# Tag Info

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Quick summary: Your model should still be well specified, as long as: 1) You do the analysis on a heavily traded asset, e.g. IBM on NYSE, and 2) You use heteroskedasticity-consistent standard errors in your estimation framework, e.g. White's standard errors. I'm going to start the long answer by re-stating the question to make sure I've got it right. Let ...

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1) Gatheral expresses everything in forward terms: forward value of the spot and of the call. Consider an asset $A$. You need to hold $A$ at time $T$ but since you don't need it now you don't want to buy it now. Instead you enter a forward contract with someone that says that at time $T$ you will pay the amount $K$ and get the asset in exchange. What ...

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Even though it's a straightforward extension, it took me a while (a year? yikes!); but now you can easily incorporate Bayesian ar(1) (or more generally, Bayesian regression) in joint estimation by using designmatrix = "ar(1)" as an argument to svsample. It's not well documented yet (except in the help files), but I nevertheless hope easy to use. From the ...

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I am not sure if I understood your question correctly but I will try to answer it anyway. If you have a standard normal random vector $z \sim N(\mathbb{0},I_n)$ (where $z,0 \in \mathbb{R}^{n\times1}$ and $I_n \in \mathbb{R}^{n\times n}$ is the identity matrix) and you want to transform it into a multivariate normal $x \sim N(\mu,\Sigma)$ you do it the ...

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Not sure your question is about having a process for covariance or to have multivariate GARCH. The standard viewpoint on a stochastic volatility for covariance is to use a Whishart process. See for instance Philipov, A. and M. E. Glickman (2006, July) Multivariate stochastic volatility via wishart processes. Journal of Business & Economic Statistics 24 ...

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I think you're looking for multivariate GARCH models of which this is an overview paper. Multivariate GARCH models have one big drawback: they are pretty hard to estimate due to the number of correlations. This paper by Caporin and McAleer might be of interest in that regard.

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The local vol model has exactly enough freedom to match the individual densities $X_t.$ There is no additional freedom in the local vol model to match even a joint density for a pair of times $(X_t,X_s).$ When you ask about the joint density across the continuum of times $t \in [0,T]$ it is pretty easy to show that any local vol model differs from any ...

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I think you need to ask your question differently to get better answers than mine. Your Black Scholes part has two problems. First positive infinity should be negative infinity. Second, you are assuming zero dividends in Black Scholes but you are assuming a possibly positive div yield q in the CEV part. If the div yield q is sufficiently positive in the ...

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Diffusion brings about a standard deviation which increases with the square root of time (just like in Brownian motion), while jumps add variability proportional to time (since the jump times are a Poisson process). So they are quite different. Experience shows that sharp stock market moves do occur (in connection with big news events for example), so ...

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The problem is that what some mean when they say "volatility" is BS implied vol from an option price. What some others mean when they say "volatility" is some diffusion parameter from a drift diffusion model (with or without jumps). These are the same value in the log normal model of stock prices but different for many other models including those with ...

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Different methods exists to compute implied vol from the same option prices, eventually it's prices that matters to calibration. But if you can reproduce same option prices accurate to the cent by fitting implied vol, I think it doesn't matter.

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Hans Buehler investigated this in some detail, including in his doctoral thesis. When I tried it out some years ago, back when volatility exotics were more liquid, I found the models nearly impossible to calibrate to my satisfaction, even for the SP500 complex. I think the mathematical analogy is fair, and enjoyed Buehler's work, but in practice it won't ...

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Jumps are totally different from volatility. Imagine a stock whose price has jumps but has no volatility. The asset pricing implications for options on that stock are totally different than from a stock with volatility. Below I simulated 3 stock paths: (i) Jumps and volatility, (2) Only Jumps and (3) No jumps but higher volatility. As you can imagine the ...

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I know one article (download) that explaining how to calculate local vol surface from IV surface and also chapter 18 of this book is very good In this context. However you know that Dupire’s (1994) formula for local volatility is \begin{align} \sigma_L(k,T)=\sqrt\frac{\frac{\partial C}{\partial T}}{\frac{1}{2}K^2\frac{\partial^2 C}{\partial K^2}} \end{align} ...

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The volatility of your asset $y_t$ is simply its time varying standard deviation, given by $\beta \exp(x_t/2)$. Once you've got the estimates for latent factor $x_t$ from converged MCMC chain, calculate the expected value for volatility at time $t$ using $$\hat{v_t} = \mathbb{E}[\beta \exp(x_t/2)] = \frac{1}{R}\sum_{r=1}^R \beta \exp(x_t^{(r)})$$ where $R$ ...

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Look at Gilli & Schumann's paper. They provide a Bates' model estimates set, the way to improve such estimates calibrating those ones using an Heuristic model and, lastly, the relative codes in matlab, in order to be able to replicate the model. Unfortunately, there are not available the relative call prices estimated time series; I think that noone ...

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alpha + beta < 1 is the stationary condition for GARCH. If alpha and beta are low that means volatility of the stock does not have clustering behaviors. I think you can have a look at ADF and PACF of Return^2 time series first. If the first order autocorrelation is very significant but alpha is not, then perhaps you can check on the parameter calibration. ...

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There's no best method. The question is : what is the behavior of the volatility structure (atm and skew) when the underlying moves? Each method assumes something different. In the real market, one method might work well for a period of time (in the sense that it minimizes residual p/l), but then another method might take over as best. Practitioners ...

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I cannot guarantee that it is error-free, but this paper (appendix A) has a relatively straightforward derivation of the Heston price for a european call.

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The Feller condition applies without modification. That is under the assumption that $v$ is square-root process with poisson-arrival jumps (as you wrote), and assuming the jump distribution is strictly positive and initial level $v_0>0$. The reason is, conditional on no jumps occuring, the process is just a square root process, for which the references ...

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Show that the discounted expectation price of the new security is the same as the solution of the PDE. Once this is done all three assets have discounted price processes which are martingales so there can be no arbitrage.

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