Description
Describes different types of option positions, factors affecting option premium, black scholes options pricing formula, naked options, flex options, american vs european option, put call parity, impact of dividend on option prices, options on futures, put call parity for future options, exotic options, different types of exotic options, interest rate swaps, currency swaps.
Non-Linear Derivatives: Options and Swaps
AGENDA
• Introduction to Linear and Non-Linear Derivatives • Options Valuation, Pricing And Implied Volatility • Option Greeks
2
American vs European Options
?An American option can be exercised at any time during its life ?A European option can be exercised only at maturity ?Is the value of an American Option higher or lower than a European Option? ?A Bermudan option is a call or put option which can be exercised on prespecified days during the life of the option. It is reasonable to say that Bermudan options are a hybrid of European options, which can only be exercised on the option expiry date, and American options, which can be exercised at any time during the option life time. ?As a consequence, under same conditions, the value of a Bermudan option is greater than (or equal to) a European option but less than (or equal to) an American option.
3
Option Positions
• Long call
• Long put
• Short call • Short put
4
Payoff for Call Option Buyer
Gain Profit Gain
Strike Price
0 Loss
Market Price
Premium Paid (Price to purchase Option)
• Note: Upside potential is unlimited, Downside risk is limited
5
Long Call
Profit from buying one European call option: option price = $5, strike price = $100, option life = 2 months
30 Profit ($) 20 10 70 0 -5 80 90 100 Terminal stock price ($) 110 120 130
6
Payoff for Call Option Seller
Gain
Premium Received Market Price Strike Price
0 Loss
Profit Loss
• Note: Upside potential is limited to the premium received. Downside risk is unlimited.
7
Short Call
Profit from writing one European call option: option price = $5, strike price = $100
Profit ($)
5 0 -10 -20 -30
8
110 120 130 70 80 90 100 Terminal stock price ($)
Payoff for Put Option Buyer
Gain Strike Price 0 Loss Market Price
Premium
• Note: Upside potential is limited to the price of the security. Downside risk is limited to the premium.
9
Long Put
Profit from buying a European put option: option price = $7, strike price = $70
30 Profit ($) 20 10
0
-7
Terminal stock price ($)
40 50 60 70 80 90 100
10
Payoff for Put Option Seller
Gain Premium 0 Loss Strike Price Market Price
Loss
• Note: Upside potential is limited to the premium. Downside risk is limited to the price of the security.
11
Short Put
Profit from writing European put option: option price = $7, strike price = $70
Profit ($) 7 0 -10 -20 -30
12
40
50
60 70 80
Terminal stock price ($) 90 100
Payoffs from Options
What is the Option Position in Each Case?
X = Strike price, ST = Price of asset at maturity
Payoff
Payoff X
X ST Payoff X X ST ST ST
Payoff
13
Specification of Exchange-Traded Options
• Underlying Asset
• Expiration date • Strike price • European or American • Call or Put
14
Specification of OTC Options
• Expiration date – (e.g. for currencies, specify “New York cut” or “Tokyo Cut”)
• Strike price • European or American • Call or Put
• Underlying asset
15
Terminology
• Option class • Option series • Intrinsic value • Time value
16
Option Terminology (Moneyness)
In-the-money Option For a call option, strike price < Market Price For a put option, strike price > Market price At-the-money Option
For both call and put options
Strike Price= Market Price Out-of-the-money Option For a call option, Strike Price > Market Price For a put option, strike Price < Market Price
17
Intrinsic Value – e.g. of Call Option on Commodity Futures
18
Intrinsic Value – e.g. of Put Option on Commodity Futures
19
Time Value
20
Factors Affecting Option Premium
21
Black-Scholes Options Pricing Formula
22
Notation
• c : European call option
price
• p : European put option price • S0 : Stock price today
• C:
• • • • P:
American Call option price American Put option price
ST :Stock price at option maturity D : Present value of dividends during option?s life r: Risk-free rate for maturity T with cont comp
• K:
• T: • ?:
Strike price
Life of option Volatility of stock price
23
Effect of Variables on Option Pricing
Variable S0 K T ? r D
c
+ – + + + –
– + + + – +
24
p
C
+ – + + + –
– + + + – +
P
Naked Options and Covered Calls
• Naked options are unhedged positions • Covered calls involve writing calls when the underlying asset (to be delivered in case the option is exercised against the option writer) is owned
25
Flex Options
• CBOT offers flex options on equity and equity indices • These are options where traders on the floor of the exchange agree to nonstandard terms on strike price or exercise dates, American vs European, etc. • Attempt by exchanges to regain business from OTC markets
26
Warrants
• Warrants are options that are issued (or written) by a corporation or a financial institution • The number of warrants outstanding is determined by the size of the original issue & changes only when they are exercised or when they expire • In International markets, Warrants are traded in the same way as stocks • The issuer settles up with the holder when a warrant is exercised • When call warrants are issued by a corporation on its own stock, exercise will lead to new treasury stock being issued
27
Executive Stock Options
• Option issued by a company to executives • When the option is exercised the company issues more stock • Usually at-the-money when issued • They become vested after a period of time (usually 1 to 4 years) • They cannot be sold • They often also last for as long as 10 or 15 years
28
Convertible Bonds
• Convertible bonds are regular bonds that can be exchanged for equity at certain times in the future according to a predetermined exchange ratio • Very often a convertible is callable • The call provision is a way in which the issuer can force conversion at a time earlier than the holder might otherwise choose
29
American vs European Options
An American option is worth at least as much as the corresponding European option C?c P?p
30
Upper Bound for Options
• Upper bound for call options c<=S; C<=S • Upper bound for put options
p<=K;
p<=K.e(-rT)
P<=K
For European put option: • If above is violated, then arbitrage would arise
31
Calls: An Arbitrage Opportunity?
• Suppose that c=3 T=1 S0 = 20 r = 10%
K = 18
D=0
• Is there an arbitrage opportunity?
32
Puts: An Arbitrage Opportunity?
• Suppose that p T =1 = 0.5 S0 = 37
r
K
= 5%
= 40
D =0 • Is there an arbitrage opportunity?
33
Lower Bound for European Call Option Prices; No Dividends
c ? S0 –Ke -rT
Lower Bound for European Put Prices; No Dividends
p ? Ke
-rT–S
0
34
Put-Call Parity; No Dividends
• Consider the following 2 portfolios:
• Portfolio A: European call on a stock + PV of the strike price in cash • Portfolio C: European put on the stock + the stock
• Both are worth MAX(ST , K ) at the maturity of the options
• They must therefore be worth the same today. This means that
c + Ke -rT = p + S0
• This is similar to a range forward (long call and short a put), where
c - p = S0 - Ke -rT
35
Arbitrage Opportunities
• Suppose that c T K =3 = 0.25 =30 p = 2.25 ? S0 = 31 r = 10% D=0 p=1?
