Questions tagged [mathematics]
Used for question on application of mathematics in finance - from interest calculation to mathematical description of random processes.
192
questions
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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 ...
- 121
0
votes
0
answers
9
views
How to calculate non-linear intervals for a grid given high, low and number of intervals required
I would like to create a grid where the X axis is a function of time and the Y Axis represents the Range of a Stock Price and then I need to calculate the percentage gap for a specified number of ...
- 1
9
votes
5
answers
2k
views
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 ...
- 188
0
votes
0
answers
28
views
What is the Accumulation/Distribution formula doing geometrically?
My understanding of the essence of the Accumulation/Distribution Index is that it tracks the closing price of a security during each period relative to its price range for that period, something like ...
6
votes
0
answers
116
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 $S$ ...
- 128
2
votes
0
answers
85
views
Zero coupon price using Vasiceks model under the Real-world P measure model
I'm wondering if there is a way to work out the formula for the price of the zero-coupon bond using the Vasicek's model (P measure). I have tried to find reference on it but could not, I don't know if ...
- 21
1
vote
0
answers
60
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Sharpe ratio and uniformly distributed random portfolio
I am currently working on this paper which derives the Sharpe ratio distribution of uniformly random porfolios:
https://www.researchgate.net/publication/...
- 23
2
votes
1
answer
226
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Reflection principle of the Brownian motion
really appreciate some guidance on how to get the following equality:
- 21
1
vote
0
answers
67
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Bergomi's model normalisation
On his book https://www.amazon.fr/dp/B019FNKQS8/ref=dp_kinw_strp_1 Bergomi derives a multifactor mean reversible volatility of the volatility such that :
\begin{equation*}
d \xi_{t}^{T}=\omega(\tau) \...
- 436
0
votes
0
answers
40
views
how can properties of transition matrix be applied in the transcation cost of option
I am currently reading the PP BOYLE's article ' Option Replication in Discrete Time with Transaction Costs' written in 1992. Here is one place i couldn't figure out:
Where does that $\widehat{p}$ ...
3
votes
2
answers
816
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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 ...
- 437
-2
votes
1
answer
59
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One-Period Binomial Model
So, I'm required to consider the one-period Binomial market model for a particular question. We're told that the savings account is \$1 at time 0 and \$β at time 1. The stock price is given by S0 = 1 ...
2
votes
0
answers
268
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What is the relationship between Vanna and Gamma?
I'm trying to build a crude model for the effects of delta hedging on major indices like the S&P 500. My background is more in pure mathematics so a lot of this stuff is new to me. That said I ...
0
votes
1
answer
83
views
Calculate resistance / support on 5 minutes timeframe
I'm starting to learn resistance / support.
I'm trying to calculate it, but i'm not sure to understand something.
Let say i have an array of 5 last trades done (i can have much more, it's just for the ...
- 101
2
votes
1
answer
691
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Optimal Strategy in 3 Dice Game
In a recent interview I received the following question (an optimisation/strategy game)...which left me a bit stumped.
The rules of play, you start with 0 points, then:
Roll three fair six-sided dice;...
- 29
1
vote
0
answers
104
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Prove norm $\frac{1}{p}\sum_{i=1}^n |w_i|^p$ of min-variance portfolio $\leq$ max-Sharpe portfolio
The minimum-variance portfolio weight vector is
$$\boldsymbol{w}_{MV} = \frac{\boldsymbol{\Sigma}^{-1} \boldsymbol{1} }{\boldsymbol{1}' \boldsymbol{\Sigma}^{-1} \boldsymbol{1}}$$
whereas the maximum ...
- 2,885
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0
answers
37
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Calculating currency indexes weights?
I was looking at this formulas:
USD_INDEX= 50.14348112 × EURUSD^-0.576 × USDJPY^0.136 × GBPUSD^-0.119 × USDCAD^0.091 × USDSEK^0.042 × USDCHF^0.036
and
...
0
votes
1
answer
489
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Mathematical proof that the covariance between two portfolios is $w_A^\top\Sigma w_B$
How to prove in a line-by-line derivation that the covariance between two mean-variance efficient portfolios is equal to
$$w_A^\top\Sigma w_B$$
where $w_i$ is a unique portfolio weight vector, and $\...
