25
If you need a primer covering various domains of math then Dan Stefanica's text will do the job. The text covers multivariable calculus, lagrange multipliers, black scholes PDF, greeks & hedging, newton's method, bootstrapping, taylor series, numerical integration, and risk neutral valuation. It also includes a mathematical appendix.
If you want an ...
24
Two aspects of statistical learning are useful for trading
1. First the ones mentioned earlier: some statistical methods focused on working on live datasets. It means that you know you are observing only a sample of data and you want to extrapolate. You thus have to deal with in sample and out of sample issues, overfitting and so on... From this viewpoint, ...
20
Variance ratio tests have been used numerous times to show that financial asset prices do not follow a random walk. You can for example look at
-Lo and MacKinlay : Stock market prices do not follow a random walk : http://press.princeton.edu/books/lo/chapt2.pdf (US Stocks)
-Hoque, Kim, Pyun: A comparison of variance ratio tests of random walk: A ...
16
I doubt you will find one book that covers everything you need, but here are a few that I continually come back to whenever I have some questions on the mathematics.
Analysis of Financial Time Series by Ruey Tsay
An Introduction to High-Frequency Finance by Dacorogna et al
Probability and Statistics by DeGroot and Schervish
Statistical Inference by Casella ...
14
A function $f : \mathbb R^n\backslash\{0\} →\mathbb R$ is called (positive) homogeneous of degree $k$ if
$$f(\lambda \mathbf x) = \lambda^k f(\mathbf x) \,$$
for all $\lambda > 0$. Here $k$ can be any complex number. The homogeneous functions are characterized by
Euler's Homogeneous Function Theorem. Suppose that the function $f : \mathbb R^n \...
13
I think he was jokingly suggesting to breed top PhD candidates in pure mathematics. I often heard complaints that a lot of PhDs in Mathematical disciplines lack a rigorous base in pure Mathematics. Obviously the hedge fund manager was not suggesting that the proof of the hypothesis will be in any way relevant to trading or financial pricing applications. On ...
12
If you asked me for a single book as a starting point I'd probably go for:
Frequently Asked Questions in Quantitative Finance by Paul Wilmott
12
I think there are a lot of different ways to specify this problem. For simplicity, consider independent Garch processes
$$
r_{1,t} \sim N\left(0,\sigma_{1,t}^{2}\right)
$$
$$
\sigma_{1,t}^{2} = \beta_{1,1}+\beta_{1,2}\varepsilon_{1,t-1}^{2}+\beta_{1,3}\sigma_{1,t-1}^{2}
$$
and
$$
r_{2,t} \sim N\left(0,\sigma_{2,t}^{2}\right)
$$
$$
\sigma_{2,t}^{2} = \beta_{...
12
At the top of this list I still recommend you to seek employment in order to learn from others in QF space. Could you possible work in a quant team within an investment bank where you currently reside?
Start to reach out to the quant finance community so you are connected once you decide to locate to where you can practice this discipline.reach out to alumni,...
12
Edit: Freddy's answer is good -- we wrote concurrently. He rightly points out that QF is a broad field, and that it is among other things a community. Here, I describe a practical, down-to-earth path for getting your feet wet in one key piece of it -- software and model development for derivatives analysis, starting with vanilla options. Your best bet ...
11
My understanding is because the Ito's integration definition keeps the martingale property.
With Brownian motion $W(t, \omega)$ defined, to define stochastic integration in a Riemann–Stieltjes style:
$$\int_0^t f(t, \omega) d W(t, \omega) = \lim_{\| \Delta_n\| \to 0 } \sum_{i=1}^{n} f(\tau_i,\omega) \left ( W(t_i, \omega) - W(t_{i-1}, \omega) \right ) $$
, ...
9
Why don't you try it and report back?
Recall, though, that while a random walk is often a rather competitive forecast, realized data is understood to have weak dependence especially in higher moments.
Having worked a bit with DieHarder, I'd suspect it to reject a number of series. But the proof is in the pudding...
9
In fact Ito and Stratonovich calculus are both mathematically equivalent. In the following paper you can e.g. see that both derivations lead to the same result, i.e. the Black-Scholes equation:
Black-Scholes option pricing within Ito and Stratonovich conventions by J. Perello, J. M. Porra, M. Montero and J. Masoliver
From the abstract:
Options financial ...
8
I'd say to read Prof. Shreve's well-known two-volume textbook Stochastic Calculus for Finance I and II.
7
At the first glance, what you are asking for is a model admitting arbitrage, so there is a zero chance of losing money and positive chance of yielding profits. Well, many equilibrium models start with assuming arbitrage is not possible (otherwise it would be trivial wouldn't it).
