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The best paper is probably Relative Volume as a Doubly Stochastic Binomial Point Process - James Mcculloch. In this paper the volume is modelled via a Point Process, and theoretical laws are derived (with confident intervals, etc). And if you can wait few days (it will be available very soon), we put elements about this in Market Microstructure in Practice, ...

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To recover the Black-Scholes pricing equation, you should first express the standard normal cdf in terms of its characteristic function analogous to the Heston solution: $$N(x) = \frac{1}{2} - \frac{1}{\pi} \int_0^{\infty} Re [\frac{e^{-i\phi x} f(\phi)}{i\phi}] d\phi$$ where $f(\phi)$ is the characteristic function of the standard normal distribution: $$... 4 You can use \sin or \cos to model seasonality. If all you have is a calculator it might be the most practical way. 4 One can use the Karhunen–Loève expansion to approximate a trajectory of a Wiener Process, which can be used to model the evolvement of returns in time. (http://en.wikipedia.org/wiki/Karhunen%E2%80%93Lo%C3%A8ve_theorem#The_Wiener_process) Though the Karhunen–Loève expansion has theoretical advantages to other variants to generate a trajectory of a Wiener ... 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\\ ... 3 Fourier methods use sine and cosine functions, and are used in calculating option prices, VaR, time series analysis etc. It is an alternative process for doing many things in finance. Some links Fourier Methods in trading on StackExchange and Wiki 3 I have asked myself the very same question when I first read the book. As far as I can tell, the "scalability" condition is only imposed for technical reasons. It simplifies the subsequent proof of the Fundemental Theorem of Asset Pricing in constrained markets. There are several papers that have shown that the theorem is valid for conic constraints. ... 3 I deal recently with some analysis of the Volume time series, daily volume in € for European stocks. I found out that an ARIMA model works well. But, some EWMA could also provide good forecast if it's well parameterized. You can also face some seasonality effect due to macroeconomic events, some you may need to clean you data and treat these days in a ... 2 Try the following : perform the logarithmic transformation of the volume data. check if the transformed data fits the normal distribution nicely. if you are working with intraday volume, then adjust for the seasonality for time of the day effect, if using daily data, in some cases some special seasonalities like expiry day, etc might be applied but it may ... 2 From my point of view, dynamic models like the one developped in Relative Volume as a Doubly Stochastic Binomial Point Process - James Mcculloch to provide a dynamic forecast of the volume does not improve significantly the forecasting comparing to a static volume curve forecast using historical data (last month intraday data, and an EWMA algorithm). I've ... 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 These games are usually won by luck. If there is no fee for buying stocks I'd diversify, i.e. buy many different stocks, to get stable returns. After some weeks you'll see which profit you'll need to beat. Depending on the rules if options are allowed you could invest in highly leveraged derivatives and hope you win. As there is no point not to try to win I ... 2 Trigonometric functions show up in econometric models for business cycles. For example: the average length of a cycle of an AR(2) process is  k = \frac{2 \pi}{\cos^{-1}( \phi_1/ (2 \sqrt{-\phi_2}))} For an AR(2) model given by  r_t = \phi_0 + \phi_1 r_{t-1} + \phi_2 r_{t-2} + a_t with complex roots, \phi_1^2 + 4\phi_2 <0  2 When you do Monte Carlo simulation and would like to draw sample from the normal distribution \mathcal{N}(\mu,\sigma^2), you may use Box-Muller transform and come up with formulas using \sin and \cos. 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 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. $$To show the details, \mathbb{E}^Q\left[S_T\middle\vert \mathcal{F}_0\right] = S_0 e^{(r-\sigma^2/2) T} \mathbb{E}^Q\left[e^{\sigma W_T} \middle\vert ... 1 Solving some heat/diffusion equations under certain conditions needs trigonometric functions. Black-Scholes reduces to a heat/diffusion equation by a change of variables. 1 If you mean by fat tails just fatter tails than the gaussian distribtuion, i.e. a distribution with finite variance, for instance the Student's t-distribution has fatter tails than the normal distribution. If you mean distributions with infinite variance, you have to have a look at Lévy distribution. In a first attempt you could just substitute the standard ... 1 Trigonometric functions are WAVE phenomena. As such, they are best used to model so-called periodic functions, that is, functions with cycles of a fixed period in length. That's why they are good for modelling, seasonal, annual, "blue moon" (once every two and half years), or other functions with set "periods." 1 Assuming the underlying mortgages that have been pooled into a Mortgage-Backed Security (MBS) are freely prepayable, the notional of the interest swap is unknown at inception. Therefore, you have two options - estimate a notional schedule to the best of your ability assuming some future evolution of interest rates (which are an important driver of ... 1 If you need data for the EU, you can look here - http://www.eba.europa.eu/ 1 I got a solution to this problem by posting an excerpt of it at math.stackexchange: http://math.stackexchange.com/questions/716242/equation-involving-expectations-of-levy-processes 1 In the paper OPTION PRICING BY ESSCHER TRANSFORMS the authors explore this topic extensively and provie equations that enable the calculation of the risk neutral \theta. Also note that you can easily deal with the expectation in$$ e^{\theta} e^{(T-t)\psi(u)} = e^{t\psi(u)}E(e^{-X_T}|\mathcal{F}_t)  if the process $X_t$ itself has nice properties. ...

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I dont know what you want to hear, but i have several points for you: The main driver of uncertainty is a Wiener process, which goes back to the discrete binomial model for stock prices. In reality the main stochastic source could be something completly different. $\alpha$ and Vola $\sigma$ are depending directly on your stockprice. Why should they? the ...

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