15 votes
Accepted

Is volatility for the next day forecastable? To any extent?

Upon close reading, this appears to be 3 (interesting) questions, not one. I'm not sure if the mods have the tools needed to split it up, so I'm just going to write down the three questions as I see ...
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9 votes
Accepted

Copulas simply explained

I found Coping With Copulas by Thorsten Schmidt really helped me to get a more basic understanding of copulas. As well as looking at some simple examples in R and thinking about different directions ...
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  • 206
7 votes

Copulas simply explained

In the theory of copulas you want to model a multivariate (often bivariate) distribution and keep the marginals fixed. Thus you have random variables $X$ and $Y$ with cdf $F_X(x) = P[X \le x]$ and $...
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  • 13.2k
7 votes

Copulas simply explained

The best introduction to copulas I know, i.e. with rigour and intuition, is the following. THE QUANT CLASSROOM BY ATTILIO MEUCCI A Short, Comprehensive, Practical Guide to Copulas Visually ...
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  • 26.7k
7 votes
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Is it possible to deal with non-normal distribution in Black-Litterman model?

Well there are two main things to consider here. Many implementation of Black-Litterman use the market portfolio and the ex post volatility and correlation structure to back out implied returns to ...
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  • 2,874
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 ...
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  • 118
6 votes
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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 ...
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  • 13.9k
6 votes
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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 ...
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  • 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 ...
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  • 7,598
5 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 ...
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  • 4,066
5 votes
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Calculate VaR for a liabilty taking a exponential distribution?

The VaR of level $\alpha$ a loss random variable (the bigger the worse) is the quantity $q$ such that the loss is bigger with probability $1-\alpha$. Thus we need a $q$ such that $$ P[L>q] = 1-\...
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  • 13.2k
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...
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  • 758
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 ...
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  • 186
5 votes
Accepted

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, ...
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  • 1,242
5 votes
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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 ...
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  • 5,368
4 votes

Can Gaussianity of returns depend on the time frame?

Surely, there is; search for aggregational gaussianity in Google Scholar or ScienceDirect. In fact, 5 minutes returns are leptokurtic and fat-tailed; then as you increase timeframe, returns become ...
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4 votes
Accepted

Density of Geometric BM via Fokker-Planck

Hi bcf: This is a good question. As you pointed out below, \begin{align*} p_0 &= \delta(y-y_0)\\ &=\delta(e^w-y_0). \end{align*} Then, \begin{align*} p_0 * g &= \int_{-\infty}^{\infty}\...
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  • 20.4k
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 ...
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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 ...
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  • 2,362
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, ...
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  • 7,997
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 ...
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  • 2,356
4 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 ...
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  • 1,242
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}\...
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  • 5,368
4 votes
Accepted

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 ...
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  • 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 ...
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  • 4,208
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, ...
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  • 155
3 votes
Accepted

Can Gaussianity of returns depend on the time frame?

My main reference will be "Dan Xu, Christian Beck - Transition from lognormal to chi-square superstatistics for financial time series" Non-equilibrium statistical mechanics (more specifically, ...
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  • 335
3 votes

How to combine Gaussian marginals with Gaussian copula to obtain multivariate normals?

You can express the Normal distribution by Sklar's Theorem in terms of Gaussian Marginals and Gaussian Copula as follows: $$F(x_1,...,x_n)=C(F(x_1),...,F(x_n))=C^{Gau}(N(x_1),...,N(x_n))$$ So the ...
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  • 5,629
3 votes
Accepted

Brownian Bridge's first passage time distribution

What about this sketch of an answer: Let's put $T=1$ in your formula to simplify the notation. Then $Y_b(t)$ is a Brownian bridge where $Y_b(0)=0$ and $Y_b(1)=b$. This can be written as $Y_b(t) = b\ ...
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  • 4,217

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