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# Tag Info

6

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 ...

6

"Skewness" quantifies how asymetric a distribution is about the mean. "Kurtosis" quantifies how peaked or flat the distribution is. Skewness is defined as: $E[ (X - mean)^3 ] = \frac{(\sum (x_i - x_{mean})^3 )}{N}$ and Kurtosis as: $E[ (X - mean)^4 ] = \frac{(\sum (x_i - x_{mean})^4 )}{N}$ where X is your distro values (x_1, x_2, ... x_N), mean is the ...

4

That can be a somewhat difficult question to answer, given that the context may yield different distributions. Nevertheless, I think that you could try to fit the best distribution algorithmically. For instance, lately I found this package at Matlab file exchange: Finding the best distribution that fits the data Link (...) This is where Mike's allfitdist ...

4

In my opinion you have two choices: You calculate annual returns from the daily returns that you have - I guess it is clear how. Subsequently you calculate your statistics on these $11$ data points. When I look at your comment above, this could be what you want to achieve. Then you have the ex-post statistics on your data. The drawback is that $11$ data ...

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

Assuming you have return time series $$r_1(1), r_1(2), \ldots, r_1(T) \qquad \text{and} \qquad r_2(1), r_2(2), \ldots, r_2(T)$$ for the 2 assets and asset weights $w_1$ and $w_2$, we can follow the calculation of the $N$-asset portfolio skewness laid out in another answer for a similar question. To extend it to include portfolio kurtosis, we need the co-...

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

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 ...

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

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 ...

1

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 ...

1

What is the data basis that you start from? If you just have the covariance matrix, then you can only calculate portfolio variance or volatility by $$w^T \Sigma w$$ where $w$ are the portfolio weights and $\Sigma$ is the covariance matrix. If you have the individual asset continuously compounded returns $r^j_t$ where $j$ indexes assets, $j=1,\ldots,N$, and $... 1 I think the best answer is to test different distributions with your specific data set and see which fits the data the best... skewness and kurtosis are just a small piece of information , there is still a good deal of information we don't know and won't have with out having the data set in front of us 1 Selling 2 ATM calls against 100 underlying shares result in Delta neutral. "Given the extra downside protection, and potential need for a stop order if the asset price rises too high, is the added risk of the naked leg justified?" The risk is if the move is more than extrinsic premium collected. One thing to watch out this type of trade is skewness. ... 1 Maybe you like working with coskewness. But it is not needed if you just want to estimate the skewness of the portfolio. If you have retunr times serise$(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 ...

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