27

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,...


15

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.


11

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: $$ ...


11

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 ...


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

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 ...


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

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

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

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

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

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

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 ...


5

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- ...


5

Since $dW_A$ and $dW_B$ are already correlated as per the way you construct it, your portfolio being the sum of the two is already correlated. If you want it very explicitity written out, then you could rewrite $dW_B = \rho dW_A + \sqrt{1-\rho^2}dW_Z$ where $dW_Z$ is independent of $dW_A$. More generally (higher dimensions) you can use Cholesky. Now with ...


5

Assume that your stationary time series (here a daily close-to-close log-returns' series) is modelled as follows $\forall t \in \mathcal{T}=\{1,...,N\}$ \begin{align} r_t &= E_{t-1}[r_t] + \epsilon_t \\ &= E_{t-1}[r_t] + \sigma_t z_t \end{align} with $z_t \sim N(0,1)$ and $\{z_t\}_{t \in \mathcal{T}}$ are IID. The above equations suggest that, ...


5

One approach that I have seen being used is to try to model the (joint) dynamics of the forward at-the-money volatility as well as its first one or two derivatives. The idea is to find a parametrization for each of these quantities that you can easily estimate from historical data. You will generally find that the sensitivities themselves have a significant ...


5

There is large literature on MIDAS (mixed-frequency data sampling) models, the leading scholars being Eric Ghysels and Rossen Valkanov — google their research for references. However, the motivation for these models has mostly been to forecast low-frequency stuff with high-frequency variables, updating, say, quarterly GDP predictions as weekly ...


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