Questions tagged [distribution]

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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 ...
Bill Zissimopoulos's user avatar
1 vote
1 answer
110 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 ...
nzc's user avatar
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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 ...
Mango's user avatar
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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 ...
SuperCodeBrah's user avatar
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58 views

Taking a set of normally distributed random variables as the sample space to fitting an exponential distribution

Disclaimer, this is my first question/interaction in this forum. Let's assume I have random variables that are normally distributed. Then, say I take the observations that are greater than the mean, i....
ak10's user avatar
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1 vote
1 answer
148 views

Option pricing under distribution assumption

For simplicity assume zero interest rates in the following. Given the price of a (European) put option with strike K and maturity T at time point t. $P_t(K, T)$ for a given underlying S with values $...
MrLCh's user avatar
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1 answer
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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 ...
bond-pricer's user avatar
4 votes
1 answer
236 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?
keon6's user avatar
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1 answer
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How to find out if Asymmetric Laplace Distribution is having Finite/Infinite Variance?

I was fitting the NIFTY 50 Daily Log Returns (To be more precise Returns in this case refers to the Log of 1+Returns rather than Log of Returns as Log cannot be taken of negative values which returns ...
Anon9001's user avatar
0 votes
1 answer
175 views

Student-t measure of return volatility and time scaling

I have a series of price returns of an asset (4 days worth of data). They are relatively high-frequency. My ultimate goal is to calculate realized volatility, but using a student's t-distribution. I ...
Bob Dobbs's user avatar
1 vote
2 answers
279 views

Can I apply the Kelly criterion directly, without fitting any distributions?

Problem I want to apply the Kelly criterion to asset returns, so that I know how much to hold of each, ideally (and how much I should keep as a cash reserve). As far as I understand the Kelly ...
danuker's user avatar
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Pareto comparison of return distributions

In making a choice among financial strategies, each of which has some estimated return distribution, some strategies will clearly be better than others. But many times, the choice is a question of ...
dmgreenwald7's user avatar
1 vote
1 answer
527 views

Integral of brownian motion wrt. time over [t;T]

From the post Integral of Brownian motion w.r.t. time we have an argument for $$\int_0^t W_sds \sim N\left(0,\frac{1}{3}t^3\right).$$ However, how does this generalise for the interval $[t;T]$? I.e. ...
Landscape's user avatar
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How to compute the combined probability of loss for 2 time series (consisting of historical stock prices)?

May I please ask the community's support with the following problem? I have 2 time series, with approximately 1000 observations each (same number of observations for both). They represent the daily ...
mihnea_11235's user avatar
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1 answer
140 views

Terminal wealth distribution from dollar cost averaging

If monthly stock market returns follow an IID lognormal distribution, the terminal wealth distribution of investing a lump sum for many years is also lognormal. What is the terminal wealth ...
Fortranner's user avatar
1 vote
0 answers
44 views

Assymetric Rate Distribution

The pandemic has disavowed any notion of nominal rate distributions to being truncated at 0%. However, if Central Banks at Debtor nations are conflicted in that they are incented to suppress interest ...
AlRacoon's user avatar
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1 answer
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Why stock prices changes don't follow Pareto Distribution?

I calculated the distribution of the stock price changes (diffs). The diffs are multiplicative, $d_t=p_{t} / p_{t-1}$. As far as I know the distribution should look like Power law distribution (Pareto ...
Alex Craft's user avatar
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basic numerical integration question related to case of high positive volatility skew

is the below equation true irrespective of if that 2nd derivative turns out to be negative or >1 , (ie even if theres an arbitrage) ? the reason i ask is that i am writing a single asset montecarlo ...
Randor's user avatar
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1 answer
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Fat tailed can be estimated through a t-distributions?

I have a simple question that makes me doubt a bit. In a multiple choise exam I ecountered this question: "if the stocks returns are not normally distributed, the fat tail effect can be estimated ...
gabriele's user avatar
9 votes
0 answers
387 views

On a time integral of Brownian motion up to the hitting time

Just come up with a 'simple' and interesting problem that I've been struggling to deal with for some time. Consider a filtered probability space $(\Omega, \mathcal{F}, \{\mathcal{F}_t\}_{t\in[0,T]},\...
FoolAlex's user avatar
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2 answers
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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 ...
Řídící's user avatar
4 votes
1 answer
198 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 ...
user1337's user avatar
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2 answers
284 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 ...
Vivek Subramanian's user avatar
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0 answers
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 ...
Dmitriy's user avatar
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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 ...
torino's user avatar
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1 vote
0 answers
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How can I find the distribution function of the following random variables?

