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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.

Certificates are structured financial instruments that aim to provide investors with investment solutions tailored to their needs. Certificates can be modeled using a bond component and a derivative component, typically an options strategy. The pricing of certificates is typically performed using the Monte Carlo numerical methodology. Such method allows for projections of the underlying using series of random numbers. The results obtained display an error (standard deviation) that depends on the number of simulations used and on the specific characteristics of the structured product. This work has the objective of minimizing the experimental error, and, consequently, of accelerating the speed of convergence using statistical techniques known in the literature as variance reduction methods. The most popular stochastic dynamics have been analyzed, like the classical Black and Scholes model, the Local Volatility model and the Heston model. Three certificates are analyzed in the paper and they are characterized by different payoffs. The variance reduction techniques, implemented in different programming languages (Python, Matlab and R), are: Latin Hypercube, Stratified Sampling, Antithetic Variables, Importance Sampling, Moment Matching and Control Variates.

The Monte Carlo method, thanks to its flexibility in designing even extremely complex payoffs, is assuming an increasingly important role in quantitative analysis. Its main limitation is the high computational cost linked to its modest speed of convergence to the fair value of the product. One of the best-known statistical techniques is to replace the random number generator with "low discrepancy" deterministic numerical sequences, producing a Quasi-Monte Carlo. Through its implementation for the analysis of three investment certificates featuring different characteristics and different stochastic processes used for the underlying simulation, the study demonstrates the possibility of achieving interesting results in terms of performance even for pricing these structured products ever more popular in the financial industry. [ABSTRACT FROM AUTHOR]