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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Ito Process: How to calculate expected return?
On page 300 of Hull's Options, Futures and Other Derivatives
It is tempting to suggest that a stock price follows a generalized Wiener process; that is, that it has a constant expected drift rate and ...
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Incomplete market
How to prove that market with one risky asset $S_t$ and interest rate $r = 0$ is incomplete:
$$dS_t = S_t (\mu dt + \sigma_t dW_t^{1}), \quad S_0 = 1,$$
$$\sigma_t = 1 + |W_t^{2}|,$$
$W_t^{1}$ and $...
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State space equation of CARMA(p,q) processes
Thanks for visting my question:)
I am currently working on Carma(p,q) processes and do not understand how to derive the state equation. So the CARMA(p,q) process is defined by:
for $p>q$
the ...
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Heston model using YUIMA package
I am trying to estimate a Heston model using the Yuima package, but i am in trouble.
This is my script:
...
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Ito formula and confusion with the differential operator $d$
Thanks for visiting my question.
Im am currently working on this paper (https://arxiv.org/abs/2305.02523) and I am stuck at page 21 (Theorem 14 proof).
First these SDE's were defined:
\begin{align*}
...
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Show discrete market is arbitrage free if and only if there exist no admissible arbitrage portfolios
Problem: Let S be a discrete market. Show S is arbitrage free if and only if there exist no admissible arbitrage portfolios.
Definition of Discrete Market: Let $T$ be a positive real number and
$N$ ...
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Binary Signals and Combined Price Predicitions
Consider a binary signal $s(t)\in\{0,1\}$ for times $t\in\mathbb{R^+}$. Also define an asset price $X(t)$. Suppose that the curve,
$E(r(t+h)\space|\space s(t) = 1) = \alpha (1 - e^{-\delta h})$
where $...
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Empirical Evidence for Support/Resistance Levels in Martingale Price Processes and Its Impact on Option Volatility Surfaces
In financial mathematics, the martingale property often serves as an essential foundation for the stochastic processes that underlie securities pricing models. According to martingale theory, the most ...
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Financial software: academia vs. real world [closed]
I am looking for resources (if they exist) that explain the differences between quant finance software in academia and the real world, or explain how quant software is implemented in practice.
For ...
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Optimal Fitting Criteria of SABR
I was reading about SABR Model and curious about this.
The process of fitting the SABR model involves finding values for the parameters α, β, ρ, ν that minimize the difference between model-implied ...
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Resource recommendations: Levy process estimation using programming languages
Perhaps this type of question is not very suitable for this forum, but I'll try to make my question a little useful.
I'm studying stochastic processes, more precisely, Levy processes. A Levy process $...
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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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Is homogeneity preserved under change of measure?
In a paper, Joshi proves that the call (or put) price function is homogeneous of degree 1 if the density of the terminal stock price is a function of $S_T/S_t$. In the paper I think Joshi is silently ...
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Integrated Brownian motion
I occasionally see a post here: Integral of brownian motion wrt. time over [t;T].
This post has the conclusion that $\int_t^T W_s ds = \int_t^T (T-s)dB_s$.
However, here is my derivation which is ...
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Is it possible to calibrate Mertons Jump Diffusion Model such that it matches mean and vola from a normal process without jumps? [closed]
I'm currently playing around with Mertons version of jump diffusion processes where i'm testing the predicitions of a trading model given a time series with and without jumps to isolate the effects of ...
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Time-shifted power law in path dependent volatility
I can't understand a function which is part of a volatility model.
This is all explained in an open access paper titled "Volatility is (mostly) path-dependent" by Guyon and Lekeufack. My ...
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Backward induction: equation including expected values of stochastic process
Given the following SDE: $$ d\psi_t = \rho dt + \mu \psi_t dX_t$$, where $G(t) = \rho t$, $\rho = \frac{1}{T}$ $\psi_0 =0$, $T=1$, $\mu > 0$ and $X_t$ is a standard Brownian Motion (assume we know ...
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Necessary conditions to ensure that stochastic integral is a normal variable
Let $\left(W_t\right)_{t\geq 0}$ be a Brownian motion with respect to filtration $\mathbb{F}=\left(\mathcal{F}_t\right)_{t\geq 0}$. Let $\left(\alpha_t\right)_{t\geq 0}$ be an $\mathbb{F}$-adapted ...
