stochastic processes is a collection of random variables representing the evolution of some system of random values over time.

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Statistical properties of stochastic processes for moving average trading to work

Common wisdom holds it that a moving average approach is more successful than buy-and-hold. There is quantitative evidence for that across different asset classes (see e.g. this book, or this paper ...
17
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5answers
6k views

Is the stock price process a martingale or a Markov process?

Some people claim that the data-generating process for stocks is a "martingale" and that is has the "Markov property". Are they unrelated? Is it that the Markov property implies some sort of ...
16
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3answers
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How to simulate stock prices with a Geometric Brownian Motion?

I want to simulate stock price paths with different stochastic processes. I started with the famous geometric brownian motion. I simulated the values with the following formula: $$R_i=\frac{S_{i+1}-...
15
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4answers
2k views

Strictly local martingales: what is the intuition behind them?

A process $X_t$ is a local martingale if for each increasing sequence of stopping times $\{\tau_k,k=1,2,...\}$ the stopped process is a martingale. All true martingales are local martingales, but the ...
15
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2answers
2k views

Parameter estimation of Ornstein–Uhlenbeck and CIR processes

I would like to estimate Ornstein–Uhlenbeck process' parameters via Kalman filter. My process is the following one: $\text{d}x_{t}=\alpha(\theta-x_{t})\text{d}t+\sigma\text{d}W_{t}$ I'm interested ...
14
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3answers
703 views

Discrete-time model: stock dynamics

I am working in the area of probability theory and for a case study I would like to make some calculations in finance. Since I am developing theory for the discrete time, I am interested in models for ...
12
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1answer
435 views

Consistency of economic scenarios in nested stochastics simulation

I am interested in references on research regarding the consistency of economic scenarios in nested stochastics for risk measurement. Background: Pricing by Monte-Carlo: For pricing complex ...
11
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1answer
387 views

Are BSDE's used in practice?

In the academic applied probability/math finance community, Backwards Stochastic Differential Equations (BSDE's) are extremely popular, and they provide a single framework for several different ...
11
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2answers
277 views

Deriving the definition of stochastic integrals with respect to Ito processes from first principles

When I first encountered the definition of integrals with respect to Ito processes (Shreve's Stochastic Calculus for Finance Vol II), I didn't think twice. However, I wanted to see if the definition ...
10
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1answer
447 views

Processes used in quant finance

What are the main stochastic processes (and their SDE) used in quant finance? For example to model currency prices, stock prices, etc.
10
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2k views

What are some examples of Compound Poisson processes in insurance?

I'm writing the Bachelor thesis but I need some information. I need to find some practical examples and applications of the Compound Poisson Process in insurance. Does anyone have any good examples?
10
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4answers
483 views

Why is Brownian motion merely 'almost surely' continuous?

Why is Brownian motion required to be merely almost surely continuous instead of continuous? For example, this is stated as condition 2 in this article in section 1, Characterizations of the Wiener ...
9
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1answer
151 views

Simulating property price index

I am trying to write a Monte Carlo simulation to calculate risk associated with some property based products. What is the most reasonable stochastic process to model property price index? Do people ...
9
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1answer
417 views

Is there a closed-form solution for the partial autocorrelation function of a Markov regime-switching process?

Consider a Markov Regime-switching process $X_{t}$ with $k$ regimes represented by $s_{t}$ such that $$X_{t}=\mu\left(s_{t}\right)+\epsilon_{t}$$ and $$\epsilon_{t}\sim N\left(0,\sigma^{2}\left(s_{...
9
votes
3answers
374 views

Why do people always seek finite-variance models for option pricing

For the purpose of getting fatter tails than the Guassian, I have seen people for example use $\alpha$-stable processes to model the stock. But in that case they end up using 'tempered' versions of ...
9
votes
2answers
349 views

Why does Black-Scholes equation hold on continuation region of American Option?

Explanation for Put Option: $ \frac{\partial V}{\partial t}+ \mathcal{L}_{BS} (V) = 0 $, where $\mathcal{L}_{BS} (V) = \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + (r-q) S \frac{\...
9
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1answer
153 views

How to test that a distribution has infinite mean?

I observe a sample from a distribution that I expect to be the hitting time $$\tau = \inf\{t>0| X(t)>a\}$$ where $X(t)$ is a Lévy process with $X(0)=0$ and $a$ is some constant. $X$ is not a ...
9
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0answers
122 views

2-state HMM / ARMA process?

