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 functions. This method uses the error between quoted market prices and model prices, or between market and model implied volatilities. The parameter estimates $\hat\Theta=(\hat\kappa, \hat\theta,\hat\sigma,\hat\upsilon_o,\hat\rho)$ are those values which minimize the value of the loss function, so that the model prices or implied volatilities are as close as possible to their market counterparts. A constrained minimization algorithm must be used in this regard so that the constraints on the parameters \begin{align} \kappa>0\ ,\ \theta>0\ , \ \sigma>0 \, \ ,\upsilon_0>0 \, \ , \,\rho\in[-1,1] \end{align} are respected. Since loss functions use market option prices (or implied volatility derived from those prices) as inputs, they produce estimates of the risk-neutral parameters of the Heston model.Suppose we have a set of $N_T $ maturities $\tau_i$($ i=1,2,...,N_T)$ and set $N_K$ strikes $K_k$ ($k=1,2,...,N_k$).For each maturity For each maturity-strike combination $(\tau_t,\ K_k)$ we have market price $P(\tau_t , K_k)$ and a corresponding model price $P(\tau_t , K_k;\Theta)=P_{t,k}^{\Theta}$ generated by the Heston model. Attached to each option is an optional weight $w_{t,k}$ . There are many possible ways to define a loss function, but they usually fall into one of two categories: those based on prices, and those based on implied volatilities. The first category of loss functions are those that minimize the error between quoted and model prices. The error is usually defined as the squared difference between the quoted and model prices, or the absolute value of the difference; relative errors can also be used. For example, parameter estimates obtained using the mean error sum of squares (MSE) loss function are obtained by minimizing \begin{align} \frac{1}{N}\sum_{t,k}w_{t,k}(P_{t,k}-P_{t,k}^{\Theta})^2 \end{align} with respect to $\Theta$ where $N$ is the number of quotes. The relative mean error sum of squares (RMSE) parameter estimates are obtained with the loss function \begin{align} \frac{1}{N}\sum_{t,k}w_{t,k}\frac{(P_{t,k}-P_{t,k}^{\Theta})^2}{P_{t,k}} \end{align} The second category of loss functions are those that minimize the error between quoted and model implied volatilities. Again, the error is usually defined as the squared difference, absolute difference, or relative difference, between quoted and model implied volatilities. This category of loss function is sensible, since options are often quoted in terms of implied volatility, and since the fit of model is often assessed by comparing quoted and model implied volatilities. Hence, for example, the implied volatility mean error sum of squares (IVMSE) parameter estimates are based on the loss function \begin{align} \frac{1}{N}\sum_{t,k}w_{t,k}(IV_{t,k}-IV_{t,k}^{\Theta})^2 \end{align} where $IV_{t,k}$ and $IV_{t,k}^\Theta$ are are the quoted and model implied volatilities, respectively. The relative and absolute versions can also be used.Estimation of the Heston model parameters by loss functions has been used by Bakshi, Cao, and Chen (1997), Bams et al. (2009), Christoffersen and Jacobs (2004), Mikhailov and No¨ gel, (2003), and many others. There is no consensus on which loss function is the best one, but Christoffersen and Jacobs (2004) point out that the same loss function should be used for parameter estimation and for evaluating model fit.
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 functions. This method uses the error between quoted market prices and model prices, or between market and model implied volatilities. The parameter estimates $\hat\Theta=(\hat\kappa, \hat\theta,\hat\sigma,\hat\upsilon_o,\hat\rho)$ are those values which minimize the value of the loss function, so that the model prices or implied volatilities are as close as possible to their market counterparts. A constrained minimization algorithm must be used in this regard so that the constraints on the parameters \begin{align} \kappa>0\ ,\ \theta>0\ , \ \sigma>0 \, \ ,\upsilon_0>0 \, \ , \,\rho\in[-1,1] \end{align} are respected. Since loss functions use market option prices (or implied volatility derived from those prices) as inputs, they produce estimates of the risk-neutral parameters of the Heston model.Suppose we have a set of $N_T $ maturities $\tau_i$($ i=1,2,...,N_T)$ and set $N_K$ strikes $K_k$ ($k=1,2,...,N_k$).For each maturity For each maturity-strike combination $(\tau_t,\ K_k)$ we have market price $P(\tau_t , K_k)$ and a corresponding model price $P(\tau_t , K_k;\Theta)=P_{t,k}^{\Theta}$ generated by the Heston model. Attached to each option is an optional weight $w_{t,k}$ . There are many possible ways to define a loss function, but they usually fall into one of two categories: those based on prices, and those based on implied volatilities. The first category of loss functions are those that minimize the error between quoted and model prices. The error is usually defined as the squared difference between the quoted and model prices, or the absolute value of the difference; relative errors can also be used. For example, parameter estimates obtained using the mean error sum of squares (MSE) loss function are obtained by minimizing \begin{align} \frac{1}{N}\sum_{t,k}w_{t,k}(P_{t,k}-P_{t,k}^{\Theta})^2 \end{align} with respect to $\Theta$ where $N$ is the number of quotes. The relative mean error sum of squares (RMSE) parameter estimates are obtained with the loss function \begin{align} \frac{1}{N}\sum_{t,k}w_{t,k}\frac{(P_{t,k}-P_{t,k}^{\Theta})^2}{P_{t,k}} \end{align}
