Introduction

Definitions

A. Option Contract- Obtain rights in exchange for monetary transfer

a. Call- right to purchase underlying security at a specific price

b. Put- right to sell c. American vs. European Optiond. Writer- seller of option contracts.

B. Futures Contracta. Delivery of a specified good by the seller for

payment of an agreed amount at some fixed future date.

b. Both sides of an obligationc. Cash does not change hands between buyer and

seller until delivery.

C. Option on futures Contract- Option to buy or sell a futures contract. Allows for standardization of underlying asset.

D. Swaps- Exchange of cash flows between two parties. Generally fixed for flexible.

E. Arbitrage-Simultaneously purchase and sell the same good at different prices. Forces prices to converge.

Why these markets are valuable

a. Modification of risk-return opportunity set.a. Hedging- reduces risk of holding the underlying

asset

b. Speculation- highly levered portfolio that is easily obtainable.

b. Provides information concerning the underlying asset displayed in an organized market.

c. Provides for more efficient markets by creation of arbitrage opportunities.

Options market Overview

Variables that describe an options contract.

1. Type (put, Call)2. Underlying asset (IBM, GE etc)3. Exercise or strike price

4. Expiration date 5. Example IBM March 85 call

Types of Underlying assets

1. Stock Options (65%)2. Stock Index Options (15%)3. Commodity Options (5%)4. Futures Options (5%)

Typical Specifications on American Stock Options

1. Options traded in 3-month cycles with up to 2 adjacent months.

2. Generally 3 to 5 dates traded at one timeNeed choices for spreads and various risk trade-offs.Also want liquidity.

3. Strike prices in $2.50 increments till $25/share then $5.00 increments

4. New strikes prices open up as prices change.5. 100shares/option

Open Interest

1. Number of contracts outstanding2. Up if both buyer and writer are establishing new position.3. No change if either side is liquidating position4. Down if both sides are liquidating position5. example

Market Mechanics

1. Exchange Floor participantsa. Floor brokers- Commission traders who cannot

trade for own acct.b. Market makers- Trades only for own accountc. Order Book Official- Keeps track of all limit orders

2. Clearing housea. Matches buyers and sellers each day.b. Assigns writer in delivery process.c. Serves regulatory function (sets position limits,

guarantees writer performance etc.)

3. Exercise procedurea. Buyer has right to exercise.b. Writer is assigned obligation from clearing housec. Options expire on 3rd Friday of delivery month.

All outstanding options with value are exercised at that time.

d. Opening rotation - starts earliest month low strike to high. May take as long as 10 minutes to establish prices.

Introduction to Valuation

Terminology

1. Intrinsic valuea. Call = Security Price minus Exercise Price

if positive otherwise 0.b. Put = Exercise Price – Security Price if

positive otherwise zero.

2. Time value = Observed option price minus its intrinsic value.

3. In-the-money = option with positive intrinsic value

a. Stock price > exercise price for callsb. Stock price < exercise price for puts.

4. Out-of the money = intrinsic value is zero.5. At expiration an Option is always worth its

intrinsic value.

Graphical depiction.

1. Maximum value for put but not for call2. Minimum value zero for both given limited

liability of option.3. See graph

Notation to be used

1. S = Stock price2. E = Exercise or Strike Price3. T = Time till Expiration4. i = Risk Free interest rate. 5. C = Call Price

6. P = Put Price7. PV( ) = Present Value of what’s in Parenthesis

Boundary Limit Propositions

Assumptions

1. Call options will be considered first2. More is preferred to less3. No taxes or transactions costs4. American Option unless otherwise specified.5. No Dividends

Proposition 1.

C > 0 this is the minimum call priceProof:

Since there is limited liability for the option it can never be worth less than zero at expiration. Therefore it can’t be worth less than zero today.

Proposition 2.

C > Max (S-E ,0) Call must be worth its intrinsic value.

Proof:If not buy option and exercise.Example

S = 25 E = 20 C = 4Buy Option -4 Must pay for the optionExercise -20 Must pay for the stock

Sell Stock +25 market priceProfit +1

Graph of new boundary

Proposition 3.

C(E1) > C(E2) given E1 < E2. Lower exercise price worth at least as much as higher exercise price all else equal.Proof:

Consider a portfolio of buying E1 option and selling E2 option.Look at all possible portfolio values at expiration.

1 2 3S* < E1 < E2 E1 < S* < E2 E1 < E2 < S* C(E1) = 0 C(E1) = S*- E1 C(E1) = S* - E1C(E2) = 0 C(E2) = 0 C(E2) = S* - E2

Port = 0 Port = S*- E1 Port = E2 –E1

Therefore since portfolio will never have negative value at expiration it can’t have negative value today which it would if C(E1) < C(E2). If it did you could buy the portfolio today collect the cash flow and never have to pay it back.

Also since the maximum value of the port at expiration is E2-E1, the maximum difference is price has to be the present value of E2 – E1. If

not sell C(E1) buy C(E2) and put proceeds in bank. Bank value at expiration will be greater than E2 –E1, which is the most you will have to pay at expiration.

Show on Graph.

Proposition 4.

C(T2) > C(T1) given T2 >T1. Options with a longer time to expiration cannot have less value. Proof:At expiration time T1, C(T1) = max S*- E or 0 and from Prop 2. C(T2) must be worth at least the same so it has to be worth at least the same today.

Proposition 5.

C < S Proof:Stock can be considered an option with infinite time to expiration and zero expiration price. From Prop 3 and 4 its must be worth at least as much as any option.

Proposition 6.

C > Max(0, S-PV(E)) A Call option must be worth its discounted intrinsic value.Proof:Consider two portfolios: Port A: Own a CallPort B: Own stock

Borrow PV(E)Compare all possible values of each portfolio at expiration

S < E S >EPort A C =0 C= S*- EPort B S= S* S=S*

Must pay back E Must pay back E

S*- E <0 S*- E Port A more valuable Portfolios have

equal value

Therefore, since at expiration Portfolio A will always be worth at least as much as Portfolio B, it must be worth at least as much today.

Example of an arbitrage situation:C = 3, S =55, E = 50, PV(E) = 49Call is below discounted intrinsic value which is $5.Buy Call –3 (Buy Port A) todaySell Stock +55 and Lend $49 = -49 (Sell Port B) todayPositive cash flow of 55-49-3 = $3 today.

Since we know the call will be worth at least the other two at expiration we keep the $3 and maybe more if S*<50.

Show graph with new boundary condition.

Implications1. longer time to maturity implies higher

lower boundary since PV(E) is smaller i.e S- PV(E) larger.

2. higher interest rate implies higher lower boundary. Same reasoning.

4. both shift lower boundary to the left in graph.

Variables used to determine the Call Price

Variables determining Intrinsic Value1. Exercise Price Negatively related to call price2. Stock Price Positively related to call Price

Time Value3. Interest rate Positively related to Call price

Think of owning a call as not having to purchase stock till some future date. Therefore higher interest rate implies larger savings.

4. Volatility Positively related to Call Price

Profit on gain but limited liability on loss.

5. Time to maturity Positively related to Call priceBetter chance to profit on gain and more savings by not having to buy till later.

Show relationship movements on Graph.

Proposition 7:

Given on Dividends, never exercise an American Call early.

Proof:If exercised buyer receives intrinsic value i.e. Max(0, S-E)However before expiration from Prop 6

C > Max (0, S – PV(E)) which is greater than intrinsic value. Therefore never exercise early. Hence American Call price = European Call Price

Intuitively, this is because as seen above calls always have positive time value before expiration and exercising foregoes that value.

