Mathematics of Options, Futures and Derivatives Securities

Mathematics of Options, Futures,and Derivative Securities I.

Daniel Spirn,School of Mathematics,University of Minnesota.

Options, Futures, Derivatives / September 5, 2007 back to start 1

Syllabus

Semester I

1. Mechanics of Futures Markets

2. Hedging Strategies Using Futures

3. Interest Rate Markets

4. Determination of of Forward and Future Prices

5. Interest Rate Futures

6. Swaps

7. Mechanics of Options Markets

8. Properties of Stock Option Prices

9. Trading Strategies Involving Options

10. Binomial Trees

11. Wiener Processes and Ito’s Lemma

12. The Black-Scholes Model

13. Options on Stock Indices, Currencies and Futures

14. The Greek Letters

15. Volatility Smiles

Semester II

1. Review of Black-Scholes

2. Review of Greeks

3. Volatility Smiles

4. Basic Numerical Procedures

5. Value at Risk

6. Time Series

7. Estimation of Volatilities and Correlations

8. Credit Risk

9. Credit Derivatives

10. Exotic Options

11. Weather, energy, and insurance derivatives

12. More on models and numerical procedures

13. Martingales and measures

14. Interest Rate Derivatives

15. Convexity, timing, and quanto adjustments

16. Models of Short Rate

17. Heath-Jarrow-Morton

18. Swaps revisited

19. Real Options


Class Information

• Lecture: Mondays & Wednesdays 4:30PM–6:00PM.

• Lecture Room: Vincent Hall 20

• Office Hours: Mondays 3:30PM–4:30PM & Wednesdays: 2:30PM–3:30PM

• Office: Vincent Hall 112b

• Contact: [email protected] & 612-625-1349

• Textbook: Options, Futures, and Other Derivatives, 6th Edition, John Hull,Prentice Hall.

• Grade Information:

– Homework: 50 %– Midterm: 20 %– Final: 30 %


Mathematics Involved

Control Theory

Probability Theory

Partial Differential Equations


What is a derivative?

What is a derivative?

A financial instrument whose value derives from the value of underlying variables.

or

Financial instruments whose price and value derive from the value of assets underlying them. 1

or

Financial contracts whose value derive from the value of underlying stocks, bonds, currencies,

commodities, etc.

Examples:

Future contract for orange juice. Not the orange juice itself.

Option to buy/sell a stock. Not the stock itself.

1Edmund ParkerOptions, Futures, Derivatives / September 5, 2007 back to start 5

Asset Derivatives

Examples

• Commodity Derivatives - Pork Bellies (Trading Places), Precious Metals

• Equity Derivatives - Stocks / Bonds

• Interest Rate Derivatives - Interest Rates

• Currency Derivatives - Currency Exchange Rates (Yen vs. Euro)

• Property Derivatives - Real Estate

• Other More Exotic Derivatives


Why trade derivatives?

• Used to protect assets from drastic fluctuations

• Covers many kinds of risk.

• Allows access to financial instruments with considerably larger risk thanconventional assets (speculation).


Types of Markets trading derivatives

• Exchange-traded Markets:Examples: Chicago Board of Trade & Chicago Mercantile Exchange.This is an open-outcry exhange where traders meet physically and negotiatetrades via signals and shouts.There are also electronic exchanges where traders enter desired trades andcomputers match the trades.

• Over-the-counter (OTC) Markets:These are telephone and computer run exchanges. Large financial companiesact as market makers for commonly traded instruments. This requires them toquote both a bid and offer price.

Markets are huge.OTC market on the order of $ 298 Trillion in 2005.

The Exchange-traded market was on the order of $ 49 Trillion in 2004.


Financial instruments used in derivative transactions

Forwards

Futures

Options

Swaps


Forward Contracts

Definition: A forward contract is an agreement to buy or sell an assetat a certain time for a certain price. Usually traded in the OTC market.

One party assumes a long position by agreeing to buy the underlying asseton a specified future date for a specified price.

The other party assumes a short position by agreeing to sell the underlying asseton the same date and at the same price.

Definition: A spot contract is an agreement to buy or sell an asset today.

Spot contracts are for immediate delivery of the asset.Common among currency derivatives.


Example of a useful forward contract

Spot and forward quotes for USD/GBP exchange rate, June 1, 2007

Bid OfferSpot 1.6281 1.6285

1-month forward 1.6248 1.62533-month forward 1.6187 1.61925-month forward 1.6094 1.6100

• Suppose today is June 1, 2007. US corporation will need to pay £1 million onDecember 1, 2007. Company wants to hedge against a change in the exchangerate.

