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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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 ...
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9 votes
4 answers
1k 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 ...
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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 ...
5 votes
0 answers
112 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 $S$ ...
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2 votes
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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 ...
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0 answers
29 views

fulfilling the condition for backtesting scalping technique

I am backtesting scalping technique with the 3 sma, in aligator indicator. One of the conditions for scalping long, would be to see alll the three ema's are radiating out from one another as in the ...
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1 vote
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60 views

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/...
2 votes
1 answer
217 views

Reflection principle of the Brownian motion

really appreciate some guidance on how to get the following equality:
1 vote
0 answers
61 views

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) \...
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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
734 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 ...
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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 ...
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175 views

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 ...
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1 answer
79 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 ...
2 votes
1 answer
593 views

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;...
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1 vote
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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 ...
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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 ...
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1 answer
454 views

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 $\...
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1 vote
0 answers
71 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. ...
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53 views

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, ...
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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 ...
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1 vote
1 answer
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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 ...
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2 votes
1 answer
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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,...
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1 vote
1 answer
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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
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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 ...
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2 votes
0 answers
526 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 ...
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1 vote
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100 views

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 ...
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-1 votes
2 answers
105 views

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 ...
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0 answers
99 views

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 ...
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193 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 ...
1 vote
1 answer
120 views

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 ...
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1 answer
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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 ...
2 votes
1 answer
156 views

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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7 votes
1 answer
2k 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?
2 votes
1 answer
3k views

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.
2 votes
1 answer
167 views

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 ...
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4 votes
1 answer
873 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 ...
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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 $...
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2 votes
2 answers
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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 = ...
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2 votes
0 answers
50 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. ...
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1 answer
481 views

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 views

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
265 views

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
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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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4 votes
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+(...)...
2 votes
0 answers
56 views

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 $g$ ...
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0 votes
1 answer
128 views

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 ...
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1 vote
1 answer
265 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 ...
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0 votes
1 answer
102 views

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 ...
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2 votes
1 answer
167 views

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 $W$ ...
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