Computational Finance

Lecture 2

Markets and Products

Review of Last Class

Interest theory Simple interest Discretely compounded interest Continuously compounded interest

Basic financial products: Equities Bonds

Derivatives: Forward

Review of the Last Class

Determination of the Delivery Price:

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

Review of the Last Class

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

maturity.

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

forward contract at the maturity.

Agenda

Forward contracts on cum dividend stocks

Futures Options

Cum Dividend Stocks?

Consider a 6-month forward contract on a stock with a price of $50. We assume that the risk-free interest rate is 8% per annum. A dividend of $0.75 per share is expected in 6 months. What should the delivery price of this forward be?

Hint: suppose that the delivery date is right after the dividend payment date.

Forward Contracts on Stocks with Dividends

Short position: Borrow $50 from a bank at interest rate 8%; Buy 1 share of stock now; Deliver the share to the long position at the mat

urity. Cash flows for Short position:

Liability: loan Income: Delivery price Dividend $0.75

Forward Contracts on Stocks with Dividends

Long position: Short selling 1 share to get $50; Deposit $50 in a bank to earn interest; Repurchase 1 share back at the maturity.

Cash flows: Liability: Delivery price Dividend: $0.75 Income: Deposit

Forward Price

The delivery price of a forward contract is chosen according to the information when the contract is entered.

As time goes by, the delivery price will not be fair any longer.

Stock Price Forward 1 Forward 2 Day 0 -

Day 1

Forward Price

Suppose that Mr. A enters a forward contract on day 0.

On day 1, the fair payment changes to If , then his promise payment for

the underlying asset at the maturity is more than what the market believes that he should pay. This contract causes him a potential loss worthy of at the maturity. It should be valued as

Forward Price

If , then his promise payment is less than what the market believes that he should pay. This contract brings him a potential profit worthy of . Thus it should be valued as

Forward Price

In general, a forward contract should have a positive/negative value after it is signed.

Suppose that the delivery price is , the time to maturity is , and the current price of the underlying is . The value of this forward contract is given by

Futures

The trading of forward contracts involves risks.

Sometimes one of the parties may not have enough financial resources, or may regret the deal, to honor the agreement.

To avoid the risk, futures contracts are introduced.

Futures

Like a forward contract, a futures contract is also agreement between two parties to buy or sell an asset at a certain time in the future for a certain price.

Futures are traded in the exchanges. To avoid contract defaults, the exchanges require both parties to deposit funds when the contract is entered.

The Operation of Margins:Margin Account and Initial Margins

The initial amount that must be deposited is known as the initial margin. It is deposited into a margin account.

For example, an investor wants to buy two March gold futures contracts on the New York Commodity Exchange. Each contract size is 100 ounces. The current futures price is $400 per ounces and the initial margin is $2,000 per contract.

The Operation of Margins:Marking to Market

To start, this investor should set up a margin account with the exchange and deposit initial margin $4,000 in it.

Like forward contracts, futures may have a positive/negative value after they are entered.

The exchanges settle down margin accounts at the end of every day to reflect the gain/loss of investors.

The Operation of Margins:Marking to Market

On the next day, the March gold futures price drops down to $397 per ounce.

The investor has a loss of 2 ($400-$397) 100 =$600. $600 is deducted from the margin of

the investor and the total balance is reduced to $4,000-$600=$3,400.

The $600 will be passed on to another investor with a short position.

The Operation of Margins:Maintenance Margin

Usually the exchanges will set a lower bound for each margin account, known as the maintenance margin to prevent the balance in the margin account from being negative.

Once the balance is below the maintenance margin, the investor will receive a margin call to top up the account to the initial level.

Options

Options give the holders a right to buy or sell the underlying asset by a certain date for a certain price.

The difference of forward contracts and options: Forward contracts: the obligation Options: the right

Call and Put Options

Call options: buy Put options: sell The price in the contract is known

as the exercise price or strike price; the date is known as the expiration date or maturity.

Long position and short position

European and American Options

European-style options only can be exercised on the date of the maturity of the contract.

American-style options can be exercised at any time up to the maturity of the contract.

Example

As an example, consider an option on Intel’s stock. Suppose that the strike price is $20 per share and the maturity is May 21, 2007.

If this is a European call option, the long position is entitled a right to buy Intel shares at the price of $20 per share on May 21, 2007.

If this is an American call option, the long position has a right to buy Intel shares at $20 per share any time before May 21, 2007.

Payoffs of European Call Options

Suppose that Intel stock price turns out to be $25 per share on May 21, 2007.

The long position buys shares at the price of $20 per share by exercising the option. He/She buys shares at lower price than the spot price. The gain he/she realizes is 25-20 = $5 per share.

