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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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....
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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 ...
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how to use ratio spread?
If I sell more options, then my gamma risk will be more difficult to control, but if I sell too few options, then when I judge the wrong direction, I will leave the market with a loss. I try to ...
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How useful is Markov chain Monte Carlo for quantitative finance?
Naively, it seems that Bayesian modeling, structural models particularly, would be quite useful in finance because of their ability to incorporate market idiosyncrasies and produce accurate ...
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Estimating implied probability based on prediction betting odds
I am attempting to estimate prediction betting market efficiency for a project, and I am hoping for assistance with a couple of questions.
The prediction market makers add a commission to the betting ...
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What is the probability of an asset trending or ranging
Some assets are know(or at-least assumed)to trend more than others. Is the probability of an asset trending equal to the probability of that same asset ranging(i.e 50-50)?
Is there a mathematical ...
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State Price Densities vs PDF of Payoffs in Ait-Sahalia (1998)
At the start of section I in the paper, the authors talk about the difference between the SPD/risk-neutral PDF/equivalent martingale measure vs the PDF of payoffs. I understand that the SPD is used in ...
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Оptimal strategy when throwing dice [closed]
Given a dice, you can throw it no more than three times, and you can stop at any time. How should you act so that on average you get as many as possible in the last throw?
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Expected Loss on a Portfolio, which contains an asset and a default protection contract, due to credit defaults
A portfolio consists of one (long) 100 million asset and a default protection contract on this asset. The probability of default over the next year is 10% for the asset, 20% for the counterparty that ...
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Imperfect Competition among Informed Traders - Back, Chao and Willard
The following assumptions are part of the paper of Back, Chao and Willard and I can not solve for the statistic that is denoted as $\phi$ in the sequel. I would be glad if anyone could help me. Below ...
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Probability Puzzle from a Quant Interview
An urn contains 20 balls colored each of the 7 colors of the rainbow (140 total balls). We select balls one-by-one without replacement. Given that in the first 70 draws we selected 5 more red balls ...
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How to calculate VaR on loss distribution
I sorted simulated portfolio losses in ascending order (sorted_losses variable). X-axis is loss, Y-axis probability of loss. I want to calculate 95% Value at risk in R.
I used the below code, but am I ...
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Credit value at risk
In my dataset I have probabilities of default for each borrower, loan amounts. I have calculated expected loss= EAD * PD * LGD.
How would I calculate unexpected credit losses of a portfolio and in ...
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Are there optimal portfolio theories than instead of the expected value they were based on the Mode of distributions
Are there optimal portfolio theories than instead of the expected value they were based on the Mode of distributions?
During my engineer student days I saw the Markowitz theory for portfolio selection ...
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How to calculate expected return on a loan while having probability of default
I have data on loans including: initial loan taken by each borrower, their probability of default. Is there a way to calculate VaR that would show potential loss of that asset?
First what I need is ...
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Curve optimization to predict monetary policy path (OIS Curve)
This is a question about a relatively undeveloped market (Chile) in which Camara the O/N rate is daily compounded (OIS Curve).
The available instruments in the market are short term rates ie 1m 2m 3m ...
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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 ...
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Market Making Card Sum Game
I am preparing for an interview with a prop trading firm and wanted to discuss potential strategies for the classic market making games. I have seen similar posts on the forum, but a lot of the ...
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How to calculate the variance of this coin flip?
I am reading the article “Shannon’s Demon & How Returns Can Be Created Out of Thin Air” by Richmond Quantitative Advisors (2021).
The main premise is a fair coin flip. If heads, you gain 50%. If ...
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Design a bilateral price negotiation for assets swap
Person alpha owns asset A.
Person beta owns asset B.
Alpha and beta wish to swap their assets A and B and settle the net cash of agreed price of A and B.
What's the best mechanism to agree on price A ...
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Solving Equation for estimation risk averse parameter
Let the portfolio value follow the SDE:
$$V_t=(\mu w+r(1-w))\cdot V_t\cdot dt +\sigma \cdot w\cdot V_t \cdot dB_t $$
where $\mu$ = drift of the portfolio,
$\sigma$=standard deviation of the portfolio, ...
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Find probability of stock reaching certain price in the future given current price today based on historical data [closed]
Say I have historical data of a ticker for the past 5 years. I look at the price on the current date for each of the five years (e.g. today is 20 Jul 2023 so I will look at 20 Jul 2022, etc.) and then ...
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Bayes' rule for conditional expectations (Proof review)
The Baye's rule for conditional expectations states
$$ E^Q[X|\mathcal{F}]E^P[f|\mathcal{F}]=E^P[Xf|\mathcal{F}] $$
With $f=dQ/dP$ - thus being the Radon-Nikodyn derivative and $X$ being
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Variance of the price from returns variance
Let's say that we have the variance of the daily return at $t_0$:
$$\sigma_{r_{t_0}}^2=\text{Var}[r_{t_0}]=\text{Var}[\frac{S_{t_0}-S_{t_0-1}}{S_{t_0-1}}]$$
for price process $S_t$. Is there a way to ...
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Bloomberg FXFM: what is the point of knowing risk neutral probabilities?
