Questions tagged [mathematics]
Used for question on application of mathematics in finance - from interest calculation to mathematical description of random processes.
195
questions
143
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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 ...
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8
answers
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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?
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5
answers
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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 ...
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6
answers
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What is a martingale?
What is a martingale and how it compares with a random walk in the context of the Efficient Market Hypothesis?
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votes
7
answers
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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 ...
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16
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0
answers
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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....
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15
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1
answer
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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 ...
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14
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2
answers
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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?
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14
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3
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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 ...
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13
votes
2
answers
3k
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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 ...
user8040
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4
answers
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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 ...
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1
answer
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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)}{\...
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2
answers
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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 ...
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votes
5
answers
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Does anyone know where to practice mental math for trader interviews?
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 ...
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2
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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 ...
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8
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7
answers
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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 ...
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3
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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
...
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1
answer
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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 ...
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votes
1
answer
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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 ...
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votes
1
answer
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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
...
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votes
2
answers
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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 ...
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2
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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 ...
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1
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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?
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4
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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 ...
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votes
2
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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 ...
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0
answers
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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 $S$ ...
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3
answers
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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?
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3
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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 ...
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5
votes
1
answer
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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)}...
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5
votes
2
answers
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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 $- ...
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0
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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 ...
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votes
0
answers
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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 ...
5
votes
0
answers
564
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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}$ ...
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4
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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 ...
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2
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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\...
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1
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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 ...
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4
votes
1
answer
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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? ...
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4
votes
1
answer
402
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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.
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1
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derivation of heston pde in gatheral
Following Gather (the volatility surface, chapter 2) we assume the following process:
$$ dS_t = S_t(\mu_t dt+\sqrt{\nu_t}dZ^1_t)$$
$$ d\nu_t= -\lambda(\nu_t-\bar{\nu})dt+\eta\sqrt{\nu_t}dZ^2_t$$
...
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1
answer
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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:
$$\frac{\partial \sigma(x)}{\partial x_i} = \frac{1}{\sigma(x)} \left( w_i \sigma_i^2 + \sum_{\...
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1
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European call delta derivation
Let's write $S(T) = S_T$ and $S(0) = S_0$. We want to compute $\frac{d}{dS_0}\mathbb{E}[f(S_T)]$. From a previous discussion this is equal to $$\mathbb{E}_{S_0}\left[f(S_T)\frac{g'_{S_0}(S_T)}{g_{S_0}(...
user26372
4
votes
1
answer
668
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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 ...
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votes
1
answer
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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 ...
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votes
1
answer
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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 ...
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0
answers
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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+(...)...
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0
answers
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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 ...
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0
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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 ...
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votes
2
answers
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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 ...
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votes
2
answers
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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 ...
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votes
3
answers
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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 ...
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