I have issues with this problem: Let $\{X_t, t\in \Bbb N\}$ be a 2-state stationary Markov chain, with transition $M$ (and $M(1,2)\neq 0 \neq M(2,1)$), let $\{W_t, t\in \Bbb N\}$ be a strong Gaussian ...
8
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3answers
581 views

How to test for and how to simulate price rise/fall asymmetry in the stock market

One of the stylized facts of financial time series seems to be a fundamental asymmetry between smooth upward movements over longer periods of time followed by abrupt declines over relatively shorter ...
8
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4answers
1k views

How to simulate stock prices using variance gamma process?

I want to simulate stock prices with the variance gamma process. The model is given by: $S_T=S_0 e^{ {[}(r-1)T + \omega + z{]}} $ where $S_0= $ starting value $T= $ Time $\omega=\frac{T}{\nu}ln(1-...
8
votes
2answers
3k views

What is the mean and the standard deviation for Geometric Ornstein-Uhlenbeck Process?

I am uncertain as to how to calculate the mean and variance of the following Geometric Ornstein-Uhlenbeck process. $$d X(t) = a ( L - X_t ) dt + V X_t dW_t$$ Is anyone able to calculate the mean ...
7
votes
5answers
836 views

Geometric Brownian motion - Volatility Interpretation (in the drift term)

A Geometric Brownian motion satisfying the SDE $dS_t = rS_t dt+\sigma S_t dW_t$ has the analytic solution $$S_t = S_0\exp\left\{\left(r-\frac{\sigma^2}{2}\right)t\right\}\exp\{\sigma W_t\}$$ Recently ...
7
votes
2answers
2k views

Simulation of GBM

I have a question regarding the simulation of a GBM. I have found similar questions here but nothing which takes reference to my specific problem: Given a GBM of the form $dS(t) = \mu S(t) dt + \...
7
votes
2answers
335 views

Filtration and measure change

I asked this question in math stackexchange but to no avail. So i'm trying the luck here. I'm reading Steven E. Shreve's "Stochastic calculus for finance II", and find myself not really understand ...
7
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6answers
10k views

Why non-stationary data cannot be analyzed?

Searching online, i found out that non-stationary cannot be analyzed with traditional econometric techniques as in case of non-stationarity some basic model assupmtions are not met and correct ...
7
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1answer
316 views

Use of Girsanov's theorem in bond pricing

Assume that we want to calculate the time $t=0$ price of a bond: $B(0,T) = E_P[\exp(-\int_0^T r_s ds)]$, where $r$ is the interest rate following the SDE $dr_t=k(\theta-r_t)dt+\sigma dB_t=b(r_t)dt+\...
7
votes
1answer
596 views

How to compute the Radon-Nikodym derivative?

Suppose $B(t)$ is a standard Brownian motion, and $B_{1}(t)$ is given by $dB_{1}(t)=\mu dt+dB(t)$. Suppose $P$ is the Wiener measure induced by $B(t)$ on the $C[0,\infty)$, and $P_{1}$ is the Law ...
6
votes
8answers
5k views

Why should we expect geometric Brownian motion to model asset prices?

Disclaimer: I am a complete ignoramus about finance, so this may be an inappropriate forum for me to ask a question in. I am a mathematician who knows nothing about finance. I heard from a popular ...
6
votes
4answers
123 views

Stochastic process with non-independent increments

All stochastic process I see always have independent increments. It is true for: standard brownian motion geometric brownian motion (?) Ornstein Uhlenbeck (?) in general, Levy process etc. What ...
6
votes
2answers
1k views

What is the average stock price under the Bachelier model?

Let's say stock price follows following process: $$dS(t) = \sigma dW(t)$$ where $W(t)$ is Standard Brownian motion. The initial level for the stock is $S(0)$. Define the average of stock price $Z(t)...
6
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6answers
922 views

Why the expected return rate of a stock has nothing to do with its option price?

OK, I admit that this is a frequently asked question. But I couldn't find a satisfying answer after I read the explanations of books, went through the derivations of B-S formula, and searched answers ...
6
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3answers
4k views

How to use Itô's formula to deduce that a stochastic process is a martingale?

I'm working through different books about financial mathematics and solving some problems I get stuck. Suppose you define an arbitrary stochastic process, for example $ X_t := W_t^8-8t $ where $ W_t ...
6
votes
2answers
3k views

How does one go from measure P to Q(risk-neutral) when modeling an asset paying dividends?

I am really having a terrible time applying Girsanov's theorem to go from the real-world measure $P$ to the risk-neutral measure $Q$. I want to determine the payoff of a derivative based an asset ...
6
votes
1answer
99 views

Proof that the stopping time for a Brownian Motion is finite for given target levels

Given a standard brownian motion $W_t$ and defining $\tau$ as: $\tau :=inf\{t\geq0:W_t=1$ or $W_t=-2\}$ The proof below shows that the stopping time is finite: $P(\tau < t) \geq (|W_t|>2)\\$ ...
6
votes
4answers
2k views

Is a stationary process necessarily mean-reverting?

