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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A question in information strucutres and probability measures - How are they connected?
Suppose that $\mathcal{I}=(X,\sigma^{\mathcal{X}},\mu)$ is an information strucutre, which is a probability space, where
$X=X^1\times X^2$ is the cartesian product of the individual finite sets of ...
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Requesting for price?
Just for education purpose. Assuming I have some trading ideas that involves the use of OTC derivatives but I may not be able to put them into practice due to regulatory issues and huge minimum ...
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Can I combine the exotics for a payout?
Can I combine a one touch option(barrier lower than current price) and no touch option(barrier higher than current price), so that I get a payout immediately only if the one touch barrier is breached ...
user59094
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What is the probability of touching point A first?
The probability of a stock touching a point A which is below the current spot price is 35%, and the probability of the stock touching a point B which is above the current spot price is 20%.
How can I ...
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Compare errors in estimating a probability
Let $X_t$ be a geometric Brownian motion: $dX_t = \mu(X_t,t)dt + \sigma(X_t,t)dW_t$ with $W_t$ a standard Brownian motion.
Given the intervals $[t_{j-1}, t_{j}]$ for $j\in {1,...,U,...,N}$, let $M_j$ ...
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Variance of Random Walk with Drift
For Gaussian random variables $\xi_t$ with mean $\mu_t$ and standard deviation $\sigma$, consider the random walk with initial condition $P_0=100$, such that
\begin{equation}
P_t=P_{t-1}(1+\xi_t).
\...
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Expected stock price range using implied volatility calculated by Black-Scholes
What's the correct way to calculate the expected stock price range using implied volatility, without the simplifying assumption that the stock price follows a normal distribution?
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Identity of recent books on stock market & risk
Apologies if this seems out of place, but a couple years ago I read several popular books written in the last decade by a single author who was trying to disabuse readers of several fallacies ...
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Finding Option Probability Density Using Local Volatility from Dupire Model
This question is different than pricing using dupire local volatility model and Is Dupire's local volatility model path independent to recover historical option price?
I also asked this on Math ...
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Trading a Bouncy Stock
I came across the following question and am trying to understand it better. I was hoping you could share your intuitions.
For a given stock, you are certain that for the next 100 days, it will
move ...
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is the concept of skew observed in fixed odds betting markets?
Bear with me if this sounds a little flippant, but this has got me curious. I know "sports arbitrage" is an active economic activity, although the arbitrage arguments, I think, are not ...
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Statistical significance in the context of financial data?
I understand statistical significance in the general sense: we take a sample from a population and compute some parameter from the sample to infer what is the propulsion parameter to some degree of ...
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Ito Lemma for Poisson Process
I'm new to stochastic calculus on jump processes and encountered a difficulty. I would appreciate some clarification from the community on the following question.
Let $g_t$ be a $\mathcal{F_t}$-...
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Structuring and Customization
It seems complex derivatives in particular exotic options are not available at any retail broker. Can a regular retail trader get access to these instruments? Maybe through prop firms or banks? ...
user57027
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Does time remaining matter in NO Touch-ONE Touch probabilities?
I asked a question some days back and got an answer which I understand and make sense:
Probability of touching short call strike and not touching touching short put strike of a short strangle?
However,...
user56826
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Probability of touching short call strike and not touching touching short put strike of a short strangle?
I just came across a blog post. I believe the answer is a correct approximation:
http://tastytradenetwork.squarespace.com/tt/blog/probability-of-touching-both-sides
I modified the question in the post ...
user56826
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Drift Term in Black-Scholes Model Martingale
How would I prove that a Black-Scholes Model is not a Martingale if it has drift. In many cases it is just stated as a fact (without proof).
For instance if Im looking at:
$$dS_{t} = \mu S_{t} + \...
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Risk-Neutral Probability in a Binomial Tree
This question is probably very simple and I'm just missing the easy solution but I'm a bit confused so I thought I might as well try ask here.
I've been given this question:
When I tried to calculate ...
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Why autocall probabilities are decreasing with time
I am wondering why autocall probabilities decrease with observation dates. Intuitively, I understand that as time goes, if the spot has not breached the barrier, it would need more and more kind of ...
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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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If arbitrage can happen exactly at one moment, is it really arbitrage?
There are many "interpretations" of what no-arbitrage means in mathematical finance, the most well known is no free lunch with vanishing risk:
If $S=\left(S_{t}\right)_{t=0}^{T}$ is a ...
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Query on Lebesgue Measure
I am reading Steven E. Shreve's book, titled "Stochastic Calculus for Finance II". I have a query w.r.t. an example given in the book which is as follows:-
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How to prove that the following is still a Brownian motion [closed]
Given a Brownian motion $B_t$ on a filtered probability space, how can I prove that $W_t=B_t+\alpha t$ is still a Brownian motion, with $\alpha \in \mathbb{R}$? Is it always true? Do I need necessarly ...
