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This is an interesting and not so easy question. Here's my 2 cents: First, you should distinguish between mathematical models for the dynamics of an underlying asset (Black-Scholes, Merton, Heston etc.) and numerical methods designed to calculate financial instruments' prices under given modelling assumptions (lattices, Fourier inversion techniques etc.). ...

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$X_t$ being a stochastic process, one cannot use ordinary calculus to express the differential of a (sufficiently well-behaved) function $f$ of $t$ and $X_t$. Instead one should turn to Itô's lemma, one of the key results of stochastic calculus, which stipulates (assuming $X_t$ is here a continuous, square integrable stochastic process) $$df(t,X_t) = ... 5 This effect is coming from the supply and demand in the options markets. Many portfolio managers want (or need) to buy out of the money put options, and many are willing to sell out of the money call options (thereby funding the purchase of put options). Now, when the market goes down, dealers find themselves short vol and they need to buy options to cover ... 4 Let$$q (S) := \frac{d\mathbb {Q}(S_T \leq S)}{dS} denote the probability density function of the stock price at time T>0 under the risk-neutral measure. By definition, the price of a European call then writes \begin{align} C (K,T) &= P (0,T) E_0^{\mathbb {Q}}[(S_T-K)^+] \\ &= P (0,T) \int_K^\infty (S - K) q (S) dS \end{align} with P ... 4 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 ... 3 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 ... 3 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 ... 2 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 ... 2 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 ... 2 It is difficult to gain intuition by just looking at the price surface, and it is also easier to calibrate models on the volatility surface rather than on the price surface because with the later you are dealing with numbers of very different sizes (depending on the moneyness and maturity) which is not good for minimization algorithms. However low and high ... 1 Your adjusted scheme is correct. Basically, taking a maturity T, you can consider the forward price process F_t^T = S_t e^{r(T-t)}. You apply the Andersen scheme to F_t^T and then note that \begin{align*} S_{t+\Delta} &= F_{t+\Delta}^T e^{-r(T-(t+\Delta))}\\ &=F_t^T \exp(\ \Box \ ) e^{-r(T-(t+\Delta))}\\ &=S_t e^{r(T-t)}\exp(\ \Box \ ) ... 1 Chan, Karolyi, Longstaff, and Sanders (1992) compares empirically the performance of the main interest rates model. The first page also provides the main references on those models. 1 I suggest taking a look at optimal hedge Monte Carlo and by extension the garam model from http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1428555. The basic issue is that a risk premium exists in option markets which is like Vol also unobserved. The better your guess of the future realized vol distribution, the better your guess of the risk premium ... 1 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. 1 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 ... 1 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 ... 1 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 ... 1 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} ... 1 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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Do these work for you? P34 of http://web.mit.edu/junpan/www/SVJ.pdf P1360 of http://www.darrellduffie.com/uploads/pubs/DuffiePanSingleton2000.pdf P2045 of http://www.math.ku.dk/~rolf/bakshi.pdf

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