Questions tagged [stochastic-processes]
stochastic processes is a collection of random variables representing the evolution of some system of random values over time.
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PDF Calculation by Fourier Inversion of Characteristic Function for Affine Intensity Process in Matlab
I'm trying to use the Fourier inversion formula to plot the PDF of an Affine Stochastic Intensity Reduced Form Credit Model, given its characteristic function.
The characteristic function of an ...
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Using a Constant as a Numeraire
Please provide steps to justify the below.
1) Can we use a constant as a numeraire?
Related Question: Scaling Stock Price and Strike etc. by a Constant
The rest of standard Geometric Brownian ...
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Transformation of local volatility model
Assume we have an SDE
$$dX_t=\mu(X_t)dt + \sigma(X_t)dW_t$$
where $\sigma>0$ and $W_t$ is a Wiener process. Is there a transformation $y(X_t)$ that will make the dynamics of the transformed process ...
4
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Expectation of $\int_0^t \frac{1}{1+W_s^2} \text dW_s$ [duplicate]
I am trying to calculate the expectation of
$$\int\limits_0^t \frac{1}{1+W_s^2} \text dW_s,$$
where $(W_t)$ is a Wiener process.
I was told that the value of this expectation is zero. Can someone ...
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2
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Integration on Wiener Process
How can I show that below equation holds ?
$\int\limits_{0}^{t} f \left( s \right)W_s ds = W_t \int\limits_{0}^{t}f \left( s \right)ds - \int\limits_{0}^{t}\int\limits_{0}^{s} f\left( u \right)dudW_s $...
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Show that $E[B_t|\mathscr{F}_s] = B_s$ for $B_t = W_t^3 - 3 t W_t$
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 $B_t = W_t^3 - 3tW_t$. ...
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List: Behavioural characteristics of key Ito processes used in finance
My hope from this question is to become a repository of the behavioural characteristics, use-cases and interesting features of the key Ito processes used in quantitative finance - examples being GBM, ...
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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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Determine $E[W_p W_q W_r]$
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 0 < p < q < r. Determine $E[W_p W_q W_r]$.
...
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what does the cover page of Guyon and Labordere's Nonlinear Option Pricing represent?
It could be a bit offtopic, but I don't see the link between the contents of the book and the cover page.
Thanks
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Solution to SDE being Evolution of Price Process
I am trying to explain the concept of a solution to SDE being the model for the evolution of a price process. How would you do this to someone who doesn't have a financial engineering background?
...
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Application of Ito's lemma
Let $X_t$ be some stochastic process driven by wiener process ($W_t)$ so it can be expressed as:
$$dX_t=(...)dt+(...)dW_t$$
Let $f(t,x)$ be some $C^2$ function. Define the process $Z_s=f(t-s,X_s)$ ...
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Speed of mean reversion of an interest rate model
I would like to have a bit more of intuition about the concept of "speed of mean reversion" for an interest rate model, e.g. Vasicek or CIR. In particular, is a negative speed of mean reversion ...
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Variable Drift Ornstein–Uhlenbeck Process
The Ornstein–Uhlenbeck process is defined as the stochastic process that solves the following SDE:
$dx_t = \theta (\mu-x_t)\,dt + \sigma\, dW_t$
where $\theta>0$, $\mu$ and $\sigma>0$ are ...
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Fractional Brownian motion
In Mandelbrot(1968)'s paper, the fractional brownian motion, denoted by $B_{H}(t,\omega)$,(t>0) is defined by
$$B_{H}(0,\omega)=b_{0}$$
$$B_{H}(t,\omega)-B_{H}(0,\omega)=\frac{1}{\Gamma(H+\frac{1}{2})...
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What the expectation of S^2 is from GBM? [closed]
I was at an interview and was asked to write down the SDE for GBM.
$$
dS = S\mu dt + S\sigma dX
$$
Then I was asked how I would compute the expectation of S^2. I didn't know where to start. Any ...
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How can I calculate $Cov\left(\int_{0}^{s}W_u\,du\,\,\,,\,\int_{0}^{t}W_v\,dv\right)$
How can I calculate?
\begin{align}
Cov\left(\int_{0}^{s}W_u\,du\,\,\,,\,\int_{0}^{t}W_v\,dv\right)
\end{align}
Thank you for your attention.
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Why does the short rate in the Hull White model follow a normal distribution?
