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7 votes
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Reconciling Two Claims About Volatility Under Fat Tails

I don't think the claim that "Lévy alpha-stable distributions are better descriptors of returns" is universally accepted. While Mandelbrot (and others before him) has correctly identified ...
Adam N.'s user avatar
  • 233
6 votes

Consensus on Cauchy distribution for stock prices

I wrote a proof deriving the distribution of returns for all asset and liability classes. If there were no budget constraint, limitation on liability and liquidity had no cost, then you can prove the ...
Dave Harris's user avatar
  • 4,299
6 votes
Accepted

Quantile normal and lognormal

Quantiles are preserved under monotonic transformations, hence the quantile for $Y$ is simply the exponential of the quantile of $X$, no need for corrections whatsoever (see here for instance). Put ...
Quantuple's user avatar
  • 14.7k
6 votes
Accepted

Produce the random variable for an asset from a uniformly distributed random varible

The question requires you to provide a method which uses uniform random variables and transforms them to generate realizations of the described asset values. To give a bit more general answer: this ...
Oskar's user avatar
  • 76
6 votes
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Is it always better to use the entire distribution of a financial returns series, not just $\mu$ and $\sigma$?

It depends. For example, if you're doing option pricing in the log normal world returns are completely described by the mean and standard deviation. If you add jumps, you would also need to ...
Bob Jansen's user avatar
  • 8,562
6 votes
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What is the distribution of the risk-free asset?

The standard way to think about this is that at time $t$ the riskless asset gives you known return of $r_{f,t}$ over a short time period. However, this rate may itself be time-varying and stochastic ...
fes's user avatar
  • 1,727
5 votes

Why worry about fat tails, if you can use stoploss?

Because we are modelling the underlying price process, not the value process of your stop-loss portfolio...
Oscar's user avatar
  • 902
5 votes
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Change of measure

You can't have a precise argument without a precise definition. In general, the appropriate notion of integral here is the Lebesgue-Stieltjes integral. In a fairly general setup, let $F: \mathbb R \to ...
jacques's user avatar
  • 186
5 votes
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Which financial time series have a PDF and/or CDF?

For a continuous variable the PDF is the derivative of CDF. So returns or prices don't have a pdf if the cdf is not differentiable, e.g. it "jumps" at some point. The simplest models we use, ...
fes's user avatar
  • 1,727
5 votes
Accepted

Terminal wealth distribution from dollar cost averaging

I understand that you assume multiplicative gross returns, $W_t=W_{t-1}R_{t-1,t}=W_{t-2}R_{t-1,t}R_{t-2,t-1}$ and so on. Let's assume that you are investing $I$ at the onset, and increase your ...
Kermittfrog's user avatar
  • 6,832
4 votes

How to simulate the exponential law over an interval of the form [0,T]?

You should post on mathematics.stackexchange I answer but I should not. Let $X $ be an exponential r.v. of parameter $\lambda $ $$P (X<u|X <T)=\frac {P (X<min (u,T))}{P (X <T )} $$ So ...
M. Jeunesse's user avatar
  • 2,432
4 votes
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What the implied distribution really is?

In this related question How to derive the implied probability distribution from B-S volatilities?, it is shown how to infer the implied probability density of the future prices of a risky asset from ...
Quantuple's user avatar
  • 14.7k
4 votes

Compare two distributions for forecasting returns

Answer If you assume your returns are independent (yes your models might loosen this assumption) then the two models, $Q_1$ and $Q_2$ assign probability distributions to the returns on any given day, ...
Attack68's user avatar
  • 10.8k
4 votes
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Sampling from an empirical distribution

Financial returns exhibit well known time-dependancy in its higher conditional moments. For starters, just about no matter how you produce a time series of conditional volatility, it will be exhibit ...
Stéphane's user avatar
  • 2,486
4 votes

What is the distribution of the risk-free asset?

