Advanced Derivatives:
(plain vanilla to Rainbows)
advanced swaps
Structured notes
exotic options
S. Mann, 2006
1
Equity Swaps
Example:
Thai Bank prohibited from holding domestic equity
Bank circumvents regulation with total return swap:
Thai bank buys US government securities
Tiger fund buys Thai equity
Enter into total return swap: returns swapped, not asset.
Thai
Financial
Institution
Thai equity return
Tiger Fund
or other
Hedge Fund
US Bond return
Return details (what currency?) denoted by distinct swap names
2
Asset swaps: Quantos
Total return swap with exchange rate risk eliminated
Payments determined by total return on different assets,
multiplied by notional principal in one currency.
Example: swap S&P 500 for CAC-40 (France) + spread
(CAC-40 return + spread) x Notional principal
U.S.
Global
Portfolio
French
Pension
Fund
S&P 500 total return x Notional Principal
Payment details on next slide
3
Quanto swap outcome example
A possible sequence of events
Quanto swap: Pay S&P 500 return, receive CAC-40 + swap spread
Notional principal ($millions)
25
payments all in dollars
swap spread (basis points)
date
days
2/17/98
5/15/98
8/17/98
11/16/98
2/16/99
88
92
89
90
70
S&P 500
S&P 500
total return
index % ret
payment
955
964
0.94%
235,027
986
2.24%
558,759
1032
4.65% 1,162,832
1012
-1.86% -463,847
day count = actual/360
CAC-40 (France)
CAC-40
total return
index % ret
payment
2230
2179
-2.3% -564,964
2536
16.4% 4,093,328
2514
-0.9% -215,181
2681
6.6% 1,653,370
spread
payment
42,778
44,722
43,264
43,750
net
payment
(757,213)
3,579,291
(1,334,749)
2,160,967
4
Zero-cost collar:
sell call to pay for put:
choose put so that loss
possibility at least 10%.
(Investor is “at risk”, not an
IRS “constructive sale”).
Collar value
Monetarize position without
realizing gain.
(% of original stock price)
Equity Collars
+25%
Long
Stock
Stock plus collar
-10%
ST
Borrow against hedged position at advantageous rate (Libor + 100 bp).
Standard contracts available for large ($2 million) positions in liquid stock.
Longer the term, higher upside percentage available.
Cite: Braddock, 1997, “Zero-cost Collars,” Risk, November 1997.
5
Swap floating for floating
Basis Swap:
Libor - spread
T-bill
Payer
Libor
payer
T-bill rate
Constant Maturity swap
Constant
Maturity
Payer
Libor + spread
Libor
payer
Five-year T-note
Constant maturity yield
6
Amortizing swap
Notional principal reduced over time (e.g. mortgage)
N1
N2
N3
N4
T1
T2
T3
T4
Valuation:
0 = B(0,T1)(SFR - F1)N1 + B(0,T2)(SFR - F2)N2
+ B(0,T3)(SFR - F3)N3+ B(0,T4)(SFR - F4)N4
where
Ft
SFR
= appropriate forward rate
= swap fixed rate
7
Diff swaps: (currency hedged basis swap)
Floating for floating swap
Floating rates are in different currencies
All swap payments in one currency
Example: swap 5 year gilt (£) yield for 5 year CMT T-note yield
swap payments in $
(5-year £ gilt yield) x Notional principal ($)
U.S. Firm
desiring
exposure
to UK yield
U.S Firm
reducing
exposure
to UK yield
(5 -year CMT yield) x Notional principal ($)
8
Commodity derivatives
Commodity-linked loans
Merrill Lynch - $250 mil Aluminum-linked bond for Dubal
(Barrick)
Price protection standard for project financing
hedging to assure break-even as loan requirement.
Gold hedging used to raise LBO funds.
Gas swaps
Basis swaps (Enron)
Oil swaps
Crack Spread swaps
9
Credit derivatives
First generation:
Bankers Trust (BT) and Credit Suisse (CS) notes (Japan 1993)
objective: free up credit lines to Japanese financial sector
note payoffs:
coupon = Libor + 100 bp ;
but: coupon and principal reduced if defaults occur.
one lego (building block) is credit default swap:
Notional Principal x (40 bp)
Protection
Buyer
Protection
Seller
Floating payment contingent on defaults;
payment mirrors loss incurred by creditors
Contingent payment based on post-default value of reference security 10
Enron Credit default swaps – Fall 2001
Enron Credit Default Swaps
Bid/Offer Band (basis points)
1400
1200
1000
800
600
400
200
0
11/21/01
11/14/01
11/7/01
10/31/01
10/24/01
60 Month Maturity
10/17/01
10/10/01
10/3/01
9/26/01
9/19/01
9/12/01
9/5/01
8/29/01
8/22/01
8/15/01
8/8/01
8/1/01
120 Month Maturity
12 Month Maturity
11
GM Credit default swaps: 2002-2004
12
GM Credit default swaps – Fall 2005
13
Structured notes: Range Floaters
(Range contingent accrual bonds)
Bonds that accrue interest only on days when range
conditions satisfied.
