach you can follow to generate two correlated random variables. Let's suppose, X and Y are two random variable, such that:
$$X \sim N(\mu_1, \sigma_1^2)$$
$$Y \sim N(\mu_2, \...
- 2,218
12
votes
Is it really possible to create a robust algorithmic trading strategy for intraday trading?
Here's my favorite example of an intraday strategy on S&P500 futures that at least used to work:
Intraday Share Price Volatility and Leveraged ETF Rebalancing
I pull it out whenever people start ...
- 693
12
votes
Accepted
Correlation coeffitiont between two stochastic processes
if you talk about correlation then:
compute expectation:
$$\mathbb{E}(W_t)=0\text{ and }\mathbb{E}(\int_0^tW_d ds)=0$$
variance:
$$\text{Var}(W_t)=t\text{ and }\text{Var}(\int_0^tW_s ds)=\frac{t^3}{...
- 2,412
12
votes
Finding distribution of $\int_0 ^T uW_u du$
Another approach consists in using the Fubini theorem to write that
\begin{align}
\int_0^T u W_u du &= \int_0^T \int_0^u u\, dW_v\, du \tag{$W_u = \int_0^u dW_v$} \\
&= \...
- 14.5k
11
votes
How to get the probability of exercise call option in Black-Scholes model?
With the underlying asset price $S_t$ following a geometric Brownian motion with drift $\mu$ (risk-neutral or otherwise) , we have at time $t = T$,
$$S_T = S_0e^{(\mu- \frac{\sigma^2}{2})T}e^{\sigma ...
- 3,595
11
votes
Accepted
Expectation of $\int_0^t \frac{1}{1+W_s^2} \text dW_s$
By construction, the Itô integral, $I_t=\int_0^t X_s\text{d}W_s$, is a martingale if $\int_0^t \mathbb{E}[X_s^2]\text{d}s<\infty$.
The martingale property, $\mathbb{E}_s[I_t]=I_s$ implies $\mathbb{...
- 15.2k
11
votes
Expectation of exponential of 3 correlated Brownian Motion
You need to rotate them so we can find some orthogonal axes.
A simple way to think about this is by remembering that we can decompose the second of two brownian motions into a sum of the first ...
- 2,996
11
votes
Accepted
Expectation of exponential of 3 correlated Brownian Motion
Besides @StackG's splendid answer, I would like to offer an answer that is based on the notion that the multivariate Brownian motion is of course multivariate normally distributed, and on its moment ...
- 6,425
10
votes
Accepted
Is it really possible to create a robust algorithmic trading strategy for intraday trading?
Such a complex question...
Geometric Brownian Motion (GBM) will not typically work to aid one finding strategies based on technicals, as the pursuit of the technical trader is to find market ...
- 391
10
votes
Accepted
Mathematical proof of $g = \mu - \frac{\sigma^2}{2}$ relationship between CAGR and average returns
Here is a straightforward derivation of this approximate relationship for a discrete sample of returns that does not require specifying an underlying probability distribution or a continuous ...
- 3,595
10
votes
Why is Brownian motion useful in finance?
Physical objects move according to simple smooth curves that can be represented by low order polynomials: a straight line, a parabola, an ellipse, etc.
Financial market prices move in a completely ...
- 9,332
9
votes
Accepted
How are Brownian Bridges used in derivatives pricing in practice?
Yes, the term Brownian Bridge seems to be used loosely. I assume you are talking about continuously monitored barriers by the way, since you mention the probability of the barrier being crossed in ...
- 521
9
votes
Accepted
Correlation between stock prices given correlation between returns
We can obtain a closed-form expression for price correlation given (log) return correlation when the two stocks follow geometric Brownian motion:
$$S_1(t) = S_1(0)e^{(\mu_1- \frac{1}{2} \sigma_1^2)t}...
- 3,595
9
votes
More questions about integral of Brownian Motion w.r.t time
It is indeed Riemann integrable, so you don't need stochastic integration. For a given path, you can interpret the integral in the Riemann sense. For a given t, the paths are random, so it is a random ...
- 6,204
9
votes
Accepted
More questions about integral of Brownian Motion w.r.t time
As usual with those kind of integrals, another way to reach the result is to:
Express $W_s$ in integral form as $\int_0^s dW_u$
Use Fubini theorem to change the integration bounds of the resulting ...
- 14.5k
9
votes
Accepted
What is a Brownian motion "under the risk-neutral measure"?
A Brownian motion is always defined with repect to a given probability space. Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and $X_t=W_t^\mathbb{P}$ a Brownian motion, i.e. a stochastic ...
- 15.2k
8
votes
Accepted
2 Ito processes - $d(X_{t} + X^{'}_{t})^2 = (Y_t + Y^{'}_{t})^2 dt$ why it is true?
$X_t$ being a stochastic process, one cannot use ordinary calculus to express the differential of a (sufficiently well-behaved) function $f$ of $t$ and $X_t$.
Instead one should turn to Itô's lemma, ...
- 14.5k
8
votes
Accepted
Geometric Brownian Motion: percentage returns vs log-returns
The percentage return over the infinitesimal interval $[t, t+dt]$ is given by
\begin{align*}
\frac{S_{t+dt} - S_t}{S_t} \approx \mu dt + \sigma \sqrt{dt} \xi,
\end{align*}
where $\xi$ is a standard ...
- 21k
8
votes
Accepted
Integral of Wiener process w.r.t. time
@Ivan's comment regarding the covariances is the key.
Consider an equally spaced partition $\Pi_n = \left\{ t_0 = 0, t_1 = \Delta_n, \ldots, t_n = t \right\}$ of the interval $[0, t]$, where $t_i = i ...
- 5,959
8
votes
Difference between $W_t$ and $X_t= \sqrt{t}Z$
The means are equal
Suppose $f$ is analytic so that we can give it a Taylor series that works everywhere such that $f(x) = \sum a_n x^n$, and then let us let this be bounded too. To show that the ...
- 1,389
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