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Simulation of Gamma process (distribution of increments)
The gamma process is a Levy process $X$, where $X_t$ has gamma distribution with parameters $at,b>0$ and density
$$f\left(x\right)=\frac{b^{at}}{\Gamma\left(at\right)}x^{at-1}e^{-bx}$$
I want to ...
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Conditional probability of Brownian motion (with drift and scaling) hitting barrier
I am trying to understand the pricing of barrier options, and am considering the Brownian motion $\mathrm{d}X_t=a\mathrm{d}t+b\mathrm{d}W_t$, $a$ and $b$ constant. I am trying to:
derive the ...
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Best way to trade probability density
From the option chain of a security, we can calculate the implied probability density at the maturity
$T$ (assume the options are European. Now suppose we have our own view/prediction on the ...
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Testing the fit of an Ornstein-Uhlenbeck process
I would like to check if a time-series follows an Ornstein-Uhlenbeck process defined by an SDE:
$$dX_t - \lambda (\mu - X_t) dt = \sigma dW_t$$
where
$\lambda > 0$ is the mean-reversion ...
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Real world probabilities from option implied risk neutral density?
The work of Breeden and Litzenberger-formula (https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2642349) gives us a risk neutral probability distribution of a stock price, depending on the option ...
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What day of a week should we pick something to happen to minimize it happening on the fourth business day of the month?
This is an extension of problem 3.16 in Mark Joshi's book. My answer is to avoid Thursday, and all other weekdays are equally good. The probability that the fourth business day is Thursday is 3/7 (...
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Optimal Strategy in 3 Dice Game
In a recent interview I received the following question (an optimisation/strategy game)...which left me a bit stumped.
The rules of play, you start with 0 points, then:
Roll three fair six-sided dice;...
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Why are prediction markets based on logarithms when a linear solution can suffice?
For example, take a binary outcome; A coin toss, heads or tails.
If heads, then those that picked heads receive \$1 and tails receive \$0.
To quote the prices for each bet Hanson's LMSR uses ...
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Risk neutral probabilities in binomial option pricing with discrete dividends — whose argument is correct?
In trying to discover more about pricing American options with dividend payouts, I found the the post linked here. I notice two disagreeing answers when it comes to determining the replicating ...
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Interpretation of Value at Risk
Let $X$ be a Loss random variable (Positive values of X represents Losses) and let $p \in (0,1)$. I know that the Value at Risk at level $p$ of $X$ is defined as:
$$VaR_p(X) = inf{\{x \in \mathbb{R} : ...
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Given the density function of $S^{1}$ in one-period model, find the risk-neutral measure
Consider the one period market model $\left(\overline{\pi},\overline{S}\right)$ consisting of a risk-free asset $\left(\pi^{0},S^{0}\right)=(1,1+r)$ and a risky $\left(\pi^{1},S^{1}\right)$
Let $ r &...
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Physical Probability Measure vs. Risk Free Probability Measure (State Contigent Claims)
currently I am working on a problem regarding state contingent claims.
I have 5 securities (one of the security is a risk-free security) and in the next period, these securities will end up in one of ...
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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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Calculation Expecting Credit Loss from a Portfolio
I have the following question:
An investor holds a portfolio of 50 million dollars. This portfolio consists of 'A' rated bonds (30 million dollars) and 'BBB' rated bonds (20 million dollars). Assume ...
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Calculating the cumulative probability of default from recovery rate, yield and coupon rate
I have the following details:
A 10-year U.S.Treasury strip has a yield of 6% and a 10-year zero issued by XYZ Inc, rated A by S&P and Moody's, has 7% (semi-annual compounding). Assuming a recovery ...
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Is the portfolio return distribution a weighted combination of individual asset return distributions?
We know that the portfolio expected return is a weighted sum of the individual assets' expected returns (asset means). We also know that the portfolio variance is a weighted combination of the ...
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EMM, Supremum and Expectation
I asked this question on MSE recently.
https://math.stackexchange.com/questions/3922347/supremum-and-expectation
I want to prove this when $\mathcal{M}$ is a set of equivalent martingale measure.
...
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What's the interpretation of the probability of default implied from CDS spreads?
What's the time horizon of the probability of default implied from a CDS spread? Given CDS = PD*(1-R), if I use a 5yr CDS spread in the formula, is the implied PD the probability that that name ...
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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&...
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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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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 ...
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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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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 ...
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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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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 ...
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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_{...
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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 ...
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