• Using the table above, the company agrees to buy £1 million 6-month forwardcontract at the exchange rate of 1.6100 . The company has a long forwardcontract on GBP, i.e. the bank will pay $ 1.610 million for £1 million.

• Participating bank has a short forward contract to sell £1 millionfor $ 1.610 million on Dec. 1, 2007.


Payoffs from Forward Contracts

Outcomes

• Suppose spot exchange rises to 1.7000 by Dec. 1, 2007, then the forwardcontract would be worth:

$1, 700, 000− $1, 610, 000 = $90, 000

to the US corporation. Enables the corporation to pay at a cheaper rate.

• If the spot exchange falls to 1. 500 on Dec. 1, 2007, then the forward contractwould be worth:

$1, 500, 000− $1, 610, 000 = $− 110, 000

so the corporation would be paying more for pounds than the current market.


Payoffs for Forward Contracts, cont.

In general if K is the delivery price ($1.61 million) and ST is the spot price of theasset at maturity ($1.5 million) then we define the following payoffs for forward

contracts.

The payoff for a long forward contract is

ST −K.

The payoff for a short forward contract is

K − ST .


Futures Contracts

• Similar to a forward contract - it is an agreement between two parties to buy orsell an asset at a certain time in the future for a certain price.

• However, futures are usually traded in exchanges.

• Mechanisms by the exchange to guarantee that the contract will be honored.

• Example: Chicago Mercantile Exchange (CME) trades futures on

– commodities such as pork bellies, orange juice, copper, sugar, etc.– financial assets such as stock indices, currencies, Treasury bonds, etc.

• Mechanisms for such exchanges will be explained next week.


Options Contracts

The third type of derivative we will discuss is an options contract. These aredivided into two types:

• A call option entitles the holder the right to buy the underlying asset by acertain date for a certain price.

• A put option entitles the holder the right to sell the underlying asset by acertain date for a certain price.

Some more terminology:

• The price in the contract is the strike price or the exercise price.

• The date in the contract is known as the expiration date or maturity.


Types of options

• American options can be exercised at any time up to the expiration date.

• European options can be exercised only on the expiration date.

Mathematics involved in American options is more difficult than European options.

One contract is usually an agreement to buy or sell 100 shares.

Note an option is exactly that - an option to exercise the right to buy or sell.The holder of the option does not need to exercise the option.


Example of Stock Options

Calls Calls Calls Puts Puts PutsStrike Price($) June July Oct. June July Oct.

20.00 1.25 1.60 2.40 0.45 0.85 1.5022.50 0.20 0.45 1.15 1.85 2.20 2.85

Prices of options on Intel, May 29, 2003. Stock price = $20.83

Consider the following scenario.

• Investor instructs broker to buy one October call option contract on Intel withstrike price of $ 22.50. This is negotiated at CBOE with the price $1.15.

• This is the price for an option to buy one share. The US contract reflects 100shares; therefore, the investor must pay $115 to the exchange through thebroker. The exchange passes this amount to the party on the other side.

• Summary: our investor has paid $115 for the right to buy 100 shares of Intel at$22.50 each by the October maturity. The party on the other side received$115 to agree to sell 100 shares of Intel at $22.50 (if the investor chooses).


Example, cont.

Possible scenarios:

• If the price of Intel remains below $22.50 before the October maturity date, theoption is not exercised and the investor loses $115.

• If the price of Intel rises to $31 before the October maturity date then the callis exercised and the investor makes

[$31.00× 100− $22.50× 100]− $115 = $735.


Example, cont.

Calls Calls Calls Puts Puts PutsStrike Price($) June July Oct. June July Oct.

20.00 1.25 1.60 2.40 0.45 0.85 1.5022.50 0.20 0.45 1.15 1.85 2.20 2.85

Consider a new scenario with Intel options.

• Purchase one July put option contract with strike price of $20.00 for $85. Theinvestor now has the right to sell 100 shares of Intel at $20.00 each prior to theJuly exercise date.

• If Intel stays above $20 then the option is not exercised and the investor loses$85.

• If Intel drops to $12 then the investor earns

[$20.00× 100− $12.00× 100]− $85 = $715


Participants in Options Markets

Four types of participants in the Options markets, as yet:

• Buyers of calls a long position

• Sellers of calls a short position

• Buyers of puts a long position

• Sellers of puts a short position

Selling an option is also known as writing the option.


Types of derivative traders in the market

• Hedgers - use derivatives to reduce risk in the market from potential futuremarket movements.

