BU 423 Options, Futures & Swaps
Table of Contents
Derivative: financial instruments that depend on an underlying asset. Everything in finance could be considered a derivative. In finance, there is no wealth generation, only wealth redistribution.

Textbook: Fundamentals of Futures and Options 9th edition by John C. Hull

Four Quizzes 40%

Midterm Exam 25%

Final Exam 35%
Naturel of Derivatives
 Return = Risk Free + Risk Premium
 Risk can be transferred for a price. One sufficient manner is through the use of derivatives
Building Blocks / Examples
 Futures Contracts
 Forward Contracts
 Swaps
 Options
Why are Derivatives Used
 Hedging
 Speculation
 Locking in arbitrage
 Changing nature of a liability
 Fixed rate loan and you want to change that into a variable one
 You can either pay back the loan and take a variable rate (transaction costs)
 Or you can use a derivate
 Change the nature of an investment without incurring the costs of selling one portfolio and buying another
 Something about using a forwards contract
Futures Contracts Introduction
 Agreement to buy or sell underlying asset at some point in the future for a certain price
 Spot contract is buying or selling immediately
 An example. Spot price of $100, riskfree of 5%, Expected price in a year is $110. F0 is 105 if there is no risk?
 Selling = short position. Buying = long position
 Exchanges Trading Futures
 CME Group
 Intercontinental Exchange
Overthe Contract markets
 More important alternative than exchanges because anything goes
 In the exchangetraded market, there are standardized contracts
Lehman Bankruptcy
 Active participant in OTC derivative market
 Took high risk and couldn’t roll into shortterm funding
 Sold protection for debt instruments for mortgage backed securities
 Sold Credit Default Swaps which is a protection instrument for mortgage backed securities
 Long Term Capital (hedge fund). Proprietary trading models, promising investors 20%+ but as it got larger it was harder to attain the returns
 Then they levered their fund
 Systematic Risk
 Placed bet on USSR and USA interest model
 When USSR collapsed, they went bankrupt
 1998 Russian financial crisis
New Regulations for OTC
 Standard OTC products must be traded on swap execution facilities
 Central clearing party must be used
 Trades must be reported to a central registry
OTC Systemic Risk
 Default by one large financial institution can lead to losses in other financial institution
Forward Contracts Introduction
 Trade in OTC market
 Popular on currencies and interest rates
 Forward Price = delivery price if negotiated today
Options
 Call = option to buy at strike price by a certain date
 Put = option to sell at strike price by a certain date
 American style = exercised any time during its life
 European style = exercised only at maturity
Profit = (Stock Price  Strike Price  Purchase Price)
Examples:
 100 put options short
 Losses limited to strike price + price paid for the option
 100 call options long at strike price of $550 for $29 per option
 Profit increases as stock price increases
Options vs. Futures/Forwards
Holder has an obligation vs. option
Hedge Funds
 Not same regulations as mutual funds
 Mutual funds
 Disclose investment policies
 Redeemable at any time
 Limited use of leverage
 Hedge fund fee: 2 plus 20%
Arbitrage Examples
 stock price is quoted in 100 pounds in London and $152 in New York
 exchange rate is 1.55 for pounds to USD
 therefore, short the stock in london, buy in new york
 riskfree rate is 5%
 spot price of gold is US$100 per ounce
 1year futures is 105
 expected spot price in 1 year is $110 (does not matter)
 F = S (1 + r) ^ T
 spot price of oil s US$40
 1year futures price of oil is US$35 (or $50)
 riskfree is 2% per annum
 storage cost is 1% per annum
Futures Contracts (Chapter 2)
 Exchange traded
 Applicable to a wide variety of underlying assets
 Specs need to be defined
 What can be delivered
 Where to deliver
 When to deliver
 For many futures contracts, the delivery period is the whole month
 Settled daily (mark to market)
 Closing out futures is easy, it’s just an opposite trade
 Most contracts are closed out before maturity
 When used for hedging, profits are not recorded for accounting purposes until the contract is closed (pg. 59). If not for hedging, then the books would note the gain/loss in the year end even if the contract will not be closed until the next year
Convergence
As time goes on, the future’s price converges to the spot price
Margin
 Margin is cash or marketable securities deposited by an investor with his or her broker
 The balance in the margin account is adjusted to reflect daily settlement
 Margin minimizes the possibility of a loss through a default on a contract
 Retail traders provide initial margin and, when the balance in the margin account falls below a maintenance margin level, they must provide variation margin bringing balance back up to initial margin level.
 Short futures contract
 Want to sell the asset later before maturity
 Revenue = Spot Price + Price at T = 0 MINUS Price at T = 1
 Example sold asset at spot price of 110 + shorted future at price of 105  bought back future at price of 112 = 103
OTC Markets: Bilateral Clearing
 Transaction between two parties typically governed by ISDA Master Agreement
 Credit Support Annex (CSA) defines the collateral that has to be posted
 After Financial Crisis of 20072007, centralized counterparties (CPP) have to be used to avoid financial institutions from chain reacting defaults
 Companies can clear its transaction through a member
Terminology
 open interest: outstanding contracts
 settlement price: price before final bell
 volume of trading: number of trades for a day
Forward Contracts
 Private contract between 2 parties
 Nonstandard contract
 Usually 1 specified delivery date (futures are a range)
 Settled at end of contract
 Delivery usually occurs
 Some credit risk (risk of counterparty default)
Hedging Strategies
 Long hedge is when you know you will lock in the price for an asset you know you will buy in the future
 Short hedge is when you know you will sell an asset in the future and want to lock in the price
Arguments For Hedging
 Take steps to minimize risks arising from interest rates, exchange rates, and other market variables
Arguments Against Hedging
 Shareholders can make their own hedging decisions
 May increase risk if competitors do not hedge
 Loss on hedge but gain in the underlying is hard to explain
Basis Risk
 Difference between spot and futures
 Arises because uncertainty about basis when hedge is closed out
 An improvement (increases) in the basis benefits the short position
Long Hedge for Purchase of an Asset
 F1: Futures price at time hedge is set up
 F2: Futures price at time asset is purchased
 S2: Asset price at time of purchase (Cost)
 b2: Basis at time of purchase
 Net Amount paid =S2  (F2  F1) = F1 + b2
 Net Amount received = S2 + (F1  F2) = F1 + b2
Choice of Contract
 Choose delivery month as close to life of the hedge
 When there is no futures contract on the asset being hedged, choose teh most highly correlated with the asset price. 2 basis components
Optimal Hedge Ratio

change in standard deviation of change in spot price

change in standard deviation of the future price

price coefficient between two assets

Airline will purchase 2 million gallons of jet fuel in one month and hedges using heating oil futures