• What are the arbitrage possibilities when
36
Early Exercise
• Usually there is some chance that an American option will be exercised early • An exception is an American call on a non-dividend paying stock
• This should never be exercised early
37
An Extreme Situation
• For an American call option:
S0 = 100; T = 0.25; K = 60; D = 0
Should you exercise immediately?
• What should you do if
You want to hold the stock for the next 3 months? You do not feel that the stock is worth holding for the next 3 months?
38
Reasons For Not Exercising a American Call Early (Non Dividend paying stock)
• No income is sacrificed • We delay paying the strike price • Holding the call provides insurance against stock price falling below strike price • Alternative to exercising is to sell the American call option itself
39
Should American Puts (on Non-Dividend paying stocks) be Exercised Early ?
Are there any advantages to exercising an American put when S0 = 60; T = 0.25; r=10% K = 100; D = 0
40
The Impact of Dividends on Lower Bounds to Option Prices
c ? S 0 ? D ? Ke
p ? D ? Ke
? rT
? rT
? S0
41
Extensions of Put-Call Parity
• American options; D = 0
S0 - K < C - P < S0 - Ke -rT
• European options; D > 0 c + D + Ke -rT = p + S0 • American options; D > 0 S0 - D - K < C - P < S0 - Ke -rT
42
Introduction to Binomial Trees
A Simple Binomial Model
• A stock price is currently $20
• In three months it will be either $22 or $18
Stock Price = $22
Stock price = $20 Stock Price = $18
44
A Call Option
A 3-month call option on the stock has a strike price of 21.
Stock Price = $22 Option Price = $1
Stock price = $20 Option Price=?
Stock Price = $18 Option Price = $0
45
Setting Up a Riskless Portfolio
• Consider the Portfolio: long D shares short 1 call option
22D – 1
18D
• Portfolio is riskless when 22D – 1 = 18D or
D = 0.25
46
Valuing the Portfolio
(Risk-Free Rate is 12%)
• The riskless portfolio is: long 0.25 shares and short 1 call option
• The value of the portfolio in 3 months is 22 ? 0.25 – 1 = 4.50
• The value of the portfolio today is 4.5e – 0.12?0.25 = 4.3670
47
Valuing the Option
• The portfolio that is
long 0.25 shares short 1 option
is worth 4.367
• The value of the shares is 5.000 (= 0.25 ? 20 )
• The value of the option is therefore 0.633 (= 5.000 – 4.367 )
48
Scenario
A derivative lasts for time T and is dependent on a stock
S ƒ
Su ƒu Sd ƒd
49
Generalization
• Consider the portfolio that is long D shares and short 1 derivative
SuD – ƒu SdD – ƒd
• The portfolio is riskless when SuD – ƒu = Sd D – ƒd or ƒu ? f d D? Su ? Sd
50
Risk-Neutral Valuation
• ƒ = [ p ƒu + (1 – p )ƒd ]e-rT
• The variables p and (1 – p ) can be interpreted as the risk-neutral probabilities of up and down movements
• The value of a derivative is its expected payoff in a risk-neutral world discounted at the risk-free rate
S ƒ
Su ƒu Sd ƒd
51
Binomial
Su = 22 ƒu = 1
Sd = 18 ƒd = 0 • Since p is a risk-neutral probability
20e0.12 ?0.25 = 22p + 18(1 – p ); p = 0.6523 • Alternatively, we can use the formula
e rT ? d e 0.12?0.25 ? 0.9 p? ? ? 0.6523 u?d 1.1 ? 0.9
S ƒ
52
Valuing the Option
Su = 22 ƒu = 1 S ƒ Sd = 18 ƒd = 0
The value of the option is e–0.12?0.25 [0.6523?1 + 0.3477?0] = 0.633
53
A Two-Step Example
24.2 22
20 18
19.8
16.2 • Each time step is 3 months
54
Valuing a Call Option
D
22 20 1.2823
A
24.2 3.2
B E C
2.0257
18
0.0
19.8 0.0
16.2 F • Value at node B 0.0 = e–0.12?0.25(0.6523?3.2 + 0.3477?0) = 2.0257
• Value at node A = e–0.12?0.25(0.6523?2.0257 + 0.3477?0)
= 1.2823
55
Implied Volatility
• The implied volatility of an option is the volatility for which the Black-Scholes price equals the market price • The is a one-to-one correspondence between prices and implied volatilities • Traders and brokers often quote implied volatilities rather than dollar prices • The implied volatility calculated from a European call option should be the same as that calculated from a European put option when both have the same strike price and maturity
56
Volatility Smile
• A volatility smile shows the variation of the implied volatility with the strike price
• The volatility smile should be the same whether calculated from call options or put options
57
The Volatility Smile for Foreign Currency Options
Implied Volatility
Strike Price
58
The Volatility Smile for Equity Options
Implied Volatility
Strike Price
59
Possible Causes of Volatility Smile
• Asset price exhibiting jumps rather than continuous change
• Volatility for asset price being stochastic
(One reason for a stochastic volatility in the case of equities is the relationship between volatility and leverage)
60
Volatility Term Structure
• In addition to calculating a volatility smile, traders also calculate a volatility term structure • This shows the variation of implied volatility with the time to maturity of the option • The volatility term structure tend to be downward sloping when volatility is high and upward sloping when it is low
61
Black Scholes on Stock Indices
c ? S 0 e ? qT N ( d1 ) ? Ke ? rT N (d 2 ) p ? Ke ? rT N ( ? d 2 ) ? S 0 e ? qT N ( ? d1 ) ln( S 0 / K ) ? ( r ? q ? ? 2 / 2)T where d1 ? ? T ln( S 0 / K ) ? (r ? q ? ? 2 / 2)T d2 ? ? T
62
LEAPS
• Leaps are options on stock indices that last up to 3 years • They have December expiration dates • They are on 10 times the index • Leaps also trade on some individual stocks
63
Currency Options
• Currency options trade on the Philadelphia Exchange (PHLX)
• There also exists an active over-the-counter (OTC) market • Currency options are used by corporations to buy insurance when they have an FX exposure
64
The Foreign Interest Rate
• We denote the foreign interest rate by rf • When a U.S. company buys one unit of the foreign currency it has an investment of S0 dollars
• The return from investing at the foreign rate is rf S0 dollars
• This shows that the foreign currency provides a “dividend yield” at rate rf
65
Valuing European Currency Options
• A foreign currency is an asset that provides a continuous “dividend yield” equal to rf • We can use the formula for an option on a stock paying a continuous dividend yield : Set S0 = current exchange rate Set q = rƒ
66
Formula for European Currency Options
c ? S0e
?rf T
N ( d1 ) ? Ke ? rT N (d 2 )
?rf T
p ? Ke ? rT N ( ?d 2 ) ? S 0 e where d1 ? d2 ?
