The stochastic-processes tag has no wiki summary.
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0answers
36 views
Exact value of mean reversion rate knowing terminal value of the process
Let you have the following mean reverting process:
$\text{d}x_{t}=a(\theta-x_{t})\text{d}t$,
where the diffusion term is absent, that is this process is not stochastic.
Let you know the value of ...
3
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0answers
62 views
Is it random walk?
I would like to ask a question about random walk. Campbell, Lo & Mackinlay defined the random walk, in the following way (RW3):
$$
cov[f(r_{t}),g(r_{t+k})]=0,\qquad k\neq0
$$
for all $f(\cdot)$ ...
5
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3answers
435 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 ...
10
votes
1answer
206 views
State-space representation 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 ...
5
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2answers
341 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 ...
4
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1answer
142 views
Non-arbitrage theory and existence of a risk premium
Consider a probability filtred space $(\Omega, \mathcal F, \mathbb F, \mathbb P)$, where $\mathbb F = (\mathcal F_t)_{0\leq t\leq T}$ satisfing the habitual conditions and isgenerated by $1 d $- ...
3
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2answers
89 views
Transformation to reduce standard deviation without changing median
Consider some negative skew and high kurtosis return time-series $X_t$. I do not know the functional form of the pdf of $X_t$ and have about 150,000 data points.
Suppose that I was to create an ...
3
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1answer
121 views
Foward-start option pricing
Consider a probability filtred space $(\Omega, \mathcal F, \mathbb F, \mathbb P)$, where $\mathbb F = (\mathcal F_t)_{0\leq t\leq T}$ satisfing the habitual conditions and is generated by $1 d $- ...
3
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1answer
163 views
Stochastic modeling of stock price process
Apart from the model of Geometric Brownian motion is there any other "widely accepted" stochastic model to characterize the dynamics of a stock price process?
2
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0answers
140 views
Does the geometric Ornstein-Uhlenbeck process have stationary variance?
I know that the long run variance of the standard OU process is
$\lim_{s\rightarrow \infty}\mbox{Var}(P_{t+s}|P_t) = \frac{\sigma^2}{2\theta}$
I'm using the geometric version of the process. I ...
11
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1answer
219 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 ...
1
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0answers
139 views
Call options portfolio: what would the underlyings' moments to be maximized?
Let you have only three underlyings, like SPY, TLT and GLD, and you want to buy $n_{1}$ Call options on SPY, $n_{2}$ Call options on TLT and $n_{3}$ Call options on GLD... with a limited budget, that ...
3
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3answers
312 views
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 $\phi(T/N)$ ...
6
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1answer
172 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 ...
4
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5answers
700 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 ...
2
votes
1answer
105 views
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. ...
2
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1answer
195 views
What are $d_1$ and $d_2$ for Laplace?
What are the formulae for d1 & d2 using a Laplace distribution?
0
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1answer
125 views
how to quantify non-fundamental risk if variance is 100% discounted?
If there's better vocabulary, forgive me.
If you were required to ignore variance as risk, how would you quantify non-fundamental risk?
Many thanks in advance!
4
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1answer
137 views
Certain probability statement in discrete mathematical finance
Le'ts suppose the following setting:
We have a filtred probability space $(\Omega,\mathcal{F},P,\{\mathcal{F}\}_{k=0,1})$ and an adapted $\mathbb{R}^d$ valued process $S=(S^1,\dots,S^d)$. Let ...
4
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0answers
80 views
Simple question concerning Jump process (Lévy process) model for a risky actif price process [closed]
Consider $X= \left( X_t \right)_{t\geq 0}$ is a Lévy process whose characteristic triplet is $\left( \gamma, \sigma ^2, \nu \right)$ and where its Lévy measure is
$$ \nu \left( dx\right) = A ...
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0answers
117 views
Consistency of economic scenarios in nested stochastics simulation
I am interested in references on research regarding the consistency of economic scenarios in nested stochastics for risk measurement.
Background:
Pricing by Monte-Carlo:
For pricing complex ...
5
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2answers
145 views
Simulation of GBM
I have a question regarding the simulation of a GBM. I have found similar questions here but nothing which takes reference to my specific problem:
Given a GBM of the form
$dS(t) = \mu S(t) dt + ...
12
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5answers
2k views
Is the stock price process a martingale or a Markov process?
Some people claim that the data-generating process for stocks is a "martingale" and that is has the "Markov property".
Are they unrelated? Is it that the Markov property implies some sort of ...
5
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2answers
289 views
What is the mean and the standard deviation for Geometric Ornstein-Uhlenbeck Process?
I am uncertain as to how to calculate the mean and variance of the following Geometric Ornstein-Uhlenbeck process.
$$d X(t) = a ( L - X_t ) dt + V X_t dW_t$$
Is anyone able to calculate the mean ...
2
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1answer
237 views
Regime switching in mean reverting stochastic process
Let you have a mean reverting stochastic process with a statistically significant autocorrelation coefficient; let it looks like you can well model it using an $ARMA(p,q)$.
This time series could be ...
5
votes
2answers
247 views
How to simulate stock prices using variance gamma process?
I want to simulate stock prices with the variance gamma process. The model is given by:
$S_T=S_0 e^{ {[}(r-1)T + \omega + z{]}} $
where
$S_0= $ starting value
$T= $ Time
...
3
votes
1answer
286 views
How to simulate a Merton Jump Diffusion process?
I am talking about the Merton Jump Diffusion model, on this page, where they give the following formula:
$$ dS_t = \mu S_t dt + \sigma S_t dW_t + (\eta-1) dq$$
where $W_t$ is a standard brownian ...
4
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1answer
1k views
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:
...
2
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0answers
101 views
Measure change in a bond option problem
This is not a homework or assignment exercise.
I'm trying to evaluate $\displaystyle \ \ I := E_\beta \big[\frac{1}{\beta(T_0)} K \mathbf{1}_{\{B(T_0,T_1) > K\}}\big]$, where $\beta$ is the ...
3
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2answers
309 views
How to create a Stochastic Process through pre specified points?
I want to create a random (quasi random) process which goes through pre determined points and constraints. E.g. I have a daily price series but want to generate intra-day prices with the same OHLC ...
3
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1answer
198 views
How to calculate probability of touching a take-profit without touching a stop-loss?
How to calculate probability of touching a take-profit without touching a stop-loss (no-dividend stock, infinite time)?
2
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0answers
75 views
Difference between kappa and delta in mixed-effects model
(This question is a crosspost from Cross Validated)
I have a following stochastic model describing evolution of a process (Y) in space and time. Ds and Dt are domain in space (2D with x and y axes) ...
8
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2answers
757 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?
3
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1answer
171 views
How to measure a non-normal stochastic process?
If I understand right, Itô's lemma tells us that for any process $X$ that can be adapted to an underlying standard normal Wiener measure $\mathrm dB_t$, and any twice continuously differentiable ...
6
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0answers
126 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) = ...
5
votes
2answers
590 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 ...
8
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1answer
282 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 ...
3
votes
2answers
127 views
What mathematical characteristics are required from the asset price process in order to stay within the RNP framework?
I'm currently doing a course in derivatives pricing and I'm having some trouble wrapping my head around the sweet spot where theory meets reality in terms of Risk Neutral Pricing.
I know that the ...
8
votes
3answers
375 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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1answer
117 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 ...
15
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2answers
765 views
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 ...
12
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3answers
624 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 ...
