Mathematical Methods for Finance: Tools for Asset and Risk Management

CHAPTER 2

Differential Calculus

Financial market instruments can be divided into two groups. The first group includes cash market instruments, such as stocks, bonds, commodities, and foreign currencies, which are referred to as the primary set of assets. The second group includes financial derivatives, such as options, futures, forwards, and swaps, which are written on the primary set of assets. Financial derivatives are claims that promise some payment or delivery in the future contingent on the underlying asset’s behavior. Differential calculus is useful to understand and investigate the changes in prices and riskiness of these financial instruments. Using differential calculus, one can:

• Determine the sensitivity of bonds to changes in interest rates.
• Measure the sensitivity of an individual stock (or stock market index) to changes in cash flows (e.g., dividend yields).
• Investigate the sensitivity of an individual stock (or stock market index) to changes in discount rates (e.g., expected returns).
• Estimate the sensitivity of an individual stock (or stock market index) to changes in discount rates (e.g., expected returns).
• Estimate the sensitivity of bonds and individual stocks (or stock market indexes) to changes in macroeconomic variables (e.g., default spread, term spread, inflation rate, growth rate of industrial production, and consumption-to-wealth ratio).
• Investigate how the prices of options change as a result of changes in the price of the underlying asset.
• Investigate ...