This handbook presents the current state of practice, method and understanding in the field of mathematical finance. Each chapter, written by leading researchers, starts by briefly surveying the existing results for a given topic, then discusses more recent results and, finally, points out open problems with outlines for possible solutions. The primary audiences for the book are doctoral students, researchers and practitioners who already have some basic knowledge of mathematical finance. This comprehensive reference work will be indispensable to readers who need a quick introduction or references to specific topics within this cutting-edge material.