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5

Brownian motion - because it is simple, and results in intuitive closed form solutions, and it's not a terrible description of asset prices, especially when employed in high-frequency event time. Geometric - because the returns compound, and equities cannot go below zero due to the fact that they are limited liability corporations There are many, many ...


5

If at first you don't have a model at all, then geometric Brownian motion is not bad. As others before me said: log-returns are normally distributed in this model. This is debatable and there are times and markets where this is not true. There is more than enough research about this. But why is a model based on Brownian motion not that bad? The reason is ...


4

To provide a straight forward answer: It is not a good model. It never was, it never will be. Until we all do not come up with a better model that provides better modeling accuracy while it is equally intuitive and makes similarly simplifying assumptions the BS model with its geometric brownian motion component is here to stay. It actually does not matter ...


3

I would say Take log of first equation to get rid of dependence on $x_t$ Apply Kalman filter equations to estimate parameters I believe Conrad and Kaul (1988) J of Business do exactly what you describe.


3

To solve the expectation directly, you need to remember that a density function is not the same as the probability of the event. We have, $\frac{S_1}{S_0} \sim \ln \mathcal{N} \left(-\frac{\sigma^2}{2},\sigma\right)$, therefore, \begin{eqnarray} \mathbb{E}\left(\frac{S_1}{S_0}\right) &=& \int_{-\infty}^\infty x\, f_{\frac{S_1}{S_0}}(x)dx\\ ...


2

First of all, GNP and GDP are economic time series and they are not economic model. Secondly, you can also get these time series with different frequency, as quarterly data, avalaible on OECD website. In the case you need for lower frequency data you can get it by interpolation (as, for instance, the cubic spline interpolation); This is the Matlab tutorial ...


2

I am also not aware of any papers in this area. But having developed many such models, I can list the important steps: Decide on the target variable: usual choices are historical default data, agency ratings and expert rankings Create a sample containing the possible predictors Reduce the list with the help of some expert, e.g. exclude all the predictors ...


2

Most of the papers concern CDS spreads which you will need to convert to a PD. Paper using country specific fundamentals: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2517018 This paper uses leverage: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2361872 Another one that decomposes them against peer groups: ...


2

Basically, Black-Scholes is an "industry standard" formula. It is widely used by practitioners and usually augmented with extra specifications or intuition. It has a closed form solution, which is rare in option pricing models. It is also relative to simple to understand. Otherwise, you usually need to rely on Monte Carlo simulation or some other way. And ...


2

This is what Moody's does to calculate default probabilities, but I don't believe they give a whole lot of detail on their exact methodology because they sell their models as software. I quickly found this which gives a brief overview: http://www.moodysanalytics.com/~/media/Brochures/Enterprise-Risk-Solutions/RiskCalc/RiskCalcPlus-Fact-Sheet.ashx Edit- ...


2

Note that \begin{align*} E\bigg(\frac{S_{i+1}}{S_i}\mathbb{I}_{\frac{S_{i+1}}{S_i} < z}\bigg) &=zE\bigg(\mathbb{I}_{\frac{S_{i+1}}{S_i} < z}\bigg)-E\bigg(\Big(z-\frac{S_{i+1}}{S_i}\Big)\mathbb{I}_{\frac{S_{i+1}}{S_i} < z}\bigg) \\ &=zP\bigg(\frac{S_{i+1}}{S_i}<z\bigg)-E\bigg(\Big(z-\frac{S_{i+1}}{S_i}\Big)^+\bigg). \end{align*} Then you ...


2

In Andersen & Piterbarg's book, LGM is referred to as "The Hagan and Woodward Parameterization" and treated separately in 11.3.2.6. The fact that this practice-oriented book devotes a couple of pages would imply LGM is of practical use in the real market. I know two large software providers adopt LGM.


1

This is a very broad question and a large number of issues have been discussed in the literature. As such, it's hard to give specific advice except that it is better to model returns instead of prices directly. What I would do if I were you: Read some of the available literature to get a good overview. This is an interesting paper but many more exist. ...


1

What you are looking for is the partial expectation of $\frac{S_{i+1}}{S_i}$. Since $\frac{S_{i+1}}{S_i}$ is lognormally distributed, you can use the following result: For a lognormal random variable $X \sim LND(m,v^2)$, $$ E(X | X < z) = E[X] \Phi\left( \frac{\log(z)-m-v^2}{v} \right) $$ In your case, $m = (r-\frac{1}{2}\sigma^2) (t_{i+1}-t_{i})$, $v^2 ...


