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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 ... 4 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 ... 3 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 ... 3 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 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. ... 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 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 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. 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 ... 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 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. $$To show the details, \mathbb{E}^Q\left[S_T\middle\vert \mathcal{F}_0\right] = S_0 e^{(r-\sigma^2/2) T} \mathbb{E}^Q\left[e^{\sigma W_T} \middle\vert ... 1 If you mean by fat tails just fatter tails than the gaussian distribtuion, i.e. a distribution with finite variance, for instance the Student's t-distribution has fatter tails than the normal distribution. If you mean distributions with infinite variance, you have to have a look at Lévy distribution. In a first attempt you could just substitute the standard ... 1 Trigonometric functions are WAVE phenomena. As such, they are best used to model so-called periodic functions, that is, functions with cycles of a fixed period in length. That's why they are good for modelling, seasonal, annual, "blue moon" (once every two and half years), or other functions with set "periods." 1 Solving some heat/diffusion equations under certain conditions needs trigonometric functions. Black-Scholes reduces to a heat/diffusion equation by a change of variables. 1 Assuming the underlying mortgages that have been pooled into a Mortgage-Backed Security (MBS) are freely prepayable, the notional of the interest swap is unknown at inception. Therefore, you have two options - estimate a notional schedule to the best of your ability assuming some future evolution of interest rates (which are an important driver of ... 1 If you need data for the EU, you can look here - http://www.eba.europa.eu/ 1 I got a solution to this problem by posting an excerpt of it at math.stackexchange: http://math.stackexchange.com/questions/716242/equation-involving-expectations-of-levy-processes 1 In the paper OPTION PRICING BY ESSCHER TRANSFORMS the authors explore this topic extensively and provie equations that enable the calculation of the risk neutral \theta. Also note that you can easily deal with the expectation in$$ 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. ...