# Questions tagged [distribution]

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### Peaks and gaps in log-return of XAUUSD 1-minute log-return density

I'm tinkering around a 1-minute XAUUSD data from March 2009-December 2023 to see if I can model it with a log-normal or log-t distribution and I happen to notice some interesting properties in the log-...
32 views

### Distribution fitting to data with (isolated) extreme observations

Let's assume I have 2 time series of daily observations of a given experiment. The data of one time series show a very long tail (either side) and in absolute sense the difference between the lowest ...
1 vote
1k views

### Two Probability Questions from Quantitative Finance Interview Book

I posted the two questions in math stack exchange one month ago but cannot get an answer, so I post it here and appreciate your advice:) I'm reading an interview book called A Practical Guide to ...
311 views

### Estimating distribution of rate of return

Let $f[t]$ be the price of a stock at time $t$. We can calculate the rolling rate of return of the stock in a window of length $n$ by computing: $$r[t] = \frac{f[t] - f[t-n]}{f[t-n]}$$ $r[t]$ is ...
107 views

### Probability Theory: Maximizing the difference between distribution functions

Given a sample of observations $X$, by changing a parameter $p$ we can divide $X$ into two subsamples $X_1$ and $X_2$ (this division is done in a non-trivial way which is nonetheless irrelevant to ...
38 views

### Taking skewness into account when determining daily expected ranges

I use a method to determine daily expected ranges by combining both daily IV and daily realized vol. with different weights to get the expected range, and it worked pretty accurately. However I want ...
76 views

### Probability Distribution of Stock Returns [closed]

Is there a modern theory for the probability distribution of stock returns? It is relatively easy to deduce that under idealized conditions stock returns follow a log normal distribution. One arrives ...
1 vote
137 views

### Implied Distributions from forward prices

I understand that the common way to arrive at an implied distribution for an underlying is through the price of its call options as per the Breeden-Litzenberger formula. I am wondering if its possible ...
12 views

### constrains of return distribution and risk return trade off

Suppose we have a portfolio $V$, we are only allowed to invest in one stock $S$, its price movement follows the geometric brownian motion, i.e. $dS=S(\mu dt+\sigma dW)$. We are allowed to choose ...
351 views

### Modeling orderbook shapes as distribution

What are different distribution models typically used for generating orderbooks under high volatility, illiquidity, and multiple exchanges with different fees?
1 vote
58 views

### Determing "fair" implied volatilities for SPX options

I'm trying to come up with a method to calculate fair IVs for SPX options based on historical data. I can't find much information on this so here's how I've thought to do it: Determine a metric for ...
1 vote
153 views

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### Creating a set of histories that satisfies certain statistics

I'm looking at a download of BlackRock's capital market assumptions, which gives a bunch of statistics, such as expected and quartiles for asset classes' returns for different timeframes, volatilities ...
476 views

### Minimizing variance vs. expected shortfall: distributions where the difference is salient

In portfolio theory in finance, given a set of $n$ assets to choose from, one often selects portfolio weights so as to maximize expected return and minimize some measure of risk, e.g. variance or ...
219 views

### Reconciling Two Claims About Volatility Under Fat Tails

I have read the Wikipedia article on volatility, and Nassim N. Taleb's Incerto, and found two statements attributed to Mandelbrot's views, which appear to be in contradiction. Taleb (who was mentored ...
80 views

### non gaussian distributions with higher moments and time scaling properties?

If we assume a portfolio comprised of n asset classes, whose log returns can be modeled with a distribution. I am interested in finding a distribution that: incorporates higher moments (skewness and ...
572 views

### Probability Distribution at each Simulation Period using Geometric Brownian Motion

I am using the equation $S_t = S_0e^{(\mu-\frac{\sigma^2}{2})t+\sigma\epsilon\sqrt{t}}$ to simulate a financial metric at each $t$, where $t=1$ and $T=5$. Stated in plain English, I am trying to ...
1 vote
41 views

### How can I find the distribution function of the following random variables?

Suppose that the random variables $Z_i$ are defined as follows: $$Z_i = D(0, t_i)(R_{i-1} +c)\Delta N,$$ where $D(0, t_i)= \exp\{-\int_{0}^{t_i} r_u du\}$ for which $r_u$ ...
1 vote
298 views

### Taleb's Black-Swan: interpretation of the exponent

I am reading Taleb's "Black Swan" (revised 2020th edition). In chapter 16 "The Aesthetics of Randomness" he describes the meaning of the exponent in the context of extrapolation. ...
141 views

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### Large deviations theory in finance

In probability theory, the theory of large deviations concerns the asymptotic behavior of remote tails of sequences of probability distributions. A related post says: Large deviations theory is ...
1 vote
195 views

### Do portfolio mean and portfolio variance have probability distributions?

If $X$ is a $T\times N$ matrix of multivariate asset returns, and $w$ is some optimal portfolio weight vector, then the portfolio return series is $r_p = X w \in\mathbb{R}^{T}$. This return series ...
973 views

### law of absolute of max of brownian motion

What is the law of $\max\left(|B_t|\right)$ for $t$ in $[0,T]$ and $B_t$ is a Brownian motion? Any references for properties of this process?
1k views

### Is volatility for the next day forecastable? To any extent?

In a more general way: is there 1) a methodological approach to quantify the correctness of a model that produces a probability distribution for the, say, S&P 500 index return for the next ...
1 vote