Payoffs of European Call Options

Suppose that Intel stock price turns out to be $15 per share on May 21, 2007.

The contract charges a higher price than the spot market. Of course, the holder would not like to exercise it.

Options are rights. The holders are not required to exercise them if they do not want to. The contract will be left to mature without exercising.

Payoffs of European Call Options

In general, suppose that the strike price is , and the underlying asset price at the maturity is . Then, the payoff of the long position of the option should be

Diagram of Payoffs for Long Positions

Payoff

K Stock Price

Writing Options

The seller of options (short positions) are called the writer of the options. The writer has liability to satisfy the requirement of the long position if he/she asks to exercise options.

In the previous example, If = $25 per share, the option is

exercised. The writer loses $5 per share. If = $15 per share, the option is not

exercised.

Diagram of Payoffs for Short Positions

Payoff

K Stock Price

Options Premium (Option Price)

The long position of an option always receive non-negative payoffs in the future while the writer always has non-positive payoffs.

The long position must give a compensation to the writer of an options. The compensation is known as the options premiums or options prices.

Intrinsic Values and Time Values

The values of an option contract comes from the potential positive payoffs Call options: when the underlying asset

price goes above the strike price; Put options: when the underlying asset

price goes below the strike price. Intrinsic values:

Call options: Put options:

Intrinsic Values and Time Values

Suppose that you hold a European call option on a stock. It expires in 1 year. Today’s stock price is $50 per share, and the strike price is $65 per share. Then, the intrinsic value is

“Time value”

In the Money, At the Money and Out of the Money

For call options, Out of the money: if the current stock

price is lower than the strike price; At the money: if the current stock

price is equal to the strike price; In the money: if the current stock

price is higher than the strike price. Put options

Factors Affecting Option Prices

Factors affecting option prices: Underlying asset price; Strike price; Time to maturity; Uncertainty of asset price (volatility); Interest rate.

Factors Affecting Option Prices:Current Underlying Price

The current underlying asset price: The higher the current underlying

asset price, the higher the asset is expected to be at the maturity: Call option: more valuable; Put option: less valuable.

Factors Affecting Option Prices:Strike Prices

The strike price: The higher the strike price,

Call options: less valuable; Put options: more valuable.

Factors Affecting Option Prices:Time to Expiration

The effects of time to expiration is subtle.

The longer the time to expiration, the more time there is for the asset to rise or fall;

The longer the time to expiration, the less valuable will the payoff be, if taking interest into account..

Factors Affecting Option Prices:Volatility

Example: Call option with strike price $30. Two stocks, A and B. A is more volatile and B is more placid.

A: Price at maturity $10 $20 $30 $40 $50

Payoffs $0 $0 $0 $10 $20

B: Price at maturity $20 $25 $30 $35 $40

Payoffs $0 $0 $0 $5 $10

Factors Affecting Option Prices:Risk Free Interest Rates

Risk free interest rates: The interest rates in the economy

affect the stock market. Usually stock prices tend to fall when interest rates rise.

On the other hand, the present values of exercise prices decrease when interest rates rise.

Put-Call Parity

Imagine you buy one European call option with strike price and maturity date . Meanwhile, you write a European put on the same underlying stock with the same strike price and the same maturity date.

What is the payoff for you at the maturity?

Put-Call Parity

The payoff is

It is the same as the payoff of a forward contract with the delivery price

Recall the value of such forward contract should be

Put-call parity:

American Options and Early Exercises

American options owners are entitled the right to exercise the options at any time up to the maturities of the options.

The main point of interest with American options is of course deciding when to exercise.

Bermuda options: options are allowed to be exercised on specified dates before the maturity.

Options Strategies

Bull spread: Suppose that I buy a call option

with strike price $100 and write another with strike price $120. Both of them have the same maturity.

Diagram of Payoffs for Bull Spreads

Payoff

20

100 120 Stock Price

Options Strategies

Bear spread: Suppose that I write a put option

with strike price $100 and buy another with strike price $120. Both of them have the same maturity.

Diagram of Payoffs for Bull Spreads

Payoff

20

100 120 Stock Price

Options Strategies

Straddles: A straddle consists of two options:

a call and a put. Both of them have the same maturity and the same strike price.

For example, you buy a call and a put with strike price $100.

Diagram of Payoffs for Straddles

Payoff

100 Stock Price

Options Strategies

Butterfly spread: Buying a call with strike of $90,

writing two calls stuck at $100 and buying a $110 call.

Diagram of Payoffs for Butterfly Spreads

Payoff

90 100 110 Stock Price