• Speculators - trade derivatives to bet on the future direction of a marketvariable.

• Arbitrageurs - take offsetting positions in two or more instruments to lock in aprofit. (usually short-lived)


Hedgers - reducing risk exposure

Example: Using forward contracts. Recall

Spot and forward quotes for USD/GBP exchange rate, June 1, 2007

Bid OfferSpot 1.6281 1.6285

1-month forward 1.6248 1.62533-month forward 1.6187 1.61925-month forward 1.6094 1.6100

• June 1, 2003. ImportCo knows it needs to pay £10 million on September 1,2007 for goods it purchased from a British supplier. ImportCo can hedge itsforeign exchange risk by buying points from the financial institution in the3-month forward market at 1.6192. This fixes the cost at $1,6192,000.


Example, cont.

• Next, another US company ExportCo, is exporting goods to the UK. It willreceive £30 million 3 months later. ExportCo can hedge its foreign exchangerisk by selling £30 million in the 3-month forward market at an exchange rateof 1.6187. Then ExportCo is assured of $ 48,561,000.

• A company may do better without hedging - consider ImportCo. If it does nothedge then the £10 million may cost $ 15 million if the exchange rate drops to1.500, :) . On the other hand if the rate increases to 1.7000 then the costbecomes $17 million, :( .

Hedging forward contracts locks in particular rates that may or may not be betterthan the spot contract. However, it reduces uncertainty.


Another example of Hedging

Example: Using option contacts.

• Consider Fred who in May 2003 owns 1000 shares of Microsoft. Current price is$28.

• Concerned about possible decline in next 2 months and wants protection.

• Investor buys ten July put option contracts on Microsoft on the CBOE withstrike price of $ 27.50. This would give Fred the right to sell 1000 shares ofMicrosoft for a price of $27.50 per share. If the option price is $1, then eachoption contract costs $1× 100 = $100 and the total cost of the hedgingstrategy is $ 1000, where the value of the portfolio is $27,500.

• If the stock falls below $27.50, then the options can be exercised. When theoption is exercised the amount realized is $26, 500. If the market stays above$27.50, then the options are not exercised and expire.

• The portfolio always retains a value of at least $26,500 .


Summary: hedging

• Forward contracts neutralize risk by fixing prices that the hedger will pay orreceive for the underlying asset.

• Options contracts offer insurance for investors to protect against adverse pricemovements.

• Options involve upfront fee, unlike forwards.


Speculators - increasing risk

Example of speculating using stock options.Using $2000 to speculate on Amazon.com stock.

Dec. Stock price Dec. Stock priceInvestor Strategy $15 $27Buy 100 shares ($500) $700

Buy 2000 call options ($2000) $7000

• The stock is currently selling at $20, and 2 month call option with $22.50 strikeprice is currently selling at $1. Two scenarios above for investing $2000.

– Alternative 1: purchase 100 shares. Suppose stock rises to $ 27, then profitof 100× [$27− $20] = $700. If the stock drops to $ 15, then the investorloses 100× [$20− $15] = $500.


Speculators, cont.

• Alternative 2: purchase 2000 call options. If the price rises to $ 27, then profitof $4.50 per share yields

2000× $4.50 = $9000.

However, we paid $ 2000, so we profit $7000. This is ten times more profitable!

• On the other hand if the price drops to $15, then loss of $ 2000, since theoptions are not exercised, but we paid $ 2000 for the options (compared to($500) in the stock transaction).

Futures and options speculation both offer leverage (high risk, high return).However, forward contract speculation can create very large losses; whereas,

options contract losses are limited to the price of the contract.


Arbitrageurs

Third important group of traders in derivatives. Involves locking in riskless profitby simultaneously entering into transactions in two or more markets.

Mostly possible when future prices become out of line with spot prices.

Example:

• Consider a stock being traded on NYSE and LSE (London Stock Exchange).Suppose the price of the stock is $ 172 in New York and £100 in London at atime when the exchange rate is 1.7500 per pound.

• Arbitrageur simultaneously buys 100 shares of stock in New York and sellsthem in London to obtain a riskless

100× [($1.75× 100)− $172] = $300.

• Transaction costs may gobble up most of the profit for small investments, butlarge financial institutions could profit. Furthermore, arbitrage opportunities arequickly lost.


Homework

Due Sept. 5, 5PM.

• 1.1-1.4, 1.6, 1.13, 1.19

• Graded: 1.26, 1.27, 1.28, 1.29


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Mathematics of Options, Futures and Derivatives Securities

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