- 2,885
1
vote
0
answers
75
views
Mathematical proof of out-of-sample disappointment in portfolio performance being a function of a portfolio's variance
The minimum-variance portfolio is considered more optimal than the maximum Sharpe ratio (tangency) portfolio on the grounds that its in-sample performance is less likely to disappoint out-of-sample.
...
- 2,885
0
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0
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55
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Entropy-implied volatility requires itself to be calculated?
\begin{align}
H &= \frac{1}{2} \ln (2\pi\sigma^2) + \frac{1}{2}\\
&= \frac{1}{2} \ln (2\pi e \sigma^2)
\end{align}
is the analytical solution for the entropy of a Gaussian random variable, ...
- 2,885
0
votes
0
answers
40
views
Estimating the XIRR of a very non uniform cash flows
It's my second post, so please bear my lack of experience in this field.
I've a very irregular cash flow (here you can see the set of date - cumulative cash flow)
The XIRR, calculated with Excel, is ...
- 103
1
vote
1
answer
120
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Does Value-at-Risk have any mathematical equivalence to copulas?
Portfolio Value-at-Risk estimated using the copula approach often just means generating artificial data sampled from a parametric copula('s joint multivariate distribution) as a model fit over the ...
- 2,885
2
votes
1
answer
90
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Question on the use of a limit in a proof
I ran into a step in an argument that I can't quite figure out. It's basically how they use a limit that I don't seem to understand. The context is local-to-unity asymptotics in vector autoregressions,...
- 2,386
1
vote
1
answer
85
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Option proofing: Analytical solution for option math
How do I prove the following equation:
P(X=100)≤(P(X=110)-P(X=90))/2
I am not sure how to start and whether it involves using the Black-Sholes formula or not (something like this: https://www.youtube....
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2
votes
1
answer
119
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help with derivation of equation 8 in Derman and Kani's binomial tree for local vol
in this paper "The Volatility Smile and Its Implied Tree" - Derman and Kani 1994 i understand the derivation of all equations up to 7.
But eq 8 i cannot figure out how to derive! i have ...
- 756
2
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0
answers
548
views
Why does the Hurst exponent pseudo code not match the Python implementation?
I am working on understanding the Hurst exponent calculation by Ernest Chan; however, the description of the algorithm does not match the Python implementation.
Chan [Algorithmic Trading: Winning ...
- 23
1
vote
0
answers
109
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The distribution of mean reversion time from the OU process
I was reading the paper Statistical Arbitrage in the US Equity Market and I couldn't understand the figure that plots the histogram of the empirical distribution of characteristic time to mean ...
- 11
-1
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2
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105
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Is there a way to formulate a Martingale series that will never explode?
Martingale's betting method can be seen here:https://www.investopedia.com/articles/forex/06/martingale.asp My question is if there is a way to put a non-exploding martingale,
[There is one attempt to ...
0
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0
answers
103
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What is the differential Value-at-Risk?
I am currently working on a Machine Learning Project, implementing portfolio optimization algorithms according to different risk measures. I have found sufficient information on Sharpe Ratio ...
- 1
0
votes
0
answers
196
views
Dynamic programming and Bellman equation to obtain the maximum
This is the problem of Marhsall (1992) "Inflation and Asset Returns in a Monetary Economy" and Balvers and Huang (2009) "Money and the C-CAPM"
Suppose an endowment economy where the representative ...
- 193
1
vote
1
answer
120
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Is the variance calculation correct in the book?
I'm reading the book "Financial Markets Under the Microscope" for my market microstructure studies. In the book, the variance of the market maker's gain is calculated as follows:
Assume that with ...
- 321
0
votes
1
answer
53
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Contingent Claim Bounds
In my course on discrete-time finance we derived the following equality for a lower bound for the value of a not necessarily replicable contingent claim $D$. Here we are looking at a single period ...
- 205
2
votes
1
answer
190
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How can I convert rolling annual returns back to quarterly returns?
I have a series of rolling annual returns and would like to convert these back to quarterly returns, which have not been provided. Is this possible formulaically, or is something like Excel's solver ...