Proposition 7’:

If Stock pays dividend an American Call may be exercised early.

Proof :

Stock price falls on ex-dividend by the amount of the dividend. However, since option holder is not stockholder, he does not receive Dividend. Therefore, if dividend is greater then time value left in the option, he is better off exercising right before ex- dividend since lost time value will be less than lost intrinsic vale from stock price decline.

Example:

S = 52 E =50 Company will pay $1 dividend tomorrow. There is $.50 of time premium left in option.

If not exercised option will be worth $1.50 tomorrow. Therefore, better off getting $2.00 today.

If Time premium was $1.25 no need to exercise since option is worth $2.25.

Put Propositions

Proposition 1:

P > 0 Proof: Do to limited liability cannot be worth less than zero at expiration.

Proposition 2:

P > Max (E-S, 0) Put must be worth it’s intrinsic value.Proof: If worth less buy option and exercise.Example

S = 22 E = 25 P =2Buy Put -2Buy Stock -22

Exercise +25 buy delivering stockProfit +1

Graph boundary

Proposition 3:

P(E1) < P(E2) given E1 < E2Same as call example by forming portfolio of buying P(E2) and Selling P(E1).

Look at all possible portfolio values at expiration.

1 2 3S* < E1 < E2 E1 < S* < E2 E1 < E2 < S*P(E1) = -(E1- S*) P(E1) = 0 P(E1) = 0P(E2) = E2 - S* P(E2) = E2- S* P(E2) =0

Port = E2 - E1 Port = E2- S* Port = 0

Therefore since portfolio will never have negative value at expiration it can’t have negative value today which it would if P(E1) > P(E2). If it did you could buy the portfolio today collect the cash flow and never have to pay it back.

Also since the maximum value of the port at expiration is E2-E1, the maximum difference is price has to be the present value of E2 – E1. If not sell P(E2) buy P(E1) and put proceeds in bank. Bank value at expiration will be greater than E2 –E1, which is the most you will have to pay at expiration.

Show on Graph.

Proposition 4:

P(T2) > P(T1) given T2 >T1. Options with a longer time to expiration cannot have less value. (American options only.)

Proof:At expiration time T1, P(T1) = max E - S* or 0 and from Prop 2. P(T2) must be worth at least the same so it has to be worth at least the same today.

Proposition 5:

P < (E) Put cannot be worth more than its exercise price.Proof:Since stock minimum is zero maximum put intrinsic value is E. Therefore it cannot be worth than E at expiration or worth more than the Present value of E today. (American Option).

Graph.

Proposition 6:

P > Max ((PV(E) –S) , 0)

Proof: Consider Two portfoliosPortfolio A : Own a put

Portfolio B : Sell stock Lend PV(E) amount of cash

Compare all possible values of both portfolios at expiration S < E S >EPort A P = E – S* P= 0Port B Stock = -S* same as before

Gets E back E – S* for both Port A more valuable

Since 0 > E-S*

Show graph

This boundary is less restrictive then intrinsic value boundary. Proposition shows a negative component to the time value of a put.

Implications1. longer time to maturity implies lower

boundary since PV(E) –S* is smaller than E-S*.

2. higher interest rate implies lower boundary. Same reasoning.

both shift lower boundary to the left in graph.

Variables used to determine the Put Price

Variables determining Intrinsic Value1. Exercise Price Positively related to Put price

2. Stock Price Negatively related to Put Price

Time Value3. Interest rate Negatively related to Put price

Think of owning a call as not having the proceeds from selling the stock until some future date. Therefore higher interest rate implies larger foregone income.

4. Volatility Positively related to Put Price

Profit on gain but limited liability on loss.

5. Time to maturity Could be positively or negatively related to Put price

Better chance to profit on gain but more foregone income buy not selling stock till later.

Show relationship movements on Graph.

Proposition 7:

An American Put may be exercised early.

Proof:If (E – S)(FV) > E-S* then you are better off exercising today because value at expiration is greater than max value on Put.Doesn’t hold for call since max value does not exist.

Intuitively early exercise will also occur if negative component of time value dominates to positive component. This happens if there is certainty that E – S* will occur at expiration. In this case owning a put will always give you the same payout as selling stock and lending so its better to get the funds today.

Show on graph

Put- Call Parity

For European options

P = C – S +PV(E)

Proof: Consider the following PortfoliosPortfolio A: Own a PutPortfolio B: Own a Call

Sell Stock Lend PV(E)

Consider all possible portfolio values at expiration

S* < E S*>E

Port A: P = E – S* P=0Port B C= 0 C = S* -E

S = -S* S = -S*Get E back Get back E

Port B value E - S* Port B = 0Both Ports same value Both Ports

have same value

Therefore if both worth the same at expiration then they must be worth the same today. If not buy port with less value and sell one with higher value.

Implications1. If S=E Call will be worth more than Put.

Both options only have time value and Call value is higher.

2. May not hold for American options. Why?

Index Options

Underlying Asset

1. Consists of a portfolio of securities2. Gives purchaser to right to purchase an

amount of dollars equal to the index value multiplied by some multiplier.

3. Examplea. S &P 500b. Nasdaq 100

c. S &P 100 (OEX)

4. Exercise procedure consists of cash settlement as determined by multiplying the intrinsic value at the close of trading by a given multiplier.

a. Exampleb. OEX 450 Call with S* = 456.50. Cash

settlement would be $6.50 * 100 = $650.c. Can be difficult to create cash position in

an arbitrage.

Extra Risks with cash settlement

1. Timing risk – Writer doesn’t know of exercise till the following day.

i. Cash settlement in effects liquidates position in the market.

ii. Buyer can announce intent to deliver for about a half hour after the market closes.

iii. Therefore, if news comes out after the close buyer has option to liquidate position before the opening the next day.

iv. Not true with stock options since stock is received upon exercise

v. Example 1. Index value = 246 a 240 C = 7 at

the close of trading.2. Bad news comes out on the market

and the Index value is expected to go down 2 points on the open.

3. Can liquidate now and get $6 will only be worth about $5 or so tomorrow ($4 of I.V. $1 T.V.)

4. With stock options you get the stock so you still take the loss.

2. Early exercise risk during trading hours. i. Option may trade below its intrinsic value

during trading hours without pure arbitrage available.

ii. Because intrinsic value is not determined till the end of trading that day.

iii. Example

Use of Binomial Model to obtain Specific Option Price

Beginning assumptions

1. Perfect capital markets (no taxes, transactions costs, etc)

2. No dividends3. Risk free interest rate known and constant4. Distribution of stock price movements known5. European option6. Two possible outcomes for every period in

time

Steps in determining no arbitrage option price

1. From portfolio theory since options and stock are perfectly correlated a risk free portfolio can be created.

2. Buy stock and sell calls or buy stock and buy puts.

3. Assume there is one-period till expirationa. Observe possible stock and option

combos at expiration that yields the same value for both states of nature. (Sell calls and buy puts)

b. Let S + - C+ = value if Stock price risesc. Let S - - C- = value if Stock price fallsd. Find share of stock purchased per call

sold to equate both states. (Delta)S + - C+

= (Delta)S - - C-

e. Delta = (C+ - C-)/(S+ - S-) Will always be between zero and one.

f. This is a risk free portfolio, therefore value can be discounted at risk free rate to find portfolio value today.

g. Given stock price is known today can solve for option price today.

