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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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34 views

Why the variance of a process is $\left( \frac{dS_T^2}{dt}\right)^2$?

Consider an Ito process $dS_t = f(t,S_t) dt + g(t,S_t)dW_t $ What is the reason that we can compute the variance as: $\sqrt{VaR(S_t)} = \frac{(dS_t)^2}{dt}$
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61 views

Derivative of the Black and Scholes equation [on hold]

What is the financial interpretation that the derivative of the Black and Scholes equation is equal to 0? St n(d1) - Xe^-rt n(d2) = 0
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61 views

How can stationary time series data be used as input in an ML model?

I am halfway through "Advances in Financial Machine Learning" by Marcos Lopez de Prado. I understand that a time series like stock prices can be transformed to make it sufficiently stationary. ...
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24 views

Stochastic process with determinstic frequency of regime changes

Suppose that I have an OU process. For instance, assume that I want to model the interest rates. Suppose that regime change is known ex ante, and is deterministic in terms of frequency (For instance, ...
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26 views

Filtrations and the different “kinds” of pre-knowledge

I am searching for a reference I think I saw in a book by either Shreve or Oskendahl. I am struggling with a theoretical question. As I recall how it was posed, the idea of no prior information (or ...
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3answers
202 views

What stochastic process produces Student's t-distributed returns?

If I think daily log returns have a normal distribution, I can simulate intraday log returns as normal, because the sum of normal variates is also normally distributed. What if I want to simulate ...
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65 views

Different Forms of Geometric Brownian Motion [closed]

If the stock price S follows the geometric brownian motion: $$dS=\mu Sdt+\sigma Sdz$$ $$\frac{dS}S=\mu dt+\sigma dz$$ Where $dz=\epsilon\sqrt{dt}$ is a wiener process. Integrating this to get $S_T$ ...
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405 views

Ito`s Lemma problem

Can someone help me with calculus for this problem. I have these 3 equations and with Ito`s Lemma I have to find $dXt$. \begin{cases} dY= μYdt+σYdB \\ X=\frac{1}{2}cY\\ dc =-aαcdt\end{cases}
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1answer
61 views

Problem finding correct SDE for Stochastic Process

I am really struggling to come up with the correct SDE for the stochastic process: $Y(t) = a[Z(t)]^2$ where $Z(t)$ is a Brownian Motion. According to my Prof, the SDE is: $dY(t) = adt + 2aZ(t)dZt $...
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36 views

How does this transformation for Euler Scheme in mean reverting SDEs alleviate instability?

I saw this text in the book - Interest Rate Modelling by Andersen volume 1 on Page 112: I am unable to understand: How does instability arise when we use the Euler scheme on X(t)? What change does ...
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39 views

Unconditional Expectation vs. Conditional Expectation at time $0$

In most mathematical finance books I have read (all of them actually), the expectation, with respect to the sigma algebra at time $0$, $\mathcal F_0$, is considered the same as the unconditional ...
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112 views

How to determine the order of convergence of the Euler-Maruyama method?

To make this simple let us consider the Geometric Brownian Motions. My questions: 1. How can I show that the Euler-Maruyama Method is convergent using GBM? 2. How can I determine the order of ...
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1answer
90 views

Please explain Heston Model parameters meaning [closed]

The Heston Model is given by: $$ dS_t = \mu S_t dt + \sqrt{v_t}S_tdB_{1t}$$ $$ dv_t = \kappa(\theta - v_t)dt + \xi \sqrt{v_t}dB_{2t}$$. The parameters are: $\theta$ is the long term variance $\...
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53 views

Novikov condition for Vasicek process

Suppose that we have a money account $S^{(0)}$ with dynamics \begin{align} dS^{(0)}_{t} = r_{t} S^{(0)}_{t}\, dt, \end{align} where \begin{align} dr_t = a(b-r_t)\, dt + \sigma_{r} \, dW_t^{(0)}. \...
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1answer
110 views

Bitcoin dynamics - C++ Simulation

I would like perform a simulation of Bitcoin future prices given a sample of the 4 past years (2014-2018). My problem is that I do not know what model to use! For common stocks I used the geometric ...
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2answers
137 views

What is “Lambda” in Heston's original paper on stochastic volatility models?