N (?d1 ) f ? ? 2 / 2)T
ln( S 0 / K ) ? (r ? r ? T f
ln( S 0 / K ) ? (r ? r ? T
67
? ? 2 / 2)T
Options on Futures
Mechanics of Call Futures Options
When a call futures option is exercised the holder acquires 1. A long position in the futures
2. A cash amount equal to the excess of
the futures price over the strike price at previous settlement
69
Mechanics of Put Futures Option
When a put futures option is exercised the holder acquires 1. A short position in the futures 2. A cash amount equal to the excess of the strike price over the futures price at previous settlement
70
The Payoffs
If the futures position is closed out immediately:
Payoff from call = F0 – K
Payoff from put = K – F0 where F0 is futures price at time of exercise
71
Potential Advantages of Futures Options over Options on Physical Commodity
• Futures contract may be easier to trade than underlying asset • Exercise of the option does not lead to delivery of the underlying asset • Futures options and futures usually trade adjacent to each other on same exchange • Futures options may entail lower transactions costs
72
Put-Call Parity for Futures Options
Consider the following two portfolios: 1. European call plus Ke-rT of cash 2. European put plus long futures plus cash equal to F0e-rT They must be worth the same at time T so that
c+Ke-rT=p+F0 e-rT
73
Other Relations
Fe-rT – K < C – P < F – Ke-rT
c > (F – K)e-rT
p > (F – K)e-rT
74
Valuing European Futures Options
• We can use the formula for an option on a stock paying a continuous dividend yield
Set S0 = current futures price (F0)
Set q = domestic risk-free rate (r )
• Setting q = r ensures that the expected growth of F in a risk-neutral world is zero
75
Growth Rates For Futures Prices
• A futures contract requires no initial investment • In a risk-neutral world the expected return should be zero • The expected growth rate of the futures price is therefore zero
• The futures price can therefore be treated like a stock paying a dividend yield of r
76
Black?s Model
The formulas for European options on futures are known as Black?s model
c ? e ? rT ?F0 N (d1 ) ? K N (d 2 )?
p ? e ? rT ?K N (? d 2 ) ? F0 N (? d1 )? ln( F0 / K ) ? ? 2T / 2 ? T ? T
77
where d1 ? d2 ?
ln( F0 / K ) ? ? 2T / 2
? d1 ? ? T
Futures Option Prices vs Spot Option Prices
• If futures prices are higher than spot prices (normal market), an American call on futures is worth more than a similar American call on spot. An American put on futures is worth less than a similar American put on spot • When futures prices are lower than spot prices (inverted market) the reverse is true
78
Put-Call Parity Results
Indices : c ? K e ? rT ? p ? S e ? qT Foreignexchange : c?K e Futures: c ? K e ? rT ? p ? F e ? rT
? rT
? p?S e
?rf T
79
Market Makers
• Market makers facilitate trading • A market maker quotes both bid and ask prices when requested • The market maker does not know whether the individual requesting the quotes wants to buy or sell • Exchanges often set upper limits for bidask spreads
80
Exotic Options and Other Nonstandard Products
Types of Exotic Options
• Packages
• Nonstandard American options
• Forward start options • Compound options • Chooser options • Barrier options
• Binary options
82
Types of Exotic Options continued
• Lookback options
• Shout options
• Asian options • Options to exchange one asset for another • Options involving several assets
83
Nonstandard American Options
• Exercisable only on specific dates (Bermudans) • Early exercise allowed during only part of life (e.g. there may be an initial “lock out” period) • Strike price changes over the life
84
Forward Start Options
• Option starts at a future time, T • Most common in employee stock option plans • Often structured so that strike price equals asset price at time T
85
Compound Option
• Option to buy / sell an option
• Call on call • Put on call
• Call on put
• Put on put
• Very sensitive to volatility
86
Chooser Option “As You Like It”
• Option starts at time 0, matures at T2 • At T1 (0 < T1 < T2) buyer chooses whether it is a put or call
87
Barrier Options
• In options: come into existence only if asset price hits barrier before option maturity • Out options: die if asset price hits barrier before option maturity
88
Barrier Options (continued)
• Up options: asset price must hit barrier from below • Down options: asset price must hit barrier from above
• Option may be a put or a call
• Eight possible combinations
89
Binary Options
• Cash-or-nothing: pays Q if S > K at time T, otherwise pays 0. • Asset-or-nothing: pays S if S > K at time T, otherwise pays 0.
90
Lookback Options
• Lookback call pays ST – Smin at time T • Allows buyer to buy stock at lowest observed price in some interval of time • Lookback put pays Smax– ST at time T • Allows buyer to sell stock at highest observed price in some interval of time
91
Shout Options
• Buyer can „shout? once during option life • Final payoff is either
• Usual option payoff, max(ST – K, 0), or
• Intrinsic value at time of shout, St – K
• Payoff: max(ST – St , 0) + St – K • Similar to lookback option but cheaper
92
Asian Options
• Payoff related to average stock price
• Average Price options pay:
• max(Save – K, 0) (call), or • max(K – Save , 0) (put)
• Average Strike options pay:
• max(ST – Save , 0) (call), or • max(Save – ST , 0) (put)
93
Exchange Options
• Option to exchange one asset for another • When asset with price U can be exchanged for asset with price V payoff is max(VT – UT, 0) • min(UT, VT) =VT – max(VT – UT, 0)
• max(UT, VT) =UT + max(VT – UT, 0)
94
The Greek Letters
Option Greeks Option Premiums are directly related to relative magnitude of the price of the underlying It is calculated by using the following factors
elta-Change in price of premium corresponding to change in price of the underlying
Gamma-Change in value of Delta corresponding to change in underlying
Theta-Measure of change in premium corresponding to a one day change in its time to expiration Vega- Change in premium corresponding to 1% change in volatility in futures price of the underlying Rho- Measures the sensitivity of the portfolio to interest rates
96
Discrete Greeks
• Vanna is Change in Delta for unit change in Volatility (Ddelta/Dvol) or Change in Vega for unit change in underlying asset price (DVega/Dspot) • Charm is change in Delta for unit change in time to maturity (Ddelta/Dtime) • Speed is Change in Gamma fo unit change in underlying asset price (Dgamma/Dspot) – This is Gamma convexity or 3rd derivative • Zomma is change in Gamma for unit change in volatility (Dgamma/Dvol) • Volga is change in vega for unit change in volatility (Dvega/Dvol)
