13
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 them and then deal with them one by one. Note, it is simpler for me to talk about variance instead of volatility. This has no material impact on the answer.
...
9
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 the transformations can happen.
To answer your actual question I'll attempt to describe the steps involved as simply as I can.
Let's say you use the copula ...
7
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 introducing a powerful risk management tool to generalize and stress-test correlations
7
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 use as prior. As far as I know, there is no standard way to reverse-engineer the optimization problem in the presence of nonnormal markets. (the first guess is ...
7
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 non-normality of returns in financial time series, he wasn't really equipped at the time (1963) to pursue its real nature. Appropriate models appeared only much ...
6
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 $F_Y(y) = P[Y\le y]$ and you want to find some $F_{X,Y}(x,y) = P[X \le x, Y\le y]$ such that when you look at marginals you get $F_{X,Y}(x,\infty) = F_X(x)$ and ...
6
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 otherwise, let $q$ denote the quantile $\alpha$ of $X$ i.e.
$$\Bbb{P}(X \leq q) = \alpha$$
then
\begin{align}
\Bbb{P}( X \leq q ) &= \Bbb{P}( \underbrace{\...
6
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 is solved by the inverse transform sampling method. The main idea is to obtain realizations of a random variable $x$ with any given distribution function $F(x)$, ...
6
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 parametrize the underlying Poisson process which is fully described by one parameter and the jump size. In other words, if you have a (log)normal distribution and the ...
5
It's not possible with a simple linear transformation like the one you mentioned: since scale and thus the distance between mean and median are required to change, either the mean or the median will not be preserved.
Therefore you must use nonlinear transformations, which will complicate quite a bit mantaining skew and kurtosis and imho will not be ...
5
The consensus nowadays is that stable distributions are not a well fit, although they do possess heavy tails. In particular Cauchy has too fat tails.
The reasons for this are disparate, however the first that comes to mind is that empirically longer horizons show a decrease in tail thickness, approaching normality for 1-year returns (although this has been ...
5
Because we are modelling the underlying price process, not the value process of your stop-loss portfolio...
5
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 $F: \mathbb R \to \mathbb R$ be a right-continuous function that is of locally bounded variation, that is
$$V_F([a,b]) := \sup\lbrace \sum_{i=1}^n \vert F(x_{i+1}) - F(x_i ) \...
5
For a continuous variable the PDF is the derivative of CDF. So returns or prices don't have a pdf if the cdf is not differentiable, e.g. it "jumps" at some point. The simplest models we use, like normally distributed log-returns, imply that returns, cumulative returns and prices all have a pdf.
4
Hi bcf: This is a good question. As you pointed out below,
\begin{align*}
p_0 &= \delta(y-y_0)\\
&=\delta(e^w-y_0).
\end{align*}
Then,
\begin{align*}
p_0 * g &= \int_{-\infty}^{\infty}\delta(e^z-y_0) g(w-z) dz\\
&=\int_{0}^{\infty}\delta(u-y_0) g(w-\ln u) \frac{1}{u}du\\
&= \frac{1}{y_0}g(w-\ln y_0).
\end{align*}
Consequently, your last ...
4
Surely, there is; search for aggregational gaussianity in Google Scholar or ScienceDirect.
In fact, 5 minutes returns are leptokurtic and fat-tailed; then as you increase timeframe, returns become more and more normal. Yearly data is almost normal, if you have enough points.
4
The VaR of level $\alpha$ a loss random variable (the bigger the worse) is the quantity $q$ such that the loss is bigger with probability $1-\alpha$.
Thus we need a $q$ such that
$$
P[L>q] = 1-\alpha,
$$
where we can imagine $\alpha=99\%$ and thus we need the starting point of the $1\%$ tail. Because we have a probability of a loss of size $0$ of $75\%...
4
In this related question How to derive the implied probability distribution from B-S volatilities?, it is shown how to infer the implied probability density of the future prices of a risky asset from a continuum of call prices written on that asset (Breeden-Litzenberger identity).
The developments, which I invite you to read, basically rely on the fact ...
4
You should post on mathematics.stackexchange
I answer but I should not.
Let $X $ be an exponential r.v. of parameter $\lambda $
$$P (X<u|X <T)=\frac {P (X<min (u,T))}{P (X <T )} $$
So for $0\leq u\leq T$
$$P (X<u|X <T)=\frac {1-\exp (-\lambda u)}{1-\exp (-\lambda T)} $$
So if $U $ is an uniform on $ [0,1]$ then
$$Y= -\frac {1}{\lambda }\...
4
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, $i$: $q_1^i(r^i)$ and $q_2^i(r^i)$.
Presumably you are interested in the model that can more accurately predict the state of the market over subsequent ...
4
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 clustering patterns and almost always a high degree of persistence. So, regardless of what you want to do here, avoid sampling from the unconditional ...
4
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 so that we don't know its futures values, say $r_{f,t+s}$. E .g. a common assumption is that the rate follows an Ornstein Uhlenbeck process (implying that the ...
4
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 $M_X(t=1)=\mathbb{E}\left(e^X\right)$, i.e. the expected future spot price. Thus, at least theoretically, we are not able to pinpoint the expectation of the future asset value, thereby ...
4
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 more. Following the proposed power law by Taleb, we have:
$x_{k}:=x_02^k$ and $b_k:=b_0\left(\frac{1}{2}\right)^{ka}$, where $a\geq1$ and $k\in \mathbb{N}$.
The ...
3
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 distribution of returns follows a Cauchy law. The budget constraint triggers skew that becomes larger and larger the higher the return. The reason is that ...
3
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 $Y_b(t) = b\ t + Y_0(t)$, that is to say the standard Brownian bridge (from zero to zero) with an added drift $b\ t$.
The standard Brownian bridge can be written in terms ...
3
Maybe this could also be a comment but I think an it is not possible to answer this question with a 'yes and here is how you do it'.
It has been tried, e.g. by me for a university research project. In this research we focused primarily on aggregation of returns and the main problem was the tractability of the resulting distributions and expressions, also ...
3
the risk neutral drift is needed for pricing of derivatives. For a $100\%$ equity portfolio you can take the real world drift - sometimes a good guess is a drift of zero.
For fixed-income you could do the same and might need more sophistication for the variance term. If you have short-dated bonds then you will need a special model for the pull-to-par.
For ...
3
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, superstatistics) gives some ideas of explaining the relation between time frame and its distribution: "...to regard the time series as a superposition of local ...
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