From historical data
h^* = 0.928 * 0.0263 / 0.0313 = 0.78
 Therefore optimal number of contracts is 0.78 times 2,000,000 / 42,000 [units per contract] = 37
Hedging Using Index Futures
Beta times value of the portfolio divided by value of one index futures contract.
Example 1
 1,000 S&P futures price
 $5MM portfolio
 1.5 Beta
 One contract is $250 times the index
 1.5 * 5,000,000 / 250,000 = 30 contracts short
Example 2
 Index level of 1,000. Future price of 1,010, 250x per contract
 Portfolio value of $5,050,000
 riskfree = 4%
 Dividend = 1% per year
 Beta = 1.5
 3 month horizon
 Future exit in 4 months
1.5 \* (5,050,000 / (250 * 1,010)) = 30 contracts
 in 3 months, index is at 900 and the future is at 902
 What’s the net value?
 R = (P1  P0 + D) / P0 = (P1  P0) / P0 + D / P0
 R = Rf + Beta (Rm  Rf)
 Gain from short position =
(1010  902) * 30 * 250= 810,000
Loss from index = (900  1000 + 1% * 1000 / 12 * 4) / 1000 = 97.5 / 1000 = 9.75%
Rp = 4% / 4+ 1.5 (0.0975  0.04 / 4) = 15.125%
 Notice that we divided risk free rate to account for 3 month
Vp = 5,050,000 (1  15.125%) + 810,000 = 5,096,187.50
Hedging to NonZero beta
 Previous section, we got a beta of 0, what about nonzero?
 If Beta > desired Beta, Number of contracts to sell = (Beta  desired Beta) Vp / Vi
 If Beta < desired Beta, Number of contracts to buy = (Desired Beta  Beta) Vp / Vi
Reasoning
 You think your stocks will outperform the market
 Hedging ensure the return you earn is the riskfree rate plus the excess over the market (or minus the underperformance over the market)
Stack and Roll
 Roll futures forward to hedge future exposures
 Just before maturity, close them out and replace them with new contract reflect the new exposure
Liquidity Issues
 In any hedging situation there is a danger that losses will be realized on the hedge while the gains on the underlying exposure are unrealized
 Example: Metallgesellschaft which sold long term fixedprice contracts on heating oil and gasoline and hedged using stack and roll
Interest Rates
Treasury Rates
 Instruments issued by government in its own currency
London Interbank Offered Rate (LIBOR)
 London Interbank Offered Rate
 Based on submissions by banks
 Why would banks collude for this?
U.S. Fed Funds Rate
 Unsecured interbank overnight rate of interest
 Allows banks to adjust the cash (i.e., reserves) on deposit with the Federal Reserve at the end of each day
 = average rate on brokered transactions
 central bank can intervene
Repo Rates
 Financial institution agreeing to purchase back securities it sold now for a higher price
 It’s a loan?
 The rate of interest is calculated between X and Y
LIBOR Swaps
 LIBOR is exchanged for a fixed rate
 3 month LIBOR swap is same risk as continually refreshed 3 month AArated banks
OIS Rate
 Overnight Indexed Swap
 Fixed rate for a period is exchanged for the geometric average of overnight rates
 Single exchange for up to one year maturity
 Periodic exchange for over one year, e.g. quarterly
 OIS rate is a continually refresh overnight rate
RiskFree Rate
 Treasury rate is artificially low because Banks do not keep capital for Treasury instruments
 Treasury instruments have favourable tax treatments
 OIS rates is a proxy for the riskfree rates
For example, suppose that in a U.S. threemonth OIS the notional principal is $100 million and the fixed rate (i.e., the OIS rate) is 3% per annum. If the geometric average of overnight effective federal funds rates during the three months proves to be 2.8% per annum, the fixed rate payer has to pay 0.25 × (0.030 − 0.028) × $100,000,000 or $50,000 to the floating rate payer. (This calculation does not take account of the impact of day count conventions.)
Measuring Interest Rates
 Principal e^(RT)
 R = continuously compounded rate for time T
Conversion Formulas
 Rc : continuously compounded rate
 Rm: same rate with compounding m times per year
 Rc: m ln ( 1 + Rm / m)
 Rm: m (e^(Rc / m)  1)
import math
def convert_interest_rate(m_old, m_new, rate):
rc = math.log(1 + rate/m_old) * m_old
print(f'Continuous compounding rate: {rc * 100:.4f}')
rm = m_new * (math.e ** (rc / m_new)  1)
print(f'New rate: {rm * 100:.4f}')
10% with semiannual compounding is equivalent to 2ln(1.05)=9.758% with continuous compounding
8% with continuous compounding is equivalent to 4(e0.08/4 1)=8.08% with quarterly compounding
Rates used in option pricing are usually expressed with continuous compounding
Zero Rates
 A zero rate (or spot rate), for maturity T is the rate of interest earned on an investment that provides a payoff only at time T
Maturity (years)  Zero Rate with Continuous Compounding 