97
98
99
100
Bucketing of Greeks, Analysis of Discrete Greeks
• Based on different maturity buckets, greeks need to be tracked • This is especially required in cases of high volatility • Focus on Discrete Greeks – Gamma rebalancing and Shadow Gamma • Forward volatility can also be computed • Delta Bleed = Delta today – Delta Tomorrow • Gamma Bleed = Gamma today – Gamma Tomorrow
101
Speed vs Asset Price vs Time to Maturity
Time to Maturity
Speed
102
Steepness in Zomma
103
Track the Delta and Volatility
104
Volga is negative when option is ATM
105
INTEREST RATE SWAPS, CURRENCY SWAPS, COUPON SWAPS, FRA, SWAPTIONS, RANGE ACCRUALS
V. VENKAT GIRIDHAR FT KNOWLEDGE MANAGEMENT COMPANY LIMITED Mumbai, 29th March 2008 www.ftkmc.com
Nature of Swaps
• A swap is an agreement to exchange cash flows at specified future times according to certain specified rules • Stream of Cash Flows
107
An Example of a “Plain Vanilla” Interest Rate Swap
• An agreement by ABC to receive 6-month LIBOR & pay a fixed rate of 5% per annum every 6 months for 3 years on a notional principal of $100 million • Floating rate in “advance” or “arrears”? • Interest Rate Swap – same currency on both sides
108
Cash Flows to ABC
---------Millions of Dollars--------LIBOR FLOATING Date Rate FIXED Net Cash Flow Cash Flow Cash Flow +2.10 –2.50 –0.40
Mar.5, 2001
Sept. 5, 2001
4.2%
4.8%
Mar.5, 2002
Sept. 5, 2002 Mar.5, 2003
5.3%
5.5% 5.6%
+2.40
+2.65 +2.75
–2.50
–2.50 –2.50
–0.10
+0.15 +0.25
Sept. 5, 2003
Mar.5, 2004
5.9%
6.4%
+2.80
+2.95
–2.50
–2.50
+0.30
+0.45
109
Typical Uses of an Interest Rate Swap
• Converting a liability from
• fixed rate to floating rate • floating rate to fixed rate
• Converting an investment from
• fixed rate to floating rate • floating rate to fixed rate
110
Transform a Liability
5% 5.2%
XYZ
LIBOR
ABC
LIBOR+0.1%
111
Financial Institution is Involved
4.985% 5.2%
Financial Institution
5.015%
XYZ
LIBOR
ABC
LIBOR+0.1% LIBOR
112
Transform an Asset
5% 4.7%
XYZ
LIBOR-0.2% LIBOR
ABC
113
Company AA & Co. BBB & Co. Difference
Fixed rate 5.00% 5.85% 85 bp
Floating Rate Libor + 30 bp Libor + 85 bp 55 bp
Benefit to be obtained out of Swap transaction is 30 bp
114
Floating rate of Libor
AA & Co.
Fixed rate of 4.85%
BBB & Co.
Paying Fixed rate of 5%
Paying Floating rate of Libor + 85 bp
Lender
Net Payout: AA & Co. pays Libor + 15 bp BBB & Co. pays Fixed rate of 5.70%
Lender
115
Quotes By a Swap Market Maker
Maturity 2 years 3 years
Bid (%) 6.03 6.21
Offer (%) 6.06 6.24
Swap Rate (%) 6.045 6.225
4 years
5 years 7 years 10 years
6.35
6.47 6.65 6.83
6.39
6.51 6.68 6.87
6.370
6.490 6.665 6.850
116
Valuation of an Interest Rate Swap
• Interest rate swaps can be valued as the difference between the value of a fixed-rate bond and the value of a floating-rate bond
• Alternatively, they can be valued as a portfolio of forward rate agreements (FRAs)
117
Valuation in Terms of FRAs
• Each exchange of payments in an interest rate swap is a FRA
• The FRAs can be valued on the assumption that today?s forward rates are realized
118
An Example of a Currency Swap
An agreement to pay 11% on a sterling principal of £10,000,000 & receive 8% on a US$ principal of $15,000,000 every year for 5 years
119
Exchange of Principal
• In an interest rate swap the principal is not exchanged • In a currency swap the principal is exchanged at the beginning and the end of the swap
120
The Cash Flows
Year 2001 2002 2003 2004 2005 2006
Dollars Pounds $ £ ------millions-----–15.00 +10.00 +1.20 –1.10 +1.20 –1.10 +1.20 –1.10 +1.20 –1.10 +16.20 -11.10
121
Typical Uses of a Currency Swap
• Conversion from a liability in one currency to a liability in another currency
• Conversion from an investment in one currency to an investment in another currency
122
Comparative Advantage Arguments for Currency Swaps
ABC wants to borrow AUD
XYZ wants to borrow USD
USD
ABC XYZ 5.0% 7.0%
AUD
12.6% 13.0%
123
Valuation of Currency Swaps
• Like interest rate swaps, currency swaps can be valued either as the difference between 2 bonds or as a portfolio of forward contracts • Present value of all future cash flows
124
Swaps & Forwards
• A swap can be regarded as a convenient way of packaging forward contracts • The “plain vanilla” interest rate swap in previous example consisted of 6 FRAs • The “fixed for fixed” currency swap in our example consisted of a cash transaction & 5 forward contracts
125
Swaps & Forwards
• The value of the swap is the sum of the values of the forward contracts underlying the swap • Swaps are normally “at the money” initially • This means that it costs nothing to enter into a swap • It does not mean that each forward contract underlying a swap is “at the money” initially
126
Credit Risk
• A swap is worth zero to a company initially • At a future time its value is liable to be either positive or negative
• The company has credit risk exposure only when its value is positive
127
Types of Mortgage-Backed Securities (MBSs)
• Pass-Through • Collateralized Mortgage Obligation (CMO)
• Interest Only (IO)
• Principal Only (PO)
128
Variations on Vanilla Interest Rate Swaps
• Principal different on two sides
• Payment frequency different on two sides • Can be floating for floating instead of floating for fixed
129
Currency Swaps
• Fixed for fixed
• Fixed for floating
• Floating for floating
130
More Complex Swaps
•LIBOR-in-arrears swaps
•CMS and CMT swaps
•Differential swaps
131
Equity Swaps
• Total return on an equity index is exchanged periodically for a fixed or floating return
132
Swaps with Embedded Options
• Accrual swaps • Cancelable swaps • Cancelable compounding swaps
133
Other Swaps
• Indexed principal swap • Commodity swap • Volatility swap
134
Embedded Bond Options
• Callable bonds: Issuer has option to buy bond back at the “call price”. The call price may be a function of time • Puttable bonds: Holder has option to sell bond back to issuer
135
European Swap Options
• A European swap option gives the holder the right to enter into a swap at a certain future time
• Either it gives the holder the right to pay a prespecified fixed rate and receive LIBOR
• Or it gives the holder the right to pay LIBOR and receive a prespecified fixed rate
136
doc_100239214.ppt
Describes different types of option positions, factors affecting option premium, black scholes options pricing formula, naked options, flex options, american vs european option, put call parity, impact of dividend on option prices, options on futures, put call parity for future options, exotic options, different types of exotic options, interest rate swaps, currency swaps.
