Analysis of numerical integration schemes for the Heston model: a case study based on the pricing of investment certificates

The Heston model is one of the most used techniques for estimating the fair value and the risk measures associated with investment certificates. Typically, the pricing engine implements a significant number of projections of the underlying until maturity, it calculates the pay-off for all the paths thus simulated considering the characteristics of the structured product and, in accordance with the Monte Carlo methodology, it determines its theoretical value by calculating its mean and discounting it at valuation time. In order to generate the future paths, the two stochastic differential equations governing the dynamics of the Heston model should be integrated simultaneously over time: both the one directly associated with the underlying and the one associated with variance. Consequently, it is essential to implement a numerical integration scheme that allows such prospective simulations to be implemented. The present study aims to consider alternatives to the traditional Euler method with the aim of reducing or in some cases eliminating the probability of incurring unfeasible simulated values for the variance. In fact, one of the main drawbacks of the Euler basic integration scheme applied to the Heston bivariate stochastic model is that of potentially generating negative variances in the simulation that should be programmatically corrected each time such undesired effect occurs. The methods which do not intrinsically admit the generation of negative values of the variance proved to be very interesting, in particular the Transformed Volatility scheme.