0.5  5.0 
1.0  5.8 
1.5  6.4 
2.0  6.8 
Template
COUPON_FOR_PERIOD * e^(HALF_YEAR_CTN_COMPOUNDING * 0.5) + 3e^(r*1) = 97
Bond Pricing (Continuous)
 theoretical price of a twoyear bond providing a 6% coupon semiannually is:
3e^(0.05 * 0.5) + 3e^(0.058*1)+ 3e^(0.064*1.5) +103e^(0.068*2.0)= 98.39
 yield when you replace all the different rates above with a single rate to make the PV equal the price
Par Yield
 coupon rate that causes the bond price to equal its face value
 similar to solving for bond yield, but set the price to the face value
 c = (100  100d)m / A
Data to Determine Treasury Zero Curve
 Bootstrap Method
BOND_PRICE * e % (RATE * MATURITY_IN_YEARS) = FACE_VALUE
 For the coupon bonds, we can use the zero rates from before and just solve for the lumpsum rate R
Forward Rates
 (R2T2  R1T1) / (T2  T1)
 Approximately true when rates are not expressed with continuous compounding
Interest Rate Swap
 One person pays a fixed rate and the other pays a floating rate
 For the person paying a fixed rate (and receiving a floating rate) the credit risk is
 The floating rate is expected to decrease based on the term structure (upward sloping)
 Interest rates decline unexpectedly
 credit risk is greater when term structure slopes downward (market expects interest rates to decrease in the long term) and the risk exposure increases when interest rates decline
Forward Rate Agreement (FRA)
 OTC agreement that a certain LIBOR rate will apply to a certain principal during a certain future time period
 Predetermined rate RK is exchanged for interest at the LIBOR rate
 FRA can be valued by assuming the forward LIBOR interest rate RF is certain to be realized
 Value = Present Value of the difference between the forward LIBOR interest rate (RF) and the interest paid at the FRA rate RK
(RF  RK) * Principal * length of the contract
and then discount to 0 from T2 at the risk free rate? Use case: floating rate payment in the future but you want to make sure you are paying a fixed rate
 the receiver will want a premium for receiving
A company has agreed that it will receive 4% on $100 million for 3 months starting in 3 years. The forward rate for the period between 3 and 3.25 years is 3%. The value of the contract to the company is +$250,000 discounted from time 3.25 years to time zero at the OIS rate.
(0.04  0.03) * 100_000_000 * 0.25 / (1.03^3.25) = 250_000 / (1.03 ^ 3.25)
Suppose rate proves to be 4.5% (with quarterly compounding). The payoff is –$125,000 at the 3.25 year point. Often the FRA is settled at time 3 years for the present value of the known cash flow at time 3.25 years.
125_000 / (1 + (0.045 * 0.25)) = 123_609.39
 3x6 FRA: starts in 3 months (90 days) and ends in 6 months (180 days)
 6x9 FRA: starts in 6 months (180 days) and ends in 9 months (270 days)
 Question: rate of 3.10% for 6x9. 3% right now. What is the fixed rate in the agreement?
(1 + (0.031 * 0.75)) / (1 + 0.03 * 0.5) = (1 + R * 0.25)
 RF = 3.251%
 Question: 3 months have passed, and the rate has gone to 3.25% for 3 months and 3.3% for 6 months. what is the value of the FRA
(1 + (0.033 * 0.5)) / (1 + 0.0325 * 0.25) = (1 + R * 0.25)
 RF = 3.323%
FRA = (0.03323  0.03251) * 0.25 / (1 + 0.033 * 0.5)
 FRA = 0.00017710154273060686 of the loan amount
A financial manager needs to hedge against a possible decrease in shortterm interest rates. He decides to hedge his risk exposure by going short on a 3X6 FRA that expires in 90 days and is based on a 90day LIBOR. The current LIBOR spot rates are observed: 30day 5.83%, 90day 6.00%, 180day 6.14% and 360day 6.51%. What is the rate the manager would receive on this FRA:
 Interest paid on $1 for 180 days: 0.0614 * 0.5 = 0.0307
 Interest paid on $1 for 90 days: 0.06 * 0.25 = 0.015
 Expected interest paid on $1 from 3x6 (compounded from 90day): (1.0307 / 1.015  1)
 Expected interest rate for 3x6 (compounded from 90): (1.0307 / 1.015  1) / 0.25 = 6.19%
Theories of the Term Structure
 Expectations theory: forward rates equal expected future zero rates
 Market Segmentation: short, medium, and long rates determined independently of each other
 Liquidity Preference Theory: forward rates higher than expected future zero rate
 To manage these preferences, banks offer different rates for depositors and borrowers depending on the maturity
Determination of Forward and Futures Prices
Unit: Domestic Currency / Foreign Currency
Intro and Types of Contracts
 Futures contract
 Forward contract
 Even an airline ticket is a forward contract
 There should be a model/marketplace for this sort of thing for each airline and etc.
 we have three times: 0, t, and T (maturity)
Types
 forward contracts on investment assets that provide no income
 discount bills, bonds, stocks without dividends
 forward contracts on investment assets that provide a known dividend yield
 coupon bonds, indices, currency
 forward contracts on investment assets that provide a known cash income
 coupon bonds, indices
Valuing Forward Contracts
For all these equations, T is the time till maturity in years from the present. r is the continuous compounding rate for the period of time.
When first negotiated, a forward contract is worth 0 because neither party is actually paying for the contract to exist.
But later, when there is a contract with delivery price K and a contract with delivery price F0, we can show that the value of the contract is:
Long forward contract
Short forward contract
So in this case, a rate of say 8% continuous compounding for 3 month period requires multiplying the rate by the months.
The Forward Price
Forward Price with Continuous Compounding
Value:
0 + 100e^(5%)
Known Dollar Income
Where I is the present value of the income during life of forward contract
Known Yield
Where q is the average yield during the life of the contract (continuous compounding), For storage costs, use(+u) instead of (q). Use q = rf (foreign) for currencies. For cost of carry, use c (storage cost plus interest cost) in place of r.
Forward Pricing Example 1
 Stock without dividend
 spot price = $40
 risk free for 3 months is 5% per annum (5% / 4 = 1.25%)
 Ft = 40(e ^ 0.0125) = $40.50
 Suppose that F0 = $43
 How to execute the overpriced arbitrage strategy?
 Short the forward, borrow at riskfree (isn’t this the margin rate) to buy the underlying
 at T, deliver the underlying and close the short forward and pay off the loan
Futures Pricing
Example
 Spot: 400
 yield of 3% p.a
 rm = 8%
Index Arbitrage and Program Trading
 simultaneous purchase/sale of at least 15 stocks with total value > $1MM
 Black Monday: arbitrage opportunities
Interest Rate Futures
 day count convention
 unit of time for calculating accrued interest when instruments are traded
 Treasury: Actual / Actual
 Corporate Bonds: 30 / 360
 Money Market Instruments (e.g, LIBOR) Actual / 360
Bond Prices Between Coupons
 Cash Price = Quoted Price + Accrued Interest
 quoted price is flat price
