# Tag Info

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O-U is continuous time mean reverting process, hence used to model stationary series. It has closed form analytic solution. This allows insight into stationary processes and act like asymptotic limiting case for calculating coefficients that matter. EDIT: You can see AR(1) below $$x_{k+1} = c + a x_k + b\varepsilon_k$$ and by substituting c=θμΔt, a=−θΔt ...

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This is seen as a bit of a niche field, which is likely why there are not so many books and these issues are not covered in standard econometrics texts. Options pricing models are usually fitted to options data rather than estimated econometrically from historical data. For statistical models, it is often more convenient to start from a discrete model as the ...

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There is a mention of robust standard errors in "rugarch" vignette on p. 25. The robust standard errors are due to quasi maximum likelihood estimation (QMLE) as opposed to (the regular) maximum likelihood estimation (MLE). They are robust against violations of the distributional assumption, e.g. when the assumed distribution is Normal while the true ...

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This is a very serious problem. In general these results should not be used as they usually suffer from very low robustness and display butterfly effect. In other words, the parameters can change very quickly as new data flows into the model and the confidence intervals of the parameters may be so huge as to show that they are statistically insignificant. ...

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I thought I would answer the question of "what am I using." All shrinkage estimators map to a Bayesian estimator that differs only in the prior distributions. In other words, you get a point estimate that is indistinguishable from a Bayesian estimate except that the calculation rule determines the prior distribution. Stein estimators for the Gaussian are ...

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This book goes through exactly this problem in quite detail (with C++ codes included). I've worked through it in the past, but can't sum it up off the top of my head.

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$\alpha=0$ does not imply constant volatility. Consider just a simple Garch(1,1): $$\sigma^2_t = \omega + \alpha \eta_t^2 + \beta \sigma^2_{t-1}$$ Note that: $$\sigma^2_t = \omega + (\alpha + \beta) \eta_t^2 - \beta (\eta_t^2- \sigma^2_{t-1})$$ Now add $\eta_{t+1}^2$ to both sides: $$\eta_{t+1}^2 = \omega + (\alpha + \beta) \eta_t^2 - \beta (\eta_t^2-... 3 EDIT: My reasoning below seems to be wrong. The process as you write it tends to infinity if a is big enough and positive and if \lambda_0 is positive. I would not call this process non-meanreverting OU. It is just an Ito process of a simple form. If we remove the stochastic part then we get$$ d\lambda_t = a \lambda_t dt $$with the solution (if \... 2 One does not estimate the local volatility at a given T and K. Instead, Dupire's formula actually gives \sigma(T,K) for all T and K.$$ \sigma^2(t_0,S_0;T,K)= \frac{\frac{\partial C}{\partial T} + (r - q)K \frac{\partial C}{\partial K} + qC}{\frac{1}{2} K^2 \frac{\partial^2C}{\partial K^2}} $$where C(t_0,S_0;T,K) are the call prices for ... 2 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 cumulants of the target distribution. Cumulants are one way to (fully) describe a distribution function; i.e. if you know 'all' cumulants of a distribution ... 2 You mean what influence has the objective function on the results of the calibration? Perhaps look at this paper (there are free versions on the web). @ARTICLE{Detlefsen2007, author = {Kai Detlefsen and Wolfgang K. H{\"a}rdle}, title = {Calibration Risk for Exotic Options}, journal = {Journal of Derivatives}, year = 2007, ... 2 I agree with much of @user25064's answer but thought I'd add some more. To answer your questions: 1. Is it a considerable problem for the goodness of model? Must it be corrected or isn't it so important? It's a big problem. If the convergence criteria is not satisfied, then you haven't find a local optium to your optimization problem (within numerical ... 2 I see... the problem with too long a time frame is that most time-series are not stationary -- the means and the variances will tend to fluctuate. Given non-stationary in our time-series, and given that no uncertainty is never possible, we are damned if we wait for a larger sample size, and damned if we don't. But assuming that you know what the underlying ... 1 This will not help you with the actual question but I think the theta surface should look differently. I have a simplified solution where it is all with sliders to play around for teaching purposes (vol is constant throughout and simply an input from the slider into BSM). The sliders at the top are strike, interest rate, dividend and vol; bottom is just &... 1 I know kalman filters and have ... If this knowledge extends to Unscented filters, UKF, you can think of the UKF being a sparse particle filter. With a UKF you have a few sigma points which are propagated forward via your model function and then after the measurement update these sigma points are updated via covariance estimation. With a particle filter, ... 1 Using weekly data for the last 5 years for DAX (german equity index), S&P (US equity index) and EUR (value of "german" currency priced in US dollars) we have the following results: EUR fluctuates with an annualized volatility of 8.8% a year, DAX with vol of 18% a year and S&P vol 12% a year. In my experience it is generally true that a currency is ... 1 If you want to see how the VCOV estimator à la White (1982) is constructed, after running ugarchfit, in the fitted object, which we call myobj, look at myobj@fitA and myobj@fitB. These are the matrices defined in White (1982). rugarch uses solve(A) %*% B %*% solve(A) / n. Now, if the question is, what is the cause of Hill & McCullough (2019) ... 1 As pure speculative commentary and a non-quantitative answer, you may want to look at the short-term options volatility as a factor for possibly fine tuning your model. Your goal seems to be to find the elasticity or 'beta' factor of the price movement. The price drift you are looking at implies some form of information dissemination or perception change in ... 1 You can't estimate historical prices because they are given to you already. If you're trying to estimate future implied volatility based on historical implied volatility and historical volatility, then that is possible. Use any statistical package or code to calculate HV based on underlying asset Take your IV data and calibrate your HV model somehow (this ... 1 There is a simple Bayesian model, that is only slightly less simple if there is a stochastic budget constraint. The likelihood would be$$\mathcal{L}(x_1\dots{x_{n}}|\mu;\sigma)=\frac{1}{\pi}\frac{\sigma}{\sigma^2+(x_{t+1}-\beta{x_t}-\alpha)^2}.$$I have successfully implemented this. If you want to test this, you should use the Bayesian predictive ... 1 Let X^h be your hourly process Let X^d be your daily process Let \delta be one day you have$$X^d_t=\frac{1}{\delta}\int_{t-\delta}^{t}X^h_s dsdX^h_t = a(b-X^h_t)dt + \sigma dB_t\Delta X^d_t := X^d_{t+\delta}-X^d_t =\frac{1}{\delta}\int_{t-\delta}^t\left(X^h_{u+\delta}-X^h_{u}\right)du so it is a gaussian random variable by knowns ...

