Parameter Estimation

All of the pricing methodologies we have covered have assumed the Heston model parameters to be given. In this chapter, we describe how to estimate these parameters. We first present the most common estimation method, the loss function approach, which minimizes the difference between quoted and model option prices. In this approach, parameters are selected so that the quoted option prices are as close as possible to the model option prices. Alternatively, quoted and model implied volatilities can be used instead of prices. Next, we summarize the Nelder and Mead (1965) minimization algorithm and we show how to code it in C#. Then we describe the “Smart Parameter” method of Gauthier and Rivaille (2009) to select starting values and the Strike‐Vector Computation of Kilin (2007), which constructs the loss function in a way that greatly speeds up the estimation. We then present the Differential Evolution algorithm, which has been shown by Vollrath and Wendland (2009) to be effective in the Heston model. Finally, we present a method due to Atiya and Wall (2009) to obtain maximum likelihood estimates of the Heston model parameters. Throughout this chapter, the Heston parameters are represented as the vector Θ = (κ,θ,σ,v 0 ,ρ), and their corresponding estimates, as Θ.