1

Thank you guys. Sorry for the late reply, I just solved it in matlab using maximum likelihood estimation. Turns out that all we need to do is to specify a state space model, then estimate the coefficient using MLE. The linearity and normality here makes things pretty simple.


1

It depends on the use of your model as pointed out in the comments. If a discretized version is sufficient then state space models could be a solution. You can check out the free online textbook by Athana­sopou­los and Hyndman. State space model describe time series in terms of level/trend (and seasonality) on an additive or multiplicative way. There are ...


1

When I implemented a BL model, I chose to do the omega optimization using the technique Idzorek proposed here: https://corporate.morningstar.com/ib/documents/MethodologyDocuments/IBBAssociates/BlackLitterman.pdf It's a numerical procedure though.


1

In practice, $\Omega$ (the covariance of the investor views) often 'inherits' the market covariance $\Sigma$. A convenient choice is $ \Omega = \left( 1/c -1 \right) P \Sigma P^T$ where $c$ is a confidence parameter: the case $c \rightarrow 1$ corresponds to a strongly peaked distribution of views (the investor views dominate the market), while $c ...


1

I suggest you to start from the Altman's model, that is the basic model to implement the kind of econometric analysis you're looking for. You can find the original paper at my Dropbox public folder. After that reading, you can find a number of paper about scoring models on SSRN or Google Scholar. Moreover, I suggest you to look for all academic papers that ...


1

Bloomberg has a Default Risk model, which is similar to what you are querying. You can see a screenshot in this PDF. There you can also see the kind of variables they use. You can access it by typing DRSK at the CDS screen is Bloomberg. (If the screenshot in the PDF is not clear enough, let me know and I can post one with better resolution from Bbg) This ...


1

Did you try rmgarch package of R ? http://cran.r-project.org/web/packages/rmgarch/index.html http://unstarched.net/r-examples/rmgarch/mgarch-comparison-using-the-hong-li-misspecification-test/


1

Given that other corporate events are reasonably modelled through regression models (compare The Detection of Earnings Manipulation I would try for using an regression approach. I believe a more recent and related paper has been published but I don't seem to find it at this time. Edit: and now I did - Earnings Manipulation and Expected Returns That said, ...


1

I agree with the previous statement that this is more stats related than anything else (it's not quant finance). But it's still a great question! This sounds awfully similar to linear regression testing with multiple predictor variables; you're basically doing it in a "monte carlo" fashion :) Depending on how your data is formatted, you could enter it into ...


1

Choose the most robust (or insensitive) strategy. You are right that the best strategy might be overfit. So look at your parameter space and focus on the area where profitability, for example, changes least when you change the parameter value. Here is a 1D example: The most profitable strategy is that single point that unfortunately leaves no room for ...


1

Number one, the central limit theorem means a lot of things that may not be normal end up looking normal when lots of little 'experiments' or impacts are added up. Number 2, when dealing with finance you need a model that seems plausible. An arithmetic Brownian motion could go negative, but stock prices can't. On the other hand, it seems quite plausible ...


1

The normal distribution is very powerful distribution: By the central limit theorem, the mean of any large sample always converges to the normal distribution Considering the most simplistic Binomial Tree model, where price goes only up or down each period, it can be shown that the distribution of returns of this tree converges to Normal for infinetesimal ...


1

In terms of end-user applications, all trading desks and middle office places I know, use either their own proprietary or expensive third party sources. On the other hand there exists a c++ library called QuantLib that is well known among real world practitioners, probably because it contains several routines that are well tested and robust. Often pieces of ...


1

If $S_t = S_0 e^{(\mu-\sigma^2/2)t + \sigma W_t}$, we can compute $$\mathbb{E}^Q\left[S_T\middle\vert \mathcal{F}_0\right] = S_0 e^{r T} = \text{forward price of } S_T \text { at time } 0. $$ To show the details, $\mathbb{E}^Q\left[S_T\middle\vert \mathcal{F}_0\right] = S_0 e^{(r-\sigma^2/2) T} \mathbb{E}^Q\left[e^{\sigma W_T} \middle\vert ...


1

Trigonometric functions are WAVE phenomena. As such, they are best used to model so-called periodic functions, that is, functions with cycles of a fixed period in length. That's why they are good for modelling, seasonal, annual, "blue moon" (once every two and half years), or other functions with set "periods."



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