4. Examplea. S0 = $50, E = $50 Prices will rise or fall

50% next period. Risk free rate = 25%.b. Delta = (25 – 0)/ (75- 25) = ½ c. Own one stock and sell 2 calls.d. Portfolio values at expiration are

i. 75 – 2(25) =25 orii. 25 –2(0) = 25iii. worth $25 no matter what state occur

e. Value today is $25/(1 + .25) = $20f. $20 = S –2C =$50 –2C or C =$15g. show one period tree

5. A pure arbitrage exists today if call does not equal $15

a. Say C=$10 then buy two calls sell stock and lend $20.

b. To T1- T1+i. Buy 2 calls -20 0 +50ii. Sell stock +50 -25 -75iii. Lend $20 -20 +25 +25iv. Value +10 0 0v. Keep $10 for all states

c. Do Opposite if C > $15

6. Observationsa. Two assets can be combined to make a

third, therefore arbitrage forces deterministic price.

b. No investor risk preferences needed.c. Stock price is the only random variable

market risk does not matter.d. Probability of up and down movement not

used.e. Hedge ratio always between 0 and 1. f. Show graph from example.

7. Call price depends on the following variablesa. Size of stock price movement (Vol ) S+

=80 and S- = 20 implies C = 17b. E = 60 implies C = 7c. Int rate = 20% implies C = 14.58d. Longer time to expiration (must look at

two period model)

8. Two- Period Model.a. Must start at expiration and work

backwards for each state of natureb. Show two period tree and example. c. Hedge ratio must be adjusted each period

9. One Period Put a. Combo is for purchase of stock and putb. From example S+ + P+ = (Delta) S- + P-c. Delta = (P+ - P-)/ (S+ - S-) d. Delta = -25/50 = -1/2. Buy 1 stock and 2

puts e. Value at expiration is $75 in both statesf. $75/(1.25) = $60 = S + (2)P = 50 =2(P)

g. P = $5

10. Two period put is worth less why?

11. General formula for n period binomiala. C = S x F(a,n,p) – E x f(a,n.p)/(1 + r)n

b. Where i. F = binomial distribution functionii. n = number of periods till expirationiii. a = lowest # of up moves for call to

be in the money at expirationiv. p = size of the up or down movev. f(a,n,p) = hedge ration to be

readjusted each period.

Black- Scholes Formulation

Added assumption to binomial

1. Market operates continuously2. Stock price movements are also continuous3. Stock Price follows a log normal distribution

due to limited liability on the downside.4. show graph

Value of a call is the Present value of expected cash flows from holding option.

1. Find the expected area under the curve from log normal distribution.

2. If cash flows can be evaluated at some end point with specific boundary conditions then the differential equation can be solved.

3. Relate to graph

The Black- Scholes equation is C = S . N (d1) – E .

e-rt .N ( d2)

1. N( ) = cumulative normal distribution function2. d1 = (log(S/E) + (r + .5 . var) (T))/ (Stand Dev. .

T1/2)3. d2 = d1 – ((s.d.) . T1/2)

4. T = Time to maturity in a fraction of a year5. r = risk free rate6. N (1.00) implies .8413 is Prob of a

standardized variable (z- stat) is under 1.00.7. Show in graph

Interpretation of Equation

1. As variance goes to zero, N(d1), and N(d2) approach one since Z score up.

a. C = S – E e-rt b. No volatility premium only time value of

moneyc. Show on graph

2. N(d1) = hedge ratio or delta of position = C/S

a. This is the slope of the line on graph.b. Changes in S, T, or will change delta

i. S up implies delta up since N (.) up.ii. T up implies delta could go up or

down depending on in or out of the money

iii. up implies same could be up or down

iv. r up implies delta up.c. Rule of thumb is that delta is approximately .5

around PV(E).

3. Gamma = Delta / Stock pricea. Measure of portfolio stabilityb. Positive gamma implies that delta up

when stock price upc. Largest when stock price is around

exercise price.d. Also becomes larger when Time to

expiration is shorter and stock price around exercise price.

e. Becomes smaller when time to expiration is shorter and stock price is not around exercise price.

f. See graph.

4. Theta = C/T Also known as time decaya. Multiply change in time by theta to get

price changeb. Considered negative for calls since calls

lose value over time.c. Also not linear. Theta increases as

expiration approaches.d. Most negative at the money.

5. Kappa = C/a. Positive for calls. As volatility goes up call

price goes upb. Also greatest at the money.

Put Valuation

1. Black-Scholes Modela. P = C – S + PV(E)b. P = E . e-rt . (1- N ( d2)) - S . (1 -N (d1))

2. Put Delta = N(d1) – 1 a. Same as call delta –1b. Always negativec. Call delta + absolute value of put delta

always equals one.

3. Put Gamma a. Same as call gamma.b. Always positive and largest at-the-money

4. Put Theta a. Could be positive or negative.b. Depends on the various time value

components

5. Put Kappa same as for calls

Other Uses of B/S Valuation model

1. Can get implied volatility of stock given option price

2. All options should be priced with same volatility

3. Delta or Hedge ratio important to arbitrage mispriced options.

Show real world examples of option characteristics

Problems with B/S in practice

1. Can only be used if life of option knowna. Problems with puts and calls with

dividendsb. Multi-period Binomial may be a more

accurate model

2. Must take dividends into account a. Substitute S – PV(div) for S into B/S

equationb. Dividend must be knownc. Option must not be exercised early

3. Must continuously adjust hedge ratio to stay risk free if arbitrage appears available.

4. What is true measure of Volatility?

Strategies

Background

1. Types of positionsa. Naked – Purchase or sale of a single type

of optionb. Hedged- Portfolio of both underlying asset

and optionc. Spread- Purchase of one option and sale

of anotherd. Combination- Portfolio containing both put

and call options

2. Profit graphsa. Look at profits at expiration for various

potential stock pricesb. Profit on vertical axis Stock price on

horizontal. c. Example Stock ownership

3. Analyze components before expiration (delta, gamma, theta, kappa)

Naked Positions

1. Call Buyinga. Show expiration graph b. Example S =49, E = 50 C = 3 Delta =.45

Gamma = .04i. Max loss = 3ii. Breakeven at expiration is S* = 53

c. Positive Delta, gamma, Kappa

d. Negative Theta i.e. the position loses money over time

e. Show today’s graph f. Which option to pick (exercise price and

time to exp.)i. Lower exercise price implies less

leverage, more stable delta.ii. Longer time implies greater premium iii. Graph of different exercise prices

2. Call Writinga. limited profit, relatively unlimited

downsideb. Show graph and previous example for

writerc. Negative Delta, Gamma and Kappa. d. Positive Theta i.e. the position makes

money over time. e. I-t-m vs. o-t-m

3. Put Buying a. Maximum loss, limited profit b. Example E= 50 S = 49 P = 2 Delta = -.55

Gamma =.04c. Show expiration day graphd. Negative delta, Positive Gamma, Neg

Theta (Generally), Positive kappa.e. Show today’s graphf. Time premium lower with puts. g. In-the-money vs. out-of-the-money.

4. Put Writinga. limited profit, relatively unlimited

downside

b. Show graph and previous example for writer

c. Positive Delta, Negative Gamma and Kappa.

d. Positive Theta i.e. the position makes money over time.

5. Buying Stocka. Show graphb. Delta is always positive one, gamma is

zero theta is zero.

Synthetic Positions

1. Synthetic Long Stock a. Buy call + sell putb. Combine individual graphsc. Example E = 50 S = 49 C = 2 P =2

i. Deltas must add to one ii. Graph example

d. Smaller investment for synthetic i. Implies must have smaller profit at

expirationii. Position will lose money over time.

