Questions tagged [probability]
A probability expresses quantitatively how likely an event is to occur. We often encounter probabilities as conditional probabilities which express how likely an event is to occur in light of certain (given) information.
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Tactical Investment Algorithms
I am reading paper "Tactical Investment Algorithms" (link) (NOTE: you can download the paper without registration, just press "Download" and then "Download without ...
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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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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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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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$\frac{\partial}{\partial a} E [\sqrt{a+X} ]$, $X > 0$ a.s., $a \geq 0$
Although maybe this could have been posted at cross-validated, I actually have a financial application in mind.
Problem:
There is a very elementary mistake somewhere, but I can't see it:
Let $X$ be a ...
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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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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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A model for probability of credit rating change for a single issuer
I am looking to model the probability of a single issuer upgrading or downgrading it's credit rating at some time using historical data. I have done research and everything I have found so far are for ...
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Bayesian analysis in R for low default portfolios
I want to apply the knowledge of this paper (Bayesian estimation of probabilities of default for low default portfolios, by Dirk Tasche) in R, but I can't find the right bayesian package and functions ...
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How to test signifcance of a sharpe ratio
Let say you have measured a Sharpe Ratio of $S^*$. What is the simplest way (ie no fancy distributions) to do a hypothesis that this is different from $0$?
So $H_0: \text{ The sharpe ratio is equal ...
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What is the meaning of this notation, D lag t?
I'm reading the book Financial Markets Under the Microscope to study market microstructure. There is a notation that I could not understand. What is the meaning of D here? It is not used in the text ...
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Is option surface same as future price probability surface?
Let's consider the Option Chain for the Stock. There are two 3D surfaces representing the probability of the future stock price and the option prices. I wonder if they are representing the same thing?
...
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Empirical Probability Distribution
I have a dataset with 3.000 observation (price of an asset).
I want to study the empirical distribution of the logRet of that time series.
How can I do it in Excel? if not possible to do it in Excel, ...
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Geometric brownian motion and probabilities
A stock's price movement is described by the equations $dS_t=0.02S_tdt+0.25S_tdW_t$ and $S_0=100$. An investor buys a call option on said stock with a strike price $K=95$ which expires in $T=2$ years. ...
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CAPM Model, is this exercise done correctly?
Hey i need to know if the task is done correctly, please help :) Standard deviation of the rate of return on the market portfolio is equal to $\sigma_{MP}=1,5\%=\frac{15}{1000}$. I have portoflio ...
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Measure for probabilities inferred from prices of derivatives on non-traded random variables?
Are probabilities of certain events (e.g. amount of rainfall over a period, probability of a Fed rate hike) inferred from derivatives on non-tradeable random variables (e.g. Weather Futures, Fed Funds ...
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Call Probability of European callable IRS [closed]
When pricing a callable IRS (say only one call date) with a diffusion model (e.g. HW 1F) with a Montecarlo resolution, one can get the call probability on the call date versus maturing the date (which ...
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Radon-Nikodim Derivative at time 0
I have a very basic question about filtrations and Radon-Nikodym derivatives. I am reading the Andersen-Piterbarg, more in particular Eq. (1.12). They define the process $\zeta(t) = E^P_t[\frac{dQ}{dP}...
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Interpreting Autocorrelation as probability
I was recently asked:
Given a random time series of 1s and -1s. Eg of a sample = [1, 1, 1, -1, -1, 1, -1,..]. The autocorrelation of this series is Z. What can you say about the probability of a 1(or ...
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Probability and random walk
Let's says i have 10 years of daily prices on a stock ABC. I do some analysis and I realise that, for example, if the stock increases 5 days in a row (close > open), 75% of the time, the 6th day will ...
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What is the probability of a lookback option ending in the money (CRR-model)
I would like to compute the probability that a certain lookback option ends in the money, let's say that the option has the following payoff $h_N=\max\left\{0,K-\min\{S_1,...,S_N\}\right\} $ where $K$ ...
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Throwing a dice and risk neutral probability
Consider the game of throwing a "fair" dice. Not sure if the answer is obvious but is there any proof (e.g. replication argument) that under the risk neutral measure the probability of any outcome is ...
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Kupiec Test Backtesting VaR
I am currently analyzing the Kupiec test used for backtesting $VaR$.
Suppose that I backtest a $VaR$ system for $n$ days (for example 250), with a confidence interval of $1-\alpha$ (for example a $1-\...
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Zero Volatility Options Pricing
Suppose an asset evolves in time according to the SDE
$$
dS = \mu S dt + \sigma S dW,
$$
where $\mu>0,\sigma>0$ are fixed constants and $dW$ is a Wiener process. To price options for this ...
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Sample conditional multivariate random variable?
There's multivariate random variable, future prices of assets, 5 years from now: $$X = [Gold, Silver, SP500]$$
There's historical prices for $X$ available for last 50 years. It's possible to fit ...
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What is the connection between the risk neutral implied density and the real world density?
I understand that we can use option prices to imply volatilities and ultimately to imply a risk neutral density. I also understand that this implied density is not the same as the "real world density"....
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Two commodities which are normal distributed and perfectly correlated
The daily price change in commodity 1 is distributed $N(0,0.15^2)$ and the daily price change in commodity 2 is distributed $N(0,0.3^2)$. The two commodities are 100% correlated.
1) Does the relative ...
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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 ...
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What's the expected value of a repeated game with 50% chance to win 0.5 and 50% to lose 0.5?
Assume we start with 1.
In the first bet the expected value of remained balance is 1.5 * 0.5 + 0.5 * 0.5 = 1
For N times, is it still 1 according to E(XYZ)=E(X)E(Y)E(Z)?