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Characteristic function of Gamma-OU process
Consider the Gamma-Ornstein-Uhlenbeck process defined in the way Barndorff-Nielsen does, but consider a different long running mean $b$ which may be bigger than zero:
$$dX(t) = \eta(b - X(t))dt + dZ(t)...
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Aggregate Portfolio Simulation vs. Underlying Assets
Background: I am currently implementing a correlated Monte Carlo simulation model using Cholesky decomposition to create the sampling distribution.
Question: What is the difference between creating ...
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multivariate geometric brownian motion equivalent martingale measure
Suppose $W$ is a $\mathbb{P}$-Brownian motion and the process $S$ follows a geometric $\mathbb{P}$-Brownian motion model with respect to $W$. $S$ is given by
\begin{equation}
dS(t) = S(t)\big((\mu - ...
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Confusion about the formula for gain process in a financial market
In this wikipedia page, we consider the following financial market
The formulas for the stocks are given here
And the gain process of a portfolio $\pi$ is defined such that
From what I understand, ...
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Bond-pricing under the Vasicek short rate model
I'm currently studying the Vasicek model of the short interest rate
$$dr_t=a(\mu-r_t)dt+\sigma dW_t$$
I know how to solve this stochastic differential equation (SDE) and how to find expectation and ...
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Did I derive the Kelly criterion correctly?
$$\frac{dX_t}{X_t}=\alpha\frac{dS_t}{S_t}+(1-\alpha)\frac{dS^0_t}{S^0_t}$$
where $\alpha$ is proportion of the investment in the risky asset $S_t$ and $S^0_t$ is the risk-free asset. $S_t$ follows a ...
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integral of adapted process with respect to semimartingale is a martingale
Fix $T > 0$ a finite time horizon. Let $H$ be an adapted (or progressively measurable, if needed) continuous process and S be a continuous semi martingale, both on $[0,T]$. Under what conditions is ...
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Deriving an Analytical Expression for Standard Deviation of Log Returns
I am looking to find an expression for the standard deviation log returns of a stock price process.
I have a stock price which follows the following dynamics:
$dY(t) = Y(t)(r(t)dt + η(t)dW(t))$
Here,...
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Bessel Correction and Geometric Brownian Motion
Does it make sense to use bessel's correction for standard deviation and variance when fitting the drift and volatility parameters of geometric brownian motion to historical return data for a security....
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Expected value and variance of the short rate under the Vasicek model
Would be grateful for any assistance.
Below are the expected value and variance of the integral of the short rate under the Vasicek model (https://www.researchgate.net/publication/41448002):
$E\left[ \...
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Feynman-Kac formula: Ito's lemma for exponentiated integrals $e^{-\int b dr}$
Consider the stochastic process
$$
dy = f(y,s)ds + g(y,s)dw
$$
where, $w$ is Brownian motion.
Now consider the following exponentiated integral
$$
z_1(s) = \exp \left[ - \int_t^s b(y(r),r) dr \right]
$...
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What is the correlations between the Wiener processes in Heston Model?
In Heston Model we have two Wiener processes: One for the asset price, the other for the volatility. The model assumes a correlation $\rho$ between the two processes.
I ask what nature this ...
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On the operational process of fractional and delay Brownian motions (FGBM/GDBM) governing respective market scenarios
I have some knowledge about the fabrication of a stochastic differential equation (SDE) governing asset price ($S(t)$) dynamics (This answer helped me up to some extend).
For instance, I am little bit ...
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Maximum likelihood estimation of system of correlated SDEs
I have the following system of SDEs (which you can think of as 3 different stocks)
$$dX_t^1 = \mu_t X_t^1 dt + \sigma_t X_t^1 dW_t^1$$
$$dX_t^2 = \mu_2 dt + \sigma_2 dW_t^2$$
$$dX_t^3 = \mu_3 dt + \...
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Volatility of the product of two correlated asset following a log normal distribution [duplicate]
I am trying to solve the problem: Given two assets X and Y that follow a log normal distribution with volatility $\sigma_1$ and $\sigma_2$ respectively and with correlation $\rho$, what is the ...
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Heston Process: Accept-Reject Sampling to Alleviate the Problem of Negative Variances
I've read even in recent papers, and on page 21 of the book "The Volatility Surface" by Jim Gatheral (2006), all the debate over whether to reflect or truncate negative variances whilst ...