Non-Linear Derivatives: Options and Swaps
AGENDA
• Introduction to Linear and Non-Linear Derivatives • Options Valuation, Pricing And Implied Volatility • Option Greeks
2
American vs European Options
?An American option can be exercised at any time during its life ?A European option can be exercised only at maturity ?Is the value of an American Option higher or lower than a European Option? ?A Bermudan option is a call or put option which can be exercised on prespecified days during the life of the option. It is reasonable to say that Bermudan options are a hybrid of European options, which can only be exercised on the option expiry date, and American options, which can be exercised at any time during the option life time. ?As a consequence, under same conditions, the value of a Bermudan option is greater than (or equal to) a European option but less than (or equal to) an American option.
3
Option Positions
• Long call
• Long put
• Short call • Short put
4
Payoff for Call Option Buyer
Gain Profit Gain
Strike Price
0 Loss
Market Price
Premium Paid (Price to purchase Option)
• Note: Upside potential is unlimited, Downside risk is limited
5
Long Call
Profit from buying one European call option: option price = $5, strike price = $100, option life = 2 months
30 Profit ($) 20 10 70 0 -5 80 90 100 Terminal stock price ($) 110 120 130
6
Payoff for Call Option Seller
Gain
Premium Received Market Price Strike Price
0 Loss
Profit Loss
• Note: Upside potential is limited to the premium received. Downside risk is unlimited.
7
Short Call
Profit from writing one European call option: option price = $5, strike price = $100
Profit ($)
5 0 -10 -20 -30
8
110 120 130 70 80 90 100 Terminal stock price ($)
Payoff for Put Option Buyer
Gain Strike Price 0 Loss Market Price
Premium
• Note: Upside potential is limited to the price of the security. Downside risk is limited to the premium.
9
Long Put
Profit from buying a European put option: option price = $7, strike price = $70
30 Profit ($) 20 10
0
-7
Terminal stock price ($)
40 50 60 70 80 90 100
10
Payoff for Put Option Seller
Gain Premium 0 Loss Strike Price Market Price
Loss
• Note: Upside potential is limited to the premium. Downside risk is limited to the price of the security.
11
Short Put
Profit from writing European put option: option price = $7, strike price = $70
Profit ($) 7 0 -10 -20 -30
12
40
50
60 70 80
Terminal stock price ($) 90 100
Payoffs from Options
What is the Option Position in Each Case?
X = Strike price, ST = Price of asset at maturity
Payoff
Payoff X
X ST Payoff X X ST ST ST
Payoff
13
Specification of Exchange-Traded Options
• Underlying Asset
• Expiration date • Strike price • European or American • Call or Put
14
Specification of OTC Options
• Expiration date – (e.g. for currencies, specify “New York cut” or “Tokyo Cut”)
• Strike price • European or American • Call or Put
• Underlying asset
15
Terminology
• Option class • Option series • Intrinsic value • Time value
16
Option Terminology (Moneyness)
In-the-money Option For a call option, strike price < Market Price For a put option, strike price > Market price At-the-money Option
For both call and put options
Strike Price= Market Price Out-of-the-money Option For a call option, Strike Price > Market Price For a put option, strike Price < Market Price
17
Intrinsic Value – e.g. of Call Option on Commodity Futures
18
Intrinsic Value – e.g. of Put Option on Commodity Futures
19
Time Value
20
Factors Affecting Option Premium
21
Black-Scholes Options Pricing Formula
22
Notation
• c : European call option
price
• p : European put option price • S0 : Stock price today
• C:
• • • • P:
American Call option price American Put option price
ST :Stock price at option maturity D : Present value of dividends during option?s life r: Risk-free rate for maturity T with cont comp
• K:
• T: • ?:
Strike price
Life of option Volatility of stock price
23
Effect of Variables on Option Pricing
Variable S0 K T ? r D
c
+ – + + + –
– + + + – +
24
p
C
+ – + + + –
– + + + – +
P
Naked Options and Covered Calls
• Naked options are unhedged positions • Covered calls involve writing calls when the underlying asset (to be delivered in case the option is exercised against the option writer) is owned
25
Flex Options
• CBOT offers flex options on equity and equity indices • These are options where traders on the floor of the exchange agree to nonstandard terms on strike price or exercise dates, American vs European, etc. • Attempt by exchanges to regain business from OTC markets
26
Warrants
• Warrants are options that are issued (or written) by a corporation or a financial institution • The number of warrants outstanding is determined by the size of the original issue & changes only when they are exercised or when they expire • In International markets, Warrants are traded in the same way as stocks • The issuer settles up with the holder when a warrant is exercised • When call warrants are issued by a corporation on its own stock, exercise will lead to new treasury stock being issued
27
Executive Stock Options
• Option issued by a company to executives • When the option is exercised the company issues more stock • Usually at-the-money when issued • They become vested after a period of time (usually 1 to 4 years) • They cannot be sold • They often also last for as long as 10 or 15 years
28
Convertible Bonds
• Convertible bonds are regular bonds that can be exchanged for equity at certain times in the future according to a predetermined exchange ratio • Very often a convertible is callable • The call provision is a way in which the issuer can force conversion at a time earlier than the holder might otherwise choose
29
American vs European Options
An American option is worth at least as much as the corresponding European option C?c P?p
30
Upper Bound for Options
• Upper bound for call options c<=S; C<=S • Upper bound for put options
p<=K;
p<=K.e(-rT)
P<=K
For European put option: • If above is violated, then arbitrage would arise
31
Calls: An Arbitrage Opportunity?
• Suppose that c=3 T=1 S0 = 20 r = 10%
K = 18
D=0
• Is there an arbitrage opportunity?
32
Puts: An Arbitrage Opportunity?
• Suppose that p T =1 = 0.5 S0 = 37
r
K
= 5%
= 40
D =0 • Is there an arbitrage opportunity?
33
Lower Bound for European Call Option Prices; No Dividends
c ? S0 –Ke -rT
Lower Bound for European Put Prices; No Dividends
p ? Ke
-rT–S
0
34
Put-Call Parity; No Dividends
• Consider the following 2 portfolios:
• Portfolio A: European call on a stock + PV of the strike price in cash • Portfolio C: European put on the stock + the stock
• Both are worth MAX(ST , K ) at the maturity of the options
• They must therefore be worth the same today. This means that
c + Ke -rT = p + S0
• This is similar to a range forward (long call and short a put), where
c - p = S0 - Ke -rT
35
Arbitrage Opportunities
• Suppose that c T K =3 = 0.25 =30 p = 2.25 ? S0 = 31 r = 10% D=0 p=1?