Losses should equal the interest lost on buying stock relative to buying synthetic stock.

iii. Call premium is greater than put premium.

2. Synthetic Short Stocka. Sell Call + buy Putb. Delta always equals –1.

c. Profits are greater for synthetic then shorting stock.

d. Graph

3. Synthetic Puts and Callsa. They follow from Put-Call parity.b. Buy Put = Buy Call and Sell Stock

i. Max loss is the premium on Call plus (E-S)

ii. Example E= 50 S = 48 C=11. max loss is three at expiration2. breakeven is at S = 47

iii. Difference between synthetic put and actual is the interest earned on selling the stock to create put.

iv. Show graph

c. Buy Call = Buy put + Buy stocki. From above example maximum price

must the put be?ii. Synthetic Call will be more expensive

since you have to borrow to buy stock.

iii. Show graph

Hedged Positions

1. Covered Call Write (Write Call against Stock Ownership)a. Gives downside protection but limits

upside gainb. Show expiration day graphc. Example E=50 S=51 C=4 Delta = .55

i. Max profit is $3 for S* =50 or more. Breakeven is S =47

ii. Go through example in detail. d. Delta of position = Stock Delta (+1) – Call

Deltai. Delta goes from 1 to zero as stock

price risesii. What will of position do over time?

e. Gamma of position is negative since delta down as stock price rises.

i. Position gets more bearish as prices rise and more bullish as prices fall.

ii. Don’t want a lot of movementf. Theta is positive

i. Position makes money over timeii. This is because we sold time

premium on the call.g. Different exercise prices

i. O-t-m 1. higher max profit (allows stock to

appreciate) 2. higher breakeven ( Collect

smaller premium)3. Example E= 55 S =51 C =1 4. Max profit = 5 B.E. is 50

h. Check volatilities to find best option

2. Protective Put (Put Buying with Stock Ownership)

a. Limited liability with no Maximum profitb. Show expiration graphc. Example E = 50 S = 51 P = 1 ½ Delta =

-.45

i. Maximum loss is 2 ½ii. Breakeven is S = 52 ½

d. Delta of Position = stock Delta + put Delta

i. Always positive between zero and one

ii. .55 from examplee. Gamma is Positive (As stock price rises

put delta falls therefore position delta rises).

f. Graph with today’s positiong. Theta is negative

i. Position loses money over timeii. Bought option so lose value over

time.h. I-t-m vs. o-t-m same as for calls

3.Comparing Covered Calls and Protective puts

a. Graphical comparison b. From example can determine under what

various stock price each strategy dominates

i. Calls max profit is 3 and Put max loss is 2 ½

ii. Therefore Calls will dominate between S =44 ½ and and S = 55 ½ Puts otherwise

iii. Calls better for small movements Puts better for large movement

iv. What happens as volatility of Stock rises? Does it change your opinion?

Complex Hedges

1. Ratio Call Write a. a. Sell more than one Call against

stock ownershipb. b. Advantages and disadvantages over

simple covered calli. Adds more premium income therefore

increase downside protectionii. Creates exposure on upside because of

uncovered short call positionc. Construct expiration day graph for 2/1 ratio

i. Max Profit occurs where E = Sii. Breakeven points for both upside and

downsided. Example S = 49 C = 3 E = 50 Delta of Call

= .45i. Max profit = 7 Downside breakeven

=43 upside Breakeven = 57ii. Profits eared over a range

e. Delta of 2/1 position i. Delta goes from +1 to negative one.

Near +1 if options are way out of money. –1 if way I-t-m.

ii. From example the delta of position is zero i.e. “delta flat”.

iii. Position becomes bearish as stock prices rise and bullish as stock prices fall.

f. Gamma i. Negative since as stock prices rise delta

falls.ii. Don’t want movement in stock.

iii. Gamma is greater than for simple covered call since there are more options

g. Theta is positive position makes money over time

h. What happens with higher ratios?

2.Variable Ratio Write a. Sell Calls with two different exercise pricesb. Maximum profit occurs over a range

between the exercise prices.c. More stable than ratio write since gamma

is smaller especially around exercise prices. Lower max profit since one option is in-the-money and one is out.

d. Example and graph.

4. Ratio Put writea. Sell stock and sell more putsb. Same graph as ratio call writec. Therefore same characteristicsd. Graph and example

5.Reverse Ratios have exactly opposite characteristics

Spread Positions: Purchase of one option and sale of another

1. Bull Call Spread a. Buy low E sell high Eb. Always a debit transactionc. Bull spread because it makes profit if prices rised. Max profit and max losse. Example Buy C45 = 3 and Sell C50 = 1 S =45

i. Max gain = 3 and Max loss = 2ii. show graph

f. Delta is Positive to neutral since lower E has higher delta. ( Put deltas on example position)

g. Gamma is neutral goes from slightly positive to slightly negative

h. Theta also can be positive or negative. Depends on whether low E or high E option dominates

i. O-t-m spreads more aggressive max profit higher but need more movement for profits. Delta is more positive.

2. Bull Put Spread A. Buy low E and sell high EB. Credit transaction C. Same characteristics as bull call spreadD. Show graph

3. Bear Spreadsa. Sell low strikes and buy high strikesb. Profits if stock price fallc. Cash inflow for calls outflow for putsd. Graph e. Delta negative, all others similar as bull

spreads.

4. Butterfly Spreadsa. Uses three exercise pricesb. Combines a bull and bear spread

around the middle pricec. Buy low strike, sell 2 middle strikes and

buy high striked. Show basic graphe. Example of call butterfly S =49 , C45 = 6

C50 = 3 C55 = 1 i. Max loss at S = 45 is 6 –

6 + 1 =1ii. Max profit at S= 50 is –1

+ 6 –1 =4iii. Graph example

f. Delta is relatively flat. Goes from slightly positive to slightly negative.

g. Gamma is also relatively flath. Theta depends on locationi. Risk is low with limited profits. May

have high commissionj. Reverse ratio buy middle E sell outsides

6.Ratio Call Spreada. Buy Low E Calls and sell more high E Calls

b. May be either a credit or debit transaction.

c. Graphi. Below Low E keep proceeds ii. Between low and high E keep

premium from high E and get return on low E.

iii. Above High E, lose on uncovered call.d. Examplee. Buy C50 = 5 and Sell two C55 = 2 where S =

54i. Max loss at E =50 is –5 + 4 = 1ii. Max gain at E = 55 is 0 + 4 = 4iii. Draw graph

f. Delta of position goes from Positive to negative

i. From example what is Delta/ii. What is smallest delta from example

g. Gamma is negative for most of the position

i. As long as neg delta from sales are greater than delta from purchase

ii. When is gamma positive?g. Theta is positive over most of the positionh. What happens as ratio increases?i. Ratio spread vs. ratio write

i. No downside risk with spreadii. No premium with stock so higher max

profits with ratio writeiii. No dividend with Spread.

7.Ratio Put Spreada. Buy high E sell more low E

b. Graph positionc. Similar characteristics to Ratio Call spread

8. Reverse Ratio spreads (Backspreads)a. Sell low E buy more high Eb. Maximum loss is at high E c. Can flip example aboved. Graphe. Characteristics are opposite to ratio

spreads.f. Watch out for early exercise on in the

money options that are sold especially with puts.

