A Primer for the Mathematics of Financial Engineering

305 Pages·2008·2.95 MB·English

by  Dan Stefanica

FINANCIAL ENGINEERING
ADVANCED BACKGROUND SERIES
FE PRESS
New York
Financial Engineering Advanced Background Series
Published or forthcoming
1. A Primer for the Mathematics of Financial Engineering, by Dan Stefanica
2. Numerical Linear Algebra Methods for Financial Engineering Applica-
tions, by Dan Stefanica
3. A Probability Primer for Mathematical Finance, by. Elena Kosygina
4. Differential Equations with Numerical Methods for Financial Engineering,
by Dan Stefanica
A PRIMER
for the
MATHEMATICS
of
FINANCIAL ENGINEERING
DAN STEFANICA
Baruch College
City University of New York
FE PRESS
New York
FE PRESS
New York
www.fepress.org
Information on this title: www.fepress.org/mathematical_primer
©Dan Stefanica 2008
All rights reserved. No part of this publication may be
reproduced, stored in a retrieval system, or transmitted,
in any form or by any means, electronic, mechanical,
photocopying, recording, or otherwise, without the prior
written permission of the publisher.
First published 2008
Printed in the United States of America
ISBN-13 978-0-9797576-0-0
ISBN-10 0-9797576-0-6
To Miriam
and
to Rianna

Contents
List of Tables(cid:9) xi
Preface(cid:9) xiii
Acknowledgments(cid:9) xv
How to Use This Book(cid:9) xvii
0. Mathematical preliminaries(cid:9) 1
0.1 Even and odd functions (cid:9) 1
0.2 Useful sums with interesting proofs (cid:9) 4
0.3 Sequences satisfying linear recursions (cid:9) 8
0.4 The "Big 0" and "little o" notations (cid:9) 12
0.5 Exercises (cid:9) 15
1 Calculus review. Options.(cid:9) 19
1.1 Brief review of differentiation(cid:9) 19
1.2 Brief review of integration (cid:9) 21
1.3 Differentiating definite integrals (cid:9) 24
1.4 Limits (cid:9) 26
1.5 L'Hopit al's rule (cid:9) 28
1.6 Multivariable functions (cid:9) 29
1.6.1 Functions of two variables (cid:9) 32
1.7 Plain vanilla European Call and Put options (cid:9) 34
1.8 Arbitrage-free pricing (cid:9) 35
1.9 The Put-Call parity for European options(cid:9) 37
1.10 Forward and Futures contracts (cid:9) 38
1.11 References (cid:9) 40
1.12 Exercises (cid:9) 41
vii
(cid:9)
viii CONTENTS
2 Numerical integration. Interest Rates. Bonds.(cid:9) 45
2.1 Double integrals (cid:9) 45
2.2 Improper integrals (cid:9) 48
2.3 Differentiating improper integrals (cid:9) 51
2.4 Midpoint, Trapezoidal, and Simpson's rules (cid:9) 52
2.5 Convergence of Numerical Integration Methods (cid:9) 56
2.5.1 Implementation of numerical integration methods(cid:9) 58
2.5.2 A concrete example (cid:9) 62
2.6 Interest Rate Curves (cid:9) 64
2.6.1 Constant interest rates (cid:9) 66
2.6.2 Forward Rates (cid:9) 66
2.6.3 Discretely compounded interest (cid:9) 67
2.7 Bonds. Yield, Duration, Convexity(cid:9) 69
2.7.1 Zero Coupon Bonds (cid:9) 72
2.8 Numerical implementation of bond mathematics (cid:9) 73
2.9 References (cid:9) 77
2.10 Exercises (cid:9) 78
3 Probability concepts. Black—Scholes formula. Greeks and
Hedging.(cid:9) 81
3.1 Discrete probability concepts (cid:9) 81
3.2 Continuous probability concepts (cid:9) 83
3.2.1 Variance, covariance, and correlation (cid:9) 85
3.3 The standard normal variable (cid:9) 89
3.4 Normal random variables (cid:9) 91
3.5 The Black-Scholes formula (cid:9) 94
3.6 The Greeks of European options (cid:9) 97
3.6.1 Explaining the magic of Greeks computations (cid:9) 99
3.6.2 Implied volatility (cid:9) 103
3.7 The concept of hedging. A- and F-hedging (cid:9) 105
3.8 Implementation of the Black-Scholes formula (cid:9) 108
3.9 References (cid:9) 110
3.10 Exercises (cid:9) 111
4 Lognormal variables. Risk—neutral pricing.(cid:9) 117
4.1 Change of probability density for functions of random variables 117
4.2 Lognormal random variables (cid:9) 119
4.3 Independent random variables (cid:9) 121