The stochastic-processes tag has no wiki summary.
12
votes
1answer
258 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 ...
0
votes
0answers
48 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
votes
0answers
73 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
votes
3answers
442 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 ...
5
votes
2answers
350 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
votes
1answer
144 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
votes
2answers
95 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
votes
1answer
123 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
votes
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
votes
0answers
143 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
votes
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
vote
0answers
142 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
votes
3answers
313 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
votes
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
votes
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
votes
1answer
195 views
What are $d_1$ and $d_2$ for Laplace?
What are the formulae for d1 & d2 using a Laplace distribution?
0
votes
1answer
126 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
votes
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
votes
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
$$ \nu \left( dx\right) = A ...
6
votes
0answers
120 views
Consistency of economic scenarios in nested stochastics simulation
I am interested in references on research regarding the consistency of economic scenarios in nested stochastics for risk measurement.
Background:
Pricing by Monte-Carlo:
For pricing complex ...
5
votes
2answers
147 views
Simulation of GBM
I have a question regarding the simulation of a GBM. I have found similar questions here but nothing which takes reference to my specific problem:
Given a GBM of the form
$dS(t) = \mu S(t) dt + ...
12
votes
5answers
2k views
Is the stock price process a martingale or a Markov process?
Some people claim that the data-generating process for stocks is a "martingale" and that is has the "Markov property".
Are they unrelated? Is it that the Markov property implies some sort of ...
5
votes
2answers
289 views
What is the mean and the standard deviation for Geometric Ornstein-Uhlenbeck Process?
I am uncertain as to how to calculate the mean and variance of the following Geometric Ornstein-Uhlenbeck process.
$$d X(t) = a ( L - X_t ) dt + V X_t dW_t$$
Is anyone able to calculate the mean ...
2
votes
1answer
240 views
Regime switching in mean reverting stochastic process
Let you have a mean reverting stochastic process with a statistically significant autocorrelation coefficient; let it looks like you can well model it using an $ARMA(p,q)$.
This time series could be ...
5
votes
2answers
250 views
How to simulate stock prices using variance gamma process?
I want to simulate stock prices with the variance gamma process. The model is given by:
$S_T=S_0 e^{ {[}(r-1)T + \omega + z{]}} $
where
$S_0= $ starting value
$T= $ Time
...
3
votes
1answer
295 views
How to simulate a Merton Jump Diffusion process?
I am talking about the Merton Jump Diffusion model, on this page, where they give the following formula:
$$ dS_t = \mu S_t dt + \sigma S_t dW_t + (\eta-1) dq$$
where $W_t$ is a standard brownian ...
4
votes
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
votes
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
votes
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
votes
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
votes
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
votes
2answers
758 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
votes
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
votes
0answers
127 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
597 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
votes
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
377 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
votes
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
votes
2answers
766 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
votes
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 ...
