## Abstract

In this chapter we study the problem of valuing a (possibly defaultable) derivative asset contingent on the underlying economic state modeled as a Markov process. To gain analytical and computational tractability both in order to estimate the model from empirical data and to compute the prices of derivative assets, financial models in applications are often Markovian. In the Markovian framework, the key object is the pricing operator mapping (possibly defaultable) future payments (payoffs) into present values. The pricing operators indexed by time form a pricing semigroup{P_{t}, t ≥ 0} in an appropriate payoff space, which can be interpreted as the transition semigroup of the underlying Markov process with the killing rate equal to the default-free interest rate plus default intensity. This framework encompasses a wide range of Markovian financial models. In applications it is important to have a tool kit of analytically tractable Markov processes with known transition semigroups that lead to closed-form expressions for value functions of derivative assets. In particular, an analytical simplification is possible when the process is a symmetric Markov process in the sense that there is a measure m on the state space D and the semigroup {P_{t}, t ≥ 0} is symmetric in the Hilbert space L^{2} (D, m). In this case we apply the Spectral Representation Theorem to obtain spectral representations for the semigroup and value functions of derivative assets. In this Chapter we survey the spectral method in general, as well as those classes of symmetric Markov processes for which the spectral representation can be obtained in closed form, thus generating closed form solutions to Markovian derivative pricing problems.

Original language | English (US) |
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Pages (from-to) | 223-299 |

Number of pages | 77 |

Journal | Handbooks in Operations Research and Management Science |

Volume | 15 |

Issue number | C |

DOIs | |

State | Published - Dec 1 2007 |

## ASJC Scopus subject areas

- Finance
- Economics and Econometrics
- Computer Science Applications
- Management Science and Operations Research