Consider Hull White model $dr(t)=[\theta(t)-\alpha(t)r(t)]dt+\sigma(t)dW(t)$
when we solve the SDE above we have $r(t)=e^{-\alpha t}r(0)+\frac{\theta}{\alpha}(1-e^{-\alpha t})+\sigma e^{-\alpha t}\...
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Stochastic process for modelling correlation?
This question relates to Financial Machine Learning, and more specifically to competitions like Numerai.
In this competition we have a dataset X and a target y (return over a given horizon). The ...
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Ornstein–Uhlenbeck process – integration by parts
While deriving the solution for the stochastic differential equation that models the Ornstein–Uhlenbeck process, Paul Wilmott (Paul Wilmott on Quantitative Finance, chapter 4, page 87) performs the ...
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Value (price) of defaultable zero coupon bond with credit risk involved
I'm trying to derivate the Value (price) of defaultable zero coupon bond, but there some steps (math) in between I can't figure out.
From the default process modelling, we have:
$$P(t ≤ \tau < t+dt ...
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Black-Scholes formula proof, without stochastic integration
I've looked into many books at my academic library, and very often it goes like this:
Brownian motion
Then, stochastic integration (Itô's formula etc.)
Application: Black-Scholes formula for price of ...
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Central Limit Theorem and Lévy processes
Lévy processes are self-decomposable and independent on any non-overlapping interval, so how come the distribution of the process at time T,$\phi(T)$, which is the sum of N i.i.d with law $\phi(T/N)$ ...
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Understanding Monte Carlo to solve option price with local volatility
I have read this question pricing using dupire local volatility model which seems to have an answer from here https://www.csie.ntu.edu.tw/~d00922011/python/cases/LocalVol/DUPIRE_FORMULA.PDF
Both of ...
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Discretization of Wiener process
The Wiener process $(W_t)$ is a continuous stochastic process that satisfies the following there conditions:
$W_0 = 0$,
the increments $\mathrm{d}W_t = W_{t + \mathrm{d}t} - W_t$ are normally ...
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Expectation of Stochastic Differential
First of all, I am a mathematician, so I apologize for my ignorance regarding stochastic calculus. What exactly does an expression like:
$$
\mathbb{E}[dX_tdY_t]
$$
here $X_t,Y_t$ are stochastic ...
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negative values in geometric brownian motion
A GBM (Geometric Brownian Motion)
$ \frac{dx}{x} = \mu dt + \sigma dW $
solves to
$x_t = x_o e^{(\mu - \sigma^2)t + \sigma W_t}$
From the solution, it is clear that $x_t$ cannot become negative. ...
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Random Walk with normal increments and n time periods why is the increment $\sqrt{(t/n)}$?
Question is basically in the title. I have found several sources stating that $R_i = \sqrt{\frac{t}{n}}$, but I couldn't find the intuition behind taking the square root. And it seems to be crucial ...
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Show that the two solutions of the SDE are equivalent
I have a process:
$$dr_t = (W_t^1 - ar_t)dt +\sigma dW_t^2$$
where $W_t^1$ and $W_t^2$ are brownian motions with instantaneous correlation coefficient $\rho$.
I want to show that the solution of this ...
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To estimate the parameters when only the characteristic function is known to us
Recently I was working with a process named Variance Gamma with Stochastic Arrival (VGSA) and trying to fit this process on a given data.
To obtain VGSA, as explained in Carr et al. [2001], we take ...
4
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Default intensity in Black-Cox model
Consider the model by Black and Cox (Journal of Finance, 1976).
The default intensity function is defined in the usual way: $$h(t) \equiv - \frac{\partial \log P[\tau > t| \mathcal{F}_t]}{\partial ...
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Invariance Scaling of Brownian Motion
Prove $\frac{1}{\sqrt{t}}\log\left(\int_0^t \exp(B_s)\mathrm{d}s\right)$ converges to $\sup\limits_{t\in [0,1]}B_t$ in distribution as $t\to\infty$. I have a sense to use scaling invariance, but no ...
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1
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Compute distribution of a stochastic variable
$sign(x)=1$ if $x\geq0$
$sign(x)=-1$ if $x< 0$
Consider
$$
X_t = \int^t_0 sign(W_u)dW_u
$$
where $W_t$ is a wiener proces.
How can I determine the distribution of $X_t$ and compute $E[\exp(\...
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Equivalent martingale measure exists if and only if $a < S_0^1(1+r)< b$
Exercise :
We consider a market of one period $(\Omega, \mathcal{F}, \mathbb P, S^0, S^1)$, where the sample space $\Omega$ has a finite number of elements and the $\sigma-$algebra $\mathcal{F} = 2^\...
