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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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Showing a basic market admits no arbitrage

I'm learning the fundamentals of financial mathematics and came across the following problem I cannot solve Setting We work in $\left(\Omega, \mathcal{F},\left(\mathcal{F}_t\right)_{t=0}^1, \mathbb{P}\...
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Asset pricing based on stochastic inflation discounting (inflation controlled by stochastic state variable)

Suppose there is an asset that pays fixed nominal payout $\delta_t = \delta$, with a constant real discount rate $\bar{r}$ and stochastic inflation $\pi_t$. Suppose the price follows a controlled ...
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Question about the integrand space of stochastic integral wrt martinagles

I am reading the book "Introduction to Stochastic Integration" by Hui-Hsiung Kuo. In Chapter 5, he introduces the definition of stochastic integral w.r.t martingale: $$I(f) = \int_a^b f(t) ...
Mingzhou Liu's user avatar
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The continuous-time limit of asset price processes where there is more than one asset

I've read Merton's article "On the Mathematics and Economics Assumptions of Continuous-Time Models" (Reprinted in Continuous-time Finance, Chapter 3), where Merton proved that the price of ...
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Martingale property of the CEV model

I am a bit confused about the martingale property of the CEV model. Given dS(t)=σS(t)^βdW(t), is S a martingale for values of β<1?
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Time scale of standard deviation of stochastic asset prices

If I run a stochastic interest rate model that is used to price a bond that is a series of cash flows under $N$ scenarios where the price is the average of all the scenarios, $P = \sum_{n=1}^N p_n / N$...
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Areas of research in calibration of stochastic volatility models

I am working on a thesis in deep calibration of the Heston model, and I wanted to include a section on the historical work, before the use of neural networks in this area. Thus, I was wondering what ...
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Sample Wiener process constrained to open (initial), high (max), low (min), close (final)

With a Brownian bridge, one can sample a Wiener process constrained to a specified initial value and a final value. Can the same be done when the process is constrained also to have a specified ...
JoseOrtiz3's user avatar
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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 ...
Starlord22's user avatar
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Is this arbitrage? Infinite payoff / infinite loss (energy generation investment problem)

I'm a student using stochastic optimization in energy systems and I have a particular phenomena in an optimization problem that I think must occur in finance aswell, so I have been trying to find ...
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Modelling SPX implied volatilities dynamics

I'm interested in constructing a model for the daily implied volatilities of the SPX spanning from 2006 to 2024, considering various tenors (1M, 3M, 6M, 9M, 1Y) and moneyness levels (80, 90, 95, 100, ...
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Volatility of a stochastic Process given by an SDE

I am currently working on this thesis: http://arks.princeton.edu/ark:/88435/dsp01vd66w212h and i am stuck on page 199. There we have a portfolio $P=\alpha F+\beta G $ with $\alpha +\beta =1$ and ...
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Stochastic representation of a zero-coupon bond

In Chapter 9 of Shreve's book Stochastic Calculus for Finance II, the main theorem is the 9.2.1. Defining the discounting process $D(t)=\mathrm{e}^{-\int_0^t du r(u)}$ and $r(u)$ the, possibly ...
apelle's user avatar
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Geometric Brownian motion with volatility as function of time

With the following process: $$dS_t = r S_t dt + σ(t) St dW_t \tag1$$ and $$ \sigma (t) = 0.1 \ \ \ if \ \ t < 0.5 \\ \sigma (t) = 0.21 \ \ \ otherwise$$ I know the general solution should be : $$...
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Simulate Spot Process with Forward Variance (Bergomi)

I am reading Bergomi's book (Stochastic Volatility Modeling), and in section 8.7 The two-factor model (page 326), the following dynamics are given: \begin{align} dS_t &= \sqrt{\xi_t^t}\,S_t\,...
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what is $\mathcal{F}_t^0$-optional positive process $\tilde a(q)$ defined in Hayashi et al?

in the paper of Hayashi et al. in assumption D is defined the following convergence in probability $A(q)_t^n \overset{\mathbb{P}}\to \int_{0}^{t} a(q)_s ds$. what does it means the process $a(q)$ is $...
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Approximation of an Itô integral with python

Exercise 3.11 (Approximation of an Itô Integral). In this example, the stochastic integral $\int^t_0tW(t)dW(t)$ is considered. The expected value of the integral and the expected value of the square ...
Jessie's user avatar
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Change of numeraire : quotient

Let's consider $X_1(t)$ a geometric brownian motion (with variable volatility) and $X_2(t)$ a Brownian bridge : $dX_1(t) = \mu X_1(t) dt + \sigma_1(t) X_1(t) dW(t)$ $dX_2(t) = \frac{b - X_2(t)}{T - t} ...
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Distribution of Geometric Brownian with time-dependant volatility

The process $S(t) =\exp\left(\mu.t + \int_0^t\sigma(s) \text{d}W(s) - \int_0^t \frac{1}{2}\sigma^2(s)\text{d}s\right)$ where $\sigma(s) = 0.03s$ is log-normally distributed, but i'm not sure about the ...
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Analytic Hull White model with correlated stochastic processes

I am trying to price a path dependent option which uses two underlyings (a stock index and an interest rate index). I am using Hull White model for interest rate modelling and local vol for stock ...
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Why is BG porcess a pure jump process?

Recently (~10 years ago), Kuchler&Tappe have set up a new stochastic process called Bilateral Gamma process. This process is defined through its increments: $$\forall t\geq s, X_t-X_s\sim \Gamma_{...
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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 ...
user546106's user avatar
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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 ...
Valentin's user avatar
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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: ...
Luiz Araújo's user avatar
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86 views

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 $...
anonymous's user avatar
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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 ...
GotTheTrumpCard's user avatar
3 votes
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264 views

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 ...
Starlord22's user avatar
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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 $...
André Goulart's user avatar
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2 answers
338 views

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 ...
Lucas Morin's user avatar
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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 ...
Frido's user avatar
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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 ...
Wang Jing's user avatar
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152 views

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 ...
T123's user avatar
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1 answer
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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 ...
s5s's user avatar
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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 ...
5ilver4rrow's user avatar
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211 views

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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4 votes
1 answer
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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)...
Tom's user avatar
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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 ...
Shri's user avatar
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1 vote
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117 views

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 - ...
yrual's user avatar
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1 vote
1 answer
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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, ...
yrual's user avatar
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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 ...
Don Abbondio's user avatar
1 vote
1 answer
317 views

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 ...
user67303's user avatar
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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 ...
yrual's user avatar
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4 votes
1 answer
164 views

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,...
user67245's user avatar
2 votes
0 answers
46 views

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....
user3163829's user avatar
2 votes
0 answers
180 views

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[ \...
user1171853's user avatar
3 votes
0 answers
157 views

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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