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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' ...
David Addison's user avatar
8 votes
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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 ...
Quantuple's user avatar
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7 votes
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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 ...
Kermittfrog's user avatar
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5 votes
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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 }{\...
Kermittfrog's user avatar
  • 6,535
4 votes
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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 ...
Dave Harris's user avatar
  • 4,379
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 ...
LePiddu's user avatar
  • 393
4 votes
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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.
ir7's user avatar
  • 5,043
4 votes
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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.
Kermittfrog's user avatar
  • 6,535
4 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 ...
alexbougias's user avatar
  • 1,416
3 votes
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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 ...
John's user avatar
  • 5,391
3 votes
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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 = ...
Gordon's user avatar
  • 21.1k
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

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 ...
Chris Taylor's user avatar
  • 5,901
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: $$B_t\text{d}t=\text{d}\left(tB_t\right)-t\text{d}B_t$$ ...
Daneel Olivaw's user avatar
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 ...
steveo'america's user avatar
2 votes

Second Moment of Stock Process

If $D_i$ is dividend paid at time $t_i \in [t, T]$ , then future value of all dividends payments (assuming payments are known in advance), $D$, is: $$D= \sum_{i=i}^{n} D_i e^{r(T-t_i)}$$ Since all ...
Neeraj's user avatar
  • 2,228
2 votes
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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\...
ir7's user avatar
  • 5,043
2 votes
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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 ...
Kermittfrog's user avatar
  • 6,535
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 ...
Attack68's user avatar
  • 10k
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. ...
KT8's user avatar
  • 855
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 ...
Dimitri Vulis's user avatar
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 ...
develarist's user avatar
  • 3,000
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....
Attack68's user avatar
  • 10k
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 ...
Martin Vesely's user avatar
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^...
Idonknow's user avatar
  • 840
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 ...
mark leeds's user avatar
  • 1,102
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) + ...
steveo'america's user avatar

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