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.

Filter by
Sorted by
Tagged with
2
votes
1answer
92 views

Proving $\mathbb{P}(S_t<0|S_0=s_0)=0$ for Geometric BM

I am trying to prove that for the geometric Brownian motion of a stock $\textrm{d}S_t=\mu S_t\textrm{d}t+\sigma S_t\textrm{d}B_t$ with strictly positive constants $\mu$ and $\sigma$ and and $S_0=s_0&...
2
votes
0answers
83 views

Imperfect Competition among Informed Traders - Back, Chao and Willard

The following assumptions are part of the paper and I can not solve for the statistic that is denotes as $\phi$ in the sequel. I would be glad if anyone could help me. Suppose that in the market, ...
1
vote
0answers
44 views

Concentration of measure phenomena in financial mathematics

Concentration of measure is a small area of statistics and probability theory that proved inequalities regarding the statistical properties of sets of random variables that exclude one of those random ...
1
vote
1answer
57 views

sub-Gaussian random variables in financial economics

Unlike financial time series that typically possess fat tails, sub-Gaussian random variables have strong decay in the tails of their distribution. do sub-Gaussian random variables or processes appear ...
0
votes
0answers
45 views

Escape Dynamics in financial economics or time series

These slides describe escape dynamics to be a type of, or having some relation to, rare event(s). Black swan events in business cycles was also included under the definition of rare events. My guess ...
3
votes
0answers
106 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 ...
3
votes
0answers
162 views

Does the Shannon entropy of stock returns change over time?

Shannon entropy, $H(X) = -\sum_{i=1}^n p(x) \ln p(x)$ is a probabilistic measure of randomness or disorder within a random variable's probability distribution or histogram. If we take rolling window ...
3
votes
0answers
36 views

Characteristic function of time-changed Levy processes

Let $X_t$ be a Levy process, and $Y_t$ be a subordinator i.e. process with nondecreasing trajectories. I have to find characteristic function of $X_{Y_t}$. I know that I have to calculate: $$E[e^{iuX_{...
0
votes
0answers
41 views

Proof of existence of one only martingale measure

I know that: Hypothesis 1 (Girsanov Theorem) Let $\theta=\begin{Bmatrix} \theta_t \end{Bmatrix}_{t\in [0,T]}$ be a square-integrable and $\Im_t$-adapted process such that $\mathbb{E}[e^{\frac{1}{2}\...
2
votes
0answers
92 views

Recognizing a Martingale

Under which conditions is the stochastic process $\{X_t\}_{t=0,1,...,T}$ a martingale? Demonstrate and explain clearly for each case below. If it is not necessarily a martingale, provide a ...
1
vote
1answer
301 views

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 ...
1
vote
3answers
163 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 ...
1
vote
0answers
26 views

Link between cumulants and kurtosis

Hey in "Financial modelling with Jump processes" by Cont and Tankov is written that kurtosis of distribution of random variable $X$ is equal to $\frac{c_4(X)}{c_2(X)^2}$ where $c_n$ denotes $...
1
vote
1answer
94 views

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 ...
4
votes
2answers
172 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 ...
5
votes
2answers
169 views

$\frac{\partial}{\partial a} E [\sqrt{a+X} ]$, $X > 0$ 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 ...
0
votes
0answers
34 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 ...
1
vote
1answer
154 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,...
2
votes
1answer
72 views

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 ...
0
votes
1answer
100 views

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 ...
0
votes
1answer
119 views

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 ...
0
votes
0answers
32 views

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 ...
0
votes
1answer
64 views

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? ...
1
vote
1answer
96 views

Convert option inputs to standard Brownian motion

I want to know the probability that the strike price of an option is touched. My input values are: P = price S = strike v = vol t = time to expiration According ...
2
votes
2answers
65 views

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, ...
0
votes
0answers
63 views

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. ...
0
votes
1answer
39 views

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 ...
2
votes
2answers
53 views

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 ...
0
votes
1answer
32 views

Call Probability of European callable IRS

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 ...
0
votes
1answer
49 views

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}...
0
votes
1answer
58 views

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 ...
0
votes
0answers
13 views

Specify user-defined distribution for multivariate distribution in copula R package

For the copula R package, the function Mvdc allows the margins to be user-defined. ...
0
votes
0answers
61 views

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 ...
1
vote
1answer
111 views

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 $K$ ...
1
vote
1answer
205 views

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 ...
1
vote
0answers
311 views

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-\...
3
votes
1answer
122 views

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 ...
0
votes
0answers
19 views

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 ...
5
votes
1answer
271 views

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"....
0
votes
0answers
44 views

Classical Ruin Theory - Lundberg Model

In classical risk/ ruin theory, I see this formula crop up in my notes but my lecturer didn't explain to me why/ when it's employed: $M_X(r) = \int_{-\infty}^{\infty} e^{rx} f(x) dx$ I understand ...
0
votes
1answer
75 views

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 ...
1
vote
1answer
204 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 ...
0
votes
1answer
117 views

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. ...
3
votes
0answers
69 views

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 ...
2
votes
2answers
120 views

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+\...
0
votes
1answer
105 views

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\...
1
vote
1answer
72 views

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)+\...
0
votes
0answers
29 views

Interpretation of $\alpha$ (confidence level) in mean CVaR optimization

How are an investors risk preferences related to $\alpha \in (0,1)$ in a mean CVaR optimization? Would a risk averse investor choose a higher value of $\alpha$, and if so why? My understanding is, ...
-3
votes
1answer
96 views

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?
2
votes
2answers
127 views

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, ...

1
2 3 4 5 6