• What are the arbitrage possibilities when
36
Early Exercise
• Usually there is some chance that an American option will be exercised early • An exception is an American call on a non-dividend paying stock
• This should never be exercised early
37
An Extreme Situation
• For an American call option:
S0 = 100; T = 0.25; K = 60; D = 0
Should you exercise immediately?
• What should you do if
You want to hold the stock for the next 3 months? You do not feel that the stock is worth holding for the next 3 months?
38
Reasons For Not Exercising a American Call Early (Non Dividend paying stock)
• No income is sacrificed • We delay paying the strike price • Holding the call provides insurance against stock price falling below strike price • Alternative to exercising is to sell the American call option itself
39
Should American Puts (on Non-Dividend paying stocks) be Exercised Early ?
Are there any advantages to exercising an American put when S0 = 60; T = 0.25; r=10% K = 100; D = 0
40
The Impact of Dividends on Lower Bounds to Option Prices
c ? S 0 ? D ? Ke
p ? D ? Ke
? rT
? rT
? S0
41
Extensions of Put-Call Parity
• American options; D = 0
S0 - K < C - P < S0 - Ke -rT
• European options; D > 0 c + D + Ke -rT = p + S0 • American options; D > 0 S0 - D - K < C - P < S0 - Ke -rT
42
Introduction to Binomial Trees
A Simple Binomial Model
• A stock price is currently $20
• In three months it will be either $22 or $18
Stock Price = $22
Stock price = $20 Stock Price = $18
44
A Call Option
A 3-month call option on the stock has a strike price of 21.
Stock Price = $22 Option Price = $1
Stock price = $20 Option Price=?
Stock Price = $18 Option Price = $0
45
Setting Up a Riskless Portfolio
• Consider the Portfolio: long D shares short 1 call option
22D – 1
18D
• Portfolio is riskless when 22D – 1 = 18D or
D = 0.25
46
Valuing the Portfolio
(Risk-Free Rate is 12%)
• The riskless portfolio is: long 0.25 shares and short 1 call option
• The value of the portfolio in 3 months is 22 ? 0.25 – 1 = 4.50
• The value of the portfolio today is 4.5e – 0.12?0.25 = 4.3670
47
Valuing the Option
• The portfolio that is
long 0.25 shares short 1 option
is worth 4.367
• The value of the shares is 5.000 (= 0.25 ? 20 )
• The value of the option is therefore 0.633 (= 5.000 – 4.367 )
48
Scenario
A derivative lasts for time T and is dependent on a stock
S ƒ
Su ƒu Sd ƒd
49
Generalization
• Consider the portfolio that is long D shares and short 1 derivative
SuD – ƒu SdD – ƒd
• The portfolio is riskless when SuD – ƒu = Sd D – ƒd or ƒu ? f d D? Su ? Sd
50
Risk-Neutral Valuation
• ƒ = [ p ƒu + (1 – p )ƒd ]e-rT
• The variables p and (1 – p ) can be interpreted as the risk-neutral probabilities of up and down movements
• The value of a derivative is its expected payoff in a risk-neutral world discounted at the risk-free rate
S ƒ
Su ƒu Sd ƒd
51
Binomial
Su = 22 ƒu = 1
Sd = 18 ƒd = 0 • Since p is a risk-neutral probability
20e0.12 ?0.25 = 22p + 18(1 – p ); p = 0.6523 • Alternatively, we can use the formula
e rT ? d e 0.12?0.25 ? 0.9 p? ? ? 0.6523 u?d 1.1 ? 0.9
S ƒ
52
Valuing the Option
Su = 22 ƒu = 1 S ƒ Sd = 18 ƒd = 0
The value of the option is e–0.12?0.25 [0.6523?1 + 0.3477?0] = 0.633
53
A Two-Step Example
24.2 22
20 18
19.8
16.2 • Each time step is 3 months
54
Valuing a Call Option
D
22 20 1.2823
A
24.2 3.2
B E C
2.0257
18
0.0
19.8 0.0
16.2 F • Value at node B 0.0 = e–0.12?0.25(0.6523?3.2 + 0.3477?0) = 2.0257
• Value at node A = e–0.12?0.25(0.6523?2.0257 + 0.3477?0)
= 1.2823
55
Implied Volatility
• The implied volatility of an option is the volatility for which the Black-Scholes price equals the market price • The is a one-to-one correspondence between prices and implied volatilities • Traders and brokers often quote implied volatilities rather than dollar prices • The implied volatility calculated from a European call option should be the same as that calculated from a European put option when both have the same strike price and maturity
56
Volatility Smile
• A volatility smile shows the variation of the implied volatility with the strike price
• The volatility smile should be the same whether calculated from call options or put options
57
The Volatility Smile for Foreign Currency Options
Implied Volatility
Strike Price
58
The Volatility Smile for Equity Options
Implied Volatility
Strike Price
59
Possible Causes of Volatility Smile
• Asset price exhibiting jumps rather than continuous change
• Volatility for asset price being stochastic
(One reason for a stochastic volatility in the case of equities is the relationship between volatility and leverage)
60
Volatility Term Structure
• In addition to calculating a volatility smile, traders also calculate a volatility term structure • This shows the variation of implied volatility with the time to maturity of the option • The volatility term structure tend to be downward sloping when volatility is high and upward sloping when it is low
61
Black Scholes on Stock Indices
c ? S 0 e ? qT N ( d1 ) ? Ke ? rT N (d 2 ) p ? Ke ? rT N ( ? d 2 ) ? S 0 e ? qT N ( ? d1 ) ln( S 0 / K ) ? ( r ? q ? ? 2 / 2)T where d1 ? ? T ln( S 0 / K ) ? (r ? q ? ? 2 / 2)T d2 ? ? T
62
LEAPS
• Leaps are options on stock indices that last up to 3 years • They have December expiration dates • They are on 10 times the index • Leaps also trade on some individual stocks
63
Currency Options
• Currency options trade on the Philadelphia Exchange (PHLX)
• There also exists an active over-the-counter (OTC) market • Currency options are used by corporations to buy insurance when they have an FX exposure
64
The Foreign Interest Rate
• We denote the foreign interest rate by rf • When a U.S. company buys one unit of the foreign currency it has an investment of S0 dollars
• The return from investing at the foreign rate is rf S0 dollars
• This shows that the foreign currency provides a “dividend yield” at rate rf
65
Valuing European Currency Options
• A foreign currency is an asset that provides a continuous “dividend yield” equal to rf • We can use the formula for an option on a stock paying a continuous dividend yield : Set S0 = current exchange rate Set q = rƒ
66
Formula for European Currency Options
c ? S0e
?rf T
N ( d1 ) ? Ke ? rT N (d 2 )
?rf T
p ? Ke ? rT N ( ?d 2 ) ? S 0 e where d1 ? d2 ?
