# 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 ...
434 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 ...
250 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 ...
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 ... 96 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 ...
349 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 ...
158 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 ...
285 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=...
792 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 ...
294 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 ...
309 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 ...
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1 vote
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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 ...
1 vote
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1 vote
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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 ...
1 vote
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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 ...
1 vote
### 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\...