But, in my opinion, what you actually seek is the Efficient Markets Hypothesis....
6
I like Zapatero & Cvitanic, "Introduction to the Economics and Mathematics of Financial Markets";Steele, "Stochastic Calculus and Financial Applications" and also Mikosh, "Elementary Stochastic Calculus with Fincial View". Shreeve is always a good choice too. If you are working with physicists, SCHMIDT, "QUANTITATIVE FINANCE FOR PHYSICISTS: AN ...
6
Just following Musiela Rutkowski (the link redirects to Amazon). The risk neutral measure is derived form imposing that the present value of a self financed portfolio (i.e.; no infusion or withdraw of money) is a martingale. A portfolio can be seen as a stochastic process where its value at time $t$ is given by
$$
V_t = \phi^0_tP_t + \phi^1_tS_t\ ,
$$
...
6
People seem to think that using ML is going to circumvent the process of actually learning to trade, it doesn't. ML can be used to refine trading ideas, but it doesn't generate them, you need to use your brain for that.
6
I'm currently working on this task, to apply machine learning to stock trading. However, the concerns raised in other answers are major obstacles. So, I'm taking a different tact.
My strategy is more akin to teaching a car to drive - the machine learning is not based on the underlying data, but rather on the driver's reaction to the data. So based on what ...
6
I think one of the best (and very current) articles about how to break into QF (for any kind of background) is:
"On becoming a Quant" by Mark Joshi
For your special background in mathematics see this excerpt from section 9:
The main challenge for a pure mathematician is to be able to get one’s
hands dirty and learning to be more focussed on getting ...
6
The other answers are useful and sensible. I have worked full time in equity research for nearly two decades, so very much a "qualitative" rather than a quantitative approach. However, all the firms for which I have worked had quants and because of my casual interest in the area I've spent a lot of time talking to quant teams over the years, often over a ...
6
Let $\tau = T-t$. Then
\begin{align*}
S_T = S_t e^{(\mu - \frac{1}{2}\sigma^2) \tau + \sigma \sqrt{\tau}\, Z},
\end{align*}
where $Z$ is a standard normal random variable, independent of $\mathcal{F}_t$. Moreover,
\begin{align*}
E\left(S_T 1_{\{S_T >K\}}\mid \mathcal{F}_t \right) &= E\left(S_t e^{(\mu - \frac{1}{2}\sigma^2) \tau + \sigma \sqrt{\tau}\, ...
5
Sorry, but despite being used as a popular example in machine learning, no one has ever achieved a stock market prediction.
It does not work for several reasons (check random walk by Fama and quite a bit of others, rational decision making fallacy, wrong assumptions ...), but the most compelling one is that if it would work, someone would be able to become ...
5
One possibility worth exploring is to use the support vector machine learning tool on the Metatrader 5 platform. Firstly, if you're not familiar with it, Metatrader 5 is a platform developed for users to implement algorithmic trading in forex and CFD markets (I'm not sure if the platform can be extended to stocks and other markets). It is typically used for ...
5
I'll throw this in as an "application of RMT" ... EDHEC and FTSE use RMT to decide the optimal number of principal components in their covariance estimation procedure for which they use PCA (Principal Component Analysis). For details look here or here in Appendix C section 4 for details.
5
All the ideas above are great ideas.
Another kind of test would be an idea borrowed from Random Matrix Theory.
Assemble your time-series into a matrix. Evaluate the distribution of the eigenvalues of the matrix vs. the distribution of a random matrix. Turns out that the distribution of eigenvalues in a random matrix conforms to distributions such as the ...
5
When you decide if the performance improvement is worth it you can add these to the downside ow using single precision:
the result of your basic B-S pricer will eventually need to be multiplied with a notional and maybe a discount factor; For a sufficiently large notional you will see different results than the one calculated using double precision. Is that ...
5
I have tested lots of forex data for randomness. Some currency pairs are very close to random walk. And the problem is open question, because there is no uniform explanation what the random walk is. According to Mandelbrot, Taleb and some other authors randomness can be different.
Even if the data is not random it doesn't mean it can be effectively traded. ...
5
No, a sum of two GARCH processes is generally not a GARCH process.
(I am not even sure whether there exists a nontrivial special case where the opposite holds.)
By GARCH I mean the classic definition of GARCH due to Bollerslev (1986), not an arbitrary variation like EGARCH, IGARCH, FIGARCH or whatever else.
Let me provide an example. Take two ...
5
From this abstract:
The Heston stochastic volatility process is a degenerate diffusion process where the degeneracy in the diffusion coefficient is proportional to the square root of the distance to the boundary of the half-plane. The generator of this process with killing, called the elliptic Heston operator, is a second-order, degenerate-elliptic ...
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