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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 jumps....

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You should review your definition of what a stochastic differential equation is: dSt= mu*St*dt + sqrt(vt)*St*dW1t + Jt*dQt it means simply that S_t+1=St + mu*St*dt + sqrt(Vt)*St*(W_t+1-Wt) + Jt*(Q_t+1-Q_t) which has to be simulated one time step at a time. It does not follow that the solution is of the form S_t = S_0 exp( (mu - 0.5 sigma^2) 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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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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