Questions tagged [moments]

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What range of skewness values do asset returns exhibit?

If we have multivariate stock returns data (daily, weekly, or monthly), what range of skewness values would we typically observe across firms? $-1.5$ to $1.5$, for example Also can we expect the ...
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33 views

using moment matching to price spread options (multi asset)

this is my very first question in this forum, after having been a greed follower since a few years, feeling that I need your help in a topic. I need to price a multi asset option that has the ...
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1answer
91 views

Maximum skewness portfolio solution derived from its Lagrangean formulation

$$\arg \min_w \quad w^\top \Sigma w$$ \begin{align}\text{s.t.} \quad \mathbf{1}^\top w = 1 \end{align} is the optimization problem for the minimum-variance portfolio weights, whose analytical solution,...
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2answers
97 views

Why isn't the asset with minimum variance given a 100% portfolio weight? [closed]

The maximum expected return portfolio is the one that assigns a 100% weight to the asset with the highest expected return amongst all assets under consideration. Shouldn't then the asset with the ...
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1answer
55 views

Alternative low-moment measure of skewness

$$ \widehat{\text {Skew}}_{i, t}=\frac{3 \cdot\left[\hat{\mu}_{i, t}-\operatorname{median}\left(r_{i, d, t}\right)\right]}{\hat{\sigma}_{i, t}} $$ is called Low Moment Skewness by Baltas and Salinas (...
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1answer
106 views

Is there Cornish-Fisher volatility, given that there is Cornish-Fisher Value-at-Risk?

The Cornish-Fisher expansion is used to approximate the quantile $q_\alpha$ of a return distribution in order to extend the traditional Value-at-Risk (VaR) measure $$VaR = \mu(X) + \sigma(X) q_\alpha $...
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3answers
169 views

Do portfolio mean and portfolio variance have probability distributions?

If $X$ is a $T\times N$ matrix of multivariate asset returns, and $w$ is some optimal portfolio weight vector, then the portfolio return series is $r_p = X w \in\mathbb{R}^{T}$. This return series ...
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1answer
81 views

Contribution of an asset's variance, skewness and kurtosis to its portfolio weight?

The mean-variance model is known to assign higher weights to assets with high expected returns and low volatility, meaning that there is a direct link between the asset's weight within the portfolio ...
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53 views

asymptotic behavior of the pdf constraints due to Roger Lee

In a beautiful paper, http://math.uchicago.edu/~rl/moment.pdf, Roger Lee (2004) shows that implied variance is bounded above by a function linear in the log-strike k. Does anybody know how it ...
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2answers
256 views

Higher moments of a straddle

Following the logic of Ben-Meir and Schiff (2012) and this question the first, second, third and fourth raw moments of a put are: Similarity, for a call it is as follows: where and ...
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2answers
85 views

Calculate moments given density values

Suppose I have given a finite number of grid values belonging to a probability density function. Moreover, I have the associated values of the density support. For instance: ...
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1answer
37 views

Does standardizing/normalizing asset returns change their skewness and kurtosis?

Asset returns are obtained by log-differencing prices. Standardizing or normalizing/scaling asset returns can be carried out by de-meaning the returns and dividing them by their standard deviation, ...
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1answer
69 views

Kurtosis of a straddle

I want to determine the kurtosis of a straddle. My question is closely related with the following topic here. According to the following paper of Ben-Meir and Schiff (2012) the expected value of a ...
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1answer
76 views

Cornish Fisher VaR Parameters Calibration

I am trying to calculate Cornish-Fisher (modified VaR), but I am in a trouble because when I am reading some articles, some authors calculate the Cornish-Fisher expansion taking parameters S and K, as ...
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3answers
348 views

Any portfolio models not based on asset return moments?

The mean-variance model for portfolio optimization minimizes portfolio risk (covariance matrix), which is the second statistical moment of multivariate asset returns, and sometimes simultaneously ...
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1answer
82 views

Show that Riemann integral over BM is gaussian process

I am looking at the process $$X_t = \int_0^tB_udu$$ I know that this is a gaussian process with variance $t^3/3$. However, I would like to manually show the first statement directly. For this, I ...
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1answer
220 views

Ito isometry and the covariance of an Ito process

Let $(B_t)_{t \geq 0}$ et $(W_t)_{t \geq 0}$ be two independent Brownian motions and let $f: \mathbb{R} \rightarrow \mathbb{R}$ a deterministic function of time. We define the following process: \...
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1answer
50 views

ARMA moments proof

Consider a standard ARMA(1,1) process such as $$x_t - \beta x_{t-1} = \theta u_{t-1} + u_t$$ where $u_t$ is i.i.d. $u_t \sim N(0,\sigma^2)$. I know how to derive mean and variance with stationary ...
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67 views

Analytical formula for the moments of SABR model?

Do analytical formulae exist for the central moments under the SABR model? Assuming dynamics for the forward rate $\{F_t, t \geq 0\}$ under a shifted SABR model. How do we derive $\mathbb{E}[X(T)^n]$ ...
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1answer
269 views

Is positive skewness preferences rational or irrational?

Is positive skewness preference rational or irrational? I have a great trouble understanding why investors should prefer positive skewness over negative one. Sometimes it is argued that preference ...
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3answers
347 views

The possible preferences of investors for higher than first 2 moments of return distribution?

Can anyone explain in an intuitive manner a justification for possible preferences of investors for moments of return distribution beyond the first two moments (i.e. mean and variance). For example, ...
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194 views

Relationship between Implied Volatility Curve Derivatives and the Underlying's Moments

Very probably this question has been posed before, so if someone can pose the link to the relevant question, it would be appreciated. What is the relationship between the implied volatility skew and ...
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89 views

Using GO GARCH to optimize a yearly-rebalanced portfolio based on daily data

Is it reliable to optimize portfolio weights on a yearly-rebalanced portfolio based on the Generalized Orthogonal GARCH (GO-Garch) covariance, coskewness, and cokurtosis matrices with the rmgarch R-...
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1answer
2k views

How to price this basket option?

Underlying assets are three global stock index : Eurostoxx 50, HSI, KOSPI 200 Maturity: 36 months with advanced redemption date in every 6 months if prices of indexes satisfy given conditions at each ...
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1answer
141 views

Second Moment of Stock Process

I have a stock process which I have decided to model as $$S_T=S_t\exp((r-q-\frac{1}{2}\sigma^2)(T-t)+\sigma(W_T-Wt))-D_T$$ where $D_T$ is a cash dividend at time $T$. This dividend is known. I then ...
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308 views

Fourth moment of ARCH(2)

I am studying the ARCH(2) process given by $$X_t = \sqrt{h_t} \varepsilon_t$$ where $$h_t = \alpha_0 + \alpha_1 X_{t-1} ^2 + \alpha_2 X_{t-2} ^2$$ and $\varepsilon_t$ follows $N(0,1)$. ...
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260 views

Risk Neutral Variance Gamma

In the risk neutral version of the Variance Gamma model the stock dynamics are $$S_T=S_0 e^{ (r-q+\omega)t + X(t;\sigma,\nu,\theta)}$$ with $$\omega=\frac{1}{\nu}\ln\left(1-\theta \nu - \frac{\...