Market Risk Analysis Volume I: Quantitative Methods in Finance

I.1

Basic Calculus for Finance

I.1.1 INTRODUCTION

This chapter introduces the functions that are commonly used in finance and discusses their properties and applications. For instance, the exponential function is used to discount forward prices to their present value and the inverse of the exponential function, the natural logarithmic function or ‘log’ for short, is used to compute returns in continuous time. We shall encounter numerous other financial applications of these functions in the subsequent volumes. For instance, the fair price of a futures contract is an exponential function of an interest rate multiplied by the spot price of the underlying asset. A standard futures contract is a contract to buy or sell a tradable asset at some specified time in the future at a price that is agreed today. The four main types of tradable assets are stocks, bonds, commodities and currencies.

The futures price is a linear function of the underlying asset price. That is, if we draw the graph of the futures price against the price of the underlying we obtain a straight line. But non-linear functions, which have graphs that are not straight lines, are also used in every branch of finance. For instance, the price of a bond is a non-linear function of its yield. A bond is a financial asset that periodically pays the bearer a fixed coupon and is redeemed at maturity at par value (usually 100). The yield (also called yield to maturity) on a bond is the fixed rate of interest that, if used to ...