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Questions tagged [mathematics]

Used for question on application of mathematics in finance - from interest calculation to mathematical description of random processes.

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144 votes
15 answers
176k views

How can I go about applying machine learning algorithms to stock markets?

I am not very sure, if this question fits in here. I have recently begun, reading and learning about machine learning. Can someone throw some light onto how to go about it or rather can anyone share ...
zubinmehta's user avatar
  • 1,551
45 votes
8 answers
17k views

Recommendations for books to understand the math in quantitative finance papers?

Can anyone recommend books that explain the math used in quantitative finance academic papers?
32 votes
5 answers
8k views

Random matrix theory (RMT) in finance

The new kid on the block in finance seems to be random matrix theory. Although RMT as a theory is not so new (about 50 years) and was first used in quantum mechanics it being used in finance is a ...
vonjd's user avatar
  • 27.4k
27 votes
6 answers
21k views

What is a martingale?

What is a martingale and how it compares with a random walk in the context of the Efficient Market Hypothesis?
user40's user avatar
  • 2,707
21 votes
7 answers
3k views

How random are financial data series?

Pseudorandom number generators are often tested using e.g. a test suite like Diehard tests or Dieharder. If one would run these tests e.g. on stock market time series or other financial data, would ...
asmaier's user avatar
  • 563
16 votes
0 answers
399 views

Proving the asymptotic distribution of Manipulation-Proof Performance Measure (MPPM) (Paper by Goetzmann et al.)

In Goetzmann et al.'s (2007) paper, the authors derive a "Manipulation-Proof Performance Measure" (MPPM), which is a performance measure that is impervious to performance manipulation by fund managers....
SwiftMo's user avatar
  • 325
15 votes
1 answer
2k views

Any recommendations for textbooks for an undergraduate course in mathematical finance? [closed]

I'll teach an introductory course on mathematical finance in the near future. The course is intended to entertain and broaden some well-prepared advanced undergrad mathematics majors, some physics ...
Jeff Burdges's user avatar
14 votes
2 answers
564 views

Is a linear combination of GARCH processes also a GARCH process?

If two time series follow a GARCH process, and a third is a linear combination of them, is the third also GARCH process?
Qbik's user avatar
  • 1,018
14 votes
3 answers
8k views

How does one go from measure P to Q(risk-neutral) when modeling an asset paying dividends?

I am really having a terrible time applying Girsanov's theorem to go from the real-world measure $P$ to the risk-neutral measure $Q$. I want to determine the payoff of a derivative based an asset ...
John Tyree's user avatar
13 votes
2 answers
3k views

Why Ito calculus?

Coming from physics, I am used to the fact that the Ito interpretation of most natural stochastic equations is wrong, and one should be using Stratonovich calculus instead (of course they are ...
user avatar
11 votes
4 answers
2k views

What are the options for a mathematician to break into QF without working for a fund? [closed]

I have a degree in mathematics, and I've worked as a statistician and done some programming work. I'm very strong in my math/stats/programming background and have browsed some QF books, and I'm very ...
user4683's user avatar
  • 111
11 votes
1 answer
7k views

What is exactly Euler's decomposition?

I have often seen the following statement in different paper: As $\sigma$ is homogeneous and of degree 1, we use Euler decomposition and write $\sigma(x)=\sum_{i=1}^n x_i \frac{\partial \sigma(x)}{\...
SRKX's user avatar
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11 votes
2 answers
777 views

Is it possible to understand financial theory without mathematics?

I am trying to develop a short course on financial theory, covering the fundamentals of forward and options pricing, and 'efficient market' theory. I want to reduce the amount of mathematics to a ...
quis est ille's user avatar
9 votes
2 answers
620 views

The Relation Between the Ricci flow and the Black-Scholes-Merton Equation

Grisha Perelman once wrote that The Ricci-flow equation, a type of heat equation, is a distant relative of the Black-Scholes equation that bond traders around the world use to price stock and ...
Dendi Suhubdy's user avatar
9 votes
1 answer
4k views

application of lie groups in finance

Can some one kindly go over some of the applications and use of Lie groups in finance? The math is very rigorous and I don't fully understand it or the potential it could have. Let me share some ...
pyCthon's user avatar
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8 votes
7 answers
2k views

Proof that no trading system always wins

I am pondering on the existence/impossibility of a trading system (or algorithm) that ALWAYS ends up winning money, no matter how the price of a futures moves. In a context where one can go long or ...
Manuel Lafond's user avatar
8 votes
3 answers
2k views

How is stock data objectively different to this random walk?

I have a random walk that is generated as so using python, numpy, and matplotlib ...
Mark Dunne's user avatar
8 votes
5 answers
5k views

Does anyone know where to practice mental math for trader interviews in MC format? [duplicate]

I am currently using zetamac (has customizable number ranges but doesn't allow for decimals), tradermath (seems to be made to resemble the actual test for flow but costs money unfortunately), and ...
Quant In Spe's user avatar
8 votes
1 answer
448 views

When pricing options, what precision should I work with?

