7

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' willingness to play negative expectancy casino games, is consistent with many of the intuitions underlying Prospect Theory. There are several possible behavioral ...


7

I think most people agree that aggregate (index) stock returns have negative skewness. However, this does not appear to be the case for individual stock returns. These two papers find that average skewness of individual stock returns is positive though time-varying: Paper 1 R. Albuquerque, Skewness in Stock Returns: Reconciling the Evidence on Firm Versus ...


4

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 $$ \begin{align} \frac{\partial L }{\partial w_i}&=3w^TM_{3,i}w-\lambda \quad \forall i \\ \frac{\partial L }{\partial \lambda}&=w^T\mathbf{1}-1 \end{align} $$ where $M_{3,i}$ is the $i$th ...


3

The usual answer is that most risk assets tend to exhibit left-skew, with correlations ->1 into the left tail (ie diversification breaks down). And so positively skewed assets have attractive portfolio features, over and above their own intrinsic fundamentals. The more formal answer is that for two distributions with the same mean and standard deviation, ...


3

Let's derive a possible approach from utility theory. Our investor is risk averse and exhibits CARA utility using an exponential utility function with risk aversion parameter $\gamma>0$ (risk averse agent): $$u(x)=\frac{1-e^{-\gamma x}}{\gamma}$$ A 3rd order Taylor series expansion around $x=0$ yields \begin{align} u(x)\approx& x - \frac{1}{2}\gamma ...


3

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 moments (volatility, kurtosis, etc.). This comes from the Taylor expansion of the log. However, if one wishes to maximize the probability that returns over a given ...


3

The skewness and kurtosis values you obtain appear to be of realistic magnitude. In general higher frequencies are more non-normal, i.e. have higher skewness and kurtosis. If non-normal returns are aggregated the central limit theorem starts working and the return distribution coverges to a normal. Convergence can be quite slow under fat tails. You can try ...


2

I'm not sure if this would qualify as "empirical work" but you should definitely read Dynamic Hedging if you haven't already. Taleb talks a lot about this. To be more specific, have a look at page 264, the section called "Higher Moment Bets". I hope this is helpful, although I am not sure if this is what you are looking for.


2

The only thing weird is skewness not being lower for the weekly vs daily. In any case, take a look at table 1.1 from Campbell, Lo and Mackinlay, and check that your values are not far off the ballpark. Actually, with annual data, you should have nearly zero skewness and zero excess kurtosis (on the market). However, asset allocation might lead to severe ...


2

You would simply calculate the prices of various strike options using your parameters, then calculate the black scholes implied vol of each option. Did I miss the point of your question ?


2

Suppose you were to price 2 instruments: a strongly OTM put and a strongly OTM Call. In the standard BS settings, instantaneous volatility is assumed to be constant. Consequently, the implied volatility of these 2 instruments will be the same, resulting in an absence of implied volatility skew. Now, assume a negative spot/instantaneous volatility ...


2

From Daniel-Moskowitz ("Momentum Crashes") you can see that equity CSMOM has negative skewness. However, this is less clear for other asset classes. From their table 11 you can see that commodity momentum has essentially zero skewness (they report a mildly positive skewness). Also e.g. this paper (Menkhoff, L., Sarno, L., Schmeling, M. and Schrimpf,...


2

Maybe you like working with coskewness. But it is not needed if you just want to estimate the skewness of the portfolio. If you have retunr times serise $(r^i_t)_{t=1}^T$ for each asset $i$ and the weights $w_i$ that these assets have in your portfolio then you can form $$ r_t = \sum_{i=1}^6 w_i r^i_t \quad \text{for each } t, $$ and you simple estimate all ...


2

Conceptually, if you want constant conditional skewness, you could simply choose an error distribution that is skewed for your ARMA model. ARMA only restricts the conditional mean of the time series to vary in a certain way, but all the other parameters such as skewness or kurtosis can be chosen freely. In practice, you need a way to estimate such a model. ...


2

Instead of starting from a CARA utility function like how the other answer does, an alternative for incorporating portfolio skewness in the mean-variance model's objective function, without risk-aversion parameter $\gamma$ or going through a Taylor series expansion of some arbitrarily asserted utility function, could be $$\arg \max_w \enspace w^T\mu-\frac{1}...


1

It's a little simplistic to say that positive skew is better, you could for example have a return distribution which is negatively skewed but has a mean of 10%, versus a positively skewed one with a mean of 5%. That said, negative skew has a serious downside when it comes to risk and estimation. At any given point in time, you will generally only have a ...


1

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 to the Theory of Statistics, 3rd edition, Harper Publishing Company, 162-163.


1

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 0.4, 0.6. These are independent of the 3rd and 4th moments, suggesting that whatever the 3rd and 4th moments were in these assets the 0.4/0.6 weights would be ...


1

Another example of a thing you can do. Every day I receive a chart showing the skew, i.e the IV versus strike for S&P options. Don't throw all those out the next day, save a few of them and tape them to your cubicle wall. That way you can develop a feel for what the skew has looked like in various historical circumstances, in quiet periods and in ...


1

For example you could ask yourself what the realized volatility will be if the stock were to be at 80 at some time in the future. If the answer is ‘much higher than the realized volatility when the stock is at 100’, then that would be a reason why the implied vol of the 80 strike is higher than the at the money. The reason for this is that options derive ...


1

Mean-variance efficiency generally only considers the first two moments. Skew and kurtosis are outside of it, and Sharpe is commonly criticized for not accounting for skew and kurtosis (leading to metrics like Omega as a replacement). Regarding your specific question, it depends entirely on what risks a given investor/trader wants to take. One person ...


1

I suggest you have a look at the paper: Schloegel, Erik (2010) "Option Pricing Where the Underlying Assets Follow a Gram/Charlier Density of Arbitrary Order", Journal of Economic Dynamics and Control, Vol. 37, No. 3, pp. 611-631 available on SSRN. A random variable $Y$ that follows a Gram/Charlier Type A series distribution has the probability density ...


1

The formula in your skew function is one of skew normal distribution. That distribution has a limit on skew parameter, while in the real world there is no such limit. From personal experience, few years ago I tried doing exactly what you described in your question. After comparing skew normal distribution on SPX with the real world, I concluded that there ...


1

Selling 2 ATM calls against 100 underlying shares result in Delta neutral. "Given the extra downside protection, and potential need for a stop order if the asset price rises too high, is the added risk of the naked leg justified?" The risk is if the move is more than extrinsic premium collected. One thing to watch out this type of trade is skewness. ...


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