Mathematical Techniques in Finance. Amir Sadr

standard normal random variable,

BMBrownian motion

Brownian motion at time

along sample path

proportional, lognormal volatilityATM, ATMFat‐the‐money spot, at‐the‐money forward

Dirac's delta function,

absolute, normalized volatilityCMSconstant maturity swap rateAD,

Arrow‐Debreu price, today's price of unit payoff at state

on future date

i.i.d.independent and identically distributed

CHAPTER 1 Finance

      While economics as a social science studies the behavior of economic agents in the generation, acquisition, and expenditure of goods and services, finance is focused on the acquisition and management of capital in financial markets.

      Focusing on the end user, finance can be divided into personal finance, corporate finance, and government finance. Savings, investments, and loans, such as credit card, student, automobile, and home mortgage, insurance products, and estate planning are examples of personal finance. The raising of capital by borrowing and debt or selling shares and equity by a company and the management of a company's funds are the focus of corporate finance. Monetary policy, central banking, tax systems, and the oversight of the banking sector and financial markets fall under government finance.

1.1 FOLLOW THE MONEY

Using the reductive definition of finance as the study of money, we follow the money to get our bearings. In accounting, a balance sheet is a snapshot of an entity's (person, corporation, country) net worth or equity: assets minus liabilities equals equity. Table 1.1 shows a snapshot of the net worth of the three dominant players in the U.S. economy: households, firms, and government.

As the table shows, households hold the largest amount of equity, followed by firms, while the U.S. government runs a negative balance and is in debt. Indeed, the U.S. government is the world's biggest borrower and routinely borrows money to finance its expenditures. The breakdown of households' net worth is shown in Table 1.2.

Bonds, stocks, foreign exchange, commodities, and their derivatives are the major sectors of financial markets. Tables 1.3 through 1.5 show the market size as of year end 2020.

TABLE 1.1 Balance sheet of the United States at 2020 year end.

Sector	Net Worth ($Trillions)
Households	123.35
Firms	33.9
Government	−26.8
Total	130.46

      Source: U.S. Federal Reserve Z.1 Statistical Release.

TABLE 1.2 Households sector balance sheet as of 2020 year end.

Category	$Trillions	Percentage
Real estate	32.8	24%
Consumer durable goods	6.1	4%
Checking, savings, money market accounts	15.2	11%
Debt securities, bonds	5.1	4%
Equities, mutual funds, investments	79	57%
Misc	1.3	1%
Total assets	139.6	100%
Home mortgage loans	10.9	67%
Credit card, auto loans	4.2	26%
Other loans	1.1	7%
Total liabilities	16.2	100%
Net worth	123.35

TABLE 1.3 Market size ($Trillions).

	U.S.	World
Bonds	50.1	123.5
Stocks	40.7	93.6
Derivatives		15.8
Foreign Exchange		6.6/day

      Sources: World Bank, BIS, SIFMA.

TABLE 1.4 U.S. bond market ($Trillions).

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Type	Outstanding debt
Treasury	21.0	42%