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You can aggregate your starting hourly data to obtain daily data and re-estimate the parameters, then simulate. Alternatvely, with your parameters already obtained, you can simulate hourly data and make a post-simulation aggregation to have daily data.

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There are two cases depending on what P is in your formula. P is the market price of the bond: There exists an idealized, true but unobserved yield curve, but we observe bonds that have some measurement error. This can be because of microstructure imperfections (bid/ask spreads, liquidity and other premia, non synchroneity of quotes since bond markets are ...

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$v$ should be the total number of parameters (constants + AR + MA + GARCH + ARCH). I disagree with @kiwiakos, the student t(df) distribution is used because we are using standard errors which are estimates of standard deviations (and not true standard deviations) to compute the statistic. That is the reason why we use student t test eventhose the ...

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Before you start asking about the number of dof, how do you know that the finite sample distribution of parameters is student-t? I don't think it is. In linear regression they are student-t because of linearity and under assumption for the residual distribution. In Garch you can just say that if you estimate using max-likelihood then asymptotically (not ...

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Here is an MLE I built that uses logistic mapping. %MLE iterator: for cxm = 1:cxmax for cxth = 1:wx; %thx %Incr. theta within asymptotic min and max. thi1 = thA1(cxth,1); mint = thA1(cxth,2); maxt = thA1(cxth,3); thix = -log((maxt - mint)/(thi1 - mint) - 1); %Logistic inverse. if rand > 0.5; signx = -1; ...

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Here are two nice references: https://www.sitmo.com/?p=134 https://commoditymodels.files.wordpress.com/2010/02/estimating-the-parameters-of-a-mean-reverting-ornstein-uhlenbeck-process1.pdf

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The log likelihood function is indeed rather flat in the $\mu$-direction, for small time horizons (you used $T = 1$ it looks like). As you may have noticed, increasing the number of observations but keeping the time horizon the same DOES NOT IMPROVE the accuracy of the estimate of $\mu$ - this is a bit counterintuitive, if you ask me. But, increasing the ...

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This is what often happens in optimization problems, i.e. some direction is almost flat. Google 'preconditioning'. Basically the idea is to rescale the variables, so that the Hessian has approx. same order of magnitude values on the diagonals. Also, that's not a stationary process, so estimation of mu can be difficult. BTW not sure if it's a very good idea ...

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