8
votes
The possible preferences of investors for higher than first 2 moments of return distribution?
Investor preferences for higher level moments are probably most easily explained by behavioral finance. Investors' tendency to overvalue out-sized positive and negative outcomes, such as gamblers' ...
- 3,015
8
votes
Accepted
How to price this basket option?
No offense but it will be much more complicated than what you think... I'm not even sure that you are familiar with risk-neutral pricing in the first place? I'll try to give you some clues.
This ...
- 14.7k
7
votes
Accepted
Why isn't the asset with minimum variance given a 100% portfolio weight?
Diversification is key.
The clear cut answer is diversification. A weighted combination of assets will more often than not show a lower return variance than even the asset with the lowest variance ...
- 7,035
5
votes
Accepted
Maximum skewness portfolio solution derived from its Lagrangean formulation
Unfortunately, there exist no closed form for this.
The Lagrangean reads
$$
L(w,\lambda)=w^TM_3(w\otimes w)-\lambda(w^T\mathbf{1}-1)
$$
with first order conditions
$$
\begin{align}
\frac{\partial L }{\...
- 7,035
5
votes
implied-information in american option
I observe that Christoffersen et al. (2012) consider the implied volatility from European options, as calculated under the BS model and other extensions of it. Therefore, implied volatilities from ...
- 1,426
4
votes
Accepted
Calculate moments given density values
The key is:
$$ \mathbf{E}[X^k] = \sum_{i=1}^n x_i^k p(x_i) $$
($X$ discrete variable, $x_i$ realizations, and $p(x_i)$ realization probabilities)
See this link for further details.
- 5,113
4
votes
Accepted
Does standardizing/normalizing asset returns change their skewness and kurtosis?
From the wikipedia on skewness and kurtosis, both are defined as expectations of standardised moments of the respective distributions. Hence, no.
- 7,035
4
votes
Accepted
Any portfolio models not based on asset return moments?
The answer is sort of. I am going to provide you a history of the mathematics so that you will understand why this discussion is challenging to have in economics. Also, you are probably going to ...
- 4,319
4
votes
Any portfolio models not based on asset return moments?
Remember that asset returns are there because of the expected utility theory. More precisely, as long as you can assume a "reasonable" expected utility function to be approximated by a quadratic ...
- 393
3
votes
Accepted
Is there Cornish-Fisher volatility, given that there is Cornish-Fisher Value-at-Risk?
The motivation of the Cornish-Fisher expansion is to approximate quantiles when the data is not normally distributed.
It may help to think about parameters of a probability distribution and the ...
- 5,421
3
votes
Is positive skewness preferences rational or irrational?
I think the usual argument is that if an investor is maximizing expected log wealth, then this implies preference for higher odd order moments (mean return, skew, etc.) and for lower even order ...
- 547
3
votes
Any portfolio models not based on asset return moments?
Some allocation approaches that are not based on moments -
Fixed weight strategies (e.g. 60/40 or equal weight)
Allocation proportional to market capitalisation (often called passive investing or ...
- 5,931
3
votes
Show that Riemann integral over BM is gaussian process
We assume we work on a probability space $(\Omega,\mathcal{F},\mathbb{P})$ equipped with the filtration $\{\mathcal{F}_t\}_t$. By Itô's Lemma:
$$B_t\text{d}t=\text{d}\left(tB_t\right)-t\text{d}B_t$$
...
- 8,169
3
votes
Accepted
Higher moments of a straddle
Let $C=(S-K)^+$ and $P=(K-S)^+$. Then it is clear, for any positive integers $i$ and $j$,
\begin{align*}
C^i P^j = 0.
\end{align*}
Consequently, for any positive integer $n$,
\begin{align*}
(C+P)^n = ...
- 21.2k
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 ...
- 5,101
2
votes
Accepted
Kurtosis of a straddle
Even if you assume null cokurtosis terms, your equality is still off:
\begin{align}
\operatorname{Kurt}[X+Y] = {1 \over \sigma_{X+Y}^4} \big( & \sigma_X^4\operatorname{Kurt}[X] + \sigma_Y^4\...
- 5,113
2
votes
Accepted
Cornish Fisher VaR Parameters Calibration
The method
The Cornish-Fisher expansion is a method that helps us to approximate the quantile of a target distribution $F$ in terms of another support distribution $\tilde{F}$, using the so-called ...
- 7,035
2
votes
Do portfolio mean and portfolio variance have probability distributions?
Given a set of returns, say 500 days, and a fixed portfolio construction, you can derive the 500 daily portfolio valuation changes.
You can easily measure the mean and variance of these valuation ...
- 11.2k
1
vote
BKM risk neutral moments in python
I think the reason to interpolate using a delta grid is twofold.
First, note that the delta of an option (or lets talk about a given strike) gives you some information about how OTM/ITM the option is. ...
- 853
1
vote
Why isn't the asset with minimum variance given a 100% portfolio weight?
If you start out with a matrix specifying the covariance of every pair of assets, and an alpha for every asset (because people usually do), and define an objective function that maximizes the alpha ...
- 12.6k
1
vote
Accepted
Alternative low-moment measure of skewness
Pearson 2 skewness, which compares mean and median, lying between $-3$ and $3$ while being zero for symmetric distributions, was introduced by
Yule, G. U. and Kendall, M. G. (1950), An Introduction ...
- 3,080
1
vote
Contribution of an asset's variance, skewness and kurtosis to its portfolio weight?
It is not clear that this allocation would be useful or even possible.
Suppose you had a portfolio of two assets and that the optimal weights you had derived, based on a mean-variance approach were 0....
- 11.2k
1
vote
Calculate moments given density values
Just to add, you did not mention which kind of momement. These calculated by formula in ir7 are called general moments. However, there are also:
Central moments defined as $E[X-EX]^k$
Standardized ...
- 793
1
vote
Ito isometry and the covariance of an Ito process
Recall that for any deterministic function $g,$ Ito's integral follows a normal distribution:
$$\int_0^t g(u) dW_u \sim N\left(0,\int_0^t g^2(u) du\right).$$
Therefore, since
$$X_{t+s} - X_t = \int_t^...
- 850
1
vote
ARMA moments proof
For the first, where $|\beta| < 1.0$, you can write it using the lag operator.
$x_t (1 - \beta L) = (1 + \theta L) u_t $
$X_t = \frac{(1 + \theta L) u_t}{(1- \beta L)} $
Since $|\beta| < 1.0 ...
- 1,160
1
vote
The possible preferences of investors for higher than first 2 moments of return distribution?
The argument I have seen for higher order moments follows from an expansion of log wealth:
\begin{align}
log(W) &= log(W_0 (1+r))\\
&= log(W_0) + log(1 + E[r] + r - E[r])\\
&= log(W_0) + ...
- 547
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