Suppose that the random variables $Z_i$ are defined as follows: \begin{equation} Z_i = D(0, t_i)(R_{i-1} +c)\Delta N, \end{equation} where $D(0, t_i)= \exp\{-\int_{0}^{t_i} r_u du\}$ for which $r_u$ ...
user53249's user avatar
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1 vote
2 answers
284 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. ...
Philipp's user avatar
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2 votes
0 answers
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The distribution of the jump diffusion process

In the Merton jump diffusion model the process of the share price can be expressed as $$S_{t}=S_{0}\cdot\exp\left\{ X_{t}\right\} ,$$ where $$X_{t}=\mu t+\sigma W_{t}+\sum_{i=1}^{N_{t}}Y_{i}.$$ Here $...
Kapes Mate's user avatar
0 votes
1 answer
155 views

Calculating the Value-at-Risk when changing the confidence level

If I have a VaR estimate at a 95% confidence interval is 10, how do I calculate the approximate level of the VaR if the confidence level was raised to 99%, assuming a one-tailed normal distribution?
May's user avatar
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-3 votes
1 answer
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Should stock return series be modeled with a parametric distribution, or an autoregressive function? [closed]

If I have prior knowledg that a stock return series follows a parametric distribution, such as a Student t-distribution with 4 degrees of freedom, without actively looking for prior knowledge of ...
develarist's user avatar
  • 2,990
2 votes
1 answer
187 views

What does this absolute return distribution chart show?

I was reading some pages in Professional Automated Trading by Eugene Durenard when I came across this chart: The caption says: "S&P Absolute Return Distribution: Log-Log Scale". The ...
Flux's user avatar
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0 votes
1 answer
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FX spot distribution with student-t returns

If I am modelling my returns as $\sim N(0, \sigma^2)$, then I can evolve my spot distribution as: $$S_{t} = S_{0}e^{(\mu - \frac{1}{2}\sigma^{2})t + \sigma dW_{t}}$$ where $S_{0}$ is the spot, $\mu$ ...
sumit_uk1's user avatar
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0 answers
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What should degrees of freedom $\nu$ be set to when modeling financial returns that follow the t-distribution?

The closer the t-distribution degrees of freedom ($\nu$) is to 0, the more heavy are the tails, whereas high degrees of freedom recovers the normal distribution. In finance, what value is usually used ...
develarist's user avatar
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3 votes
0 answers
326 views

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 ...
develarist's user avatar
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1 vote
3 answers
188 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 ...
develarist's user avatar
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2 votes
0 answers
847 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?
user40929's user avatar
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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 ...
develarist's user avatar
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5 votes
3 answers
517 views

What is the distribution of the risk-free asset?

If the risk-free asset has a volatility of $0$, therefore making its mean equal to the risk-free rate, $r_f$, does this mean that it has no probability distribution, and therefore there is no reason ...
develarist's user avatar
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0 votes
1 answer
135 views

How important is the chronological ordering of historical returns?

The returns of asset $A$ in chronological order are 0.03 0.01 -0.04 0.02 0.05 -0.10 0.02 The expected return, or sample mean, is $-0.00143$ while its sample ...
develarist's user avatar
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3 votes
2 answers
278 views

Interpretation of a uniform asset return distribution

Typically asset return distributions are bell-shaped with most mass occurring in and around the center, 0% returns, and less so in the tails, with the left tail representing the probability of large ...
develarist's user avatar
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0 votes
1 answer
174 views

Monte Carlo approach and methods for generating random returns

Recently I found myself reading more about Monte Carlo approach in m.v. portfolio optimization framework. I already discuss the topic on this forum (if interested please consider the following links - ...
Nipper's user avatar
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3 votes
1 answer
444 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 ...
Richard Hardy's user avatar
0 votes
3 answers
276 views

Are mean-variance efficient portfolio weights random variables with probability distributions?

The mean-variance model outputs a portfolio weight vector whose elements are individual asset weights that sum to 1. Regardless of which portfolio along the efficient frontier is being solved, the ...
develarist's user avatar
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2 votes
1 answer
224 views

Is it always better to use the entire distribution of a financial returns series, not just $\mu$ and $\sigma$?

In finance models that use historical returns for inputs, including option pricing models, forecasting and portfolio optimization, only the statistical moments of the returns distribution, $\mu$ and $\...
develarist's user avatar
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0 votes
1 answer
229 views

Which financial time series have a PDF and/or CDF?

Consider the following types of financial time series for a single publicly-listed stock: Price data Log returns Cumulative returns Each is computed from the item listed before it: log returns are ...
develarist's user avatar
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0 votes
0 answers
66 views

Density of a portfolio's returns is the weighted average of asset distributions?

The expected return of a portfolio can be formulated as a weighted average of the constituent assets' returns: $$r_p = w_1 r_1 + w_2 r_2 + \dots + w_N r_N + \epsilon$$ Does it also follow that the ...
develarist's user avatar
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1 vote
1 answer
389 views

Why do cumulative returns have a bimodal distribution?

Regular returns (log-differenced prices) have statistical distributions that are bell-shaped and unimodal (one mode/peak) despite being non-normal and fat-tailed. Cumulative returns, on the other hand,...
develarist's user avatar
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0 votes
1 answer
183 views

Option implied distributions

I am having a bit of trouble understanding how to obtain the option implied distributions. I have strike levels, deltas and implied vols for a call option that expires in 6 months. Roughly 40 data ...
user avatar
1 vote
1 answer
115 views

Kurtosis of a straddle

I want to determine the kurtosis of a straddle. My question is closely related with the following topic here. According to the following paper of Ben-Meir and Schiff (2012) the expected value of a ...
HJA24's user avatar
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4 votes
1 answer
245 views

Change of measure

I am looking at the derivation of the Hill estimator. It is $ \bar{F}(x) = 1 - F(x)$ the right tail of the distribution. In the derivation they use the equation $$ \frac{1}{\bar{F}(u)}\int\limits_u^\...
SmurfAcco's user avatar
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