 invoice or total price paid is called the dirty price or cash price
Accrued Interest = Coupon * (days since last coupon / coupon period)
 Actually count the days in the coupon period instead of dividing by two
A semiannual coupon bond with 8% coupon rate
Days passed since last coupon payment is 30
Accrued interest = $80/2 * (30/182.5) = 6.58
coupon rate * par value * (days / 365)
Invoice = 990 (quoted) + 6.58 = $996.58
Treasury Bill Prices in the US
P = 360/n (100  Y)
Quoted based on annualized discount. So if the quoted price is 8 for 3 month, then the cash price is 100  (8/4) = 98.
Canadian Treasury Bills
Quoted on yield. Actual/Actual
365/n * (100  Y) / Y
Treasury Bond Futures
For each $100 face value of bond,
Cash Price received by short party = most recent settlement price * conversion factor + accrued interest.
 10year Treasury note futures contract quotes are to the nearest half of a thirtysecond (0.5/32ths). 127015 means 127 + 1.5/32 and 9008 means 90 + 8/32.
 5year and 2year Treasury note contracts are quoted to nearest quarter of a thirtysecond (0.25/32ths). 119197 means 119 + 19.75 / 32
Settlement is priced on a 6% bond and delivery can be any bond with a maturity of more than 15 years but less than 25 years. The conversion factor is unique to each bond.
Example
Each contract is delivery of $100,000 face value of bonds. Suppose recent settlement price is 9000, there’s a conversion factor of 1.3800, and the accrued interest is $3 per $100 face value.
Therefore, (1.3800 × 90.00) + 3.00 = $127.20. Since $100,000 face value is delivered (x1000), the total cash received is $127,200.
Conversion Factors
 Quoted price the bond would have on the first day of delivery month assuming interest rate is 6% with semiannual compounding and the maturity is rounded down to a multiple of 3 months. If the maturity is not a multiple of 6 months, assume a coupon is paid in three months meaning that accrued interest of 3 months has to be subtracted.
Example
As a first example of these rules, consider a 10% coupon bond with 20 years and two months to maturity. For the purposes of calculating the conversion factor, the bond is assumed to have exactly 20 years to maturity. The first coupon payment is assumed to be made after six months. Coupon payments are then assumed to be made at sixmonth intervals until the end of the 20 years when the principal payment is made. Assume that the face value is $100. When the discount rate is 6% per annum with semiannual compounding (or 3% per six months), the value of the bond is
 Sum from i=1 to i=40 { 5/1.03^i } + 100 / 1.03^40 = 146.23
 Divided by the face value to get a conversion factor of 1.4623
Consider an 8% coupon bond with 18 years and 4 months to maturity. For the purposes of calculating the conversion factor, the bond is assumed to have exactly 18 years and 3 months to maturity. Discounting all the payments back to a point in time three months from today at 6% per annum (compounded semiannually) gives a value of
 3 months from today, the value is 4 (for the last coupon?) + sum from i=1 to i=36 {4 / 1.03^i} + 100 / 1.03^36 = 125.83
 Discounting to today is 125.83 / (1.03^0.5) = 123.99. Subtract the accrued interest of 2 (3/6 * 4) to get 121.99
Determining Treasury Futures Price
 115 quoted bond price, 12% coupon, conversion factor of 1.6, 60 days since last coupon payment, 122 till next coupon payment, 148 after that till contract Maturity
 S0 is the CASH VALUE not the quoted value
 S0 = 115 + (60/182) * 6 = 116.978
 I = 6e^(0.1 * (122/365)) = 5.803
 F0 = (S0  I)e^(rT)
 F0 = (116.987  5.803)e^(0.1 * (270/360)) = 119.211
 Quoted F0 = 119.711  total accrued interest = 119.711  148/183 * 6 = 114.851
 Now we need to divide by the conversion factor to get 71.79
Eurodollar
 a eurodollar is a dollar deposited in a bank outside the USA
 futures on 3month LIBOR rate (eurodollar deposit rate)
 rate earned on $1 million
 a change in one basis point (0.01) in a eurodollar futures quotes corresponds to a contract price change of $25 (x2500)
 final settlement price is 100 minus actual 3 month LIBOR rate
 quoted on a value of 100
 long position = receive a rate
 for eurodollar futures lasting beyond two years, forward rates != future rates
 futures settled daily where forward is settled once
 futures settled at the beginning of threemonths, FRA settled end of 3 month period
Swaps
 OTC agreement to exchange cash flows in the future. Calculation usually involves the future value of an interest rate, an exchange rate, or another market variables
 You agree to pay a fixedrate and get paid back a floating rate (e.g. LIBOR or OIS)
Suppose Apple has an obligation to pay LIBOR + 0.1%. If they purchase a SWAP with CitiBank, they pay CitiBank a fixed rate, say 3%, and receive LIBOR. Therefore, there’s a fixed rate of 3.1%.
Can also convert a fixed rate to a floating if they think interest rates will come down.
What if we made swaps available for mortgage payers as well?
Swap Market
 Maturity in years
 Bid: how much you would get if you pay the floating
 Ask: how much you would pay to get the floating
 For floating rates, the rate at the beginning of the period determines the rate for the payment at the end of the period
Confirmations
 International Swaps and Derivatives has Master Agreements
Comparative Advantage Example
 AAACorp wants to borrow floating (4% fixed, 6month LIBOR  0.1% Floating)
 Pays 120 less in fixed and 70 less in floating
 BBBCorp wants to borrow fixed (5.2% fixed, 6month LIBOR + 0.6%)
 Spread is 70 basis points in floating compared to 120 in basis
 Swap designed:
 The benefit that needs to be split is: 120  70 = 50 basis points
 Think: one corporation has to pay floating to the other, so calculate the fixed rate paid to each other which is the fixed rate + half the benefit.
 BBBCorp borrows floating at +0.6% and pays fixed 4.35% and receives floating
 benefit = 5.2%  4.95% = 25 basis points
 AAACorp borrows fixed at 4% and pays floating and receives 4.35%
 spread = 0.1% + 0.35% = 25 basis points
 With a financial institution, there is a cut that is taken. That cut is basically split in two.
FixedforFixed Currency Swap
 Pay 3% on a US dollar principal of 15,000,000
 Receive 4% on a pound sterling principal of 10,000,000
Example
 GE wants to borrow AUD
 Current rates are 5% for USD and 7.6% for AUD
 Quantas wants to borrow USD
 Current rates are 7% for USD and 8% for AUD
 Swap
 Benefit is (2  0.4) = 160 basis points
 Therefore, GE borrows USD at 5%, pays 8% for AUD and gets 6.2% in USD
 Therefore, Quantas borrows AUD at 8%, pays 6.2% USD, and gets 8% AUD
Example 7.1
 swap 3% per annum and receives LIBOR every six months on $100million
 swap has 15 months remaining (3, 9, 15)
 Rate applicable to exchange in 3 months is 2.9%
 Forward LIBOR rates for 39 month period and 915 month periods are 3.429%, 3.734%
 OIS zero rates are 2.8% for 3 months, 3.2% for 9 months, and 3.4% for 15 months
Period  3 months  9 months  15 months 