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Ito's Lemma: Multiplication Rule
I have a conceptual question about Ito's lemma, in particular, the multiplication.
Ito's multiplication rule states, that multiplying dt by itself or by dx (the stochastic differential) equals zero. ...
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clarification to log-stock price formula
Having financial market with safe rate r and risky asset S with dynamics under physical measure P $$\frac{dS_t}{S_t}=\mu dt +\sigma dW_t$$
what is the log-stock price?
Using Ito formula it is ...
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The relation between exchange rate SDE and respective interest rates
The exchange rate between a domestic currency money market and a foreign currency money market can be expressed as
$$
dQ(t) = (r_d - r_f)Q(t)dt + \sigma Q(t)d\tilde{W}(t)
$$
where $r_d$ is the ...
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Extended CIR and discretization
Did someone know how to discretize this process efficiently :
$dX(t) = \kappa [\theta(t)-X(t)]dt + \sigma \sqrt{X(t)}dW(t)$
I am looking for something more sophisticated than the trivial Euler ...
4
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1
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Discounted risky asset stochastic process problem
$S_t$ is the random variable representing the risky asset price at time $t$.
M_t is the riskless asset. They are governed by the equations
$\frac{dS_t}{dt}=\mu dt + \sigma dZ_t$ and
$dM_t = rM_t ...
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How to apply Levenberg Marquardt to Max Likelihood Estimation
In this paper on p315:
http://www.ssc.upenn.edu/~fdiebold/papers/paper55/DRAfinal.pdf
They explain that they use Levenberg Marquardt (LM) (along with BHHH) to maximize the likelihood.
However as I ...
4
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1
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Simulating Iterated Brownian Motions
I was going through an interesting article (https://arxiv.org/pdf/1112.3776.pdf) while I was trying to read about subordinated processes. I wanted to simulate subordinated processes (in R or python) ...
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Probability distribution of the stochastic process $\int_{0} ^{t}\frac{u}{t}dW_{u}$
I am wondering about the probability distribution of the stochastic process
$$X_t=\int_0^t \frac{u} {t} dW_{u}$$
I thought of using the Kolmogorov equation but after converting this into An SDE
$$...
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3
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Are all changes of measures for continuous diffusion processes given by the change of drift?
In elementary discussions on change of measure for geometric Brownian motion, one often find statements like "change of measure = change of drift". Given a general continuous diffusion process of the ...
4
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1
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Expected payoff at future time
Let $a$, $b$, $c$, and $e$ be constants, $W_1$ and $W_2$ be Brownian motions with correlation $\rho$, and $f(t)$ and $g(t)$ be deterministic functions of time. Let $X$ satisfy $$d(X(t))=(aX(t)+ef(t)g(...
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How to price a stock under Q and stochastic interest rates?
I am interested in pricing a stock under $\mathbb{Q}$ when I assume that
$$dS(t) = \mu(S(t))dt + \sigma(S(t))dW(t)$$
where $W(t)$ is a Wiener process under $\mathbb{P}$ and
$$dr(t) = a(b-r(t))dt +...
4
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2
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Stationary distribution for square root process
Consider the process,
$$
dX_t=(-aX_t+b(1-X_t))dt + \sqrt{X_t(1-X_t)}dW_t
$$
How do I show that the stationary distribution for the transition density is a beta distribution?
I tried expanding the ...
4
votes
1
answer
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American Option price formula assuming a logLaplace distribution?
What are $d_1$ and $d_2$ for Laplace? may be running before walking.
When I tried to use the equations provided, the pricing became extremely lopsided, with the calls being routinely double puts. ...
4
votes
1
answer
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Covariation of these processes
Let $N_t \sim \text{Poisson}(\lambda t)$ and $M_t \sim \text{Poisson}(\theta \lambda t)$.
We know that if $N$ and $M$ were independent, $dNdM = 0$ using polarization identity. We also know that $(dN)^...
4
votes
1
answer
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Differential of time over Browninan motion
I know that $\frac{dW_t}{dt}$, with $W_t$ a brownian motion, does not exist. However, does $\frac{dt}{dW_t}$ exists? Or does it even make sense? I am trying to calculate the quotient of two ...
4
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1
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Bond-price dynamics in the Vasicek model
Hello I am studying about interest rate modeling
There is one good source about Vasicek (link: https://web.mst.edu/~bohner/fim-10/fim-chap4.pdf). However there is one equation that I try but unable ...