# Tag Info

4

You don't need to use the Sobol sequence to generate quasi-random numbers in MATLAB. We know the Heston model is represented by the bi-variate 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) ...

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

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 ...

3

I do not know what are the exact steps you followed, but here's my 2 cents: \begin{align*} C(S_t,t) &= P(t,T) E^Q [ (S_T - K)^+ ] \\ & = P(t,T) E^Q [ S_T 1_{S_T \geq K} ] - K P(t,T) E^Q [ 1_{S_T \geq K} ] \end{align*} Which can be re-written (using a change of numéraire) \begin{align*} C(S_t,t) &= S_t E^{Q^S} [ 1_{S_T \geq K} ] - K P(t,T) E^Q ...

3

V(t) is the variance process of the stock price, not volatility process. Cox-Ingersoll-Ross demonstrated that that specific process can be non-negative under certain conditions, which is what you want for variance.

3

In the Heston Model we have \begin{align} C(t\,,{{S}_{t}},{{v}_{t}},K,T)={{S}_{t}}{{P}_{1}}-K\,{{e}^{-r\tau }}{{P}_{2}} \end{align} where,for $j=1,2$ \begin{align} & {{P}_{j}}({{x}_{t}}\,,\,{{v}_{t}}\,;\,\,{{x}_{T}},\ln K)=\frac{1}{2}+\frac{1}{\pi }\int\limits_{0}^{\infty }{\operatorname{Re}\left( \frac{{{e}^{-i\phi \ln K}}{{f}_{j}}(\phi ;t,x,v)}{i\phi ...

2

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 ...

2

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 ...

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

In options pricing language, the probability of a spot process being above a given level $K$ at time $T$ is the undiscounted price of a digital call option on that spot process. In the Heston model, there is an analytic expression for this in terms of Fourier transform. You can find this in various standard references, e.g. Alan Lewis's book "Option ...

1

In this paper http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2626552 the authors compare the Heston model with volatility given by $dV_t = \kappa_V(\bar{V}-V_t)dt+\sigma_V\sqrt{V_t}dW_t$ with the a model where the volatiltiy is given by $dV_t = \kappa_V(\bar{V}-V_t)dt+\sigma_VV_tdW_t$. They show that the latter is inverse gamma distributed and ...

1

The reason is that Heston managed to solve the case with square root. The log-normal vol process leads to nasty properties. The 3/2 model is another case that have been solved.

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

COS method is an efficient way to recover the distribution function from the characteristic function in the Heston model. For other methods, you may refer to "Inverting Analytic Characteristic Functions and Financial Applications".

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