LIBOR  2.9%  ~3.429%  ~3.734% 
PAY  1.5M  1.5M  1.5M 
RECEIVE  1.45  1.745  1.867 
Calculation  2.9%/2 * 100  3.429%/2 * 100  3.734%/2 * 100 
Discount by the OIS Rate using the continuous compounding formula (Pe^(rT)).
Alternatively, value both cashflows as Bonds.
The value of a swap, is the difference between what you receive and what you pay.
Example 7.3 and 7.4
 Japanese interest rates are 1.5% per annum (continuous)
 USD interest rates are 2.5%
 3% yen, 4% dollars
 Principals are $10M and 1,200M yen
 Swap lasts more than 3 years
 Exchange rate is 110 yen per dollar
 Get the PV of hte cashflows for each currency and then convert one to the other
Mechanics of Options Markets
 Call = option to buy at strike price by a certain date
 Put = option to sell at strike price by a certain date
 American style = exercised any time during its life
 European style = exercised only at maturity
Payoffs
 Long Call: max loss is the premium, max win is unlimited
 Short Call: max loss is unlimited, max win is the premium
 Long Put: max loss is the premium, max win is the share price  premium
 Short Put: max loss is the drop is share price + premium, max win is the premium
Intrinsic Value
 Max{ Strike minus Stock Price, 0 }
CBOE and OTC
 Flex options
 Binary options
 Credit event binary options
 Doom options
Dividends & Stock Splits
 stripe price K to buy/sell N shares
 nform stock split
 Strike price is mK/n
 no shares is increase to nN/m
 stock dividends is similar manner
Example
 call option to buy 100 shares for $20 per share
 2for1 stock split
 strike price of $10 to purchase 200 shares
 5% stock dividend
 Equivalent to a 1.05for1 stock split
 Strike price is 20/1.05 to purchase 105 shares
Market Makers
Options Margin
 naked option
Warrants
 right to purchase new shares issued to the right holder
Convertible
 the straight bond cannot be higher than the treasury bond but will approach it as the firm’s value rises
 MAX(Value of the bond, value of the shares you could get) + conversion premium
Swaptions
Midterm Questions
 FRA 5% LIBOR and receive 7%, semiannual compounded
 forward rate is 5%
 5.1% semiannual
1000 * (0.07  0.051) / 2 * e^(0.05(3.5)) = 7.88%
Properties of Stock Options
Effect of Variables on Option Pricing
Variable  c  p  C  P 