I'm wondering if there's any point at all in double-precision calculations, or whether it's ok to just do everything in single-precision, seeing how the difference on non-Tesla GPUs for single and ...
Dmitri Nesteruk's user avatar
8 votes
1 answer
1k views

Modified Durations of Different Noncallable Bonds and function of Maturity

I'm hoping someone could help me understand this subject better. Basically I am reading a book and it shows a table ...
Matt's user avatar
  • 183
8 votes
2 answers
330 views

Why won't Bjork ever show that the integrability condition is satisfied?

A major technique employed throughout Bjork's "Arbitrage theory in Continuous Time" is that when taking the expectation of a stochastic integral, the result is 0. This is based on a result presented ...
Saku's user avatar
  • 81
7 votes
4 answers
2k views

Implications of the Riemann hypothesis in finance?

I was recently at a seminar by a top hedge fund manager at a top university for finance students, when one of the finance professors asked him what do you think are important areas of research for my ...
pyCthon's user avatar
  • 2,111
7 votes
2 answers
416 views

What is Heston's equation?

This paper mentions the elliptic Heston operator: $Av:= -\frac y2(v_{xx}+2\rho\sigma v_{xy} + \sigma^2v_{yy}) - (c_0 - q - \frac y2)v_x + \kappa(\theta -y)v_y + c_0v$. Then boundary value problem ...
Nikita Evseev's user avatar
7 votes
1 answer
3k views

Abstract algebra in economics and finance

Are there any applications of abstract algebra (group theory, rings, fields etc.) in any branch of either economics or finance?
SlavicDoomer's user avatar
6 votes
2 answers
1k views

Book recommendation: math toolkit for quantitative finance and statistics

I am looking for a book which teaches mathematical topics which are relevant to master quantitative finance and statistics. Please note, I do not mean a book which would explain how math is applied ...
zesy's user avatar
  • 161
6 votes
0 answers
123 views

What are the requirements for no arbitrage to exist in a chaotic/dynamical system?

Consider the continuous dynamical system $$\alpha\ddot{S}+\dot{S}=\mathcal{F}(S,t),$$ such that $\alpha\in\mathbb{R}$ and $\mathcal{F}$ is real and analytic. We assume that if a solution for $S$ ...
UNOwen's user avatar
  • 128
5 votes
4 answers
2k views

Examples of discrete math and graph theory within quantitative finance

The Wikipedia article on quants mentions discrete mathematics as a possible piece of their mathematical background. Are there good examples of problems within quantitative finance that are heavily ...
Juho's user avatar
  • 153
5 votes
3 answers
2k views

Is linear programming important for quant?

I'm thinking about taking a course on Linear and Convex Programming, but I don't know how useful it is in the real world finance. Which areas in finance is mathematical programming used?
mathsam's user avatar
  • 53
5 votes
3 answers
557 views

Is there an intuitive explanation for why Kelly gambling ignores odds?

I have just learned about Kelly gambling from Chapter 6 of Cover & Thomas' Introduction to Information Theory. The mathematical setup is that we have a horse race, with horse $i$ winning with ...
Michael S.'s user avatar
5 votes
1 answer
501 views

How to find interesting open math problems in quantitative finance that I could publish articles about?

Which books on financial mathematics would you recommend for people with good background in probability, statistics and stochastic processes but without any background in financial mathematics? The ...
Botnakov N.'s user avatar
5 votes
1 answer
749 views

The Heston Solution For European Option - Jim Gatheral

I have this equation (Eq. (2.4) "The Volatility Surface - A Practitioner's Guide" by Jim Gatheral (Ed. 2006)): $$-\frac{\partial C(v, x, \tau)}{\partial \tau}+\frac{1}{2}v \frac{\partial^2 C(v,x,\tau)}...
AndrewTG's user avatar
5 votes
2 answers
534 views

Foward-start option pricing

Consider a probability filtred space $(\Omega, \mathcal F, \mathbb F, \mathbb P)$, where $\mathbb F = (\mathcal F_t)_{0\leq t\leq T}$ satisfing the habitual conditions and is generated by $1 d $- ...
Paul's user avatar
  • 608
5 votes
0 answers
140 views

Is the exponential Shannon entropy sub-additive?

In a recent paper of Salazar et al. (2014), The Diversification Delta: A Different Perspective, forthcoming in the Journal of Portfolio Management , the authors propose to use the exponential Shannon ...
Nathan L's user avatar
5 votes
0 answers
616 views

Show that in an arbitrage-free and non-redundant market a certain set is compact

Some notation: We consider a financial market with $d+1$ assets, the $0$-th asset is considered the risk-free asset, the others are the risky ones. The vector $\overline \pi \in \mathbb R^{d+1}$ ...
StefanH's user avatar
  • 201
4 votes
2 answers
2k views

Understanding the solution of this integral

The following integral represents an expected value of a geometric brownian motion for $S_T>K$ (i.e. part of the Black-Scholes call option price): $$\int_{z^*} (S_te^{\mu\tau-\frac{1}{2}\sigma^2\...
emcor's user avatar
  • 5,795
4 votes
1 answer
1k views

