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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 ...


3

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 ...


3

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.


2

Time is expressed in fractions of year in the GBM formula. Therefore, $T=1$ year and $\Delta t = 1/m$. Considered that you have $253$ observations, I would use $m = 253$, so the second option as Drew suggested. In general, using 253 or 365 days in a year depends on how you consider reality: do you think that when markets are closed (i.e. weekends) the price ...


2

The second one will be the best estimate. Also, a smaller timestep usually corresponds to a smaller bias. But I agree, the answer is not obvious. You should be careful about increasing $T$ though, because for negative drifts there is a threshold value ($2\mu + \sigma^2 < 0$) beyond which the variance of the price process stops increasing. It's an ...


2

I would suggest you to add spreads to the implied hazard rates, spreads that you regress on the macroeconomic factors. Then you stress by calculating the spreads corresponding to the stressed factors.


2

It seems that implicitly you have a multi-objective optimization in mind, hence of course it may happen that you are not able to achieve all the objectives simultaneously. Let's say that output of a more general model is $f(x,y)$ so that the output of the first model is $f(x,0) = f_0(x)$. Denoting market prices by $m_k$ which in your case means $m_1 = A$ and ...


2

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 ...


1

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.


1

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


1

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: $\sigma^2_{ATM}\approx E^Q\left(\frac{1}{T}\int_0^T\sigma^2_t dt\right)$ ...


1

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 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 ...


1

Given that you have swap rates and Cap prices (ATM, I assume), you can back out the IVs for the time periods using by bootstrapping. Strictly speaking, you would need Caplet prices for the given strikes. In such a case, You would look at the shortest dated cap and (assume) it is made up of only one caplet. You can then use black's formula and back out ...


1

Thanks to my research leader, I found what I missed. $V_{0,1}$ is vol of swaption that matures at $T_0$ which is not 0 (as I thought), rather it is maturity of the first libor. So $V_{0,1}$ is the closest available point on market. And now this is all clear with table on page 323 in section 7.4. $V_{0,2}$ is realy vol of swaption that matures at $T_0$=1y ...



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