S_{0}  +    +   
K    +    + 
T  ?  ?  +  + 
σ  +  +  +  + 
r  +    +   
D    +    + 
Essentially, the european options differ in one way which is that the longer the time to expiration does not guarantee a higher price.
Lower Bound for European Call Option Prices; No Dividends
Is there an arbitrage opportunity if c = 3, T = 1, K = 18, S_{0} = 20, r = 10%, D = 0?
3 >= 3.71
Strategy: Short stock to get $20. Buy call for $3. Invest $17 at the risk free rate.
17^(e(0.10)(1)) = 18.79
Is there an arbitrage when the put premium is $1, T = 0.5, S = 37, r = 5%, K = 40, D = 0?
p = 1 >= 2.01
Borrow 38 to purchase put and stock.
Is there an arbitrage when the call premium is 3, put premium is 1 or 2.25, T = 0.25, S = 31, r = 10%, K = 30, D = 0?
C + ke^{rT} = p + S<sub>0</sub>
3 + 30e^{0.1(0.25)} != 2.25 + 31
32.26 < 33.25
Strategy
Short stock @31
sell put 2.25
+33.25
Buy call 3
Buy Bond 32.2
Left with: .99
American Put Exercised Early
S = 60, T = 0.25, r = 10%, K = 100, D = 0. What if K = 50?
 Advantages? Riskfree rate on the current payout
 Disadvantages?
American Put Options (No Dividends)
Impact of Dividends on the Lower Bounds to Option Prices
Need to read Chapter 10 again.
Chapter 11  Trading Strategies Involving Options
 Bond plus option to create principal protected note
 Stock plus option
 options of the same type (spread)
 Different types (combination)
Principal Protected Note
 $1000 instrument consisting of
 3year zerocoupon bond with principal of $1000
 3year atthemoney call option on a stock portfolio currently worth $1000
 Play: sell portfolio, buy ATM call, buy bond
Bull Spread Using Calls
 Buy ITM call
 Sell OTM call
 Maximum gain is the higher strike minus the ITM lower strike minus net premium paid
 Maximum loss is the net premium paid
Bull Spread Using Puts
 Buy OTM put
 Sel ITM put
Bear Spread Using Calls
 Buy OTM call
 Sell ITM call
 Net Premium received is the maximum gain
Bear Spread Using Puts
 Buy ITM put
 Sell OTM put
Box Spread
 combination of a bull call spread and a bear put spread
 if european, use present value of strike prices difference
 not necessarily so for american
Box Spread Example
Long C1 ST <= 40 40<ST<60 ST>60
Short P1 0 St40 ST40
Short C2 (40ST) 0 0
Long P2 60ST 60ST 0
20 20 20
Therefore Cost = 20e^(0.05)(0.5)
Two options, call and a put, with same underlying asset, strike. maturity.
At what strike price would they have the same value?
Answer: graphically or using putcall parity
PutCall Parity
 Putcall parity theorem is an equation representing the proper relation between put and call prices
 violation implies arbitrage opportunities
 sell high side, buy low side
 invest cash from sell
Butterfly Spread Using Calls
 Long call ITM
 Long call OTM
 Short 2 calls ATM
 Benefit from flat stock
 Butterfly using Puts
 Max revenue = Difference between centre and lowest price
Calendar Spread Using Calls
 Short on shorterdated call
 Long on longerdated call
Strangle or Straddle Combination
 Make money when volatility is higher
 Purchase OTM Put and OTM Call
 Breakeven stock price is the strike price(s) + net premiums paid (+ for call and  for put)
 long straddle: buying a call and put at the same strike price
 short straddle: writing the call and put at the same strike price
Strip & Strap
 Strip: Long call and Two long puts
 Strap: Two long calls and one long put
Chapter 12  Binomial Trees
Series of events where there are two possible outcomes.
Stock price is currently $20. In three months it will either be $22 or $18. Suppose call option has strike price 1.

