A standard problem in mathematical finance is the calculation of the price of some financial derivative such as various types of options. Since there exists analytical solutions in only a few cases it will often boil down to estimating the price with Monte Carlo simulation in conjunction with some numerical discretization scheme. The upside of using what we can call standard Monte Carlo is that it is relative straightforward to apply and can be used for a wide variety of problems. The downside is that it has a relatively slow convergence which means that the computational cost or complexity can be very large. However, this slow convergence can be improved upon by using Multilevel Monte Carlo instead of standard Monte Carlo. With this approach it is possible to reduce the computational complexity and cost of simulation considerably. The aim of this thesis is to introduce the reader to the Multilevel Monte Carlo method with applications to European and Asian call options in both the Black-Scholes-Merton (BSM) model and in the Heston model. To this end we first cover the necessary background material such as basic probability theory, estimators and some of their properties, the stochastic integral, stochastic processes and Ito’s theorem. We introduce stochastic differential equations and two numerical discretizations schemes, the Euler–Maruyama scheme and the Milstein scheme. We define strong and weak convergence and illustrate these concepts with examples. We also describe the standard Monte Carlo method and then the theory and implementation of Multilevel Monte Carlo. In the applications part we perform numerical experiments where we compare standard Monte Carlo to Multilevel Monte Carlo in conjunction with the Euler–Maruyama scheme and Milsteins scheme. In the case of a European call in the BSM model, using the Euler–Maruyama scheme, we achieved a cost O(ε-2(log ε)2) to reach the desired error in accordance with theory in comparison to the O(ε-3) cost for standard Monte Carlo. When using Milsteins scheme instead of the Euler–Maruyama scheme it was possible to reduce the cost in terms of the number of simulations needed to achieve the desired error even further. By using Milsteins scheme, a method with greater order of strong convergence than Euler–Maruyama, we achieved the O(ε-2) cost predicted by the complexity theorem compared to the standard Monte Carlo cost of order O(ε-3). In the final numerical experiment we applied the Multilevel Monte Carlo method together with the Euler–Maruyama scheme to an Asian call in the Heston model. In this case, where the coefficients of the Heston model do not satisfy a global Lipschitz condition, the study of strong or weak convergence is much harder. The numerical experiments suggested that the strong convergence was slightly slower compared to what was found in the case of a European call in the BSM model. Nevertheless, we still achieved substantial savings in computational cost compared to using standard Monte Carlo.

Many phenomena in our world can be described as differential equations in high dimensions. However, they are notoriously challenging to solve numerically due to the exponential growth in computational cost with increasing dimensions. This thesis explores an algorithm, known as deep BSDE, for solving high-dimensional partial differential equations and applies it to finance, namely European option pricing. In addition, an implementation of the method is provided that seemingly shortens the runtime by a factor of two, compared with the results in previous studies. From the results, we can conclude that the deep BSDE method does handle high-dimensional problems well. Lastly, the thesis gives the relevant prerequisites required to be able to digest the theory from an undergraduate level.

This thesis investigated Stochastic Runge–Kutta Lawson (SRKL) schemes and their application to the Heston model. Two distinct SRKL discretization methods were used to simulate a single asset’s dynamics under the Heston model, notably the Euler–Maruyama and Midpoint schemes. Additionally, standard Monte Carlo and variance reduction techniques were implemented. European and Asian option prices were estimated and compared with a benchmark value regarding accuracy, effectiveness, and computational complexity. Findings showed that the SRKL Euler–Maruyama schemes exhibited promise in enhancing the price for simple and path-dependent options. Consequently, integrating SRKL numerical methods into option valuation provides notable advantages by addressing challenges posed by the Heston model’s SDEs. Given the limited scope of this research topic, it is imperative to conduct further studies to understand the use of SRKL schemes within other models.