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 functions. This method uses the error between quoted market prices and model prices, or between market and model implied volatilities. The parameter estimates $\hat\Theta=(\hat\kappa, \hat\theta,\hat\sigma,\hat\upsilon_o,\hat\rho)$ are those values which minimize the value of the loss function, so that the model prices or implied volatilities are as close as possible to their market counterparts. A constrained minimization algorithm must be used in this regard so that the constraints on the parameters \begin{align} \kappa>0\ ,\ \theta>0\ , \ \sigma>0 \, \ ,\upsilon_0>0 \, \ , \,\rho\in[-1,1] \end{align} are respected. Since loss functions use market option prices (or implied volatility derived from those prices) as inputs, they produce estimates of the risk-neutral parameters of the Heston model.Suppose we have a set of $N_T $ maturities $\tau_i$($ i=1,2,...,N_T)$ and set $N_K$ strikes $K_k$ ($k=1,2,...,N_k$).For each maturity For each maturity-strike combination $(\tau_t,\ K_k)$ we have market price $P(\tau_t , K_k)$ and a corresponding model price $P(\tau_t , K_k;\Theta)=P_{t,k}^{\Theta}$ generated by the Heston model. Attached to each option is an optional weight $w_{t,k}$ . There are many possible ways to define a loss function, but they usually fall into one of two categories: those based on prices, and those based on implied volatilities. The first category of loss functions are those that minimize the error between quoted and model prices. The error is usually defined as the squared difference between the quoted and model prices, or the absolute value of the difference; relative errors can also be used. For example, parameter estimates obtained using the mean error sum of squares (MSE) loss function are obtained by minimizing \begin{align} \frac{1}{N}\sum_{t,k}w_{t,k}(P_{t,k}-P_{t,k}^{\Theta})^2 \end{align} with respect to $\Theta$ where $N$ is the number of quotes. The relative mean error sum of squares (RMSE) parameter estimates are obtained with the loss function \begin{align} \frac{1}{N}\sum_{t,k}w_{t,k}\frac{(P_{t,k}-P_{t,k}^{\Theta})^2}{P_{t,k}} \end{align} The second category of loss functions are those that minimize the error between quoted and model implied volatilities. Again, the error is usually defined as the squared difference, absolute difference, or relative difference, between quoted and model implied volatilities. This category of loss function is sensible, since options are often quoted in terms of implied volatility, and since the fit of model is often assessed by comparing quoted and model implied volatilities. Hence, for example, the implied volatility mean error sum of squares (IVMSE) parameter estimates are based on the loss function \begin{align} \frac{1}{N}\sum_{t,k}w_{t,k}(IV_{t,k}-IV_{t,k}^{\Theta})^2 \end{align} where $IV_{t,k}$ and $IV_{t,k}^\Theta$ are are the quoted and model implied volatilities, respectively. The relative and absolute versions can also be used.Estimation of the Heston model parameters by loss functions has been used by Bakshi, Cao, and Chen (1997), Bams et al. (2009), Christoffersen and Jacobs (2004), Mikhailov and No¨ gel, (2003), and many others. There is no consensus on which loss function is the best one, but Christoffersen and Jacobs (2004) point out that the same loss function should be used for parameter estimation and for evaluating model fit.
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 functions. This method uses the error between quoted market prices and model prices, or between market and model implied volatilities. The parameter estimates $\hat\Theta=(\hat\kappa, \hat\theta,\hat\sigma,\hat\upsilon_o,\hat\rho)$ are those values which minimize the value of the loss function, so that the model prices or implied volatilities are as close as possible to their market counterparts. A constrained minimization algorithm must be used in this regard so that the constraints on the parameters \begin{align} \kappa>0\ ,\ \theta>0\ , \ \sigma>0 \, \ ,\upsilon_0>0 \, \ , \,\rho\in[-1,1] \end{align} are respected. Since loss functions use market option prices (or implied volatility derived from those prices) as inputs, they produce estimates of the risk-neutral parameters of the Heston model.Suppose we have a set of $N_T $ maturities $\tau_i$($ i=1,2,...,N_T)$ and set $N_K$ strikes $K_k$ ($k=1,2,...,N_k$).For each maturity For each maturity-strike combination $(\tau_t,\ K_k)$ we have market price $P(\tau_t , K_k)$ and a corresponding model price $P(\tau_t , K_k;\Theta)=P_{t,k}^{\Theta}$ generated by the Heston model. Attached to each option is an optional weight $w_{t,k}$ . There are many possible ways to define a loss function, but they usually fall into one of two categories: those based on prices, and those based on implied volatilities. The first category of loss functions are those that minimize the error between quoted and model prices. The error is usually defined as the squared difference between the quoted and model prices, or the absolute value of the difference; relative errors can also be used. For example, parameter estimates obtained using the mean error sum of squares (MSE) loss function are obtained by minimizing \begin{align} \frac{1}{N}\sum_{t,k}w_{t,k}(P_{t,k}-P_{t,k}^{\Theta})^2 \end{align} with respect to $\Theta$ where $N$ is the number of quotes. The relative mean error sum of squares (RMSE) parameter estimates are obtained with the loss function \begin{align} \frac{1}{N}\sum_{t,k}w_{t,k}\frac{(P_{t,k}-P_{t,k}^{\Theta})^2}{P_{t,k}} \end{align}