Intuitively, a stationary stochastic process needs to be mean-reverting. This should follow immediately from the definition of stationarity: the mean of the process needs to be constant over time, so ...
6
votes
1answer
277 views

Upper bound concerning Snell envelope

Consider a non-negative continuous process $X = \left (X_t \right)_ {t\geq 0}$ satisfying $ \mathbb E \left \{ \bar X \right\}< \infty $ (where $ \bar X =\sup _{0\leq t \leq T} X_t $) and its ...
6
votes
1answer
48 views

Why does jump process has to be Cadlag and not the other way around

In all books and references that I have been exposed to, the jump processes have been defined to be Cadlag(right continuous with left limits). But no one has explained why this is the preferable case, ...
6
votes
1answer
125 views

pdf of simple equation, compound Poisson noise

I would like to find the probability density function (at stationarity) of the random variable $X_t$, where: \begin{equation*} dX_t = -aX_t dt + d N_t, \end{equation*} $a$ is a constant and $N_t$ is a ...
6
votes
1answer
112 views

Modelling EUR/USD with Ornstein-Uhlenbeck + jumps?

I'm trying to simulate a process as close as possible to EUR/USD of the ten past years. I've used a Ornstein-Uhlenbeck process: $$d X_t = -\theta (X_t - \mu) d t + \sigma d B_t$$ with the ...
6
votes
2answers
112 views

How to price an European Call/Put Option of a jump difussion Process?

Lets have the next jump difussion Stochastic Process: $$S_t = S_0 e^{\sigma W_t + (v-\frac{\sigma ^2}{2})t}\prod_{i=1}^{N_t}(1+J_i)$$ where $W_t$ is the Brownian Motion, hence $G_t \equiv e^{\sigma ...
6
votes
2answers
234 views

How to get an analytic result for option price based on this model?

I defined such a model for stock price (1).... $$dS = \mu\ S\ dt + \sigma\ S\ dW + \rho\ S(dH - \mu) $$ , where $H$ is a so-called "resettable poisson process" defined as (2).... $$dH(t) = dN_{\...
6
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0answers
87 views

Expectation over Markov Process and discrete Ito integral (discrete stochastic calculus)

I am doing a research on communication protocol design. A file of $n$ blocks is transferred in several rounds and $R_i$ denotes the number of blocks received in the $i$-th round. The sender sends $n-...
6
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0answers
119 views

Why is it useless to model stochastic volatility when pricing Vanilla style derivatives?

With respect to the answer by user AFK in Ideas about Stochastic volatility models. I am specifically interested in interest rate options (IR Caps/Floors and Swaptions).
6
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2answers
312 views

Ito, Stochastic Exponential and Girsanov

This is a two-part question relating to the change of measure density used in Girsanov and secondly to the Stochastic Exponential. Whilst reading notes relating to Girsanov it is stated that the ...
6
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0answers
95 views

Transition densities in the Heson model

Knowing the Characteristic function $\Phi_{T,t} = \mathbb{E} [ e^{i u S_T} | S_t, V_t]$ (or equivalently, the Laplace transform) of an affine process, it's possible to know the distribution of the ...
6
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0answers
69 views

What kind of errors arise when I fit ARMA(1,1) to data generated from ARMA(1,1)-GARCH(1,1) process?

As far as I know estimates of parameters of ARMA(1,1) are asymptotically optimal when fitted to data from ARMA(1,1)-GARCH(1,1) process, and only their variance increase, so when we assume large ...
5
votes
3answers
577 views

Why do we usually model returns and not prices?

I think this is a quite similar question for most of you, however it is not completely understandable for me at the moment: Why do we usually use returns and not prices to model financial data in ...
5
votes
5answers
1k views

How to improve the Black-Scholes framework?

Since the distribution of daily returns are obviously not lognormal, my bottom line question is has BS been reworked for a better fitting distribution? Google searches give me nada. The best dist I'...
5
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1answer
164 views

Why is GARCH more often applied in risk analysis than stochastics?

I am trying to look out for something I can engage in for my final year project (M.Sc) but my interests lie more in risk analysis (specifically credit risk). I have tried searching the web but really ...
5
votes
2answers
1k views

Monte Carlo simulating Cox-Ingersoll-Ross process

The CIR process is given by the SDE $$ \mathrm dr_t = \theta(\mu-r_t)\mathrm dt + \sigma\sqrt{r_t}\mathrm dW_t $$ where $W_t$ is a Brownian motion. I am interested in finite-difference schemes of ...