9.Time Spreada. Sell Nearby option and buy Option with

longer time to maturity.b. Cash outflow since longer term has higher

price.c. Example E = 50 S=48 C(t1) = 2 C(t2) = 5

i. Max loss at T1 is equal to initial investment since worst case is all options have zero value

ii. Max Profit is uncertain because price of C(T2) is not certain at T1.

d. Graph at expiration of nearby optioni. Max profit takes place at S* = E.

1. Since at prices below E keep premium on nearby while deferred increases in value as S increases.

2. While at prices above E the position loses dollar for dollar on

nearby option but only makes delta on deferred.

ii. Max losses at the tailse. Delta could be positive or negativef. Gamma is generally negative except in

the tails g. Theta is generally positive except again in

the tails.

Combination Positions

1. Straddlea. a. Buy a Put and a Call with the same

characteristics.b. b. One will be in-the-money and one will be o-t-

m.c. Max loss is S* = E since neither option will

have any value.d. Example E = 45 S = 44 P =2 C = 2

i. Max loss is 4 at 45ii. Show graph

e. Delta could be positive or negative. From example?

f. Gamma Positive, Theta Negative, Kappa Positive

2. Stranglea. Different Exercise Prices of Put and Callb. Less initial investment if both o-t-m but

max loss is greater.c. Example of o-t-m strangle S= 37 C40 = 2, P35 = 2

ii. Max loss is $4 between S*= 35 and 40 since both options are worthless.

iii. Breakeven is at 31 and 44.iv. Show graph

d. Example of I-t-m strangle S= 37 C35 = 4, P40 = 4

i. max loss between S* = 35 to 40 is $8 (initial Investment) - $5(value of put plus call) = $3.

ii. Breakevens are 32 and 43. iii. Show graphiv. Compare the two.

3. Sell either just the opposite.

Arbitrage Positions

1.Conversion (Buy stock and buy put, sell call)

a. Position is simultaneously buying and selling stock.

b. Profit if cost of position is less than the present value of the exercise price.

c.Example S = 53 C50 = 5 P50 = 1 i. Cost of position is 53 – 5 +1 = 49ii. At expiration position is always

worth 501. S* > 50 implies S* - (S*- 50) + 0

= 502. S* < 50 implies S* - 0 +(50- S)

= 50

iii. Since its always worth E or 50 at expiration it must be worth the present value of E today.

iv. Otherwise but position finance and collect 50 at expiration for arbitrage profit.

2.Reversal (Sell stock, sell put and buy call)

a. Profit if credit from position is greater than the present value of E.

b. Position will cost E at expiration for all states of nature. Therefore if the credit today plus interest on the credit held till expiration is greater than E an arbitrage profit exists.

c. Example

3.Debit Box Spread (Buy Bull call spread and Bear Put Spread)

a. Buy low E and sell High E Call, and Sell low E and Buy high E put.

b. Position must be worth the difference in exercise prices at expiration. Therefore it must be worth the present value of the difference today. If less, then buy borrow and collect value at expiration for a profit.

c. Proof. Buy C50 ,Sell C60 , Sell P50 , Buy P60

i. If S*>60 call spread worth 10 Put spread worth 0 at expiration

ii. If S* < 50 Call spread worth 0, Put spread worth 10 at expiration.

iii. If 50 < S* < 60 then the sum of the two spreads worth 10 at expiration.

iv. Therefore position must be worth the present value of 10 today. If less than buy spread and borrow. Will pay back less than 10 at expiration and collect 10 for position.

4.Credit Box Spread (Buy Bear call Spread and Bull Put spread)

a. Sell low E and buy High E Call, and buy low E and Sell high E put.

b. Position must cost the difference in exercise prices at expiration. Therefore it must be worth the present value of the difference today. If more , then buy lend credit and collect value at expiration for a profit.

c. Proof. Sell C50 ,Buy C60 , Buy P50 , Sell P60

i. If S*>60 call spread cost 10 Put spread worth 0 at expiration

ii. If S* < 50 Call spread worth 0, Put spread costs 10 at expiration.

iii. If 50 < S* < 60 then the sum of the two spreads costs 10 at expiration.

iv. Therefore position must be worth the present value of 10 today. If more than buy spread and lend credit. Will be worth more than 10 at expiration and repay 10 for position.

Value of the Firm in an Options Framework

I. Assumptions A. Pure discount Bonds issuedB. No DividendsC.Capital Structure UnimportantD. Other B/S Assumptions applyE.Let Vo = So + Bo

1. Where Vo is the value of the firm today

2. So is the value of equity today

3. Bo is the value of debt today

II. Value of EquityA. A. Consider equity an option with Exercise being

the face value of the debt and the time to expiration the maturity date of the debt.

B. B. Intrinsic Value of So > ( V* - Bd , 0) Where Bd is face value of debt.

C. C. GraphD. D. Price today can be obtained from Black/Scholes

model.1. So = Vo * N(d1) - Bd e-rt * N(d2)2. Same characteristics as call option 3. Delta positive, Gamma Positive, Theta

Negative, Kappa Positive.

III. Value of Debt

A. Intrinsic value of B = Min (Bd , V*) Either the bond gets paid off or the debt holders get the firm.

B. Graph at expirationC.Bond purchase could be considered as buying

the firm and writing a covered call to sell it back at the face value of the debt.

D. Can find Bo from total value and B/S equation.a. Bo = Vo - So

b. = Vo - (Vo * N(d1) - Bd e-rt

* N(d2))c. = Vo * N(-d1) + Bd e-rt *

N(d2) E.Delta is Positive, Gamma negative Theta

Positive, Kappa Negative.F.Show today’s graph

IV. ImplicationsA.V* up implies B up and S up. B.Vol up implies B down and S up.

i. Especially around the exercise priceii. Limited liability for stockholder, Max Profit

for bond holder.iii. Shifts value from bondholder to stockholder

C.T down implies B up and S downD.i up implies B down and S up.

V. Convertible Bonds

Futures Markets Introduction

I. Definitions

A. Futures Contract- Delivery of a specific good by the seller for payment of an agreed amount at some fixed future date.1) Both sides have an obligation2) Everything but price fixed by exchange3) Cash does not change hands till delivery

B. Forward vs. Futures Contracts1) Individual Agreement vs.

Organized Exchange 2) Implies only futures markets

are standardized contracts.3) Clearinghouse settles al futures contracts

4) Security Deposit vs. Marked-to-market a. Must pay initial margin (1% to 10%)

b. Profits and losses marked-to-market every day

C. Marked-to-market example and price paid at delivery

1) Buy a futures contact promising to pay $98,000 for $100,000 face value worth of T-bonds.

2) Tomorrow value of contract becomes worth $99,000

3) Buyer gets $1,000 credit into margin account. Seller has $1,000 debited.

4) If prices don’t move again buyer pays $99,000 at delivery. But costs $98,000 due to intermittent cash flow.

5) This procedure allows every contract to be interchangeable. You don’t have to buy from original seller.

D. Margins1) Initial is 1 to 10 percent 2) Maintenance margin is 60 to 80% of

original3) Margins set by exchange and can change

E. Terminology1. Long vs. short

2. Nearby vs. deferred contract 3. Basis (futures price minus the spot price)

F. Characteristics for a successful contract 1. Non- Perishable underlying asset

2. Standardizable quality3. Large and widely held deliverable supply 4. Competitive cash market5. Large number of potential hedgers

G. Types of contracts1. Agricultural Goods (Only contracts till

1973) 2. Industrial and Precious metals3. Financial

a. Foreign exchange b. Government securities

c. Eurodollars d. Stock Indexes4. Raw materials (Heating Oil)

II. Mechanics

A. Participants1. Hedgers – Have or will have a spot market position and wish to reduce the uncertainty about futures prices.