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How is variance derived in BS?
The realized variance under classical Black Scholes where the stock price process follows a GBM is given as
$$V_T = \frac1T\int_0^T\sigma_s^2ds\qquad (1)$$
however, the texts I have been reading do ...
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Working with wide bid ask spreads in option pricing model
I'm trying to fit an Heston model to market data. But market is data has some terms (<3M) with quite wide bid-ask spreads (12%-25%). Should I just use mid volatility? Is there maybe a model to pre-...
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how to calculate pdf and cdf for an Ornstein-Uhlenbeck process
I have the
Task. For Ornstein-Uhlenbeck process generate a path and plot a)
cumulative distribution (cdf), b) density function (pdf), c) calculate the 95%-quantile.
My solution.
From the literature we ...
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Applications of a certain type of stochastic processes in quantitative finance [duplicate]
A compound Poisson random vector $Y$ is well defined in this site in wikipidia.
Nothing prevents me from compound strictly stationary stochastic processes instead of compound random vectors. The ...
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No-arbitrage framework and the coordinate process
In the paper by Beiglböck et al, I encountered the following description of the no-arbitrage framework (see screenshot below). What is the meaning of this coordinate process $S_i$? How does it relate ...
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Expectation of Bt^4 given BS [closed]
What is the expectation of Bt^4 and Bt^3 given Bs? Given t>s.
I understand that the expectation of Bt given Bs is Bs and that the expectation of Bt^2 given Bs is something like Bs - s + t.
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Standard deviation of the difference between a time series and its EMA?
I have a time series $Q={\{q_t\}}$ of known standard deviation $\sigma$, and its EMA of parameter $\alpha$ : $\{EMA_t(\alpha)\}$.
My question is : I'm looking for a formula that would give the ...
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Can I extend the private information model of Kyle in in a continuous analogue, e.g. the Ornstein–Uhlenbeck process?
Taking into account an old post of maths.stackexchange, I recall the following:
On the one hand, we know that the Ornstein–Uhlenbeck process can also be considered as the continuous-time analogue of ...
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Calibrating Hull-White model using historical data
I'm in search of a way to calibrate a very simple Hull-White model with a constant volatility and a constant mean-reversion speed, purely based on historical zero rates.
$$dr(t) = (\theta(t) - \alpha ...
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Reference request: Approximate mapping of a multi-factor stochastic volatility model to single-factor stochastic volatility model
I am looking for approaches to transform a more complicated stochastic volatility model such as the one shown in Section 2.2 of Smile Dynamics II to a single-factor model such as the one shown in ...
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Feymann Kac pde with correlated process
I have to solve the following PDE:
\begin{equation}
\begin{cases}
\dfrac{\partial F}{\partial t}+\dfrac{1}{2}\dfrac{\partial^2 F}{\partial x^2}+\dfrac{1}{2}\dfrac{\partial^2 F}{\partial y^2}+\dfrac{1}{...
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Optimal consumption process [Munk (2011)]
I'm trying to solve problem 4.4 in Munk (2011). The problem is as follows:
Assume the market is complete and $\xi = (\xi_{t})$ is the unique state-price deflator.
Present value of any consumption ...
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Dynamics of discounted prices (multi-dimensional)
My objective is to find the dynamics of the discounted prices, given by $\mathbf{y}_{t} = \mathbf{P}_{t}\mathrm{e}^{-\int^{t}_{0} r_{s} ds}$. I know the dynamics should be $d\mathbf{y}_{t} = \mathrm{...
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Any innovations in mathematical processes behind option pricing models?
I am working on my thesis about option pricing models beyond classical Black-Scholes Model by looking for some recent innovations on mathematical processes behind the pricing structures. By that I ...
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Munk (2011) exercise 3.6
I'm trying to solve the exercise in Munk (2011). The exercise reads:
"Find the dynamics of the process: $\xi^{\lambda}_{t} = \exp\left\{-\int^{t}_{0} \lambda_{s} dz_{s} - \frac{1}{2}\int^{t}_{0} \...
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Analytical expression for SDE
I'm trying to find an analytical expression for the following. Suppose $X$ is a geometric Brownian motion, such that: $dX_{t} = \mu X_{t} dt + \sigma X_{t} dW_{t}$. Suppose furthermore, that the ...
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