Questions tagged [distribution]

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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 ...
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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 ...
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1 answer
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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 ...
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1 vote
1 answer
259 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. ...
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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 ...
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50 views

Market Sentiment Concept Question

I came across an interesting concept question and was curious what other people thought: Let's say some commodity has a certain return distribution. Now, if one knows that over the next five days, ...
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1 answer
87 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 ...
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1 vote
0 answers
43 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 ...
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Probablity distributions of zero crossings in 1D random-walk

Consider a simple 1D random walk that starts at position zero, and each second changes position by either +1 or -1 with 50-50 probabalities. I know it is proven to cross zero infinitely many times, ...
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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 ...
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101 views

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 ...
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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 ...
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8 votes
0 answers
262 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]},\...
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0 votes
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 ...
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4 votes
1 answer
134 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 ...
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2 answers
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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 ...
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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 ...
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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 ...
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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 $r_u$ ...
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1 vote
2 answers
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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. ...
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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 $...
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0 votes
1 answer
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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?
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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 ...
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2 votes
1 answer
168 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 ...
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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, $\mu$ ...
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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 ...
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3 votes
0 answers
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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 ...
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1 vote
3 answers
176 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 ...
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2 votes
0 answers
364 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?
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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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5 votes
3 answers
384 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 ...
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0 votes
1 answer
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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 ...
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3 votes
2 answers
206 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 ...
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0 votes
1 answer
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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 - ...
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3 votes
1 answer
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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 ...
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0 votes
3 answers
183 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 ...
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2 votes
1 answer
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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 $\...
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0 votes
1 answer
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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 ...
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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 ...
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1 vote
1 answer
261 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,...
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0 votes
1 answer
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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 ...
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1 vote
1 answer
90 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 ...
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4 votes
1 answer
179 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^\...
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-1 votes
5 answers
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Why worry about fat tails, if you can use stoploss?

Sorry this might sound a silly question, but -humbly- I don't understand why models assume that returns range from [-∞,+∞] instead of [-stoplimit, +takeprofit]. A common objection to most models is "...
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1 vote
0 answers
83 views

Full Copula View using Meucci's Full Flexible View

I'm currently setting up an "Investment Framework" that should allow the following steps: Investment Committee (IC) has to decide on probabilities for 4 different market states. I have historical ...
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0 votes
1 answer
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What does 'near term order flow to be distributed across short term options' mean?

Please see the red phrase below. Guide to Option Pinning at Options Expiration | Investing With Options What Have Weekly Options Done To Pinning? That's a great question for a graduate student ...
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0 answers
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Degree of freedom input for Monte Carlo simulation of asset returns with multivariate t distribution

How do I calculate or estimate the degrees of freedom in order to perform a Monte Carlo simulation of asset returns with multivariate t distribution using R functions? I am able to calculate the mean ...
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1 vote
1 answer
322 views

Value at Risk (VaR): Normal distribution with gamma distributed volatility

If I was to do a 99% VaR calculation on a portfolio with normally distributed returns $\mathcal{N} (\mu,\sigma)$, the 99% VaR would be $\mu - 2.33\sigma$. Instead of having a constant volatility, let'...
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2 votes
1 answer
82 views

Sampling from an empirical distribution

I want to sample from the empirical distribution of returns. To do so, I do not want to make the preliminary assumption of which distribution the returns follow, rather I would like to sample from the ...
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1 vote
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Skewness and kurtosis measures when full distribution is not available

I have asked this question here, but did not get any answer. I was wondering if anybody knows a method of deriving skewness and kurtosis measures from different quantiles, mean, and/or variance. I do ...
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