Just to add to the previous answer, one example of such asset (returning 'risk-free rate') is a money market (or bank) account, but it is only locally risk-free, with value accruing continuously at ...
ir7's user avatar
  • 5,043
4 votes
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FX spot distribution with student-t returns

1. Theory The Student $t$ distribution does not exhibit a moment generating function $$ M_X(t)=\mathbb{E}\left(e^{tX} \right) $$ Hence, there exist no closed form solution for $M_X(t=1)=\mathbb{E}\...
Kermittfrog's user avatar
  • 6,832
4 votes
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Taleb's Black-Swan: interpretation of the exponent

I finally got the idea behind the example. To illustrate it in a more general setting I will present a rigorous proof: Let $x_k$ denote the salary and $b_k$ the number of persons that earn $x_k$ or ...
Philipp's user avatar
  • 183
4 votes

Fat tailed can be estimated through a t-distributions?

B is the correct choice. I honestly would wish multiple choice would not even exist. It is the worst way of testing knowledge in my opinion. Without knowing the details of what was taught, I would say ...
AKdemy's user avatar
  • 9,024
4 votes

Integral of brownian motion wrt. time over [t;T]

The last integral is correct as $$\int_t^T W_s ds = \int_t^T (T-s) dW_s \sim N\left(0, \int_t^T(T-s)^2ds\right) = N\left(0,\frac{1}{3}(T-t)^3\right).$$ Ref. Arbitrage Theory in Continuos Time (Björk, ...
Landscape's user avatar
  • 548
3 votes
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Verifying that the extreme value copula is indeed a copula

Note that, you only need to show that \begin{align*} A\left(\frac{\log(u_2)}{\log(u_1u_2)}\right)-\frac{\log(u_2)}{\log(u_1u_2)}A'\left(\frac{\log(u_2)}{\log(u_1u_2)}\right) \ge 0, \end{align*} or, ...
Gordon's user avatar
  • 21.2k
3 votes

Compare two distributions for forecasting returns

Your feeling that there is more to the problem than adding up probabilities is very justified. To give you the bad news first: Your problem as stated has no solution. Since probability distributions ...
g g's user avatar
  • 2,023
3 votes

How to derive the implied probability distribution from B-S volatilities?

Brian B gives the overall idea. But the use of a simple polynomial will not be appropriate in general. The paper Model-free stochastic collocation for an arbitrage-free implied volatility: Part I ...
jherek's user avatar
  • 1,414
3 votes

Produce the random variable for an asset from a uniformly distributed random varible

Say your asset can take the discrete values {1,2,3,4} with probabilities {0.4, 0.1, 0.2, 0.3}. The question is to derive a sampling procedure that returns either {1,2,3,4} with the right ...
Attack68's user avatar
  • 10.8k
3 votes
Accepted

How would a FX price probability distibution function look?

You can extract the risk neutral density implied by option prices and have a look at that. The implied probabilities are given by the prices of butterfly spreads in the market. This is common ...
roz's user avatar
  • 989
3 votes

Why worry about fat tails, if you can use stoploss?

Not sure if this question deserves to be further piled onto, but alas... Large, institutional portfolios nearly always hold relatively illiquid and OTC traded instruments. There is no stop-loss order ...
Kch's user avatar
  • 981
3 votes
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Interpretation of a uniform asset return distribution

Such assets do not exist due to market efficiency: people would trade such assets until the price was near the expected value which would tend to yield more returns near 0 and fewer returns that were ...
kurtosis's user avatar
  • 2,910
3 votes

Do portfolio mean and portfolio variance have probability distributions?

Yes, they can/do. But you have to drink the proverbial Kool-Aid(or taking the blue pill is probably the more relevant metaphor these days ;-), and approach this as a Bayesian inference problem. So ...
demully's user avatar
  • 5,071
3 votes

What does this absolute return distribution chart show?

at closer inspection of the axes I think that this is a plot of tail frequencies in basis 10. I think they sort the absolute normalized returns from low (0.0) to high ($31\approx 10^{1. 5...}$) and ...
Kermittfrog's user avatar
  • 6,832
3 votes
Accepted

Calculating the Value-at-Risk when changing the confidence level

If you assume that everything is normally distributed, then you just divide by normsinv(95%) and multiply by normsinv(99%).
Dimitri Vulis's user avatar

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