Monte Carlo methods: theory, algorithms and applications to selected financial problems

There are many mathematical problems in statistics, economy, finance, biology, physics etc. which can be solved only in an approximate way because of their complexity. Even such “simple” problem as calculation of value of the onedimensional integral is in many cases too difficult to find the relevant solution in purely analytical form. Two groups of general approaches to such challenges are commonly used. The first group consists of strictly deterministic algorithms, like the widely known Newton’s method, which finds roots of a real-valued function. These methods have many advantages, especially their fast convergence in many cases should be emphasized. However, they share also one common disadvantage – the necessity of fulfilling various assumptions before any such method could be even applied. For example, in the case of Newton’s method, derivative for the considered function should exist. The second group of methods consists of algorithms based on numerical simulations, and this group is related to random approach. Simulations are especially useful nowadays in our computer era, when relatively cheap and fast hardware and commonly accessible, special-tailored software can be easily used to solve many important practical and theoretical problems in minutes or even in seconds. In Chapter 1 we discuss some advantages and disadvantages of such numerical methods. Chapter 2 contains some basic definitions, facts and theorems of probability theory and stochastic analysis in continuous time, needed in the chapters that follow. These elements of theory are used for description of simulations algorithms. They are also very useful in Chapter 7, where methods of stochastic analysis are applied to derive and prove the valuation formula of the there described financial instrument. In that chapter there are also many references, which can help the interested readers to familiarize themselves with further details concerning defined notions. Of course, in order to undertake any random simulations, the efficient way to obtain abundant quantity of random values is necessary. These values could be acquired as an effect of some physical random event or derived directly as an output from special computer algorithm. The physical events mentioned are related to the so called hardware random number generators. A simple example of such a generator is given by tossing a coin. However, practitioners rather rely on specially devised computer algorithms, because of the previously mentioned availability of hardware and software. But, as stated by John von Neumann, “Anyone who considers arithmetical methods of producing random digits is, of course, in a state of sin.” It is, namely, so that, such special computer algorithm, known as (pseudo)random number generator, gives purely deterministic series, which only resembles (in some statistical sense) independent, identically distributed sample of random variables. In Chapter 1, the difference between the hardware and the software random number generators are considered in a more detailed way. We also discuss important problems concerning treating deterministic values as random variables. Especially, the statistical approach, based on the sets of various tests, is emphasized. There are many kinds of simulation algorithms. Some of them are related to transformations of variables between various random distributions known in statistics, like the method to change some variable, which is distributed according to the uniform distribution into the related value from normal distribution. There are also more sophisticated approaches, which are intended to solve real, complex problems, like the previously mentioned issue of the evaluation of integrals. In this case, especially Monte Carlo (MC) or Markov Chain Monte Carlo (MCMC) methods should be emphasized. The first one is based on iid sample, the second one is related to mathematical concept of Markov chains. The necessary theoretical background concerning Markov chains is introduced in Chapter 3. In Chapter 4 useful algorithms, various examples and necessary theorems, concerning both the generation of random variables from different distributions and the sampling of trajectories of some stochastic processes are provided. Also the so called curse of dimensionality problem, which arises in multidimensional settings, is discussed in a more detailed way. In turn, Chapter 5 is devoted to detailed considerations of MC methods. As previously, practical algorithms, introductory examples and theoretical background for this simulation approach are provided. The same concerns Chapter 6, where MCMC methods are discussed. Moreover, special attention is paid to the significant problem of convergence diagnosis for MCMC methods. Some possible answers to the question when the simulations should be finished are considered there. Simulation methods are used in order to solve important problems from various scientific and practical areas, like e.g. statistics, physics, biology etc. In this book, though, we focus on applying Monte Carlo methods in financial mathematics. In Chapter 7, we consider pricing of a new kind of financial instrument, which is known as catastrophe bond (or cat bond ). Nowadays, catastrophic events like hurricanes, floods and earthquakes are serious problems for the reserves of many insurers. The losses from such catastrophes can even lead to bankruptcy of insurance enterprises. Catastrophe bond is one of the solutions to deal with these problems, as this derivative is used to transfer risk from the insurer onto the financial markets. We derive and prove a generalized version of the catastrophe bond pricing formula. Applying stochastic processes in continuous time, we propose theoretical models of catastrophe losses and risk-free interest rates. Moreover, we assume a general form of cat bond payoff function. Using methods of stochastic analysis and financial mathematics, we derive the instrument’s valuation expression, which can be applied for various types of cat bond payoff functions and affine interest rate models. Because of complex pricing formulas developed in Chapter 7, simulations are necessary to estimate the relevant expected values. Numerical analysis of cat bond prices, based on Monte Carlo methods, is presented in Chapter 8. Moreover, Monte Carlo methods are also used to solve the important problem of probability of bankruptcy of the insurer in this part of the book. In order to do this, the portfolio of the insurer, which consists of a few layers is modelled there. As these layers we apply the classical risk process based on the insurance premiums, the reinsurance contract and the payments related to the specially tailored catastrophe bond. Additionally, one-factor Vasicek model of the interest rate is embedded in the structure of such portfolio. Then, by applying numerical simulations of catastrophic events calibrated to real-life data, the stability of the constructed portfolio is evaluated. Using methods of descriptive statistics, outputs of various scenarios based on simulations are analysed. In particular, probabilities of bankruptcy for some interesting cases are estimated, as the key factors for the insurer during the process of construction of the related portfolio.