N (?d1 ) f ? ? 2 / 2)T
ln( S 0 / K ) ? (r ? r ? T f
ln( S 0 / K ) ? (r ? r ? T
67
? ? 2 / 2)T
Options on Futures
Mechanics of Call Futures Options
When a call futures option is exercised the holder acquires 1. A long position in the futures
2. A cash amount equal to the excess of
the futures price over the strike price at previous settlement
69
Mechanics of Put Futures Option
When a put futures option is exercised the holder acquires 1. A short position in the futures 2. A cash amount equal to the excess of the strike price over the futures price at previous settlement
70
The Payoffs
If the futures position is closed out immediately:
Payoff from call = F0 – K
Payoff from put = K – F0 where F0 is futures price at time of exercise
71
Potential Advantages of Futures Options over Options on Physical Commodity
• Futures contract may be easier to trade than underlying asset • Exercise of the option does not lead to delivery of the underlying asset • Futures options and futures usually trade adjacent to each other on same exchange • Futures options may entail lower transactions costs
72
Put-Call Parity for Futures Options
Consider the following two portfolios: 1. European call plus Ke-rT of cash 2. European put plus long futures plus cash equal to F0e-rT They must be worth the same at time T so that
c+Ke-rT=p+F0 e-rT
73
Other Relations
Fe-rT – K < C – P < F – Ke-rT
c > (F – K)e-rT
p > (F – K)e-rT
74
Valuing European Futures Options
• We can use the formula for an option on a stock paying a continuous dividend yield
Set S0 = current futures price (F0)
Set q = domestic risk-free rate (r )
• Setting q = r ensures that the expected growth of F in a risk-neutral world is zero
75
Growth Rates For Futures Prices
• A futures contract requires no initial investment • In a risk-neutral world the expected return should be zero • The expected growth rate of the futures price is therefore zero
• The futures price can therefore be treated like a stock paying a dividend yield of r
76
Black?s Model
The formulas for European options on futures are known as Black?s model
c ? e ? rT ?F0 N (d1 ) ? K N (d 2 )?
p ? e ? rT ?K N (? d 2 ) ? F0 N (? d1 )? ln( F0 / K ) ? ? 2T / 2 ? T ? T
77
where d1 ? d2 ?
ln( F0 / K ) ? ? 2T / 2
? d1 ? ? T
Futures Option Prices vs Spot Option Prices
• If futures prices are higher than spot prices (normal market), an American call on futures is worth more than a similar American call on spot. An American put on futures is worth less than a similar American put on spot • When futures prices are lower than spot prices (inverted market) the reverse is true
78
Put-Call Parity Results
Indices : c ? K e ? rT ? p ? S e ? qT Foreignexchange : c?K e Futures: c ? K e ? rT ? p ? F e ? rT
? rT
? p?S e
?rf T
79
Market Makers
• Market makers facilitate trading • A market maker quotes both bid and ask prices when requested • The market maker does not know whether the individual requesting the quotes wants to buy or sell • Exchanges often set upper limits for bidask spreads
80
Exotic Options and Other Nonstandard Products
Types of Exotic Options
• Packages
• Nonstandard American options
• Forward start options • Compound options • Chooser options • Barrier options
• Binary options
82
Types of Exotic Options continued
• Lookback options
• Shout options
• Asian options • Options to exchange one asset for another • Options involving several assets
83
Nonstandard American Options
• Exercisable only on specific dates (Bermudans) • Early exercise allowed during only part of life (e.g. there may be an initial “lock out” period) • Strike price changes over the life
84
Forward Start Options
• Option starts at a future time, T • Most common in employee stock option plans • Often structured so that strike price equals asset price at time T
85
Compound Option
• Option to buy / sell an option
• Call on call • Put on call
• Call on put
• Put on put
• Very sensitive to volatility
86
Chooser Option “As You Like It”
• Option starts at time 0, matures at T2 • At T1 (0 < T1 < T2) buyer chooses whether it is a put or call
87
Barrier Options
• In options: come into existence only if asset price hits barrier before option maturity • Out options: die if asset price hits barrier before option maturity
88
Barrier Options (continued)
• Up options: asset price must hit barrier from below • Down options: asset price must hit barrier from above
• Option may be a put or a call
• Eight possible combinations
89
Binary Options
• Cash-or-nothing: pays Q if S > K at time T, otherwise pays 0. • Asset-or-nothing: pays S if S > K at time T, otherwise pays 0.
90
Lookback Options
• Lookback call pays ST – Smin at time T • Allows buyer to buy stock at lowest observed price in some interval of time • Lookback put pays Smax– ST at time T • Allows buyer to sell stock at highest observed price in some interval of time
91
Shout Options
• Buyer can „shout? once during option life • Final payoff is either
• Usual option payoff, max(ST – K, 0), or
• Intrinsic value at time of shout, St – K
• Payoff: max(ST – St , 0) + St – K • Similar to lookback option but cheaper
92
Asian Options
• Payoff related to average stock price
• Average Price options pay:
• max(Save – K, 0) (call), or • max(K – Save , 0) (put)
• Average Strike options pay:
• max(ST – Save , 0) (call), or • max(Save – ST , 0) (put)
93
Exchange Options
• Option to exchange one asset for another • When asset with price U can be exchanged for asset with price V payoff is max(VT – UT, 0) • min(UT, VT) =VT – max(VT – UT, 0)
• max(UT, VT) =UT + max(VT – UT, 0)
94
The Greek Letters
Option Greeks Option Premiums are directly related to relative magnitude of the price of the underlying It is calculated by using the following factors
elta-Change in price of premium corresponding to change in price of the underlyingGamma-Change in value of Delta corresponding to change in underlying
Theta-Measure of change in premium corresponding to a one day change in its time to expiration Vega- Change in premium corresponding to 1% change in volatility in futures price of the underlying Rho- Measures the sensitivity of the portfolio to interest rates
96
Discrete Greeks
• Vanna is Change in Delta for unit change in Volatility (Ddelta/Dvol) or Change in Vega for unit change in underlying asset price (DVega/Dspot) • Charm is change in Delta for unit change in time to maturity (Ddelta/Dtime) • Speed is Change in Gamma fo unit change in underlying asset price (Dgamma/Dspot) – This is Gamma convexity or 3rd derivative • Zomma is change in Gamma for unit change in volatility (Dgamma/Dvol) • Volga is change in vega for unit change in volatility (Dvega/Dvol)