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votes
1
answer
2k
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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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1
answer
3k
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What is the formula to calculate Implied Volatility Percentile [closed]
I googled and I am unable to find any formular . Can some one give me the formula to calculate IVP , based on sets of IV's given.
Thanks.
- 143
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votes
1
answer
168
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How can I express this sum in a easier way?
For instance, I know that the sum of the first $101$ natural numbers can be expressed in the following easy computation:
$\sum_{i=1}^{101}i = \frac{101*102}{2}$
One of the questions is: and what ...
- 175
4
votes
1
answer
996
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 ...
- 51
1
vote
3
answers
2k
views
How to calculate standard deviation of continuously compounded four-year stock returns?
Currently I am preparing for quant interview and I encounter the following question in Heard on the street.
Question: If the standard deviation of continuously compounded annual stock returns is $...
- 830
2
votes
2
answers
2k
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How to derive Black-Scholes equation with dividend?
Question: The Black-Scholes equation without dividend is given by
$$\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2S^2\frac{\partial^2 V}{\partial S^2} + rS \frac{\partial V}{\partial S} -rV = ...
- 830
2
votes
0
answers
51
views
Estimator for Conditional value at risk (average value at risk)
I am following a book: Advanced Stochastic Models, Risk Assessment, and Portfolio Optimization by Svetlozar T. Rachev, Stoyan V. Stoyanov, Frank J. Fabozzi
I'm learning about average value at risk. ...
- 195
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votes
1
answer
509
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Do quants need to know bloomberg terminal and VBA? [closed]
I am a Pure Maths PhD student who will graduate in 2 years time. My aim is to land a quant job after gradauation.
When collecting more information so that I can have some edges over others, I heard ...
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1
vote
1
answer
52
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How is hypothesis testing work in population sampiling? [closed]
I am learning the basics of quant trading from quantconnect's tutorial Confidence Interval and Hypothesis Testing. I understood the first part of the article but I dont understand "Hypothesis Testing"...
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2
votes
1
answer
274
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Forward Start Spread Options
Question:
We have a spread option with payoff: $\max (P_{T} - HR\times G_T, 0)$, where $P$, $G$ are underlying prices and $HR$ is a constant.
At time zero only contract $G$ is available for ...
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4
votes
4
answers
2k
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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 ...
- 143
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votes
0
answers
50
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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.
$$
where $g(t,x)$ is a given functions and $X_t$ is a given process driven by a brownian motion ($dX_t=(...)dt+(...)...
- 301
2
votes
0
answers
56
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Transformation of random variables and second-order stochastic dominance
Suppose $X$ and $Y$ are two random variables where $X$ SOSD* $Y$. Let $g(\bullet)$ be a monotonic function and $X'=g(X)$ and $Y'=g(Y)$. Under what conditions of $g$ is
$X'$ SOSD $Y'$?
I know if $g$ ...
- 21
0
votes
1
answer
173
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Why do most interest rate formulas, and indeed finance in general, add 1 to a rate and then subtract afterwards? [closed]
For example, in the formula that shows the relationship between the nominal and effective interest rate shown below, 1 is first added to in/m and then 1 is subtracted from the result. What is the ...
- 1
1
vote
1
answer
277
views
Approximation of Forward Rates in discrete time
The forward rate from time $t$ to $T$ ($f_{t,T}$) can be approximated by:
$$
f_{t,T}= \left[ \frac{(1+r_T)^T}{(1+r_t)^t} \right]^{\frac{1}{{T-t}}}-1
\sim
\frac{(1+r_T)^T-(1+r_t)^t}{T-t}
$$
Why is ...
- 75
0
votes
1
answer
103
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standard brownian vs brownian motion
We say Xt with paramters (µ,σ) is brownian process if (Xt-s - X t) ~N (µs,σ2 s) AMONG other conditons .
Here we don't speak about any particular distribution for X t. We only say it is a brownian ...
- 21
2
votes
1
answer
169
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Finding the process of $X/Y$
This comes from Mark Joshi's concepts of mathematical finance exercise 4 chapter 11.
If
$$dX_t = \alpha X_t dt + \beta X_t dW_t$$
$$dY_t = \alpha Y_t dt + \gamma Y_t d\tilde{W}_t$$
with $W$ ...
- 698