Mathematical Techniques in Finance. Amir Sadr
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