Numerical Methods in Computational Finance. Daniel J. Duffy

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СКАЧАТЬ C_; public: MatrixOde(const ublas::matrix<value_type>& A, const ublas::matrix<value_type>& IC) : A_(A), C_(IC) {} void operator()(const state_type &x , state_type &dxdt, double t ) const { for( std::size_t i=0 ; i < x.size1();++i ) { for( std::size_t j=0 ; j < x.size2(); ++j ) { dxdt(i, j) = 0.0; for (std::size_t k = 0; k < x.size2(); ++k) { dxdt(i, j) += A_(i,k)*x(k,j); } } } } };

There are many dubious ways to compute the exponential of a matrix, see Moler and Van Loan (2003), one of which involves the application of ODE solvers. Other methods include:

       S1: Series methods (for example, truncating the infinite Taylor series representation for the exponential).

       S2: Padé rational approximant. This entails approximating the exponential by a special kind of rational function.

       S3: Polynomial methods using the Cayley–Hamilton method.

       S4: Inverse Laplace transform.

       S5: Matrix decomposition methods.

       S6: Splitting methods.

      3.7.1 Transition Rate Matrices and Continuous Time Markov Chains

      An interesting application of matrices and matrix ODEs is to the modelling of credit rating applications (Wilmott (2006), vol. 2, pp. 665–73). To this end, we define a so-called transition matrix P, which is a table whose elements are probabilities representing migrations from one credit rating to another credit rating. For example, a company having a B rating has a probability 0.07 of getting a BB rating in a small period of time. More generally, we are interested in continuous-time transitions between states using Markov chains, and we have the following Kolmogorov forward equation:

      (3.30)

      where and unit matrix and is the transition rate matrix having the following properties:

      The Kolmogorov backward equation is:

      (3.31)

      The objective is to compute the transition rate matrix Q (that is, states that are in one-to-one correspondence with the integers).

      In the case of countable space, the Kolmogorov forward equation is:

      (3.32)

      where Q(t) is the transition rate matrix (also known as generator matrix), while the Kolmogorov backward equation is:

      (3.33)

3.8 SUMMARY AND CONCLUSIONS

      This chapter took over where Chapter 2 left off. We have tried to give a self-contained overview of the analytic properties of scalar ODEs and systems of ODEs as well as their numerical approximation. The topics are important in their own right, and an understanding of them is important in finance applications. We also gave a short introduction to stochastic differential equations (SDEs) in Section 3.4. We discuss SDEs and their relationship with PDEs in Chapter 13.

CHAPTER 4 An Introduction to Finite Dimensional Vector Spaces

      There's no sense in being precise when you don't even know what you're talking about.

      John von Neumann

4.1 SHORT INTRODUCTION AND OBJECTIVES

      This chapter introduces vector spaces of finite dimension. They can be seen as the n-dimensional generalisation of the two- and three-dimensional vectors that we have become accustomed to. In three dimensions, for example, a vector is a 3-tuple consisting of three components (elements), and it can be visualised as a directed line from the point (0, 0, 0) to the point . In higher dimensions this geometric analogy is lost (unless you are Albert Einstein), and we model vectors as an n-tuple of homogeneous components of a certain type (in most cases real or complex variables). In particular, we discuss the following use cases as we progress:

       U1: Addition of vectors.

       U2: Premultiplication of a vector by a scalar (an element of a field).

       U3: Inner products in vector spaces.

       U4: СКАЧАТЬ