2. Speculators No spot position a. Scalpers (in and out of positions in a few minutes) b. Day traders (hold position throughout the day) c. Position traders

B. CFTC (Regulatory Body) 1. Evaluates new and existing contracts2. Protect against market manipulation

a. Buy futures contract b. Own the deliverable supply

C. Clearinghouse1. Matches buyers and sellers each night 2. Marks positions to market each night3. Guarantees no defaults4. Inspects and certifies the deliverable

supply

D. Primary markets1. CME2. CBT

3. New York Mercentile4. COMEX5. London International Futures Exchange

III. Economic Justification

A. Provides Information: Futures price conveys info about expected Supply/demand Conditions

B. Risk shifting through hedging 1. Separates price risk from operating risk 2. Shifts risk from hedger to speculator

a. Holder of spot is sell futures contracts b. Want to hold later will buy futures

contracts

C. Increases market efficiency 1. No price risk implies firms can lower profit

margin 2. Reduces search costs for information

Valuation of Futures Contracts

I. Valuation of Forward vs. Futures Contracts

A. Price vs. Value 1. The value of futures and forward contracts

are initially zero.2. Price is some observable number used as a

benchmark to determine future value.

B. Value of a forward contract1. At expiration time T, the forward price is

equal to the spot price.

2. The value at expiration is Vt = St – F, where

F = the price of forward contract when purchased.

3. Prior to Expiration: Vt = PV( Ft –F) which is the present value of the change in price discounted by time to expiration.

C. Value of Futures Contract 1. At Expiration

a. Price is equal to spot price f t = St b. Value = 0 as soon as daily mark-to-

market occurs. 2. Before expiration value reset to 0 after

daily m-t-m.

D. Forward vs. Futures Differences 1. No difference 1 day prior to expiration.

2. Otherwise difference is function of movements in futures prices relative to interest rate movements.a. If interest rate is constant Forwards =

Futures b. Otherwise relative prices depend on

correlation between movement in interest rates and futures prices.

c. If Int rates and futures price are positively correlated then futures are worth more

d. If opposite then futures are worth less.e. Example

II. Pure Cost of Carry Model

A. Four Costs of carrying a Commodity till Expiration

1. Storage Costs2. Financing Costs (Interest charges on

Purchasing and holding asset till expiration)

3. Possible Transportation costs 4. Insurance costs

B. Cash to futures pricing relationship 1. f0,t < S0 ( 1 + Ct )T where

a. f0,t is the futures price today with expiration time t

S0 is the spot price today Ct are all carrying costs/period with T

periods till expiration. b. Otherwise arbitrage from buying spot,

financing and selling futures contractc. Example S0 = $400 (say gold)

C = 10%/ yearSix months till expirationFutures contract = $440 Cost is $400 + 20 = $420 get $440

make $20 2. Similarly f0,t > S0 ( 1 + Ct )T must hold

a. Opposite presents arbitrage (buy f, sell spot, lend)

b. Example if futures contract is $410Assumes no restrictions on short sale and can invest funds.

3. Therefore f0,t = S0 ( 1 + Ct )T must hold 4. Similar relationship for relative futures

contracts

C. Effect of Market Imperfections 1. Bid-ask Spread and other direct transaction

costs a. Pay the ask for buying spot, get bid for

selling spot b. Forms a wider no-arbitrage boundary

condition. S0 (1-Tr) ( 1 + Ct )t < f0,t < S0 (1+Tr) ( 1

+ Ct )t

Where Tr = Transactions costsc. Discuss inequality

2. Unequal Borrowing and Lending ratesa. Let CL = lending rate and CB = borrowing

rateb. then S0 (1-Tr) ( 1 + CL )t < f0,t < S0

(1+Tr) ( 1 + CB)t

3. Short Selling Restrictions a. May preclude arbitrage of f0,t < S0 (1+Tr)

( 1 + CB)t

b. due to use of funds, no short selling or availability of spot asset.

c. Widens no-arb lower bound for futures price

4. Limitations to Storagea. Pure carry model assumes storabilityb. Effects upper boundc. S0 (1-Tr) (1 + CL )t < f0,t may not hold

since can’t buy spot and deliver in future. 5. Example

a. Spot gold = $400, CB= 10%/ year, CL= 8%/year

Tr = 1% on bid-ask spread, 1% commission and storage costs

b. No-arb becomes 400(1.04)(.98) < f0,t < 400(1.05)(1.02)

407.68 < f0,t < 428.40c. Different costs for different traders

III. Breakdowns in Carry Model

A. Convenience Yield: Futures trade at less than full carry.

1. Buy futures and sell spot is not done 2. Return for holding physical product. 3. Physical Asset in Short Supply

a. Not enough traders willing to sell spot and buy back later

b. Need commodity (seasonal supply/demand conditions) with limited storage) or short selling restriction

B. Backwardation: Cash price exceeds futures price or nearby exceeds deferred futures prices

IV. Futures Prices and Risk Premia

A. With no premium futures price = E[S]

B. Risk premium may exist if all speculators are on one side of market.

C. Keynes Theory of Normal backwardation 1. Assumes speculators are net long and

hedgers net short

2. Speculators need an expected return to take on risk therefore, f< E[S] to give speculators expected return

D. Opposite of hedgers are net long. A “Contango” market may exist.

T-Bill Contract

1.Spot Commodity Characteristics

A. Pure Discount instrument

B. Issued in $10,000 Face Value Demoninations

C. Maturities of 91 and 182 days issued every Tuesday

D. Price quotes i. P = ( 1 – dt(n/360))(Face Value)ii. n = days to Maturityiii. Example dt = 5% N= 90 days FV =

$10,0001. P = (1- .05(.25))(10,000) = 9,875

2. Futures Contract Specs

A. $1 Mil Face Value for 90,91 or 92 day T-Bills

B. March, June, Sept December delivery months

C. Expiration date is the third Wed of the

delivery month. i. Deliverable only exists for three months prior to expirationii. Is created by the six month T-bill auction

3 months before expiration

D. Price Quote = 100 – annualized yield

1. Example yield of 5.20% P = 94.80 2. Easy to know what yield is being traded

3. Actual delivery price is (1- .05(.n/360))(1,000,000)

4. Minimum move is (.0001)(90/360)(1 mil) = $25

E. Margin is approx $1500 initial, $1000 maintainance

F. Uses of short term futures contracts1. Speculate on short term movements in

interest rates2. Sell futures to hedge against rising

interest rates a. Companies who may issue S. T. debt in

the future b. Underwriter protection against short

term securitysale.

3. Buy futures to protect against fall in short term rates

a. Investor wants to purchase short term securities

b. Offset lower proceeds from variable rate loans

Eurodollar Contract

I. Definition - U.S. Deposits held in a commercial bank Outside the U.S.

A. Libor used as market rate (London

Interbank offer rate) B. Contract $1mil in U.S. Deposits held in

foreign banks with 90 days to Maturity.

C. Quotes are the same as with T-bill contract D. Differences with T-Bill contract

1. Cash Settlement 2. Libor is determined by mean of a sample

found over two points within the last 90 minutes of trading from a random selection of 12 reference banks.

