7
votes
Accepted
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 ...
- 2,915
7
votes
Accepted
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 ...
- 203
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 ...
- 14.5k
6
votes
Accepted
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-\...
- 13.6k
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 ...
- 76
6
votes
Accepted
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 ...
- 8,438
6
votes
Accepted
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 ...
- 1,707
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 ...
- 4,359
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...
- 872
5
votes
Accepted
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 ...
- 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, ...
- 1,707
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 ...
- 6,470
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 )} $$
So ...
- 2,412
4
votes
Accepted
What the implied distribution really is?
In this related question How to derive the implied probability distribution from B-S volatilities?, it is shown how to infer the implied probability density of the future prices of a risky asset from ...
- 14.5k
4
votes
Compare two distributions for forecasting returns
Answer
If you assume your returns are independent (yes your models might loosen this assumption) then the two models, $Q_1$ and $Q_2$ assign probability distributions to the returns on any given day, ...
- 9,215
4
votes
Accepted
Sampling from an empirical distribution
Financial returns exhibit well known time-dependancy in its higher conditional moments. For starters, just about no matter how you produce a time series of conditional volatility, it will be exhibit ...
- 2,436
4
votes
What is the distribution of the risk-free asset?
Just to add to the previous answer, one example of such asset (returning 'risk-free rate') is a money market (or bank) account, but it is only locally risk-free, with value accruing continuously at ...
- 5,008
4
votes
Accepted
FX spot distribution with student-t returns
1. Theory
The Student $t$ distribution does not exhibit a moment generating function
$$
M_X(t)=\mathbb{E}\left(e^{tX} \right)
$$
Hence, there exist no closed form solution for $M_X(t=1)=\mathbb{E}\...
- 6,470
4
votes
Accepted
Taleb's Black-Swan: interpretation of the exponent
I finally got the idea behind the example. To illustrate it in a more general setting I will present a rigorous proof:
Let $x_k$ denote the salary and $b_k$ the number of persons that earn $x_k$ or ...
- 183
4
votes
Fat tailed can be estimated through a t-distributions?
B is the correct choice.
I honestly would wish multiple choice would not even exist. It is the worst way of testing knowledge in my opinion.
Without knowing the details of what was taught, I would say ...
- 8,193
4
votes
Integral of brownian motion wrt. time over [t;T]
The last integral is correct as
$$\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, ...
- 548
3
votes
Accepted
Density plot of the skew-t distribution
The rsgt is a skewed generalized t distribution, whereas your picture is a skewed student-t distribution. Try using fGarch ...
- 735
3
votes
Accepted
Normal Inverse Gaussian distribution - any consensus on an accurate quantile function?
When possible, I look at implementations in IMSL and the GSL for really good accuracy. Neither one appears to implement the Wald (inverse gaussian) or its quantile function.
Matlab does have the ...
- 14.7k
3
votes
Accepted
Problem with obtaining densities
You know that :
$X \sim N(\mu,\sigma^2)$.
$Z = \large\frac{X-\mu}{\sigma}$.
$\text{Var}(Z) = \large\frac{1}{\sigma^2}\text{Var}(X) = \large\frac{1}{\sigma^2}\sigma^2 = 1$.
So that $Z \sim N(0,1)$. ...
- 2,552
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 ...
- 1,367
3
votes
Accepted
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, ...
- 21k
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 ...
- 1,973
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 ...
- 9,215
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 ...
- 969
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