Example:
$10 million bond: 12% coupon, accrual range contingent;
range is ($.50, $.59) $/DM
semiannual coupon =
$10m x (.12) x (S (days within range)/365)
(this is a restart accrual; can be barrier terminal accrual)
14
Structured notes - Inverse floater
Example:
GNMA 10-year note; maturity 12/15/07
coupon paid semi-annually: 6/15 and 12/15
coupon = max(0.02, (0.18- 2xLibor)) x (180/360) x Face
coupon on $1 million note a function of Libor:
Libor coupon
Coupon
.050 40,000
40,000
.055 35,000
T-note coupon
.060 30,000
30,000
.070 20,000
20,000
.080 10,000
Floater coupon
.090 10,000
10,000
5%
6%
7%
8%
9% Libor
15
Exotic options
Binaries: Digital ; Gap ; Ranges.
Chooser (as you like it)
Rainbow (welcome to OZ) option on best of two
Asian (average price or average strike)
Bermudan (exercise windows)
Lookback (no regret)
barrier options:
knockouts:
up and out; down and out
Knockins:
up and in; down and in
many, many more, including
“Down and in” Arrow, or Arrow-Debreu (advanced*)
(* see Carr and Chou, 1997, RISK magazine, vol 10 #9)
16
Digital and Gap options
Examples:
1) European Gap call
option, with G=0
Payoffs:
ST - G
0
if ST > K
if ST < K
PayoffT
K
2) digital European call
Payoffs:
K
0
if ST > K
if ST < K
K
ST
17
Range Binary options
Example:
1) binary $/DM range option
with range = ( $.56, $.575)
PayoffT
Payoff:
3x premium if $.56 < ST < $.575
0
if ST < $.56 or ST > $.575
Typical underlying:
exchange rates, interest rates
commodity prices
3x premium
$0.56
Usage example: Corp long DM, buys put and range.
Outcomes:
1) DM up :
gain on long DM position
2) DM down: hedged with put
3) unchanged: range pays off, pays for put.
$0.575
ST
18
Quattro option (Banker’s Trust 1996)
binary quad-range option:
four ranges!
Payoff:
8x premium
6xpremium
4xpremium
2xpremium
0
PayoffT
8x premium
if all four ranges unbroken
if only one range broken
if two ranges unbroken
if only one range unbroken
if all ranges broken
All four ranges!
ST
Note this allows sale of volatility
with limited loss (as opposed to sale of straddle)
19
Rainbow Options
Rainbow option: Option on best of two assets
$180
$160
Asset A
asset prices
$140
$120
$100
Asset B
$80
$60
Option payoff = max(0, AT-K,BT-K)
if K=$100; AT = 110; BT = $143
$40
$20
Rainbow payoff = $43
$0
1
9
17
25
33
41
49
57
65
73
81
89
97 105 113 121 129 137 145 153 161 169 177 185 193 201 209 217 225 233 241 249
Time (days)
20
Asian (Average price) Options
Price history for Asian option payoff
140
Asset price
120
Average=94.75
100
80
60
40
20
Option life (averaging period= 180 days)
0
1
12 23 34 45 56 67 78 89 100 111 122 133 144 155 166 177 188 199 210 221 232 243
Time (days)
21
Barrier Options: down and out
Down and Out call option
120
Asset price
100
80
60
Lower barrier
40
Option ceases
to exist
20
0
1
12 23 34 45 56 67 78 89 100 111 122 133 144 155 166 177 188 199 210 221 232 243
Time (days)
22
Barrier Options: down and in
Down and In put option
90
80
Asset price
70
Lower barrier
60
Lower barrier
50
Option is
activated
40
30
20
10
0
1
12 23 34 45 56 67 78 89 100 111 122 133 144 155 166 177 188 199 210 221 232 243
Time (days)
23
Up and out knockout put
Up and Out Put Option
120
100
Asset price
Knockout upper barrier
80
60
Option ceases
to exist
40
20
0
1
12 23 34 45 56 67 78 89 100 111 122 133 144 155 166 177 188 199 210 221 232 243
Time (days)
24