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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Distribution of hitting time of the integrated CIR process
If an increasing process $X_t$ has a known Laplace transform $\mathbb{E} e^{-s X_t} = m_t(s)$, define its hitting time $\tau$ to some level $B$ to be
$$
\tau = \inf\{ u > 0 : X_u \geq B \}.
$$
Can ...
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Transition densities in the Heston model
Knowing the Characteristic function $\Phi_{T,t} = \mathbb{E} [ e^{i u S_T} | S_t, V_t]$ (or equivalently, the Laplace transform) of an affine process, it's possible to know the distribution of the ...
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2-state HMM / ARMA process?
I have issues with this problem:
Let $\{X_t, t\in \Bbb N\}$ be a 2-state stationary Markov chain, with transition $M$ (and $M(1,2)\neq 0 \neq M(2,1)$), let $\{W_t, t\in \Bbb N\}$ be a strong Gaussian ...
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Fitting Student t-distributions to log-returns
It seems that some tail-risk centric groups are bent on using Paretian and t-distributions to account for tail risk when fitting log-returns. It has been observed, however, that with and without ...
user6430
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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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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 do I calculate the present value of a credit default swap?
I am paid 20 million every time a bond drops to a new low over a 120 month period. I need to know how to find the present value of such an arrangement if there is a continuously compound interest of 5 ...
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4
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negative transition probabilities in the heston model
I've been trying to implement a bivariate tree for pricing american options with the heston model in R using the paper of Beliaeva and Nawalkha (http://papers.ssrn.com/sol3/papers.cfm?abstract_id=...
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Monty Hall Model
Given a fixed time period,say 3 days, the stock/market can go up,down or stay sideways. A hedge fund can long, short or use rangebound(options strategy) to bet for that 3 days closing level.
Hedge ...
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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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3
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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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What exactly is/How exactly do we interpret the binomial model's Radon-Nikodym derivative?
Related: Dumb question: is risk-neutral pricing taking conditional expectation?
Maybe there's not quite an interpretation given Lewis' triviality result if $E^Q[X]$ is a real world conditional ...
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Binary probit model: relevant which outcome is 1?
I'm currently working on predicting bear and bull phases with a dynamic probit model in the form of $y_t=\beta_1X_t+\gamma_1y_{t-1}+\epsilon_t$. So far I've written all my code in matlab and it works ...
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Is there a countably infinite Sigma-Algebra? Why?
Assume $\,\mathcal{F}$ be a nonempty collection of subsets of $\Omega$. $\,\mathcal{F}$ is called a $\sigma$-Algebra whenever
if $A\in\mathcal{F}$ then $A^c\in\mathcal{F}$, and
if $A_1,A_2,...\in\...
user16651
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Law of a geometric brownian motion first hitting time (formula dont match Monte Carlo Simulation)
I posted this question before on MSE
I need to use it in a small step in the middle of a simulation and
I think I'm not getting correct results to this probabilities and so
for my all ...
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405
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default probability
Suppose the hazard rate is $\lambda$
the default probability density function follow exponential
$f(t) = \lambda e^{-\lambda t}$
and cumulative probability function is
$F(t) = 1 - e^{-\lambda t}$
...
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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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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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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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2
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509
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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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2
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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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1
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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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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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Convolution of Dependent Random Variables with Copulas
Lets say I have 2 different observations which are fitted to a parametric distribution.
And lets say that they are dependent and can be modeled by one of the copulas.
I want to calculate “a value” ...
2
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0
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Detecting butterfly spread arbitrage for American options through European option prices
It's easy to demonstrate that if European option prices are concave with strike, then an arbitrage exists. For example, the risk-neutral probability density is the second derivative of European put ...
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Laplace Exponent of a Jump-Diffusion Process
I'm currently reading a paper (https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2543702) which uses the following process to describe the dynamics of a firm's asset value:
\begin{equation}
V_t = ...
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Portfolio diversification on default risk
A portfolio of 13 different companies have loans. Company $i$ default on their loan with probability $p_i$ and survive with prob $q_i=1-p_i$. Let $Y_i=1$ denote default. Question: How could I get to a ...
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interview question : replication strategy of a betting game
Here is a question I found in a book I am not able to finish. Your help will be much appreciated! I also included where I have been so far.
Q: Team A plays team B in a series of 7 games, whoever wins ...
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Beta distribution - Holding period
Let's say I have a risk factor that is defined between [0,1], such as recovery rates. Assuming I have daily data, I can estimate the "daily VaR", i.e. the tails over 1 day period, since the data is ...
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Probability Density of Returns of Bonus Certificates
Could anyone please help me with the following?
I need to generate a histogram (resp. probability density) of returns of a bonus-certificate.
A bonus-certificate can be replicated by an underlying ...
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Modeling asset performance to Bitcoin revenue
I'm attempting to model asset performance to Bitcoin revenue, which is a driving force in the Bitcoin community.
Question
Is there any model, or research being done that tracks "hashes per second" (...
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58
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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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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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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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vote
0
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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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0
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How much compensation need to take on risk?
Quant Firm Interview Question
We roll three, 8 sided dice. If same face appears 3 times we win 80 dollars. We have a bank of 10,000 dollars. How much are we willing to pay to play? What if we increase ...
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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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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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vote
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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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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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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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0
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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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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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How to determine the default probability of a county in a bond that is not in its native currency?
Disclaimer: This post is cross posted in here also.
Consider the following case:
Country P uses the currency Euro and gives p percent interest on a one year bond issued in Euro.
Country Q uses the ...
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0
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Prove that $F(s,x_0)=0$, $F(t,x)=1$ and $\frac{\partial F}{\partial t}+\frac{1}{2}\frac{\partial^2 F}{\partial x^2}=0$
Using the Dynkin's formula, prove that $F(s,x_0)=0$, $F(t,x)=1$ and $\frac{\partial F}{\partial t}+\frac{1}{2}\frac{\partial^2 F}{\partial x^2}=0$
where $F(s,t)=2\int_{x-x_0}^{\infty}\frac{1}{\sqrt{2\...
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How are Risk indices linked to Physical Trading returns?
Ref to my previous question here:
Physical trading spot transaction analysis-Quantified
I have been able to narrow down my aim to defining a physical trading strategy P&L. My question is, how ...
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