Delta: shares long for every options shorted

Value of the portfolio when short a call:

20 * delta  f
where f is the value of the option 
At 22,
22 * delta  1

At 18,
18 * delta

22*delta  1 = 18 * delta
→delta = 0.25

The value in 3 months is 4.5

Today,
4.5 * e^( 0.12*0.25) = 4.367
where riskfree rate is 12% 
Therefore, 20(0.25)  f = 4.367 → f = 0.633

Risk less when
Delta = (fu  fd) / (S0u  S0d)

f = (pfu + (1  p)fd)e^(rT)

p = (e^(rt)  d) / (u  d)

u = the multiplicative factor for an up movement

d = the multiplicative factor for a down movement
RiskNeutral Valuation
In a riskneutral world, stock at time T is worth S_{0}e^{rT}. In original example, p = 0.6523 and option value is e^{0.12*0.25}(0.6523 * 1 + 0.3477 * 0) = 0.633
TwoStep Examples
Valuing a European Call Option
Valuing a European Put Option
Valuing an American Put Option
Choosing u and d
 Sigma is the annualized volatility
u = e^(sigma sqrt(delta t))
d = 1/u = e^(sigma sqrt(delta t))
Options on Stock Indices, Currencies, Futures
 Same process except p is different
 Probability of an up move
 Cash settlement one day later
 S&P 100 and AMEX Major Market Index are examples of a broad indices
p = (a  d) / (u  d)
 NonDividend: a = e^(r * delta t)
 Index with yield q: a = e^((rq) * delta t)
 Currency with foreign riskfree rate rf: e^((rrf) * delta t)
 Futures: a = 1
Time Steps
 At least 30 time steps are required for good option values
 DerivaGem allows up to 500 time steps
Chapter 18  Binomial Trees in Practice (DerivaGem)
 approximate movements in the price of a stock or other asset
 for each small interval of time (delta t), stock moves up u or down d
 tree parameters for a nondividend paying stock: volatility, riskfree rate, stock price
Custom Derivative Payoff Example
 S0 = 20
 Sigma = 25%
 r = 5%
 T =6mo
 Time step = 3 mo
 Payoff = MAX(S^2  400, 50)
 Using the twostep example, we get a price of $103
Put Example Delta Shares Short
 Delta = (2.16  6.96) / (56.12  44.55) = 0.41 (the payoff from the next step)
 As time passes, delta will change (delta hedging)
Chapter 13  BlackScholesMerton Model
 Assumptions
 The assumption in equation (13.1) implies that the stock price at any future time has a lognormal distribution.
 Volatility on the underlying is known and constant
 price of a European call option as the time step tends to Zero
 mean \mu is the expected return and \sigma is volatility
 Delta S / S is the stock return which is normal distributed
Lognormal Property
Standard deviation:
Therefore the normal distribution, \phi [mean, variance], Is
or
Example
 N = 16%, std = 35%, S0 = $38
 Calculate probability that a european call option with k = %40 and maturity 6 months out will be exercised
 P(S_T > 40)
 Use online distribution calculator to figure it out
Estimating Volatility from Historical Data
 Take observation S_0, S_1, S_n at intervals of Tau years (for weekly data, Tau = 1/52)
 Calculate the continuously compounded return in each interval as:
mu_i = ln (S_1 / S_{i  1})
 Calculate the standard deviation, s, of the mu_i’s
 Historical volatility estimate is: \sigma \hat = s / (sqrt ( Tau))
 Tau decision: need as many observations as possible
 With daily, lots of noise
 Period has to be big enough period for validity
 Need more than 30 observations
BlackScholes Formulas
European Call
European Put
d1 variable
d2 variable
The variable mu does not appear in the blackscholes equation. It is independent of all variables affected by risk preference. Consistent with riskneutral valuation principle.
 N(d2) is the probability of exercising
With dividends, need to substitute the stock price with the stock price minus the dividends paid through the maturity
Implied Volatility
 The volatility that makes the model price the derivative the same as the market price.
 If two options with the same underlying have different implied volatilizes, something might be overpriced/underpriced
Chapter 15  Options on Stock Indices and Currencies
 Most popular in the U.S.A are S&P 100 (OEX, XEO), S&P 500 (SPX), DOW times 0.01 (DJX), NASDAQ 100 (NDX)
 Contracts are settled on 100 times the index in cash OEX is American whereas the others are European
Example 15.1
 Portfolio Beta of 1.0
 Value is $500,000
 Index at 1,000
 What trade is necessary to provide insurance to prevent value from falling below $450,000
Example 15.2
 Portfolio has Beta of 2.0
 Value is $500,000
 Index at 1,000
 rf = 12% per annum
 dividend yield on both is 4%
 How many put options to purchase on the index at the strike?
How to solve?
 Find relationship of portfolio to index.
 The portfolio return is the value that fell plus the prorated dividends that was received
 Then use this return to calculate the situational return on the index and subtract the dividend yield
 Do do this use CAPM formula
 This nominal value on the index is the strike price we want to purchase of the put
 Find number of puts to purchase
 Put options to purchase to cover the initial portfolio:
Beta \* ValueOfPortfolio / (ValueOfIndex * 100)
where values are the initial values
 Put options to purchase to cover the initial portfolio:
Currency Options
 NASDAQ OMX
 Used for buying insurance when exposed to FX
 Lower bound is equivalent to european options with dividends
Range Forward
Chapter 16  Futures options and Black’s Model
 American and expires a few days before the earliest delivery
 When a call futures option is exercises
 The Holder acquires
 A long position in the futures
 A cash amount equal to excess of the futures price at the most recent settlement over the strike price
 The Holder acquires
Example 16.1
 July call option on gold futures with a strike of $12000 per ounce. Exercised when futures price is 1,240 and recent settlement of 1,238. One contract is 100 ounces
 Trader receives: one long July contract on gold and (1238  1200) * 100 = 3800.
Example 16.2
 September put option on corn 300 cents per bushel
 exercised when futures is 280 cents per bushel with recent settlement of 279 cents per bushel
 Trader received: long short futures on the corn contract and 21 cents per bushel in cash
Immediately Selling the Future Payoff
 Payoff from call = F  K
 Payoff from put = K  F
Advantages of Future Options over Spot Options
 futures may be easier to trade
 no delivery
 futures and options trade on the same exchange
 futures options may entail lower transaction costs
European Futures Options
 the futures option and spot options are equal at maturity
 spot options are regarded as futures options when valued over the counter
 when futures prices decrease with maturity, American call futures are worth less than the corresponding American call on the underlying asset
 American futures options are never equal to European futures options unless it’s the last day of exercising
PutCall Parity for European Futures Options
Binomial Riskless Futures Option Pricing
 1 month out
 3Delta  4 = 2 Delta → Long Delta futures of 0.8
 With a riskfree rate of 6%, the value of the portfolio is (2 * 0.8)e^{0.06/12} = 1.592
 Value of the options must be 1.592 since the value of the futures is 0
The portfolio is riskless when
Chapter 17  The Greek Letters
 Delta = change in option price with relation to underlying
 Vega or Lambda: change in option price with relation to underlying implied volatility
 if volatility goes up and price of underlying stays the same, both the put and call options go up in price
 Theta: change in option price with relation to time
 If gamma and delta are 0, then the portfolio is riskneutral and should be appreciating at the riskfree rate
 Rho: change in option price with relation to interest rate
 Gamma: Change in option’s delta in relation to stock price change (2nd derivative)
 Positive for long puts and calls since if price goes up, delta will go up for call obviously, but also up for put since less short is needed in underlying
StopLoss Strategy
 assuming a naked call position
 buying 100,000 shares as soon as price reaches $50
 selling 100,000 shares as soon as price falls below $50
 if the stock fluctuates around the strike price of 50, then this strategy loses lots of money due to buying high and selling low
How DeltaHedging Works
If we hold a short call position and hold delta shares, why are we doing so? When the underlying’s value goes up, we lose delta in the short call options, but we also gain on the underlying shares.
 delta on a european nondividend call is N(d1)
 european nondividend put: N(d1)  1
 As time to maturity decreases, delta of out the money calls increases, and in the money calls decreases
 Buy high sell low
Example 17.1
 A bank has sold for $300,000 a European call option on 100,000 shares of a nondividend paying stock
 S0 = 49, K = 50, r = 5%, s = 20%,
 T = 20 weeks, m = 13%
 The BlackScholesMerton value of the option is $240,000
 How does the bank hedge its risk to lock in a $60,000 profit?
Gamma Addresses Delta Hedging Errors Caused by Curvature
 greatest for options at the money
 Tau = portfolio value
 change in gamma = Theta times change in time + 0.5 Tau change in share price squared
Managing Delta, Gamma, and Vega
Gamma and Vega require taking a position in the options themselves.
–  Delta  Gamma  Vega 