3. Yield is add-on vs. discount yield for Bills Add-on = (discount amount/ price)(365/N) N = days to maturity Example T-bill = 6% discount amount is $15,000 Add-on = (15,000/985,000)(4) = 6/09%

E. Largest volume of any short term instrument

T-Bond Contract

I. Spot Market A. Auctions with maturities in Feb, May, Aug.

and Sept.B. Pays principle at maturity, interest Semi-annually

C. Quoted in 32nds of a % of par example 102-05 D. Invoice price = price plus accrued interest. A.I. = N (coupon amt/365) N = days since last payment

II. Futures Contract A. $100,000 face value of 6% coupon bond

with at least 15 years to maturity B. Price quote in 32nds with 100 face value C. Delivery dates March, June, Sept and Dec. D. Can delivery anytime during delivery month

E. Can delivery other than 6% coupon bond but must be the same issue on entire amount

F. Conversion Factor adjusts for delivery off other than 6% coupons

1. price of the bond if it had a 6% y-t-m/ price 2. example a. 20 years to maturity 10% and 10% y-t-m. b. Price if 6% yield = 146. 23 Show formula c. C.F. = 146.23/100 = 1.462 G. Finding the “cheapest to Deliver” 1. Multiply conversion factor times futures price. 2. The invoice price is this amount plus A.I.

3. Compare cash market price to (C.F.) ( F.P)

4. Choose bond that gives deliverer greatest relative amt.

5. see example

H. minimum tick = (.03125)(1000) = $31.25

Stock Index Futures

I. Spot Market A. Price weighted - (DJIA) B. Value Weighted – (S&P 500, Nasdaq etc) C. Equal Weighted – (Russell 2000, Value line) II. Futures Contracts - There are many recently. A. Most have both regular and mini contracts

B. The minis are traded electronically, regulars on open outcry C. All contracts are cash settled

D. S&P 500 Specs 1. Regular contract size = 250 * Index Value

2. Minis are 50* Index value 3. March, June Sept and Dec delivery months

with expiration being Thursday before the third Friday.

4. Settlement base on closing pot price on last day.

5. Contract movement min is .10 *250 = $25 on Reg.

6. Margin approx 10% initial and 70% of initial for maintenance

E. Uses 1. Adjust portfolio exposure for little cost

2. Short term substitute for an equity position 3. Highly leveraged way the speculate on market.

Hedging

I. DefinitionsA. Short Hedge – Holder of position worried about

falling prices.

B. Long hedge- Holder of position worried about rising prices.

C. Direct Hedge – Hedger offsetting deliverable asset

D. Cross Hedge- Hedger not offsetting deliverable asset 1. Many more cash assets than futures contracts 2. Limited number of expiration dates on futures contract

3. Limited size of each contract. E. Risks in hedging and ways to reduce it

1. Basis Risk – Especially for cross hedges a. Random shocks do not always have an

equal effect on both cash and futures prices.

b. Trend movement since carry gets smaller over time c. More predictable basis implies more effective hedge d. Find the best contract for a given asset

1. horizon- futures contract should expire near end of hedging period but must also be liquid.

2. May need to rollover nearby contract 3. Must find the best hedge ratio. 2. Margin risk

a. Provision of funds for Marking-to-the-market b. Do not want to have to close futures position early c. Adequate reserve depends on Volatility and contract type 3. Quantity risk

II. Basic Naïve Hedge hedging example Hedge dollar for dollar A. In one month you will have $1mil to invest B. Interested in a 6.5% y-t-m t-bond maturing in 20 years

C. Current price = 104-08 D. What is the concern? 1. Worried about interest rates falling 2. Need to buy futures contracts E. Buy 10 June T-bond futures contracts at P = 98-10 F. Suppose interest rates fall 1. Cash price becomes 107-30 2. Futures price becomes 101-00 3. Made 2-22 on futures but lost 3-22 on cash 4. Hedge did not eliminate risk because the basis narrowed.

III. Determining the best hedge ratio to eliminate price risk.

A. Portfolio Theory Hedge ratio. (Minimum Variance Hedge) 1. Consider a two asset portfolio (futures and spot holdings)

2. sf2(f) + 2Cov(sf)*f

Where f= number a futures contracts per 1 spot

holding This follows from Portfolio theory

3. Want to find the minimum variance portfolio a. Take the derivative with respect to change

in the number of futures contracts held and set to zero

b. 2*(f(f)) + 2*cov(sf) = 0 or f = - Cov/(f) is the risk minimizing hedge ratio c. This is also Beta from the regression: 1. s = a + B(f) 2, show plot

4. A measurement of hedging effectiveness a. 1- (var(h)/var(u) =effectiveness b if var (h) =0 perfectly effective 1-0 =1 c if var(h) = var(u) hedge ineffective 1-1=0

5. Example: Beta = 1.25 implies sell 1.25 futures per spot held.

6. Problem is one of stability a. what if optimal Beta changes over time? b. Then optimal hedge ratio from data may not be applicable in future 7. If Beta =1 is optimal naïve hedge is optimal

IV. Price sensitivity hedge (Duration Model)

A. Duration: Weighted Ave. time to maturity of the PV of all Cash flows 1. D = ( (T)*((Coup)/(1 +i)t )/ (Price of bond) 2. D approximates ( i)/ i) Which is the price sensitivity of a change in interest rates 3. Example 4. (= -Dur * (i / (1 + i)) 5. -Dur * (i / (1 + i)) * B 6. From example $

7. Higher duration implies more price sensitivity to int rate change

B. Using Duration to hedge 1. f-Durf * (if / (1 + if)) * f 2. Therefore Nf = f = (Durb / Durf) (B/f) ((1 + if)/ (1 + i) 3. Example

C. Problems 1. Assumes yield curve constant over hedging period. Otherwise durations change. 2. Must estimate relative changes in yields if there are any

Hedging Examples

I. Corporate borrower- Borrows a variable rate loan A. Borrows $2 million at prime rate + 1%

Payable quarterly for next three quarters B. Borrower is worried about rates rising C. Sells futures contracts over life of loan D. If rates rise losses offset by futures gains E. Want to hedge using Eurodollar contract Sell 2 contracts F. Currents conditions 1. Prime rate = 7% 2. Spot Eurodollar = 5% 3. June Eurodollar futures contract = 95.00 or 5% Sept Eurodollar futures contract = 94.5 or 5.5%

Dec. Eurodollar futures contract = 94 or 6% 4. If prime rate is stable at Eurodollar = 2% expected Payments are: June = 8% or (2mil)(8%)(1/4) = $40,000 Sept = 8.5% (2mil)(8.5%)(1/4) = $42,500 Dec = 9% (2mil)(9%)(1/4) = $45,000 Total expected interest expense = $127,500 G. Assume in June Prime rate goes up to 8% but basis remains the same. 1. Spot Euro = 6% loan payment is 9% 2. June payment = (2mil)(9%)(1/4) = $45,000 June Futures Contract = 94 since euros at 6% Profits are (100)($25)(2) = $5,000 3. The hedge was successful since the basis didn’t move. H. Have to go through it two more times to complete hedge.

II. Investor has $1 mil to invest in one month A. Interested in a 4% coupon 7- year note selling at par B. Concerned rates will fall C. Buys 10 10-year note futures contracts.

