All Questions
Tagged with finance probability-distributions
34
questions
1
vote
1
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41
views
Find certainty equivalent $C[x]$ with respect to the utility function $u(x)=-e^{-x}$
Let $X$ - a random variable with a Poisson distribution with parameter $\lambda >0$. Find certainty equivalent $C[x]$ with respect to the utility function $u(x)=-e^{-x}$.
My try:
$$u(C[X])=\mathbb ...
- 116
1
vote
1
answer
64
views
Multivariate t distribution: Find probability of region enclosed by constant-density hypersurface
I am working with a multivariate t distribution, say of dimension p. Given a point P = (x1, ..., xp) in the sample space I need to calculate the probability of the region of the sample space enclosed ...
- 283
2
votes
1
answer
77
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Proof that $VaR_c(L)=(\Phi^{-1}(\frac{c+1}2))^2$
The loss $L$ has the $\lambda_1^2$ distribution, i.e. the distribution
of the random variable $X^2$, where $X$ has a standard normal
distribution. Proof that $VaR_c(L)=(\Phi^{-1}(\frac{c+1}2))^2$, ...
- 353
1
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1
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415
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One period Binomial model - why is return function needed at "present time" for no arbitrage condition?
One period binomial model considers asset prices $S(i)$ where $i=0,1$ where $S(0) = S$ and we have either $S(1) = Su$ or $S(1) = Sd$, where $0 < d< u$, and nominal interest rate per period is $r$...
- 167
3
votes
1
answer
165
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Distribution of a conditional expectation
I am reading a book on financial mathematics and a Theorem gives a price formula for a Call Option:
$$
\begin{aligned}
\pi_{\text {call }}(t)
&=P(t, S) q(t, S, \mathcal{I})-K P(t, T) q(t, T, \...
- 3,120
2
votes
0
answers
98
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What is the probability distribution of the Sum of continuous Unimodal RVs if nothing is known about its individual distributions?
A) What is the probability distribution of the Sum of continuous Unimodal RVs if nothing is known about its individual distributions?
I want to know if is possible to make some insight about the ...
- 1,586
3
votes
1
answer
105
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Optimal Leverage with Options
The optimal leverage which maximizes the log utility of a portfolio is well known and has a simple solution.
For example, for a Geometric Brownian Motion with drift a and volatility b, the optimal ...
- 31
0
votes
1
answer
44
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Question on two equivalent densities
I have two integrals $I_1$ and $I_2$ that are almost similar :
$$I_1=\int_K^{+\infty}(x-K)f_1(x)dx$$ $$I_2=\int_K^{+\infty}(x-K)f_2(x)dx$$ with$f_1$ and $f_2$ being two equivalent densities (so they ...
- 93
2
votes
1
answer
636
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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:
$$VaR_p(X) = inf{\{x \in \mathbb{R} : ...
- 3,067
1
vote
0
answers
38
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Determine explicitly the set of reachable $K = {(H · S)_1 : H ∈ H}.$
Let $Ω = {−1, 0, 1},\; \mathbb{P}({−1}) = 1/4,\; \mathbb{P}({0}) = 1/4,\; \mathbb{P}({1}) = 1/2,\; S_0 = 1,\; S_1(ω) = 1+ω$ and $\mathcal{F}$ from $S$ generated filtration. Determine explicitly the ...
- 1,073
1
vote
1
answer
40
views
Layperson's explanation of what it means for something to become more and more like a Gaussian
Question: I was asked by a friend what it means for something to become more and more like a Gaussian and I was unable to come up with a satisfactory answer. Therefore my question is:
How would you ...
- 4,048
1
vote
0
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50
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Heavy tail distribution in terms of the mean residual hazard function (proof)
I've been trying to prove the next characterization and looked in several books. But all of their proofs seem a little poor. In fact, many of them say that is easy and let the proof as an exercise. ...
- 119
0
votes
1
answer
65
views
Probabilistic sorting
Scalar version of the problem: in order to build a portfolio we sort list of companies by some score and choose top 10. Where the score - a single non-negative number calculated by some algorithm, (...
- 123
0
votes
0
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746
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Brownian bridge interpolation
I got stuck on a question goes like this:
Let $W$ be a standard Brownian motion path. Suppose there are three times $0\leqslant t_1<t_2<t_3$. Denote the values at these times by $W_1$,$W_2$,$...
1
vote
0
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60
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Bivariate Gaussian copula family is ordered
The bivariate gaussian copula is defined as
$$C_{\rho}(u,v)=∫_{-∞}^{Φ^{-1}(u)}∫_{-∞}^{Φ^{-1}(v)}\frac{1}{2π\sqrt{1-ρ^2}}×exp(-\frac{x^2+y^2-2ρxy}{2(1-ρ^2)})dxdy$$
where $\Phi$ is the cumulative ...
- 11