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4

I know two papers explaining how to calibrate this kind of models, and one of them explain the impact of the quality of the fit on a pricing model: Aït-Sahalia, Y. (2002, January). Maximum likelihood estimation of discretely sampled diffusions: A closed-form approximation approach. Econometrica 70 (1), 223-262. Azencott, R., Y. Gadhyan, and R. Glowinski ...


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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 ...


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I would say Take log of first equation to get rid of dependence on $x_t$ Apply Kalman filter equations to estimate parameters I believe Conrad and Kaul (1988) J of Business do exactly what you describe.


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The main difference is that one approach assumes that a certain dynamical structure properly describes the underlying instrument, while the other approach is really only a re-writing of the price in terms of an implied volatility. Implied volatility Implied volatility really only needs two things: the underlying stock price and the call option price (apart ...


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The typical approach is: you only use option data from the last day. Furthermore, you only include those points that are liquid enough. One approach to this is to weigh the modelling error of an option by its bid-ask spread and vega. Using data from multiple days is not a good approach, because you might have options with the same strike but different ...


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The Heston model is represented by the bivariate system of stochastic differential equations (SDE) \begin{align} & d{{S}_{t}}=rS_tdt+{\sqrt\upsilon_t} d{{W}_{1}}(t) \\ & d{{\upsilon}_{t}}=\kappa(\theta-\upsilon_t) dt+\sigma{\sqrt\upsilon_t}d{{W}_{2}}(t) \\ \end{align} The most popular way to estimate the parameters of the Heston model is with loss ...


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The most used equity volatility models in the industry are the Black-Scholes model (including its time dependent version) and the local volatility model. It always come along with stochastic rates, discrete dividends and quanto effects (a must-have when pricing even simple payoffs) so the calibration/pricing process is much more involved than what you might ...


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You should not use the Feller condition as a constraint. In many cases its violation will be required for a good fit to the market data.


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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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I will just answer your first question as I do not know the details of SSVI. Total variance is more intrinsic than volatility. The BS formula can be rewritten in terms of 3 parameters: the log-strike (log-moneyness would be more accurate) $k$, the total variance $w$ and the discount factor. Volatility never appears without a $\sqrt{T}$. It is just there ...


1

Yes, you are correct. Consider the following toy example: 1) Log prices follow: $dp_t=\mu dt+\sigma dW_t$ 2) Then: $r_{t+h,h}=p_{r+h,h}-p_t ~ N(\mu h, \sigma^2 h)$ 3) standard ML estimators: $\hat{\mu}=\frac{1}{nh}\sum_{k=1} r_{kh,h}$ $\hat{\sigma^2}=\frac{1}{nh}\sum_{k=1} (r_{kh,h}-\hat{\mu}h)^2$ Assymptotic distribution of estimators: $\sqrt ...


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This is related to a common misconception called the Time Diversification fallacy. Returns do not average out over longer periods of time. On the contrary, as emcor points out, the variance increases. You might find this article named Risk and Time by John Norstad interesting.


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You generated only one realization of the GBM. The variance of a GBM increases with time and so you must generate more realizations to get accurate estimates. Here see a sample of Brownian simluations: http://tex.stackexchange.com/questions/59926/how-to-draw-brownian-motions-in-tikz-pgf


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This is pretty straight forward: The market prices vanilla options via implied volatility. You can like it or not like it but that is the way it is. So, the fair price of the option is the equivalent of the implied vol via BS. Now, if you believe the true price of an option should be different from the traded market price and you figure out that you have ...


1

It depends what you want volatility for. Theory will tell you that: "Implied variance of short maturity ATM options is approximately equal to the expectation of the realised integrated variance of the underlying over the life of the option and under the risk neutral measure" In math: $\sigma^2_{ATM}\approx E^Q\left(\frac{1}{T}\int_0^T\sigma^2_t dt\right)$ ...


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Thank you guys. Sorry for the late reply, I just solved it in matlab using maximum likelihood estimation. Turns out that all we need to do is to specify a state space model, then estimate the coefficient using MLE. The linearity and normality here makes things pretty simple.


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It depends on the use of your model as pointed out in the comments. If a discretized version is sufficient then state space models could be a solution. You can check out the free online textbook by Athana­sopou­los and Hyndman. State space model describe time series in terms of level/trend (and seasonality) on an additive or multiplicative way. There are ...


1

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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