Among other things, Bloomberg FXFM function allows you to check risk neutral probabilities for currencies. For instance, you can check the probability of the euro depreciating 5% vs the dollar in 6 ...
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Gaming strategy for "closest number" game [closed]
Suppose there are 3 people A, B, C and a referee. A, B, C individually takes one number from [0,1] with the order A->B->C. B could see the choice of A, C could see the choice of A and B. After ...
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Predicting Bank of Canada Future Rate Changes Based on 3-month CORRA Futures [duplicate]
Earlier I asked a general question about how probabilities are derived from futures prices for derivatives related to the Bank of Canada's policy rate.
I have been advised the Overnight Index Swaps (...
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Incorporating Market Prices into Betting Models
In betting models, the price offered by the market is often ignored until the end. However, it seems like the price is a valuable piece of information that cannot be overlooked. Consider a ...
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Implying a probability distribution from option prices [duplicate]
I was reading this article, when I came across this text:
Without using a complex options pricing model, one can use intuition
to translate option prices into implied probabilities. For instance,
the ...
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$\mathbb{P}$ vs $\mathbb{Q}$ Probabilities - Transitioning Between Measures
I'd like this question to definitively guide a practitioner to using both $\mathbb{P}$ vs $\mathbb{Q}$ probabilities in trading and research.
Let's take only one fact as given: if I have a risk-...
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Generalizing a hidden semi-Markov model for trading
Taken from Wikipedia:
A hidden semi-Markov model (HSMM) is a statistical model with the same
structure as a hidden Markov model except that the unobservable
process is semi-Markov rather than Markov. ...
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Determining Stock Price Distribution [closed]
I am trying to derive a Stock Price Distribution for a particular time frame. Meaning thereby, let's say Market is about to close in 30 minutes and I want to calculate Stock Price Distribution for the ...
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Prove $E_{\mathbb Q}[X_t | \mathscr F_u] = X_u$ given $Y_t$ is a martingale
Edit years later: No idea why I'm upvoted. I actually am not sure how I'm correct. But maybe I haven't forgotten conditional expectation as much as I thought I have.
We are given a filtered ...
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How to fit KDE from existing probability density function values
I am working with options data, and I am using Breeden-Litzenberger formula to derive the risk-neutral terminal stock price PDF.
After applying the formula, here is a scatter plot of strike price vs ...
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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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Call probability of a callable swap
For one call date,
The call probability is just the probability that the swap rate for the remaining life of the swap is below the strike rate. This is easily obtainable in a normal vol model, it is :
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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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Default risk and stock price probability distributions [closed]
First of all, I realise this question might border on `meta-finance', so I'd be totally OK if it gets closed.
Having said that, the question itself:
Given a stock $S$, in the absence of default it is ...
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What is the meaning of the following mathematical equations? [closed]
Let's say that we have a discrete probability distribution, where
$$ x_i $$ represents each of the possible outcomes (discrete set of possible outcomes), and
$$ L $$ represents the expected value we ...
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Vol binomial tree
Suppose that we have a stock $X_t$ valued at 100 euros per share. At each time step the price can go up or down 1 euro with prob $1/2$. Assuming that interest rates are $0$ and the volatility of the ...
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calculating probability of a return below a specific value [closed]
assume a probability distribution with a mean of %10 and standard deviation of %1.5. In wanting to solve the probability being lower than %5, the normal distribution is written down and integrated as ...
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Option implied risk neutral distribution vs BKM risk neutral moments
I am doing some research on the option implied risk neutral distribution and methods calculate it, and so far have come across two ways to do so.
The first way is through the Breeden-Litzenberger ...
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Probability distribution function for stock price given many parameters
First of all, I am not in the US market.
I am trying to find out if I can do probabilistic analyses of stock price movements using the buy and sell summaries.
Kindly let me then explain my problem.
My ...
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Probability of touching
For a vanilla option, I know that the probability of the option expiring in the money is simply the delta of the option... but how would I calculate the probability, without doing monte carlo, of the ...
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Convert implied probability into real probability
In this article I have read that:
A risk-neutral world is one where all investors are indifferent to risk and don’t require any extra risk premium for the risk they bear. In this world, all assets (...
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Quantile function for fractional Brownian motion (fBm)
If anyone could help me to understand if it is possible calculate the quantile function for fBm?
I’ve checked several papers([1],[2],[3]), and although several works stated that it is centralised ...
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If $\Delta \log(V_{t})$ behaves like the increments of fractional Brownian motion, why do we model the rough volatility as follows
From Gatheral's paper, Volatility is rough and empirical evidence, it is clear that $\big\{\log(V_{t+1})-\log(V_{t})\big\}_{t}$ behaves like the increments of fractional Brownian motion $B^{H}$ with ...
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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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Inconsistency between simulation and the probability of a "stock" hitting take profit before stop loss
Let's assume a stock at time $t$ is worth $X(t)$. If the returns of $X(t)$ are i.i.d. and normally distributed,the probability of $X(t)$ hitting a value $H>X(t)$ before $L<X(t)$ is $\frac{H-X(t)}...
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Confidence in Sharpe ratio given performance
Suppose I have a strategy that I believe has a Sharpe ratio of X - not the Sharpe ratio of the backtest (this can be absolutely determined), but the ratio I expect it will actually take on over the ...
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