Hot answers tagged stochastic-processes
13
From what I remember, there is no real relation between Markov and Martingale, and my intuition was confirmed by this post.
Basically, it says that you can say neither of the following:
If A is Markov, then A is a martingale.
If A is a martingale, then A is Markov.
further down the post, you can find two counter examples:
$dX_t = a dt + \sigma dW_t$ is ...
10
I will defer to others answering the parts of your question concerning the relationship between Markov processes and martingales (@SRKX has already given a good explanation of the relationship) and concerning statistical testing. Broadly, however, it is not possible to "prove" either assumption, but only to fail to reject them. A Non-Random Walk Down Wall ...
8
"Treshold Garch" or T-Garch models are designed to capture this asymmetry. See this exposition by U. Chicago's Ruey Tsay who has a terrific text on time-series models in "Analysis of Financial Time Series".
You can use the structure of the T-Garch models to simulate data with this property.
There is a package called fGarch that creates APARCH models. A ...
7
These patterns are of course well-known enough to have been "priced in" to the financial markets. Jump diffusions are a classic way to capture the phenomenon, and often have closed-form option pricing formulas associated with them. The implied option skew, for example, gets a lot flatter when you use a JD model.
Jump diffusions are often combined with ...
7
The way you do it in the first place is a discretization of the Geometric Brownian Motion (GBM) process. This method is most useful when you want to compute the path between $S_0$ and $S_t$, i.e. you want to know all the intermediary points $S_i$ for $0 \leq i \leq t$.
The second equation is a closed form solution for the GBM given $S_0$. A simple ...
6
I like Richard's answer, but I think we can compute the mean and the variance of $\int_0^T W_t dt$ by ourselves using Ito's lemma. Let $f(W_t, t) = t W_t$.
$$
d( t W_t ) = W_t dt + t dW_t .
$$
Integrating both sides, and re-arranging the terms, we get
$$
\int_0^T W_t dt = T W_T - \int_0^T t dW_t \, .
$$
We'll be using Ito's isometry formula $\mathbb{E} ...
6
The model for the stock is the Bachelier model with the solution
$$
S(t) = S(0) + \sigma W(t)
$$
Thus the law of the stock $S(t)$ is Gaussian with mean $S(0)$ and variance $\sigma^2 t$.
For average process $Z(T)$ is thus the average of linear Brownian motion, we can rewrite this as
$$
Z(T) = \frac{1}{T} \int_0^T S(0) + \sigma W(t) dt = S(0) + ...
6
I have low frequency data (daily) from which I want to construct high frequency data, going though all the lower frequency sampling points.
Bad idea in my opinion. I don't really know why you really want to do this (what's are you going to do with the generated data). If it's for backtesting purposes, it's a really bad idea as there are so many ...
5
In general, if you have a process that you can write under the form $F(B_t,t)$ where $F$ is $\mathcal{C}^{2,1}$ then Itô's lemma gives you the drift term and diffusion term of $dF$. Then if the resulting SDE has a null drift (that's where Black Scholes PDE comes from), and you get a only local martingale. For it to be a proper martingale you can look at ...
5
The best I have seen so far is William Wheaton's work in this area. I don't know how much is described in his papers but he and Torto created a system that combined factor models for things like local and national price indexes with specific economics of commercial real estate ventures (such as balloon payments on construction milestones and the like).
The ...
5
Apparently yes, (I haven't verified the math but have no reason to doubt it).
For this simple case you can find a closed form in the following paper:
Jeff A. BILMES: What HMM can do
The closed form is given on part 4.4 of the paper but the whole thing is worth reading as it clearly shows the main properties of these models.
You can also note that ...
5
For completeness, let's restate that the discrete case goes like this:
$$\Delta S_t = S_{t+\Delta t}- S_t = \mu S_t \Delta t + \sigma \sqrt{\Delta t} Z_t $$
with $Z_t \sim \mathcal{N}(0,1)$.
What you are doing in your case (although there is a typo in your formula) is to use the exact solution of the SDE to model the move between two points of $S$.
...
4
Okay so I'll take Jase answer and format it properly so that it answers your question and it will be useful for users in the future.
For clarity, let me restate the dynamics of the Modified Ornstein-Uhlenbeck model using the more common notation:
$$dS_t = \theta (\mu-S_t)dt + \sigma S_t dW_t$$
This blog post provides a closed form solution:
$$ S_t = S_0 ...
4
Check out these resources:
The book Levy Processes in finance.
This paper basically enabling you to use any distribution for asset prices: Option Valuation Using the Fast Fourier Transform
4
I believe your problem is that you're assuming all Lévy processes are stable with exponent $2$. Here is what happens if we try to use your argument: Let $X$ be a Lévy process (that is a martingale, for simplicity). At time $t$, for any $N$, we have
$$
X_t \sim\sum_{i=1}^N X^i \left(\frac{t}{N}\right),
$$
with each $X^i \left(\frac{t}{N}\right)$ i.i.d. and ...
