# Tag Info

26

It's an interesting question. I particularly agree with the $\mathbb{Q}-\mathbb{P}$ dichotomy mentioned by many. I would add to the other answers that, come to think of it, the Black-Scholes postulated Geometric Brownian Motion could be interpreted as an AR(1) process on the logarithm of the stock price as you discretise the SDE from which it is a solution,...

14

I think you need to differentiate between Q-quants vs P-quants. The former might not use Econometrics, but P-quants use them a lot.

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Traditional econometric (time series) models are of little or no value in forecasting market prices for purposes of "making money", i.e, generating excess return over a benchmark in an asset management setting. They have some limited value in strategic and tactical asset allocation. The ineffectiveness of time-series modeling in asset management stems ...

10

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: $$... 9 Attilio Meucci does some very interesting things with PCA. See e.g. his paper on managing diversification which makes heavy use of it (and explains it very intuitively along the way): Managing Diversification by Attilio Meucci 9 GARCH will work if volume has memory with some decay. AR will work if volume has mean reversion properties. Both of these are empirical questions and depend on the market. You should also consider if there are seasonal (day-of-week, monthly, quarterly effects) in which case you would want to add dummy variables. MA models will work well if volume behaves ... 9 The best paper is probably Relative Volume as a Doubly Stochastic Binomial Point Process - James Mcculloch. In this paper the volume is modelled via a Point Process, and theoretical laws are derived (with confident intervals, etc). And we put elements about this in Market Microstructure in Practice, Chap 2.1. Volume curves are analyzed, not only during the ... 8 Fourier methods use sine and cosine functions, and are used in calculating option prices, VaR, time series analysis etc. It is an alternative process for doing many things in finance. Some links Fourier Methods in trading on StackExchange and Wiki 7 Aleš Černý has very simple examples in his book. Alternatively, this paper seems to recap part of the chapter on Fourier series: Introduction to Fast Fourier Transform in Finance - Aleš Černý 7 When the pdf of a distribution is not known analytically, it's common to compute by taking the inverse Fourier transform of its characteristic function. The same idea applies here. Consider the discounted expectation formula of a European option$$V (S,\tau) = e^{-r\tau} \mathbb{E}_{x_0} [\theta(x_T)] $$. for log prices x and time to expiry \tau=T-t. In ... 7 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 ... 7 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 ... 7 Having thought about this I think the following reason is also important and wasn't mentioned so far: When you look at the inner working of this whole class of econometric models it all boils down to the following: It is possible (under some reasonable assumptions) to express any MA(q) model as an AR(\infty) model (and vice-versa for expressing AR(p) ... 6 You could try using the Gaussian Affine Term Structure Models (GATSM), with the right boundary conditions to stop rates being negative (in the style of their Black implementation). See, for example, Monika Piazzesi, the "Affine Term Structure Models" if you want to enter/modify the basis or the work of Krippner, for example "Measuring the stance of monetary ... 6 From an academic viewpoint you do not have a lot of choices: The Rosenbaum-Robert approach, the price model with uncertainty zones is a model of trades and duration between trades (implicitly). It is worthwhile to try it. You can also use an Hawkes process, it will have the nice effect of capturing clustering effects on trades. if you want to use ... 6 I have honestly not come across a good book (or good enough review to make me buy the book) on Fund Transfer Pricing. While it is not my career focus, I had to familiarize myself a bit with the topic because of certain requirements involving funding trading operations and the performance of funding specific operations. Personally I would recommend the ... 6 Estimation of the initial states of R and particularly Q is indeed more of an art than science. The task at hand is to estimate the covariances. You have basically two main choices: Live with the fact that you will never be able to exactly pinpoint the covariance of noise in financial time series. The most often used approach is to pose the coveriance ... 6 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 ... 6 MIDAS is useful when you have a low frequency series and you want to include high frequency data in the regression. So for instance, if you want to forecast quarterly GDP data and want to include daily S&P 500 data as a regressor instead of just using the quarter end value of S&P 500. Usually we assume that the causality runs from S&P 500 to ... 6 My answer is very much in the spirit of Kiwiakos' answer. E.g. in this paper (where I am one of the coauthors) we use VMA (vector moving average) models (in the multivariate case) and AR models in the univariate case to calculate proper scaling of volatility or its contributions if there are (cross-) auto-correlations. This happens in the P world due to ... 5 Let's first restate the formula of the beta of a portfolio P relative to a benchmark B:$$\beta_P=\frac{Cov(r_P,r_B)}{Var(r_B)} $$As chrisaycock said in his comment, the key thing to understand is that the beta is a statistical measure computed relative to a benchmark. Hence, I believe that the real question you should be asking is: Which benchmark ... 5 Yes, there is a software application that you can purchase for 39.99 which stores all your tick data in a highly compressed format while still allowing maximum throughput and lowest latency data queries that I have ever seen. The package provides APIs to all languages under the sun but because they have a special sale going on it comes with the complete ... 5 You could read it like this: The typical change in equity value is equal to the typical change in asset value, adjusted for the probability of the assets surviving. Note that the formula is not specific to Merton models, it's also true for regular options and their underlyings. It's just that volatility of option prices isn't typically a concern in "... 5 I think this blog post is quite good at explaining option pricing via fourier transforms. 5 Let C be the price of the option, S_t=S_0e^{X_t} be the stock price, r be the risk-free rate, K be the strike price, T be the maturity time, m=S_0/K, f be the density of X_T and \phi be the characteristic function E(e^{i\xi X_T}) which we assume is known.$$ C = e^{-rT}E((S_T-K)^+) = e^{-rT}S_0\int_{-\infty}^\infty \left(e^x-m\right)\...

5

You can use $\sin$ or $\cos$ to model seasonality. If all you have is a calculator it might be the most practical way.

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 ...

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