Implementation of variance reduction techniques applied to the pricing of investment certificates

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

Pricing American options - aspects of computation

An American option is a type of option that can be exercised at any time up to its expiration. American options are generally hard to value, as there is no closed-form solution for the price of an American option. When there are multiple stochastic factors in the equation, the usual solution methods – binomial trees and finite difference approaches – become infeasible. Therefore, only estimators based on Monte Carlo simulation can provide good quality results. The Least-square Monte Carlo method (LSM) is the most widely used Monte Carlo-based algorithm in the financial industry. In this thesis, the LSM algorithm and associated literature are reviewed and analysed. The first major contribution is the identification of the basic powers polynomial of 4th order as the most efficient basis polynomial for the least-squares regression within the LSM simulation. The conclusion is also drawn that the performance of LSM depends on both the number of time-steps and the number of simulated paths. Another significant finding in this thesis is that, for every option being valued with a predetermined number of paths, an 'optimal' number of time-steps exists for which the estimator's mean is closest to the exact value of the option. It is proved that, in the case of the LSM algorithm, the general belief that Monte Carlo simulations become more and more efficient with the increase in the number of iterations within the simulation does not necessarily hold. The proposed Average of Batch of LSM Estimates (ABO-LSME) approach calculates the average of multiple optimal LSM estimates within the same or less time than needed for the original LSM estimate and, surprisingly, yields more precise results than the original LSM approach. The basis of the newly introduced Bundled LSM (BLSM) algorithm is an LSM algorithm in which all of the in-the-money paths at each time-step are sorted (similar to Tilley's bundling algorithm, except only in-the-money paths are sorted) and divided into a predetermined number of bundles, to which separate least-squares regressions are applied. This method provides much more stable and precise results than the original LSM algorithm. When optimal BLSM is compared to the optimal LSM algorithm, the superiority of the BLSM estimator becomes clear. BLSM provides results with lower relative errors and RMSEs, around two times faster than optimal LSM