4
Hi here are my two cents,
It is true that BSDE's framework represents a very powerful theoretical tool to attack abstract problems in mathematical finance. Nevertheless to my knowledge they are very rarely used in practice for at least three reasons. First they are very "unnatural" in their expression (integrating in the future in time and still being ...
3
Note: There is a typo in your third equations. Instead of $S(u)$ it should be $S(t_{i})$ and in place of $S(t)$ there should be $S(t_{i+1})$.
In fact, given $S(t_{i})$ we have that
$$S(t_{i+1}) = S(t_{i}) \exp\left( (\mu - \frac{1}{2} \sigma^2) (t_{i+1} - t_{i}) + \sigma (W(t_{i+1}) - W(t_{i})) \right)$$
is the exact solution of the SDE. Hence, the ...
3
Just following Musiela Rutkowski (the link redirects to Amazon). The risk neutral measure is derived form imposing that the present value of a self financed portfolio (i.e.; no infusion or withdraw of money) is a martingale. A portfolio can be seen as a stochastic process where its value at time $t$ is given by
$$
V_t = \phi^0_tP_t + \phi^1_tS_t\ ,
$$
...
3
I think a simple solution is to try to construct a Brownian motion $W_t$ through known points (e.g., $W_0 = W_1 = 0$); it is also known as a Brownian Bridge [ http://en.wikipedia.org/wiki/Brownian_bridge ].
See also question 3 in http://www.math.nyu.edu/faculty/goodman/teaching/StochCalc2012/assignments/assignment4.pdf .
3
This link is to a book that covers this exact question:
http://onlinelibrary.wiley.com/doi/10.1002/9783527610006.ch9/summary
Summary: the models that map to both stock markets and currency markets are those that have an autoregressive feature (curreny markets commonly exhibit this feature, limiting the choice of models that apply to both currencies and ...
3
In fact there is an exhaustive paper on this issue available now:
"The Trend is not Your Friend! Why Empirical Timing Success is Determined by the Underlying’s Price Characteristics and Market Efficiency is Irrelevant"
by Peter Scholz and Ursula Walther, Frankfurt School Working Paper, CPQF No. 29, 2011
Fascinating read - highly recommended!
3
Here couple points that at least helped to formulate a daily guide for myself:
Losses are just what they are, losses. You return tomorrow to play again. But bankruptcy means game over, you are done. Thus such event is to be avoided at all cost.
Long-term, equities exhibit positive drift and have outperformed other competing asset classes. However, the ...
3
It's not possible with a simple linear transformation like the one you mentioned: since scale and thus the distance between mean and median are required to change, either the mean or the median will not be preserved.
Therefore you must use nonlinear transformations, which will complicate quite a bit mantaining skew and kurtosis and imho will not be ...
2
Martingale and Markov process are both stochastic processes where the sequences of random variables are not entirely independent, and their differences are:
In martingale, the expectation of the next value IS the present value, so this property is sometimes called 'fair game'.
In Markov process, the expectation of the next value only DEPENDS ON the present ...
2
For Itô Processes $dX(t) = \mu(t) \mathrm{d}t + \sigma(t) \mathrm{d}W(t)$ you have the result that (under appropriate assumptions which ensure that the local martingale is a martingale, e.g. $E( (\int \sigma(t)^2 \mathrm{d}t )^{1/2} ) < \infty$, etc.): $X$ is a martingale $\Leftrightarrow$ $\mu(t) = 0$.
So in order to check if a process $X$ is a ...
2
An insurer might model the filing of claims as a Poisson process, but the cumulative amount of the claims as a compound Poisson process.
As an example, suppose a company has issued a large large number of auto liability policies that are geographically dispersed and have identical limits and driver risk profiles. The incidence of claims being made by ...
2
This depends. I am not aware of a general risk neutral pricing framework applying to all asset classes and/or stochastic processes. In order to reach more general statements about risk neutral pricing you need to consider jumps and autocorrelation depending on the asset regarded, maybe stochastic interest rates and/or volatility. If I remember correctly, in ...
2
ad) "Is it normal to assume no other drift?"
Under measure P you might have drift. You could use it as a working assumption, but in general indices drift every now and then. So, no, usually you do not assume away the drift.
"The index is described as "following a geometric Brownian motion", which to me says that the there is no other drift going on" ...
2
As far as I can tell, you've essentially written the model that you are concerned with. The only difference is that you would instead have $\theta_{i}$ when $s_{t}=i$ where $s_{t}$ is a latent variable that reflects the probability of being in state $i$. You would also need to include the dynamics that drive the probability transitions as another part of ...
2
a) you can run a Monte Carlo simulation in which you model stock price movements and then you can look at the future pay off as a function of path dependency into which you incorporate your stop losses and take profits. Done this over many iterations you will be able to derive your probabilities. Caveat here is your result will be strongly dependent on your ...
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