Fourier Methods for Multidimensional Problems and Backward SDEs in Finance and Economics

Abstract

In this thesis we deal with processes with uncertainties, such as financial asset prices and the global temperature. We model their evolutions by so-called stochastic processes. Many of these stochastic processes are based on the Wiener process, whose increments are normally distributed. Other models may contain jump components, to model, for example, economic disasters or degradation failures. An important class of models is the Levy class, where successive increments are independent and statistically identical over different time intervals of the same length. This may give computational advantages. A well-known application of stochastic processes is in financial mathematics, where the goal is to price financial derivatives or to estimate risk measures. The underlying asset prices may be modeled by, e.g., geometric Brownian motions. More involved models, like the Variance Gamma process, are defined by jumps. In other, for instance economic, personal, or societal, contexts one may face options in the sense of real `choices'. For example, should one build a new factory now or in the future? Or should one heighten a dike today, and by how much, or in the future? These decisions are called real options and can often be related to financial options. Similar methods can be used to value them. The numerical problems we consider deal with conditional expectations. Often these problems can be connected to a partial differential equation (PDE) by a Feynman-Kac theorem. Then we can apply PDE methods, such as finite difference schemes and finite volume methods, to approximate the solutions. From the perspective of the probabilistic representation, the class of Monte Carlo methods can be beneficial for high-dimensional problems. They are based on simulated paths of the stochastic process. Besides, the expected value can often be represented by an integral, which offers opportunities to use numerical integration techniques, such as Newton-Cotes formulas. We focus on a subclass of numerical integration methods, i.e., Fourier-based methods. These `transform methods' combine a transformation to the Fourier domain with numerical integration. The probability density function of the random variables of interest is usually unknown. Its Fourier transform, i.e., the characteristic function, is however often known and can be used to approximate the corresponding density and distribution function. Our method of choice is the COS method, which is based on Fourier cosine series expansions and the characteristic function. The matrix-vector product appearing may be computed in an efficient way by using a Fast Fourier Transform (FFT) algorithm, especially when dealing with Levy processes. Besides, the use of the discrete Fourier cosine transform helps us with the approximation of the Fourier coefficients. After a general introduction of this thesis in Chapter 1, in Chapter 2 we explain the COS formula to compute conditional expectations and provide an error analysis. Then the COS method is applied to a specific class of problems: stochastic control problems, in which an agent has the possibility to affect the trend or variation of a stochastic process in such a way that his target function is maximized. For example, he can determine his consumption or savings rate. With the COS method we approximate functions by using Fourier cosine series. Similar to Fourier series they may suffer from the Gibbs phenomenon: trying to recover a function with a jump discontinuity results in undesired oscillations, even if the number of terms in the series goes to infinity. Smooth density functions give rise to a fast exponentially converging error of the COS method, but a density function with a discontinuity in one of its derivatives results in slower algebraic convergence. This is related to the Gibbs phenomenon. A remedy to improve this is by using spectral filters, which smoothen the approximations, see Chapter 3. Vanilla call and put options are based on one underlying asset. On the other hand, rainbow options are written on multiple assets and the holder may possess a `basket' of assets. The payoff of, for example, a call-on-max option, depends on the maximum of several assets. In this way, an option holder can manage his risks. Financial options on two assets are discussed in Chapter 4. Also single-asset options under the Heston model, in which an additional stochastic process for the volatility is used, are priced by the so-called 2D-COS method, developed in this chapter. Chapter 5 deals with a problem from the field of climate change economics. The future global temperature is highly uncertain and there are different damage estimates. In our model an agent can choose the consumption level of his wealth while he is subject to these uncertain climate damages. There is a trade-off between consuming now and saving for later. Economic equilibrium conditions result in a mathematical expression for the appropriate social discount rate. Here the future temperature process and the economic wealth are the uncertain processes and we combine the methods from Chapter 2 and Chapter 4 to solve the problem. Forward stochastic processes are rather well-known. Their initial value is prescribed and a prescription for the process forwards in time is given. During the last decades the backward stochastic differential equations (BSDEs) have become popular. A BSDE is stochastic differential equation for which a terminal condition, instead of an initial condition, has been specified. Its solution consists of a pair of processes, where one of the processes `steers' the other towards the terminal condition. The value of a call option can be modeled in this way as the holder of a replicating portfolio aims to end up with a certain payoff at the terminal time. Market imperfections can also be incorporated, such as different lending and borrowing rates for money, the presence of transaction costs, or short selling constraints. In Chapter 6 we extend the COS method to solve such problems and name it the BCOS method. A forward stochastic process can be approximated by different simulation schemes. The stochastic Euler scheme is a generalization of the Euler scheme for ordinary differential equations. Higher order Taylor schemes include more stochastic terms to obtain a better convergence rate. In Chapter 7 we extend our pricing and valuation methodology by using the characteristic function of these discrete schemes within the BCOS method framework. With the second-order weak Taylor scheme and a theta-time discretization we obtain second order convergence in the number of timesteps for several problems. The techniques in Chapter 7 enable us to generalize the applicability of the BCOS method to forward SDEs for which the `continuous' characteristic function is not available.