What the expectation of S^2 is from GBM? [closed]

I was at an interview and was asked to write down the SDE for GBM. $$ dS = S\mu dt + S\sigma dX $$ Then I was asked how I would compute the expectation of S^2. I didn't know where to start. Any ...
Daniel's user avatar
  • 51
4 votes
1 answer
717 views

Mark Joshi, The concepts and practice of mathematical finance chapter 6 exercise 4

Let an asset follow a Brownian motion $$dS = \mu dt + \sigma dW$$ with $\mu$ and $\sigma$ constant. The constant interest rate is $r$. What process does $S$ follow in the risk-neutral measure? ...
Wolfy's user avatar
  • 728
4 votes
1 answer
420 views

Clarify a derivation in Pat Hagan's Convexity Conundrums

I am looking for help in understanding the algebraic derivation to go in between some of the lines in Pat Hagan's famous Convexity Conundrums paper e.g. how he goes from 3.4a to 3.5a.
Randor's user avatar
  • 786
4 votes
1 answer
2k views

derivation of heston pde in gatheral

Following Gather (the volatility surface, chapter 2) we assume the following process: $$ dS_t = S_t(\mu_t dt+\sqrt{\nu_t}dZ^1_t)$$ $$ d\nu_t= -\lambda(\nu_t-\bar{\nu})dt+\eta\sqrt{\nu_t}dZ^2_t$$ ...
math's user avatar
  • 1,718
4 votes
1 answer
233 views

What is the analytic value of an asset's risk contribution, if $n=2$?

The marginal risk contribution of asset $i$ is defined by Roncalli in his paper on ERC as follows: $$\frac{\partial \sigma(x)}{\partial x_i} = \frac{1}{\sigma(x)} \left( w_i \sigma_i^2 + \sum_{\...
SRKX's user avatar
  • 11.1k
4 votes
1 answer
704 views

European call delta derivation

Let's write $S(T) = S_T$ and $S(0) = S_0$. We want to compute $\frac{d}{dS_0}\mathbb{E}[f(S_T)]$. From a previous discussion this is equal to $$\mathbb{E}_{S_0}\left[f(S_T)\frac{g'_{S_0}(S_T)}{g_{S_0}(...
user avatar
4 votes
1 answer
831 views

How to prove we have a $\mathbb{Q}$-Brownian motion?

Background Information: This question comes from the book Financial Calculus by Baxter and Rennie. WE start with looking at the marginal of $W_T$ under $\mathbb{Q}$. We need to find the likelihood ...
Wolfy's user avatar
  • 728
4 votes
1 answer
2k views

How to price and find a replicating portfolio for a call spreads using a two-period binomial model?

Consider a two-period binomial model for a risky asset with each period equal to a year and take $S_0 = 1$, $u = 1.03$ and $l = 0.98$. a.) If the interest rate for both periods is $R = .01$, find the ...
Wolfy's user avatar
  • 728
4 votes
1 answer
344 views

backward Kolmogorov equations - Markov properties

I'm a physicist who's research has lead him into the theory of stochastic differential equations. If this question is not appropriate for this forum, please feel free to delete it. So I've been ...
user7670's user avatar
4 votes
0 answers
50 views

Stochastic integral representation of $F(T-s,X_s)$-type equations

For $T\in R$ given and fixed consider: $$ {\rm d}F(T-t,X_t)=g(T-t,X_t)\,{\rm d}W_t. $$ where $g(t,x)$ is a given functions and $X_t$ is a given process driven by a brownian motion ($dX_t=(...)dt+(...)...
econmajorr's user avatar
4 votes
0 answers
329 views

Arrow-Debreu Equilibrium Pricing

I have this problem in asset pricing that I don't know how to solve. Here it is: Consider an economy with a complete set of Securities and $N$ states of the world Tomorrow. Assume that there are two ...
james42's user avatar
  • 676
4 votes
0 answers
798 views

Monty Hall Model

Given a fixed time period,say 3 days, the stock/market can go up,down or stay sideways. A hedge fund can long, short or use rangebound(options strategy) to bet for that 3 days closing level. Hedge ...
Shelagh's user avatar
  • 131
3 votes
2 answers
1k views

How do you derive this Carr-Madan-like equation?

How do you derive equation (3) below? The equation is tagged as equation (11) in this paper: http://janroman.dhis.org/finance/IR/Heston%E2%80%93Hull%E2%80%93White%20Model%20Part%20I.pdf There are ...
user54908's user avatar
  • 437
3 votes
2 answers
3k views

Preparation for interview: influx of power of the moon

I am preparing myself for an interview for a quantitative analyst position and one of the sample questions asked in previous examinations was: "Suppose the moon were to disintegrate, and fall to ...
Giuseppe's user avatar
3 votes
3 answers
763 views

Budget Constraint in Sharpe Ratio Optimization

I am a math student and I am trying to understand the budget constraint in Sharpe Ratio optimization for portfolio design. Recall the budget constraint requires that the sum of the portfolio weights ...
Mykie's user avatar
  • 345