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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11
votes
1answer
594 views

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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338 views

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 ...
10
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0answers
233 views

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 ...
5
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0answers
1k views

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 ...
4
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0answers
85 views

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 ...
4
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0answers
127 views

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 ...
4
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0answers
263 views

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=...
4
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0answers
748 views

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 ...
3
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1answer
122 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 ...
3
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0answers
182 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
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226 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
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0answers
44 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_{...
3
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80 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 ...
3
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400 views

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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34 views

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 ...
3
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0answers
186 views

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\...
3
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0answers
969 views

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 ...
3
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0answers
268 views

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 $F(t) = 1 - e^{-\lambda t}$ ...
2
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0answers
54 views

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 ...
2
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0answers
139 views

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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45 views

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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108 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 ...
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92 views

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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0answers
600 views

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 ...
2
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0answers
72 views

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 = ...
2
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0answers
119 views

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 ...
2
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0answers
866 views

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 ...
2
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0answers
179 views

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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0answers
234 views

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 ...
2
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0answers
258 views

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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52 views

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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0answers
53 views

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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0answers
110 views

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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0answers
96 views

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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60 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 ...
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1answer
151 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 ...
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0answers
682 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-\...
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0answers
82 views

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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0answers
106 views

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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0answers
160 views

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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69 views

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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0answers
51 views

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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0answers
49 views

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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0answers
272 views

How to compute SABR's probability density function

I am trying to compute the probability density function of the forward rate implied by the SABR formula approximation in order to see how the density implied by the approximation has negative ...
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0answers
40 views

Computing the PDF of the sum of N moves of an empirical PDF for USDJPY 1-minute moves

Per-minute tick data for USDJPY is available here. Suppose we download this file to usdjpy.txt and then save it into a Numpy array in Python 3 as follows: ...
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0answers
61 views

Solving for roots of a stochastic pay-off function

I have a pay-off function for a derivative which is defined by the Heaviside difference between $G$ and $B$ shifted by $-F$. To find the value of $V_{t=0}$, I need to find $\tau$ when $\frac{dV}{dt} = ...
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0answers
1k views

Forward price - T-forward martingale

I have a problem figuring out some of the calculations in the book: Fixed Income modelling In the chapter on forwards the author makes an argument that the forward is a martingale under the T-forward ...
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0answers
83 views

Cox-Ross-Rubinstein - getting volatility

i have exam coming on financial engineering, and need help asap with this thing. Basically there's a European put option ex dividend. We know that the stock price is $S_t = 85$, the exercise price is $...
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0answers
57 views

A priori selection of acceptable backtesting errors (type I and II)

Is it possible to a priori select an acceptable values of type I and II errors in backtesting (f.e. in case of the unconditional coverage test)? Type I error is directly connected to the significance ...
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0answers
221 views

methodology confirmation for computing implied risk-neutral CDF from option prices

In this question, the risk-neutral probability distribution $q(S_T=s)$ for the underlying at time $t = T$ is given by the Breeden-Litzenberger identity as: $$ \frac{1}{P(0,T)} \frac{ \partial^2 C }{\...