Mathematical Techniques in Finance. Amir Sadr

Mortgage‐related 11.2 22% Corporate debt 9.8 8% Municipal 4 8% Federal agency securities 1.7 3% Asset‐backed 1.5 3% Money markets 1 2% Total 50.1

      Source: SIFMA.

TABLE 1.5 Global derivatives market size ($Trillions).

Market	Gross market value
Interest rate contracts	11.4
Foreign exchange contracts	3.2
Equity‐linked contracts	0.8

      Source: BIS.

1.2 FINANCIAL MARKETS AND PARTICIPANTS

Households typically earn wages and receive salary from firms, while firms earn income when households consume their goods and services. The government collects taxes from households and firms for its expenditures for defense, government services, infrastructure, public health, and transfer payments such as social security and Medicare. Banks and financial intermediaries facilitate the transfer of funds between these three sectors: households and firms deposit their excess funds in banks and earn interest, and banks avail these funds in the form of consumer and corporate loans. Other financial intermediaries such as investment banks, insurance companies, and investment companies provide capital and financial services to firms and individuals.

The capital markets and financial instruments facilitate the flow of funds between different sectors of the economy. Focusing on the United States, the bond market is the largest market with capitalization of $50 trillion at the end of 2020. The U.S. government routinely borrows by issuing debt in the form of coupon bonds. Similarly corporations finance their growth by issuing debt in the form of corporate coupon bonds. States and municipalities also raise capital by issuing debt for infrastructure and other projects.

TABLE 1.6 Market participants and financial products.

Participant	Usage	Product
Households	Custody, banking, borrowing	Checking and interest bearing accounts, credit cards
	Home mortgage, auto, student loan	Level pay loans
	Investments	Cash, options brokerage accounts, financial or robo‐advisor advice for asset allocation
	Insurance, estate planning	Auto, home, life insurance; annuities
Corporations	Financing	Bonds, stock issuance
	Cash flow management	Commercial paper, lines of credit, swaps
	Asset liability management, interest rate risk management	Derivatives, interest rate futures, swaps, options
Insurers, mortgage servicers	Rate risk	Swaps, caps, swaptions
Pension plans	Asset allocation and insurance	Derivatives
Hedge funds	Investment, speculation	Leveraged products, derivatives, statistical methods
Banks, financial institutions	Financial services	All products
States and local government	Financing	Bullet bonds, callable bonds
Fannie Mae, Freddie Mac	Financing, risk management	Swaps, swaptions, swapped issuance
U.S. government	Financing	Bills, notes, bonds
Federal Reserve	Monetary policy	Repo and reverse, quantitative easing

      Banks and other financial institutions provide home mortgage, auto, and student loans in the form of level pay loans. These loans and receivables are in turn bought and securitized as mortgage‐backed and asset‐backed securities by companies originally set up by the U.S. government to promote home ownership and student loans, prominent among them are Fannie Mae (Federal National Mortgage Association), Freddie Mac (Federal Home Loan Mortgage Corporation), and Sallie Mae (Student Loan Marketing Association).

      Corporations raise capital by issuing stock (equity), which is publicly traded. Households participate in the stock market directly via brokerage accounts or retirement plans primarily investing in mutual funds and ETFs (Exchange Traded Funds). The allocation of investments between different assets or funds is the subject of portfolio selection.

      Firms and households use insurance and derivatives markets to mitigate and manage financial risk. Consumers buy home and auto insurance to protect against loss. Corporations raise money from the capital markets and manage their interest rate exposure through interest rate swaps and derivatives. Producers use commodity futures and derivatives to manage price risk, and pension plans and investors use equity derivatives for risk management and speculation.

1.3 QUANTITATIVE FINANCE

Households, corporations, governments, and financial firms, such as commercial and investment banks, insurance companies, asset management companies, and hedge funds, all participate in financial markets and employ a variety of products and increasingly sophisticated quantitative methods (see Table 1.6). The mathematics includes results and techniques from calculus, linear algebra, probability and statistics, numerical methods, optimization techniques, stochastic processes, differential equations, and machine learning techniques. In the following chapters, we will introduce these techniques as used in different markets