Computational Finance

Lecture 1

Products and Markets

Computational Finance

Computational Finance (quantitative finance, financial engineering, or mathematical finance): A cross-disciplinary field which uses

quantitative methods developed in math or engineering to solve financial problems.

Computational Finance

Finance

EngineeringMathematics

ComputationalFinance

Agenda

Review of Interest theory Basic financial products:

Equities Bonds

Derivatives: Forward and Futures Options

Time Value of Money

$1 today is worth more than $1 in 1 year;

Interest Several types of interests:

Simple interest ; Discretely compounded interest Continuously compounded interest

Simple Interests

Notation: interest rateSuppose that you invest $1 for years w

ith interest per annum. After years, under the rule of simple interest you will have:

Discretely Compounded Interests

Suppose that you invest $1 for years with interest per annum. After 1 year, you will have:

Compound interest is interest on interest.

After 2 years, you will have

After years:

Nominal and Effective Interest Rates

The compounding periods of interest rates may not be the same as the periods that they are quoted.

For example, you receive an interest rate of per year and the rate is compounded times per year,

Nominal interests and effective interests

Continuously Compounded Interests

As compounding more and more frequent, i.e., as ,

Continuously compounded interest:Suppose that you invest $1 with

continuously compounded interest rate. After years, you will have .

Continuously Compounded Interests

Suppose that an interest rate is per annum and its compounding period is

I have an amount in the bank at time . A short while later, the amount will have increased to

Thus,

Ordinary Differential Equations

Ordinary differential equations (ODE)

The solution:

Present Values

Interest rates facilitate the comparison of money at different time epochs.

Now 1 year Time

$1000

$1100

Discounting

Financial Instruments

Two fundamental financial instruments: Bonds Equities

Bonds

Bonds are a kind of financial instruments issued by government or corporations to borrow money from the public.

Bonds

Bonds usually obligates the issuer to make periodic interest payments to the bondholder, called coupon payments.

At the maturity of a bond, the issuer repays the principal, called the par value or the face value, to the bondholder.

Fixed-income financial instruments

Bond Pricing

Interest rate: per annum Face Value: Coupon rate: Principal Coupons

Coupon

1 2 3 4 ……… maturity

Yield-to-Maturity (YTM)

In practice, people usually infer how much the “return rate” of a bond is using the bond price, maturity date and coupon payments.

This return rate is known as the yield-to-maturity of this bond.

Mathematically,

YTM and Bond Price

The yield-to-maturity of a bond has a reverse relation to the bond price: YTM Bond price

YTM Bond price

Are Fixed-Income Financial Instruments Risk Free?

Interest rate risk Credit risk

Equities

Equities (stocks or shares) entitle the holders a portion of the ownership of a company.

The shareholders have rights to determine business matters of a company.

Board of directors

Characteristics of Common Stocks

Residual claim: Stockholders are the last in line of all those

who have a claim on the assets and income of a company.

Limited liability: The greatest amount shareholders can lose i

n an event of failure of a company is their original investment. They are not personally liable for the firm’s obligations.

Dividends

When a company makes a profit, it usually reinvests some of the profit into the business as retained earnings and distributes the rest among the shareholders as dividends.

The amounts and dates of dividend payments are decided by the board of directors.

Cum dividend and ex dividend

Stock Index

Indicator of stock market performance: stock indices.

A typical index is made up from the weighted sum of a selection of representative stocks. Dow Jones Industrial Average (DJIA), Standard & Poor’s 500 (S&P 500), Financial Times Stock Exchange Index (FTSE

100), Hang Seng Index (HSI)

Short Selling

Investors are allowed to sell stocks they do not own. This strategy is known as short selling.

In the procedure, an investor borrows shares of stock from another through a broker and sells them. Later, he/she must repurchase the shares in the market to replace what he/she borrowed.

Short Selling

Suppose that the current price of IBM stock is $96 per share. You are very pessimistic on it.

You may instruct your broker to sell short 1,000 shares. Then, the broker will borrow 1,000 shares from other one and sell. $96,000 cash proceeds are credited to your account.

Short Selling

If you are right, and IBM stock falls to $86 per share, you can close your short sale by repurchasing 1,000 shares back. The profit will be

96,000-86,000=$10,000. If the prediction is not correct, IBM

stock price goes up to $106 per share, you will suffer a loss of

106,000-96,000=$10,000.

Dividends and Short Selling

The investor with a short position must pay to the borrower any dividends that would normally be received on the stocks that have been shorted.

In previous example, suppose that one dividend payment, $1 per share, is made during the short sale. Then, the investor must pay $1,000 to the borrower.

Regulations on Short Selling

Short sales can be done only after an uptick, i.e., the most recent movement of the stock price was an increase. This rule prevents investors from speculating against the stock.

Short sellers are required to set up margin accounts with the broker to ensure that they can cover any losses if the shorted stock price rises.

Forwards

A forward contract is an agreement where one party promises to buy an asset from another party at some specified time in the future and at some specified price. Underlying asset Maturity or delivery date Delivery price Long position and short position

An Example of Forward Contract

A US corporation enters a forward contract with an international bank on Jan. 8 to agree to buy 1M British pounds at a price of 1.8991 US dollars per pound in 1 month.

Payoffs from Forward Contracts

Forward contracts are obligations. Two positions must honor the agreement no matter what happens in the market.

Payoffs from Forward Contracts

Suppose the exchange rate between US dollars and British pounds turns out to be US$1.92 per pound. The corporation would

be favored by the contract and it would pay (1.92-1.8991=0.0209M) less than the market price;

US$ 1.88 per pound. The forward contract would have a negative value to the corporation because it would pay (1.8991-1.88=0.0191M) more than the market price.

Payoffs from Forward Contracts

In general the payoff from a long position in a forward contract should be

where is the delivery price and is the price of the underlying asset at the maturity.

Short position:

Determination of Delivery Price

What is a “fair” delivery price for both positions?

Information available: , ,

Determination of Delivery Price

Short position: Borrow from a bank at interest rate ; Buy the underlying asset now; Deliver the asset to the long position at the

maturity. Total loan at the maturity for the

short position is: Delivery price should not be larger

than it, i.e.,

Arbitrage Opportunities

If , the short position gets . The short position costs nothing to

set up the strategy in the last slide, and ends up with positive profit.

Arbitrage opportunities

Arbitrage Opportunities

An arbitrage opportunity exists if an investor can exploit the mispricing of assets to make a deal at no initial cost, has no risk of future loss and a non-zero probability of future profit.

Arbitrage free assumption

Determination of Delivery Price

Long position: Short the underlying asset to get ; Deposit in a bank to earn interest; Repurchase the asset back using the

forward contract at the maturity.

The total deposit will be . Delivery price should not be

smaller than it, i.e.,

Determination of Delivery Price

In general, consider a forward contract on a non-dividend stock. Suppose that the current stock price is , the time to the maturity is and the risk free interest rate is , then a fair delivery price should be