Chapter 1 Preliminaries And Review

Financial Models A model is an artificial or idealized structure describing the relationships among variables or factors. The easiest financial models to construct and analyze are often those that assume efficient capital markets, in which security prices fully reflect information. Efficient markets are characterized by: 1. No monopoly power 2. No transactions costs. 3. No taxes 4. Free markets exist for all assets. 5. Investors have equal access to costless information. 6. Investors are rational and attempt to maximize their wealth.

Financial Security And Instruments A financial security is a contract or certificate indicating a financial claim. Securities are traded in the financial marketplace.

Types of Financial Securities And Instruments Debt securities: Denote creditorship. They typically involve payments of a fixed series of coupon payments and/or amounts towards principal (often known as face value). Examples include: 1. Bonds: Long term debt 2. Treasury securities: Debt securities issued by the Treasury of the United States federal government. They are often considered to be practically free of default risk. Equity securities (stock): Denote ownership in a business or corporation. They typically permit for dividend payments.

Types of Financial Securities And Instruments (Continued) Derivative securities: Have payoff functions derived from the values of other securities, rates or indices. They include: 1. Options: Securities that grant their owners rights to buy (call) or sell (put) an underlying asset or security at an exercise price on or before its expiration date. 2. Forward and Futures Contracts: Instruments that oblige their participants to either purchase or sell a given asset or security at a settlement price on the future settlement date. A long position obligates the investor to purchase and a short position obligates the investor to sell. Swaps: Provide for the exchange of cash flows associated with one asset, rate or index for the cash flows associated with another asset, rate or index.

Types of Financial Securities And Instruments (Continued) Commodities: Contracts, including futures and options on physical commodities. Commodities are traded in spot markets, where the exchange of assets and money occur at the time of the transaction or in forward and futures markets. Currencies (FX): Exchange rates denote the number of units of one currency that must be given up for one unit of a second currency. Indices: Contracts pegged to measures of market performance such as the Dow Jones Industrials Average or the S&P 500 Index.

Review Of Matrices And Vectors

Review Of Matrices And Vectors (Continued) If a matrix A is of order m×n and a matrix B is of order n×p, then the matrix product AB exists and is a matrix of order m×p. The entry in the i th row and j th column of AB is found by multiplying the entries of the i th row of matrix A by the corresponding entries of the j th column of matrix B and summing the resulting products.

Illustration Of A Matrix Product

Solving A Matrix Equation

Illustration Of Expressing A System Of Linear Equations As A Matrix Equation

Solving The Previous Matrix Equation

Linear Independence

Determining Linear Independence

Illustration Of Determining Linear Independence

Illustration Of Determining Linear Independence (Continued)

Span And Basis

An Application Of Span And Basis

Illustration Of Payoff Vectors And Basis

Continuation Of The Illustration

Continuation Of The Illustration (Continued)

Review Of Differential Calculus

Illustration Using Derivatives

The Differential

Second Order Taylor Expansion Of A function Of Two Variables

Illustration Of A Second Order Taylor Expansion

Illustration Of A Second Order Taylor Expansion (Continued)

Review Of Integral Calculus

Illustration

The Definite Integral In order to define the definite integral of a function f(x) on the interval [a,b], some preliminary set-up is required. Divide the interval [a,b] into n subintervals of equal width Δx = (b-a)/n. Consider the values of x on the x-axis that are endpoints of the subintervals. They are: x 0 = a, x 1 = a + Δx, x 2 = a + 2Δx,…, x n = a + nΔx = b. The n subintervals are: [x 0,x 1 ], [x 1,x 2 ],…, [x n-1,x n ]. For the i-th subinterval [x i-1,x i ], choose a convenient x- value x i * in this subinterval; that is, x i-1 ≤ x i * ≤ x i.

The Definite Integral (Continued)

Applications Of The Definite Integral

Further Applications Of The Definite Integral Suppose the marginal price of wheat is p(x)=400-6x dollars per ton, where x is the number of tons of wheat that are on the market. Find the revenue from the sale of 3 tons of wheat if the farmer enters the market when there are 5 already tons on the market.

Further Applications Of The Definite Integral (Continued)