Currently yielding 4% Price = 116-11 (116,351) Show on board D. In one month rates rise to 4.5% 1. Cash bond only costs $97,026 2. Futures price is $111,973 (6% coupon, 10 year note) 3. Made 100,000 – 97,026 or $2,974/$100,000 on spot 4. Lost 116,351 – 111,351 or $ 4,378/contract on futures 5. Net loss is $1,404 * 10 or $14,040 on investment E. Needed to buy less futures contracts since duration of futures was greater than cash. (Use price sensitivity hedge ratio) (Durb / Durf) (B/f) = Nf

Strategies

I. T-bill Futures price determination A. Let P(t1) and P(t2) be the price of discount bonds paying $1 at times t1 and t2 respectively B. Let t1 be the expiration date of a t-bill futures contract and Let t2 be the maturity date of Bill delivered at t1. C. Compare the following portfolios 1. Buy time t1 security HPR = (1- P(t1))/P(t1) 2. Buy time t2 security and sell Futures contract HPR = (fp – P(t2)/ P(t2) 3. Since both portfolios held for same time HPR’s must be the same 4. Setting them equal and solving for fp fp = P(t2)/P(t1) D. Also one can substitute 1. 1/(1 + it1 ) = P(t1) therefore fp = P(t2)( 1 + it1 ) 2. This is the pure carry model carrying the deliverable to expiration. E. if fp > P(t2)( 1 + it1 ) In this case the yield from holding 2 period bill is higher than return on one period bill and

Futures contract therefore, 1. sell futures contract 2. Buy P(t2) bill 3. Finance by selling P(t1) bill F. if fp < P(t2)( 1 + it1 ) 1. Buy futures 2. Sell P(t2) bill 3. Buy P(t1) bill

II. T-Bond Futures Contract Price Determination A. f*(CF) = B( 1 + it1 - ib) 1. if f*(CF) > B( 1 + it1 - ib) Sell futures contract Buy cash bond and finance at rate it1

2. if f*(CF) < B( 1 + it1 - ib) a. Buy futures contract Sell bond and lend proceeds b. This arbitrage presents some unique problems since Short initiates delivery proceed 1. What is the holding period? Delivery can take Place anytime throughout the month 2. What bond will be delivered? B. Special “options” for short in T-bond contract 1. Quality option short delivers “cheapest” bond 2. Timing option – delivers anytime throughout the month

3. Wild card option Gets till 9:00 p.m to deliver 4. End of month option Can deliver after contract expires C. Empirical results similar to T-bill. 1. f*(CF) > B( 1 + it1 - ib) appears to be the norm 2. Again t-bill holder is better off buying bond and selling futures contract.

2. There is some reinvestment rate risk to be considered

III. Stock Index Futures A. f = S(1+ i –d) Where d = dividends paid over holding period B. Index arbitrage and program trading 1. Computer directed index arbitrage 2. Since inception could trade index via DOT a. Designated Order Turnaround b. Can be used to by S&P index with one trade c. $5 mil minimum d. Reduced the potentially large transaction costs 3. Could have caused crash in 1987 a. Futures < spot plus carry b. Used DOT to buy futures and sell cash c. Helped to push cash down further d. Eventually spiraled out of control

e. As a result DOT turned off after large move C. Problems with arbitrage 1. Transactions costs have been reduced dramatically with advent of ETFs (QQQ, SPDRS etc). 2. Short sales on spot 3. Cash settlement must leg out position like options

D. Can use futures to arbitrage stock index put-call Parity 1. P = C + (E – f)/ (1 + 1)t 2 . Consider two portfolios Port A: Own a put Port B: Own a call Sell futures contract at f Lend (E – f)/ (1 + 1)t

0 At E if S* < E Port A Port B E- S* 0 – (S* -f) + (E-f) = E –S* If S* > E Port A Port B

1 (S* -E) +(E-S*) = 0

Spreading Strategies

I. Types of spreads

A. Intermarket- Same Commodity, Different Exchange Pure arbitrage doesn’t happen very often B. Intercommodity 1. Different but economically related assets 2. Usually same maturity but could be different markets 3. Example expect the yield curve to steepen a. interest rates on t-bills will fall relative to t- notes b. Buy t-bill futures and sell t-note futures 4. Other common spreads S & p vs. small cap index Gold vs. Silver etc. C. Intramarket 1. Same commodity different maturities 2. Better if non-perishable 3. example gold contracts should equal cost of carry If you think rates will narrow buy nearby sell differed D. Intra and Intercommodity Spread 1. T-Bond Spread gives an implied forward rate. 2. Should be equal to the T-bill futures contract 3. If not buy one and sell the other

Options on Futures Contracts

I. Definition –Option to buy or Sell a Futures Contract A. Last day of trading in month prior to expiration of futures contract. B. For T-Bonds- 5 days prior to first notice day of futures cont. C. Exercised Commodity is a futures contract at E. 1. Clearinghouse creates long position for buyer, short for Seller. 2. Credits and debits must be made upon exercise 3. Example T-Bond a. Strikes are offered at 1% of par i.e 101, 102 ect. b. Options traded in 64ths of 1% c. Say futures contract is at 101 d. The June 100 –call may be selling today for 2-32 that’s $2.50/100 or $2,500/ $100,000 e. if at expiration f = 101 option buyer gets $1,000 credit and option writer owes $1,000

II. Economic Rationale A. Expands to set of hedging opportunities

1. Suppose you are a money manager who may or may not purchase T-bonds in 3 months 2. With futures only how do you hedge 3. With options let it expire if you don’t need it. B. Allows to have options on a standardized commodity C. Lets futures market speculators reduce some risk of holding position

III. Valuation

IV. Early Exercise of Call and Put options on Futures contracts. A. Both puts and call may be exercised early B. Consider deep-in the money call. a. if f* > E is assured then delta of option is 1. Then both assets move dollar for dollar b. No money tied up in futures contract however there is a premium paid on call option c. if exercised investor has the same profit opportunities with no cost d. not the same with stock options, must commit funds upon exercise.

C. Example 1. T-Bond futures option E = 101 with f = 107 2. Will get a $6,000 credit if exercised 3. If profit opportunity till expiration are identical why not. V. Binomial formulation

1. Riskless hedge consists of a position in the futures option and an opposite position in the underlying futures contract.

2. Similar to equities except that the value of the futures contract is zero at time zero

3. No initial investment required in futures contract.

4. No foregone income from holding put or savings from holding call.

5. Therefore, interest rate term not needed in finding hedge ratio.

6. Example 7. General B/S equation

a. C = F . N (d1) – E . e-rt .N ( d2)b. However no interest rate term in N (d1)

or N ( d2) terms

Exotic Options

I. Forward Start Options – Options life begins in the future but paid for now. Generally at-the money when the life begins. Often used for executive compensation.

A.3 dates to consider1. Valuation date t(o)(today’s date)2. Grant date t(g) date when the options life

begins3. Expiration date (T = time to expiration)

B. When life begins, Option has T- t(g) time left. T(o) – t(g) life gone

C. Value from B/S1. Time to expiration in model is T-t(g) 2. C = F . N (d1) – E . e-r(T-t(g)) .N ( d2) 3. Value today is C . e-r(t(g)-t(o))

D. Example

II. Compound Options – The underlying asset is also an option.

A. Buy the right to pay X for an option with Strike Price E.B. Let E and t(2) be the exercise price and

expiration date of the underlying option.C. Let t(1) be the expiration date of the

compound option.D. Pricing is from the date of expiration of the

compound option1. From t(1) to t(2) option pricing is as

before. 2. Must decide at t(1) whether or not to

exercise. 3. at t(1) C = max (Cu –X, 0 )

4. Will exercise if S at time t(1) implies Cu > X

5. Need a bivariate cumulative normal distribution since two events must occur.

6. Show example7. Four types ( call and put options on call

and put underlyings)