# Tag Info

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If you want to calibrate on time series, then you have a 'non linear filtering' problem, since volatility is latent. There have been papers from late 90s/ early 00s that do that: Google for Heston together with Ghysels, Gallant, Renault, Chernov, Tauchen, Pan, Bates, Shephard, MCMC, unscented Kalman filter/ particle filter. Given the significant complexity ...

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You might want to set $a= \epsilon - d$ and write $\epsilon>0$ as a constraint. I guess $\textbf{lsqnonlin}$ is the suitable fonction for what you intend to do. I personnally like to use and play around with $\textbf{fmincon}$, which allows more flexibility and performs well, if you are willing to provide Jacobian and/or Hessian in algorithms options

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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 \... 2 For short maturity SPX option chain, the analytic form of the V-shape volatility smile has been fully worked out in my latest paper on SSRN. You can take a look. 2 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. 2 I'm giving no assurance that this model is rigorous/functional. It also appears that time steps are severely limited. In general, though, the only way to ensure that something is created well is to create it yourself. I have been burned by canned functions/models in the past, so I avoid them whenever able or if I'm doing anything that is actually important.... 2 Given the Ho-Lee interest rate model of the form \begin{align*} dr_t = \theta_t dt + \sigma dW_t, \end{align*} the price at timet>0$of a zero-coupon bond, with maturity$Tand unit face, has the form \begin{align*} B(t, T) &=E\Big(e^{-\int_t^T r_s ds} \mid r_t \Big)\\ &=e^{-(T-t)r_t - \int_t^T (T-u)\theta_u du + \frac{\sigma^2}{6}(T-t)^3}. \... 1 Suppose we have a set ofN_T$maturities$\tau_t$and a set of$N_k$strikes$K_k$.For each maturity-strike combination$(\tau_t,K_k)$we have a market price (for example)$Caplet(\tau_t,K_k)=C_{tk}$and a corresponding model price$Caplet(\tau_t,K_k,\Lambda)=C^\Lambda_{tk}$in which$\Lambda$is Hull-Whit's Parameters. The first category minimize the ... 1 Actually, I want to have the calibration model to calibrate parameter such as "a" and "sig" based on swaption volatilities and market price of swaption. For the trinomial model, I can manage to implement it. 1 Market practitioners do the following: Correlation is calibrated most often by looking at historical correlations between liquid par swap rate pairs. One could look at implied correlations within options on the yield curve (eg 10 yr minus 2yr) also. Swaption calibration should be done by comparing straddle prices in the market to prices produced by the ... 1 Different methods exists to compute implied vol from the same option prices, eventually it's prices that matters to calibration. But if you can reproduce same option prices accurate to the cent by fitting implied vol, I think it doesn't matter. 1 I will just answer your first question as I do not know the details of SSVI. Total variance is more intrinsic than volatility. The BS formula can be rewritten in terms of 3 parameters: the log-strike (log-moneyness would be more accurate)$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 T(\hat{...

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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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