The tag has no wiki summary.

learn more… | top users | synonyms

1
vote
0answers
42 views

one-step-ahead Stochastic Volatility for 5-minute VWAP prices

I'm trying to run an SV model against prices of Euro/USD. For those not familiar with SV, its a volatility model in which each point gets its own volatility parameter $h_t$ with 3 main parameters that ...
4
votes
1answer
52 views

Discounted risky asset stochastic process problem

$S_t$ is the random variable representing the risky asset price at time $t$. M_t is the riskless asset. They are governed by the equations $\frac{dS_t}{dt}=\mu dt + \sigma dZ_t$ and $dM_t = rM_t ...
1
vote
1answer
86 views

Trading over a Ornstein/AR process

For a OU/AR(1) process is there anyway to analytically calculated most probable period of time the process is likely to diverge from the average, before turning to converge. Basically I am looking ...
1
vote
1answer
78 views

Simulate non-stationary time series with cointegration

how can I simulate/generate two non-stationary time series (with unit root) so that they can be also cointegrated (using R or Matlab). Thanks in advance.
8
votes
1answer
123 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 ...
2
votes
1answer
83 views

Ito integral approximation by Euler?

I was wondering how to find the solution of the following stochastic integral: $$dY_{t}=a(W_{t},Y_{t})dW_{t}+b(W_{t},Y_{t})dZ_{t}$$ or in integral notation ...
1
vote
0answers
55 views

Max Likelihood via Marquardt Optimisation

I asked a related question here: How to apply Levenberg Marquardt to Max Likelihood Estimation I tried the approach suggested it works for some of the parameters but not the variances. I spoke to ...
0
votes
1answer
71 views

Getting the next price of a GBM (Geometric Brownian Motion)

I am writing a program that creates realizations of a GBM. Starting from an initial price, I get the following price with this formula: ...
0
votes
1answer
40 views

Combining BHHH and Levenberg Marquardt

I already asked a question related to this here: How to apply Levenberg Marquardt to Max Likelihood Estimation I know understand how Levenberg Marquardt (LM) can be applied to the objective ...
2
votes
1answer
86 views

How to apply Levenberg Marquardt to Max Likelihood Estimation

In this paper on p315: http://www.ssc.upenn.edu/~fdiebold/papers/paper55/DRAfinal.pdf They explain that they use Levenberg Marquardt (LM) (along with BHHH) to maximize the likelihood. However as I ...
3
votes
1answer
56 views

Solving the Jamshidian Zhu (1997) PCA short rate model

This is my first time posting a question. I have very limited experience in the field of stochastic calculus and interest rate modelling. I have been tasked with implementing the short rate model ...
5
votes
1answer
141 views

PDF Calculation by Fourier Inversion of Characteristic Function for Affine Intensity Process in Matlab

I'm trying to use the Fourier inversion formula to plot the PDF of an Affine Stochastic Intensity Reduced Form Credit Model, given its characteristic function. The characteristic function of an ...
2
votes
1answer
106 views

Monte Carlo for MultiFactor Ornstein Uhlenbeck

I'm following loosely the exposition given in "Monte Carlo Methods in Financial Engineering by Glasserman. For a multifactor OU process: $dX(t)=C(b-X(t))dt+DdW(t)$ Where C and D are d*d matrices ...
2
votes
1answer
112 views

Variance of Multi-Dimensional OU process

I'm trying to implement this model shown here: http://www.sciencedirect.com/science/article/pii/S0304407611000388 As part of the modelling process I have to calculate the unconditional variance of X ...
0
votes
0answers
43 views

Bond pricing by Monte Carlo methods

Im trying to calculate monthly ZCB bond prices with a fixed maturity T, over a period of months via Monte Carlo methods. Here is my attempt: For the first month, the price is $P_{t_0}(0,T) = ...
1
vote
1answer
134 views

Stochastic Calculus in Quantitative analysis

I am an aspiring quant that would like to get a head start learning stochastic calculus, which books FROM EXPERIENCE are the most reader friendly?
1
vote
1answer
79 views

Distribution of minimum of hazard functions

Suppose I have two random variables, $X_1$ and $X_2$, that are independent (but not identically distributed) and assume both have hazard functions $\lambda_1(s)$ and $\lambda_2(s)$, for $s > 0$. ...
1
vote
1answer
80 views

Differenced Brownian Motion covariance

I am having some difficult showing what the following equals, where $x$ and $y$, $x>y$, distinct times: $\mathbb{E}[\Delta W_x \Delta W_y]$ where each $\Delta W_t = W_t - W_{t-1}$. I have ...
4
votes
1answer
169 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 ...
5
votes
3answers
213 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 ...
4
votes
1answer
177 views

Definition of orthogonality and independence for a stochastic processes

Somehow I can't find the explicit definition of when two processes are supposed to be orthogonal or independent anywhere. I think orthogonality and independence should mean the same thing in this ...
1
vote
1answer
164 views

List of financial derivatives Ito's Lemma does not apply

According to Ito's Lemma there is no restriction on the continuity of the stochastic process. The restrictions are on the continuity of the pay-off so that second derivatives with respect to ...
1
vote
3answers
432 views

What does it mean by autocorrelation coefficient near 1?