But 1.5^50 * 0.5^50 is not 1.
...
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GBM probability of hitting non constant barrier
I know there is a formula for probability of hitting a constant barrier for GBM/BM (See page 651 in Martinagle Methods in Financial Modelling).
Is there a formula for non-constant barrier?
The ...
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Question regarding No Arbitrage price of a call option
I have a question regarding how to solve the NA price for a slightly modified call option.
Say that I have a money account $B(T)=e^{r(T-t)}$ and a stock dynamic $\frac{dS(t)}{S(t)}=(r-\delta)dt+\...
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Drawing values from a lognormal distribution of a GBM
I'm looking at a GBM with parameters
$$
r=0.05 \\
\sigma=0.2 \\
K=130\\
T=0.25\\
S_0 = 100
$$
This is a process that is lognormally distributed with mean and variance given by
$
\mu = S_0e^{r T+0.5\...
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How to determine the no arbitrage price of following claim? (change of numeraire)
How do I determine the no arbitrage price for claims such as $min(S_1(T),S_2(T))$ or $max(S_1(T),S_2(T))$? We can consider a standard Black Scholes model. Hence $S_i(T)=S_i(t)e^{(r-\sigma_i^2/2)(T-t)+\...
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How to derive the CDF and the probability density function [closed]
Is there something missing in this question i dont seem to understand, can anyone help explaining what is required?
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Produce the random variable for an asset from a uniformly distributed random varible
I'm working on a quant interview question from the book called Quant Job Interview Questions And Answers (by Mark Joshi and other authors). I cannot understand the following question(not the answer, ...
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Introducting a new probability measure
I'm trying to understand what means :
$$ \frac {d \mathbb {\tilde{P}} }{d \mathbb P } \bigg\rvert_{\mathcal F_t }$$where $\mathcal F_t $ is a filtration I guess (not explicitely mentionned).
they ...
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Excel formula for Laplace distribution
I am trying to create a forecast model, projecting the number of passengers through an airport over a period of time (daily, weekly, and monthly). I've already used Excel's FORECAST and POISSON ...
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Probability Density Function of a Wiener Process Minimum
Let $W_t$ be a standard Wiener process. Find the probability density function of $m_T =
min_{t\in [0,T ]}W_t$.
I know that it is based of the concept of the reflection principle, but I wasn't too ...
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Arithmetic Brownian Motion in Market Making papers
We often consider high-frequency market maker and suppose that the reference price is the arithmetic Brownian Motion:
$dS_{t} = \sigma d W_t$
What is the difference $t_n - t_{n-1}$ in this case? Is ...
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Intuitive explanation of why ITM options have low Time/Extrinsic Values?
While brushing up on my knowledge about the Greeks, I have been struggling coming up with an intuitive, probability-based explanation behind why not only Out-of-the-Money (OTM), but also In-the-Money (...
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Event Occurs Almost Surely
Consider an uncountably infinite space, an infinite coin-tossing.
Let $(\Omega,\mathcal{F},\mathbb{P})$ be the probability space. If a set $A\in\mathcal{F}$ satisfies $\mathbb{P(A)=1},$ then we say ...
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If price is a random walk, is ok to use the binomial distribution to estimate a trading strategy?
Is it OK to assume a trading strategy should have a binomial distribution if the price is just a random walk?
using p of the event as:
$$\frac{AverageStopLossPercent}{AverageStopLossPercent + ...
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Option and probability of finishing in the money?
This seems to be another easy question but I am a bit confused. I know delta is a proxy for an option finishing ITM. Delta also happens to be N(d1) in the BSM pricing model. N(d1) usually is pretty ...
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Probability that the price of stock following a brownian motion goes under a certain value
The price of the stock XYZ follows a brownian motion pattern with
starting price = 10, μ = 0 and σ = 20 (on annual basis). What's the probability that in 6 months the price is less or equal to 8?
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stock specific volatility
I was unsure about the precise definition of "stock specific volatility". Used in this question "A stock has beta of 2.0 and stock specific daily volatility of 0.02. Suppose that yesterday's closing ...
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Show that the variance of the market portfolio is the weighted average of the ovariances between each constituent and the market portfolio itself
Let us assume that the market portfolio consists of n assets.
Given that the return of the market portfolio can be written as $r_m = \sum_{j=1}^{n} w_jr_j$, we have that $\sigma^2_m = E(\sum_{j=1}^{n} ...
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Credit spread model
Let $c(t,T):=-\frac{1}{T-t}[\mathrm{ln}(P_1(t,T))-\mathrm{ln}(P_0(t,T))]$, with:
$c$ measure of how a company is prone to fail;
$P_0(t,T):=e^{-r(T-t)}$ price of no-defaultable bond.
$P_1(t,T):=\...
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How to solve these SDE Problems
Quuestion1.
I make a solution $r(t)$ used by Ito's lemma
$r(t)=e^{-a t}r(0)+\int _{0}^{t}e^{a (s-t)}\theta (s)ds+\sigma e^{-a t}\int _{0}^{t}e^{a u}\,dB^{1}(u)$
Is this right?
and I try to make ...
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Problem in copula fitting
I have returns of 2 stocks: stock1 and stock2. And I want to fit pair copula.
I use this libraries
library(VineCopula)
library(copula)
then I select an ...
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Geometric Brownian Motion - Price Probabilities
I am modeling a stock price that follows Geometric Brownian Motion and have the following:
$E(X)$ = .16 (16%)
$\sigma$ = .24 (24%)
$X_0$ = 95
$T$ = 1 (12 months)
I am trying to find the ...
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