In his paper (link), he has the equations: b1 = k + ƛ - (ρ * σ) b2 = k + ƛ k is the rate of mean reversion, ρ is the correlation between the two Wiener processes, σ is vol of vol, what is ƛ? ...
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62 views

pricing deliverable vs non-deliverable fx forwards

I am trying to link these two questions together Pricing a regular FX forward This is a contract (say USD vs JPY) where you exchange 2 currencies at maturity at a pre-determined rate, while no ...
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1answer
137 views

When $E[f(\alpha,X)] = f(\alpha, E[X])$

When $E[f(\alpha,X)] = f(\alpha,E[X])$, where $f$ is some convex function of the first and second variables, except when the first variable takes the value $\alpha$ in which case the equality holds, ...
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2answers
123 views

Find the brownian motion associated to a linear combination of dependant brownian motions

I have $N$ correlated standard one-dimensional Brownian motions $W_1,\ldots,W_N$ with correlation matrix $\rho$ and I consider the process $Z_t \equiv \sum_{i=1}^N \mu_i (t) W_t$ where the $\mu_i$ are ...
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56 views

Brownian motion for modelling future asset values

Assume that an asset price $S$ is given by a Brownian motion. Argue from the definition why it is not possible to predict future values of the asset based on the past values of $S$. I am not sure ...
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36 views

Milstein discretization of the CIR process

Given the CIR process $\ dX_t = (a − bX_t ) dt + \sigma \sqrt{X_t}dW_t$ - I want to show that its Milstein scheme is $\ X_{i+1} - X_i = ((a − bX_i) - 0.25\sigma^2)\Delta + \sigma\sqrt{X_i}\sqrt{\...
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2answers
67 views

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 ...
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127 views

Estimation of Radon–Nikodym derivative from hisotrical returns and option price data

Say we have an estimate of empirical density function $f^{\mathbb{P}}_S(s)$ of historical log-returns on a stock $S$ over a 30-day period under the real-world objective measure $\mathbb{P}$. We also ...
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1answer
69 views

Short rate models

On the short rate model in Wikipedia https://en.m.wikipedia.org/wiki/Short-rate_model Why is the first function, the P(t,T) given? This is not the short rate model this is generating prices for a ...
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1answer
115 views

American put option. Exercise time is a random variable, calculation of expected payoff

I got an American put option, where the payoff is $V_\tau = \max(K - X_{\tau}, 0)$ and $X_{\tau}$ is the price of an underlying at the stopping time $\tau < T$. The underlying follows a standard ...
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74 views

Feynman-Kac to derive stochastic representation

$u_t + \frac{1}{2}\sigma^2x^2u_{xx} - \alpha + \lambda((K_d - x)^+ - u) = 0$ with terminal condition $u(T, X) = (K_m - X(T))^+$ $dX = \sigma X(t)dW_t$ $\alpha$ and $\lambda$ are constants Ok so ...
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64 views

For an Ito Process, $d\ln{X} \neq \frac{dX}{X}$ and $(d\ln{X})^2 = (\frac{dX}{X})^2$, but $d\ln{X} \neq \pm \frac{dX}{X}$

In normal calculus we can write $d\ln{x} = \frac{dx}{x}$ since there is no quadratic variation to deal with. This isn't true for stochastic processes, and Ito's Lemma is used to calculate $d\ln{X}$. ...
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58 views

Proving Flow Property of Stochastic Differential Equation

I am trying to show that $X_t^{s,x} = X_t^{r, X_r^{s,x}}$ for $0 \leq s \leq r \leq t$, $x \in \mathbb{R}^n$ is a given initial condition for time $s$, for some SDE: \begin{equation*} d X(u)=b(X(u))d ...
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1answer
64 views

If S(t) is geometric Brownian motion, what is the distribution of S(t+h)-S(t)?

Suppose we have a geometric Brownian $S(t)$ which follows a lognormal process. Say $$ \begin{equation} dS_t = \mu S_t dt + \sigma S_tdW_t \end{equation} $$ My question is what is the distribution of $...
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2answers
100 views

How to numerically simulate exponential stochastic integral

For example given an integral $$ \int^t_0 \exp(aW(t'))\,dt', t\in\mathbb R_+ $$ where $W(t')$ is a standard Wiener process. I've been very confused about stochastic integrals like $\int^t_0 W(t')\,...
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85 views

Uniqueness of the Hedging strategy

I am currently reading the book "Nonlinear Option Pricing" by Julien Guyon. In the book they defined an attainable payoff $F_T$ as a $\mathcal{F}_T$ measurable random variable for which there exists ...
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69 views

SDE of futures price under non-constant interest rate and volatility process

I'm trying to figure out the form of the SDE of futures price under the risk neutral measure, when stock price follows GBM:             &...
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114 views

Feller Condition (Cox-Ingersoll-Ross) source

For the Cox-Ingersoll-Ross model $$\text{d}r_t = a(b-r_t)\text{d}t+\sigma\sqrt{r_t}\text{d}W_t$$ the condition (referred to as "Feller condition") $$2ab\geq\sigma^2$$ ensures that the solution is ...
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46 views

Stochastic Long-Run Mean Instantaneous Variance in Heston Model (and extensions)?