It is said that the time series has a stochastic trend if the first autocorrelation coefficient will be near 1. Q1) What does it mean by the above statement? Q2) How do we calculate the first ...
5
votes
0answers
65 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 ...
0
votes
2answers
70 views

Wiener process proof

Can someone prove to me how $dW_t=W_t-W_s$, where $t=s+1$, the difference of the Wiener process eventually equates to $dW_t=z*(dt)^{(1/2)}$ where $z$ is standard normal, $N(0,1)$ in the following ...
0
votes
0answers
23 views

Gibson & Schwartz (1190) - Time series empirical properties and Stochastic Process assumed

Gibson and Scwhartz in their paper "Stochastic convenience yield and the pricing of oil contingent claims" assume a log normal process for the spot price. They later claim to justify this process ...
4
votes
0answers
90 views

Finding the dynamics of a dividend paying asset under arbitrary numeraire

Assuming I have a dividend paying asset $S$ with dividend process $D$. Now I would like to use the bank account process $B$ as numeraire and determine the dynamics of $S$ under the the corresponding ...
4
votes
1answer
87 views

The distribution of jump gaps for Levy processes

Assume $X_{t}$ is a Levy process with triplet $(\sigma^{2}, \lambda, \nu)$, here $\nu$ is the Levy measure of $X_{t}$. Define $\tau_{1},\tau_{2},\dots$ be the time gap between the successive jumps ...
1
vote
0answers
36 views

Weak convergence of Lookback payoff with correction term

In this article on the Multilevel Monte Carlo method on page 8, http://people.maths.ox.ac.uk/gilesm/files/mcqmc06.pdf, Giles uses a correction term to improve the weak convergence rate of the lookback ...
3
votes
0answers
145 views

Time series (stochastic process) estimating parameters using characteristic function

I have a time series of assets ${A_1, A_2, ..., A_n}$, which is described by a sophisticated distribution having the following characteristic function: $\phi(u; t;\theta)$, where $\theta$ is a vector ...
1
vote
1answer
173 views

Examples of non-increasing variance of a time homogeneous Markovian process

This is an edit to the previous question, on stationary process, which was answered by Richard below. Let $x_t$ be a zero mean, time homogeneous Markovian process over time $t$ starting from ...
4
votes
2answers
179 views

Itô diffusion processes in finance with unknown distribution at a terminal value

In several papers it is argued that for many Itô diffusion processes, $$dX_t = a(t,X_t)dt+b(t,X_t)dB_t,$$ in mathematical finance the distribution of $X_T$ for fixed $T>0$ is unknown, which makes ...
2
votes
1answer
203 views

Closed form european option prices for a variance gamma process with a randomly distributed drift, volatility, and variance rate

Does an option pricing model with a closed form European option price exist that takes into account randomly distributed drift, volatility, and variance rate? I prefer a modification to the variance ...
0
votes
0answers
39 views

Index Price Simulation Volatility Bands

I am building a simple stochastic model for learning purposes in excel. I took daily data for the SPY since 1/1/1993. I computed the daily log returns and found that the SPY has had an average daily ...
7
votes
2answers
313 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 ...
5
votes
1answer
177 views

From $AR(p)$ to SDE

Let the Vasicek model to be $$\Delta r_{t}=k(\theta - r_{t-1})\Delta t+\sigma\Delta z_{t}$$ Due to the fact that $$\Delta r_{t}=r_{t}-r_{t-1}$$ if you let $\Delta t=1$, it is easy to see by ...
4
votes
6answers
4k 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 ...
1
vote
2answers
414 views

Brownian motion - first passage time

Can anyone point me to the expression for the first passage time for a geometric Brownian motion process X(t) as a function of the starting point, threshold, drift and diffusion parameters. I am ...
4
votes
0answers
91 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 ...
1
vote
2answers
126 views

how to extend lognormal model so that $\sigma$ is correlated to $\mu$?

Consider a log-normal model, $dx / x = \mu dt + \sigma dW$, where $W(t)$ is a Wiener process. Let's say $\mu$ and $\sigma$ change with time, slowly, so we note them by $\mu(t)$ and $\sigma(t)$. ...
0
votes
0answers
98 views

close form for stochastic integral

I am new to stochastic calculus. Can I know how to compute the close-form solution for $$\int_0^t \exp(\alpha s - \sigma W_s) \; ds$$ and $$\int_0^t \exp(\alpha s - \sigma W_s) \; dW_s.$$ I encounter ...
0
votes
2answers
208 views

Change option B&S pricing

Consider a market composed by two stocks whose prices $X$ and $Y$ are given by B&S diffusion $$dX_t= \mu X_t dt+ \sigma X_tdW_t$$ $$dY_t= \mu Y_t dt+ \sigma Y_tdB_t$$ Supposing the market is ...
7
votes
2answers
279 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 ...
3
votes
1answer
477 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 ...
0
votes
0answers
118 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
136 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 $f(\cdot)$ ...
14
votes
2answers
734 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 ...
3
votes
2answers
260 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
165 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 $- ...
4
votes
1answer
214 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 $- ...