97
98
99
100
Bucketing of Greeks, Analysis of Discrete Greeks
• Based on different maturity buckets, greeks need to be tracked • This is especially required in cases of high volatility • Focus on Discrete Greeks – Gamma rebalancing and Shadow Gamma • Forward volatility can also be computed • Delta Bleed = Delta today – Delta Tomorrow • Gamma Bleed = Gamma today – Gamma Tomorrow
101
Speed vs Asset Price vs Time to Maturity
Time to Maturity
Speed
102
Steepness in Zomma
103
Track the Delta and Volatility
104
Volga is negative when option is ATM
105
INTEREST RATE SWAPS, CURRENCY SWAPS, COUPON SWAPS, FRA, SWAPTIONS, RANGE ACCRUALS
V. VENKAT GIRIDHAR FT KNOWLEDGE MANAGEMENT COMPANY LIMITED Mumbai, 29th March 2008 www.ftkmc.com
Nature of Swaps
• A swap is an agreement to exchange cash flows at specified future times according to certain specified rules • Stream of Cash Flows
107
An Example of a “Plain Vanilla” Interest Rate Swap
• An agreement by ABC to receive 6-month LIBOR & pay a fixed rate of 5% per annum every 6 months for 3 years on a notional principal of $100 million • Floating rate in “advance” or “arrears”? • Interest Rate Swap – same currency on both sides
108
Cash Flows to ABC
---------Millions of Dollars--------LIBOR FLOATING Date Rate FIXED Net Cash Flow Cash Flow Cash Flow +2.10 –2.50 –0.40
Mar.5, 2001
Sept. 5, 2001
4.2%
4.8%
Mar.5, 2002
Sept. 5, 2002 Mar.5, 2003
5.3%
5.5% 5.6%
+2.40
+2.65 +2.75
–2.50
–2.50 –2.50
–0.10
+0.15 +0.25
Sept. 5, 2003
Mar.5, 2004
5.9%
6.4%
+2.80
+2.95
–2.50
–2.50
+0.30
+0.45
109
Typical Uses of an Interest Rate Swap
• Converting a liability from
• fixed rate to floating rate • floating rate to fixed rate
• Converting an investment from
• fixed rate to floating rate • floating rate to fixed rate
110
Transform a Liability
5% 5.2%
XYZ
LIBOR
ABC
LIBOR+0.1%
111
Financial Institution is Involved
4.985% 5.2%
Financial Institution
5.015%
XYZ
LIBOR
ABC
LIBOR+0.1% LIBOR
112
Transform an Asset
5% 4.7%
XYZ
LIBOR-0.2% LIBOR
ABC
113
Company AA & Co. BBB & Co. Difference
Fixed rate 5.00% 5.85% 85 bp
Floating Rate Libor + 30 bp Libor + 85 bp 55 bp
Benefit to be obtained out of Swap transaction is 30 bp
114
Floating rate of Libor
AA & Co.
Fixed rate of 4.85%
BBB & Co.
Paying Fixed rate of 5%
Paying Floating rate of Libor + 85 bp
Lender
Net Payout: AA & Co. pays Libor + 15 bp BBB & Co. pays Fixed rate of 5.70%
Lender
115
Quotes By a Swap Market Maker
Maturity 2 years 3 years
Bid (%) 6.03 6.21
Offer (%) 6.06 6.24
Swap Rate (%) 6.045 6.225
4 years
5 years 7 years 10 years
6.35
6.47 6.65 6.83
6.39
6.51 6.68 6.87
6.370
6.490 6.665 6.850
116
Valuation of an Interest Rate Swap
• Interest rate swaps can be valued as the difference between the value of a fixed-rate bond and the value of a floating-rate bond
• Alternatively, they can be valued as a portfolio of forward rate agreements (FRAs)
117
Valuation in Terms of FRAs
• Each exchange of payments in an interest rate swap is a FRA
• The FRAs can be valued on the assumption that today?s forward rates are realized
118
An Example of a Currency Swap
An agreement to pay 11% on a sterling principal of £10,000,000 & receive 8% on a US$ principal of $15,000,000 every year for 5 years
119
Exchange of Principal
• In an interest rate swap the principal is not exchanged • In a currency swap the principal is exchanged at the beginning and the end of the swap
120
The Cash Flows
Year 2001 2002 2003 2004 2005 2006
Dollars Pounds $ £ ------millions-----–15.00 +10.00 +1.20 –1.10 +1.20 –1.10 +1.20 –1.10 +1.20 –1.10 +16.20 -11.10
121
Typical Uses of a Currency Swap
• Conversion from a liability in one currency to a liability in another currency
• Conversion from an investment in one currency to an investment in another currency
122
Comparative Advantage Arguments for Currency Swaps
ABC wants to borrow AUD
XYZ wants to borrow USD
USD
ABC XYZ 5.0% 7.0%
AUD
12.6% 13.0%
123
Valuation of Currency Swaps
• Like interest rate swaps, currency swaps can be valued either as the difference between 2 bonds or as a portfolio of forward contracts • Present value of all future cash flows
124
Swaps & Forwards
• A swap can be regarded as a convenient way of packaging forward contracts • The “plain vanilla” interest rate swap in previous example consisted of 6 FRAs • The “fixed for fixed” currency swap in our example consisted of a cash transaction & 5 forward contracts
125
Swaps & Forwards
• The value of the swap is the sum of the values of the forward contracts underlying the swap • Swaps are normally “at the money” initially • This means that it costs nothing to enter into a swap • It does not mean that each forward contract underlying a swap is “at the money” initially
126
Credit Risk
• A swap is worth zero to a company initially • At a future time its value is liable to be either positive or negative
• The company has credit risk exposure only when its value is positive
127
Types of Mortgage-Backed Securities (MBSs)
• Pass-Through • Collateralized Mortgage Obligation (CMO)
• Interest Only (IO)
• Principal Only (PO)
128
Variations on Vanilla Interest Rate Swaps
• Principal different on two sides
• Payment frequency different on two sides • Can be floating for floating instead of floating for fixed
129
Currency Swaps
• Fixed for fixed
• Fixed for floating
• Floating for floating
130
More Complex Swaps
•LIBOR-in-arrears swaps
•CMS and CMT swaps
•Differential swaps
131
Equity Swaps
• Total return on an equity index is exchanged periodically for a fixed or floating return
132
Swaps with Embedded Options
• Accrual swaps • Cancelable swaps • Cancelable compounding swaps
133
Other Swaps
• Indexed principal swap • Commodity swap • Volatility swap
134
Embedded Bond Options
• Callable bonds: Issuer has option to buy bond back at the “call price”. The call price may be a function of time • Puttable bonds: Holder has option to sell bond back to issuer
135
European Swap Options
• A European swap option gives the holder the right to enter into a swap at a certain future time
• Either it gives the holder the right to pay a prespecified fixed rate and receive LIBOR
• Or it gives the holder the right to pay LIBOR and receive a prespecified fixed rate
136
doc_100239214.ppt