I'm working on my dissertation in Financial Economics, focusing on the topic of Stochastic Volatility Jump Diffusion models; and I'm playing around with some ideas for model extensions. In particular, ...
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1answer
82 views

Getting $df(t,T)$ when given $d\ln P(t,T)$ and $f(t,T)=-\frac{\partial}{\partial T} \ln P(t,T)$

Let the HJM dynamics of $\ln P(t,T)$ (log of bond prices) given by (In the risk neutral measure ) : $$d \ln P(t,T) = \mathcal{O}( dt) - \sigma_P (t,T) dW(t)$$ Knowing that $f(t,T)=-\frac{\partial}{\...
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1answer
125 views

Bayes Theorem with change of measure

Tomas bjork- arbitrage theory in continuous time. Appendix B, proposition B41 says: The proof is not clear to me. Thanks to Gordon's comment below of $E^Q (X/G)$ being $G$ measurable, I think the ...
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1answer
179 views

How to compute the dynamic of stock using Geometric Brownian Motion?

I have been given the following question: Given that $S_t$ follows Geometric Brownian Motion, write down the dynamic of $S_t$ and then compute the dynamic of $f(t,S_t) = e^{tS^{2}}$ For the first ...
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1answer
148 views

Why don't I get this right $\frac{d}{dt}\mathop{\mathbb{E}}\left[ e^{-\int_t^Tr(s)ds}|\mathscr{F}_t \right]$

Let $r$ a random process defined by : $$dr_t=\theta(t)dt + \sigma dW_t$$ $\theta$ is deterministic in $t$ and $W$ a brownian motion. I don't know where my calculation below is going wrong : Let $...
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99 views

Ultra Powerfull Vibrato Montecarlo for delta sensitivities of a not regular payoff

Ciao, I am working on a derivative with the following payoff at time $T$: $$ \sqrt{(S_T - K)^+} $$ where $S_T$ is the value of the stock at the expiring date. As usual we will assume $S_t$ to be a ...
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64 views

kolmogorov backward equation intuition

The kolmogorov backward equation equation states that the probability density of a random variable $x$ which follows $dx= \mu dt + \sigma dw$ is given by $-p_t = \mu p_x + 0.5\sigma^2 p_{xx} $ ...
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22 views

Disalignment between global standard deviation and mean of rolling standard deviation

Ciao, I am working on proprerties of time series. I was trying to deduce an estimate of standard deviation of a process from the series of rolling standard deviation but I've got some issues when I ...
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1answer
114 views

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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2answers
74 views

Conditional Expectation with Indicator Functions for Poisson Process First Jump Time (Option Pricing PDE)

This is supposed to be for the derivation of a PDE for pricing a specific type of option, from the book 'Nonlinear Option Pricing' (Guyon). The option delivers $g(\tau, X_{\tau})$ at time $\tau$ if $\...
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237 views

Understanding and simulating the jumps in Merton's Jump-Diffusion SDE?

I found this great post deriving the solution to the Merton Jump-Diffusion SDE $$S_t = S_0\exp\left(\left(\mu - \frac{\sigma^2}{2}\right)t + \sigma W_t\right)\prod_{j=0}^{N_t}V_j$$ The first part of ...
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0answers
45 views

SquareRootProcess in QuantLib - Python

I would like to price an American put option using the SquareRootProcess class in QuantLib - Python but it seems that it does not exist. As the underlying follows the following model : $$\rm{d}S_t=...
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1answer
176 views

negative values in geometric brownian motion

A GBM $ \frac{dx}{x} = \mu dx + \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. However, it is not so clear ...
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79 views

Pre-requisites for Finance Mathematics

I would like to pursue research in the areas of Financial Mathematics. Hoping to look into Operations Research, Risk Management and Stochastic Modeling. Anyone got some suggestions on useful resources ...
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3answers
158 views

How to prove that $X_s=\int^s_0 f(u)dW_u$ is independant from $X_t-X_s$

I am asked to prove that $X_s$ and $X_t-X_s$ are independant for $s<t$ then $$X_t=\int^t_0f(u)dW_u$$ for a deterministic function $f$ and brownian motion $W_t$. For the proof I am giving a hint to ...
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0answers
34 views

Stochastic integral representation of $F(T-s,X_s)$-type equations

For $T\in R$ given and fixed consider: $$ {\rm d}F(T-t,X_t)=g(T-t,X_t)\,{\rm d}W_t. $$ where $g(t,x)$ is a given functions and $X_t$ is a given process driven by a brownian motion ($dX_t=(...)dt+(...)...