Questions tagged [moments]

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Robust approach to capture price movement of a stock universe

Let's say we have a universe of 100 stocks, S1, S2, ... S100, and their historical price data (daily, for the past 50 days). What would be the most robust/widely used method to capture the price ...
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Is there a modified Bachelier's futures spread option model with adjustments for skew and kurtosis?

I'm looking at pricing a very large deal and while the distribution is kind of "normal," there's quiet a bit of skew and kurtosis that isn't being considered when I use the normal Bachelier'...
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BKM risk neutral moments in python

I am trying to compute the BKM implied moments (Bakshi, Kapadia and Madan 2003) in python by following this paper: Neumann, Skiadopoulos: Predictable Dynamics in Higher Order Risk-Neutral Moments: ...
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Asian option analytical approximation

I'm trying to approximate the price of an Asian option via the Black-Scholes formula by considering the discrete arithmetic average as a log-normal distribution. $$ A_{T}(n):=\frac{1}{n} \sum_{i=1}^{n}...
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Power options for pricing European claims

I have the following question: Why would somebody be interested in the expression $E[S^\theta]$ for $\theta$ between zero and one. The only thing I know is that this then can be somehow used to ...
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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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2 votes
1 answer
142 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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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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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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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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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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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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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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5 votes
2 answers
292 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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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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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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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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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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3 answers
432 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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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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4 votes
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348 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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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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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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318 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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3 answers
430 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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204 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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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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1 answer
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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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2 votes
1 answer
145 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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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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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{\...
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