Financial Markets
1 Financial Assets
Financial mathematics is a branch of applied mathematics that focuses on mathematical models for financial markets and the prices of assets.
A financial asset is a tradable resource that has value and can be bought, sold, or exchanged.
A financial asset can be tangible, such as a stock certificate or a bond, or intangible, such as a digital representation of ownership in a company or a cryptocurrency. Some common examples of financial assets include:
- Stocks: Represent ownership in a company and entitle the holder to a share of the company’s profits.
- Bonds: Debt securities issued by governments or corporations that pay interest to the holder over a specified period.
- Commodities: Physical goods such as gold, oil, or agricultural products that can be traded in financial markets.
- Currencies: Foreign exchange (forex) markets allow for the trading of different currencies.
- Derivatives: Financial instruments whose value is derived from an underlying asset, index, or rate.
Liquidity refers to the ease with which an asset can be bought or sold in the market without significantly affecting its price. Highly liquid assets can be quickly converted to cash, while illiquid assets may take longer to sell and may require a discount to their market value.
Assets can be traded in two ways:
- Over-the-counter (OTC): Trading that occurs directly between two parties, without the involvement of an exchange. OTC trading is often used for customized or less liquid assets.
2 Derivatives
A financial asset is a tradable resource that has value and can be bought, sold, or exchanged.
A financial derivative is a financial instrument whose value is derived from some underlying asset, index, or rate.
Think of a derivative as a contract between two parties that specifies the terms notinf a financial transation based on the performance of the underlying asset.
Below we define some common types of financial derivatives:
Forwards: A forward contract is a customized agreement between two parties to buy or sell an asset at a specified price on a future date. Forwards are traded over-the-counter (OTC) and are not standardized.
Futures: A futures contract is a standardized agreement to buy or sell an asset at a predetermined price on a specified future date. Futures are traded on organized exchanges and are subject to daily settlement.
Options: An option is a contract that gives the holder the right, but not the obligation, to buy (call option) or sell (put option) an asset at a specified price (strike price) on or before a specified date (expiration date).
Swaps: A swap is a derivative contract in which two parties agree to exchange cash flows or other financial instruments over a specified period. Common types of swaps include interest rate swaps and currency swaps.
3 Payoff Functions
We can represent the value of an underlying asset over time using Stochastic Process. For example, let \(S_t\) denote the price of one share of Company XYZ stock at time \(t\).