Portfolio  0  5000  8000 
Option 1  0.6  0.5  2.09T11 
Option 2  0.5  0.8  1.2 
What position in option 1 and the underlying asset will make the portfolio delta and gamma neutral? Answer: Long 10,000 options, short 6000 of the asset
What position in option 1 and the underlying asset will make the portfolio delta and vega neutral? Answer: Long 4000 options, short 2400 of the asset
What position in option 1, option 2, and the asset will make the portfolio delta, gamma, and vega neutral? We solve
−5000+0.5w1 +0.8w2 =0
−8000+2.0w1 +1.2w2 =0
to get w1 = 400 and w2 = 6000. We require long positions of 400 and 6000 in option 1 and option 2. A short position of 3240 in the asset is then required to make the portfolio delta neutral
Rho
Rho is the rate of change of the value of a derivative with respect to the interest rate
Hedging in practice
 become deltaneutral at least once a day
 whenever opportunities arise, improve gamma and vega
 hedging becomes less expensive as a portfolio gets larger
Greek Letters When Underlying has a Yield
See slide 32 of slide deck 17 (or see page 381)
Futures for Delta Hedging
futures is e^{(rq)T} times the position required in the spot contract
Synthetic Option
 take positions that match the greeks of the option
Portfolio Insurance
 Sell enough of the portfolio or index futures to match the delta of the put option (e.g. October 1987)
 As portfolio value increases, delta goes down and so original portfolio is repurchased to some extent
 As portfolio value decreases, more of portfolio is sold
Chapter 22  Exotic Options
 Packages
 Nonstandard American options
 Gap options
 Forward start options
 Cliquet options
 Compound options
 Chooser options
 Barrier options
 Binary options
 Lookback options
 Shout options
 Asian options
 Options to exchange one asset for another
 Options involving several assets
Packages
 Portfolios of standard options
 Examples from Chapter 11: bull spreads, bear spreads, straddles, etc
 Example from Chapter 15: Range forward contracts
 Packages are often structured to have zero cost
Nonstandard American options
 Bermudan: exercisable on specific dates
 initial lock out period
 strike price changes over life
Gap Options
 Call option pays (ST  K1) when (S > K2)
 Put option pays (K1  ST) when (S < K2)
 Valuation formula found in Chapter 22
Forward Start Options
 Option starts at a future time T
 Often structured so that strike price equals asset price at time T
 Planning to give employees atthemoney options in each future year can be regarded as a series of forward start options
Cliquet Option
 rules determine how the strike price is determined
 for example, 20 atthemoney threemonth options (total life of five years)
 When one option expires, a similar one comes into existence
Compound Options
 An option on an option
 Call on call
 Put on call
 Call on put
 Put on put
 Very sensitive
Chooser Options
 start at 0, mature at T2
 at T1, buyer can choose whether the option is a put or a call
 this is a package
 p = c + e^{r(T2T1)}K  S1 e^{q(T2T1)}
 At T1, c + e^{q(T2T1)} max(0, Ke^{(rq)(T2T1)}  S)
 call maturing at T1 plus put maturity at T1
Barrier Options
 in: option comes into existence only if asset price hits barrier before option maturity
 out: option are knocked out if asset price hits barrier before option maturity
 up: asset price hits barrier from below
 down: asset price hits barrier from above
 eight possible combinations (put or call)
 parity
 c = cui + cuo
 c = cdi + cdo
 p = pui + puo
 p = pdi + pdo
Binary Options
 Cashornothing: pays Q if S > K at time T. Value = e^{rT}QN(d2)
 Assetornothing: pays S if S > K at time T, or nothing. Value = S0e^{qT}N(d1)
Lookback Options
 Floating call: Pays
Stock at time T – Stock minimum
at time T Allows buyer to buy stock at lowest observed price in some interval of time
 Floating put: pays
Stock max  Stock at time T
at time T Allows buyer to sell stock at highest observed price in some interval of time
 Fixed call: pays maximum observed asset price minus strike price
 Fixed put: pays strike price minus minimum observed asset price
Shout Options
 Able to lock in a price once during the life
 Usually pays like a call or a put but also the intrinsic value at the shout time
Asian Options
 Payoff related to average stock price
 average price
Options to Exchange
 One asset for another
 Payoff is price difference between the assets
Basket Options
 option on the value of a portfolio
MortgageBacked Securities
 PassThrough
 Collateralized Mortgage Obligation (CMO)
 Interest Only (IO)
 Principal Only (PO)
Variations of Interest Rate Swaps
 different principles
 different payment frequencies
 floating for floating or fixed for fixed
Compounding Swaps
 Business Snapshot 22.2
 Interest is compounded instead of paid
Complex Swaps
 LIBORinarrears swaps
 CMS and CMT swaps
 Differential swaps
Equity Swaps
 Business Snapshot 22.3
 Total return on equity index is exchanged for a fixed or floating return
Embedded Swaps
 Accrual swaps
 Cancelable swaps
 Cancelable compounding swaps
Other Swaps
 Indexed principal swap
 Commodity swap
 Volatility swap
 Bizarre deals: P&G 5/30 swap
 P&G receiving 5.3% interest on $200M for 5 years semiannually
 P&G would pay back the 30day commercial paper rate minus 75 basis points plus spread
 spread = max (0, 98.5 * (5 year commercial constant maturity rate) / (5.78%)  5 year treasury price) / 100
Chapter 8  Secularization
 traditionally loans are funded via deposits
 loans can increase much faster than deposits
 Assets are combined and sold in tranches
 Senior Tranche 80% (ABSs)  AAA
 Mezzanine Tranche (15%)  BBB  ABS CDO Created
 Senior Tranche (65%)  AAA (does not convey actual risk)
 Mezzanine Tranche (25%)  BBB
 Equity Tranche (10%)
 Equity Tranche  Not Rated
 Bankruptcies wipe out from equity first
 Asset cash flows go first to senior, then mezzanine, then equity
What Led to the Financial Crisis
 Starting in 2000, mortgage originators relaxed lending standards and created large subprime first mortgages
 demand for real estate and prices rose
 100% mortgage
 ARMs
 teaser rates
 no income, no job, no assets (NINJAs), ARMs, teaser rates, liar loans, nonrecourse borrowing (repossession)
What was not accounted for
 Default correlation increase in stressed market conditions
 Recovery rates are less in stressed market conditions
 Tranche with a certain rating cannot be equated with a bond with the same rating
 BBB tranches used to create ABS CDOs were 1% wide nad had all or nothing distributions
 not the same as the loss distribution for a BBB bond
Regulatory Arbitrage
 capital required to keep for the tranches was less than the mortgages themselves
 mortgage originators: only cared about originating mortgages that can be securitized
 Valuers: under pressure to provide high valuations to keep business
 traders: focused on yearend bonus and not long term
Aftermath of Financial Crisis
 Banks required to hold more equity capital with the definition of equity capital being tightened
 Banks required to satisfy liquidity ratios
 CCPs and SEFs for OTC derivatives
 Bonuses limited in Europe
 Bonuses spread over several years
 Proprietary trading restricted
Chapter 25  Derivative Mishaps
Losses by NonFinancial Corporations
 Allied Lyons ($150M)
 Gibsons Greeting ($20M)
 Hammersmith and Fulham ($600M)
 Metallgesellschaft ($1.8B)
 Promised client long term supply of oil at certain prices
 Sold hedge at the bottom and then the hedge was useless
 Orange County ($1.6B)
 Robert L. Citron was making excess returns
 Reverse repos
 Borrowed money in shortterm and invested in shortterm markets
 Took new assets and put them up as collateral to buy in longterm securities
 Procter and Gamble ($90M)
 Shell ($1B)
 Sumitomo ($2B)
 Trader at Sumitomo was trying to recoup losses through copper
Losses by Financial Institutions
 Allied Irish Banks ($700M)
 Amaranth ($6B)
 Barings ($1B)
 Enron’s counterparties (billions)
 Kidder Peabody ($350M)
 LTCM ($4B)
 high leverage
 exposure to 1997 Asian financial crisis and 1998 russian financial crisis
 Midland Bank ($500M)
 Societe Generale ($7B)
 Subprime mortgages (tens of billions)
 UBS ($2.3B)
Chapter 19  Volatility Smiles
For options with some maturities, the implied volatility versus the strike price makes a smile.