Option Strategies &
Exotics
1
Note on Notation
• Here, T denotes time to expiry as well as time
of expiry, i.e. we use T to denote indifferently
T and δ = T – t
• Less accurate but handier this way, I think
2
Types of Strategies
• Take a position in the option and the underlying
• Take a position in 2 or more options of the same type
(A spread)
• Combination: Take a position in a mixture of calls &
puts (A combination)
3
Positions in an Option & the
Underlying
Profit
Profit
K
K
ST
ST
(a)
(b)
Profit
Profit
K
ST
(c)
K
ST
(d)
Basis of Put-Call Parity: P + S = C + Cash ( Ke-rT)
4
Bull Spread Using Calls
Profit
ST
K1
K2
5
Bull Spread Using Calls
Example
• Create a bull spread on IBM using the following 3month call options on IBM:
Option 1:
Strike: K1 = 102
Price: C1 = 5
Option 2:
Strike: K1 = 110
Price: C2 = 2
Gamble on stock price rise and offset cost
with sale of call
+1
Long Call (at K1)
+1
0
K1
0
plus
K2
Short Call (at K2 > K1)
-1
0
Profit
equals
0
5
Call Bull Spread
+1
K1=102
K2=110
0
-3
SBE=105
Share Price
Payoff:
Long call (K1) + short call (K2) = Bull Spread:
{ 0, +1, +1} + {0, 0, -1}
= {0, +1, 0 }
= Max(0, ST-K1) – C1 – Max(0, ST-K2) + C2
= C2 - C1
if ST  K1  K2
= ST - K1 + (C2 - C1)
if K1 < ST  K2
= (ST - K1 - C1) + (K2 - ST + C2) =
= K2 - K1 + (C2 - C1)
if ST > K1 > K2
‘Break-even’:
SBE = K1 + (C1 – C2) = 102 + 3 = 105
Bear Spread Using Puts
Profit
K1
K2
ST
9
Bull Spreads with puts
& Bear Spreads with Calls
• Of course can do bull spreads with puts and bear
spreads with calls (put-call parity)
• Figured out how?
10
Bull Spread Using Puts
Profit
K1
K2
ST
11
Bear Spread Using Calls
Profit
K1
K2
ST
12
Equity Collar
You already hold stocks but you want to limit
downside (buy a put) but you are also willing to
limit the upside if you can earn some cash today
(by selling an option, i.e. a call)
COLLAR = long stock + long put (K1) + short call (K2)
{0,+1,0} = {+1,+1,+1} + {-1,0,0} + {0,0,-1}
Equity Collar: Payoff Profile
+1
+1
Long Stock
+1
plus
-1
0
0
Long Put
plus
0
0
-1
Short Call
equals
0
0
+1
Equity Collar
Equity Collar Payoffs
ST < K 1
K1  ST  K2
ST > K 2
Long Shares
ST
ST
ST
Long Put (K1)
K1 – ST
0
0
Short Call (K2)
0
0
– (ST –
Gross Payoff
K1
ST
K2
Net Profit
K1 – (P – C)
ST – (P – C)
K2 – (P – C)
Net Profit = Gross Payoff – (P – C)
K2)
Box Spread
• A combination of a bull call spread and a bear put spread
• If all options are European a box spread is worth the
present value of the difference between the strike prices
• Check it out
• If they are American this is not necessarily so
16
A Basic Combination: A Synthetic
Forward/Futures
0
+1
Short Put
plus
0
+1
Long Call
equals
+1
+1
Long Futures
Range Forward Contracts
• Have the effect of ensuring that the exchange rate paid or
received will lie within a certain range
• When currency is to be paid it involves selling a put with strike
K1 and buying a call with strike K2 (with K2 > K1)
• When currency is to be received it involves buying a put with
strike K1 and selling a call with strike K2
• Normally the price of the put equals the price of the call
18
Range Forward Contract
Payoff
Payoff
Asset
Price
K1
Short
Position
K2
K1
K2
Asset
Price
Long
Position
19
Volatility Combinations
• Mainly
• Straddle
• Strangles
• These are strategies that show the true ‘character’ of
options
• But also
• Strip
• Straps
• Etc.
A Straddle Combination
Profit
K
ST
21
Long (buy) Straddle
Data:
K = 102
P=3
C=5
C+P=8
profit long straddle:  = Max (0, ST – K) - C + Max (0, K – ST) – P = 0
for ST > K
=> ST - K – (C + P) = K + (C + P) = 102 + 8 = 110
for ST < K
=> K - ST – (C + P) = K - (C + P) = 102 - 8 = 94
Straddles and HF
• Fung and Hsieh (RFS, 2001) empirically show
that many hedge funds follow strategies that
resemble straddles:
• ‘Market timers’ returns are highly correlated with
the return to long straddles on diversified equity
indices and other basic asset classes
A Strangle Combination
Profit
K1
K2
ST
24
Strip & Strap
Profit
Profit
K
Strip
ST
K
ST
Strap
25
Time Decay Combinations
• Calendar (or horizontal) spreads
• Options, same strike price (K) but different maturity dates,
e.g. buying a long dated option (360-day) and selling a
short dated option (180-day), both are at-the money
• In a relatively static market (i.e. S0 = K) this spread will
make money from time decay, but will loose money if the
stock price moves substantially
Calendar Spread Using Calls
Profit
ST
K
27
Calendar Spread Using Puts
Profit
ST
K
28
‘Quasi-Elementary’ Securities
• Arrow(-Debrew) introduces so called ArrowDebrew elementary securities,
i.e. contingent claims with $1 payoff in one state and $0
in all other states
• These can be seen as “bet” options
• Butterflies look a lot like them
Butterfly Spread Using Calls
Profit
K1
K2
K3
ST
30
Butterfly Spread Using Puts
Profit
K1
K2
K3
ST
31
Butterflies Replication
• Butterfly requires:
• sale of 2 ‘inner-strike price’ call options (K2)
• purchase of 2 'outer-strike price’ call options (K1, K3)
• Butterfly is a ‘bet’ on a small change in price of the
underlying in either direction
• Potential downside of the ‘bet’ is offset by ‘truncating’ the
payoff by buying some options
• Could also buy (go long) a bull and a bear (call or put)
spread, same result
Short Butterflies Replication
• Short butterfly requires:
• purchase of 2 ‘inner-strike price’ call options (K2)
• sale of 2 'outer-strike price’ call options (K1, K3)
• Short butterfly is a ‘bet’ on a large change in price of the
underlying in either direction (e.g. result of reference to
the competition authorities)
• Cost of the ‘bet’ is offset by ‘truncating’ the payoff by
selling some options
• Could also sell (go short) a bull and a bear (call or put)
spread, same result
Short Butterfly Spread Using Calls
Profit
K1
K2
K3
ST
34
Variations Using Interest Rate
Options
35
Interest Rate Options
• Interest rate option
gives holder the right but not the obligation to receive
one interest rate (e.g. floating\LIBOR) and pay
another (e.g. the fixed strike rate LK)
Caps
• A cap is a portfolio of “caplets”
• Each caplet is a call option on a future LIBOR rate with the
payoff occurring in arrears
• Payoff at time tk+1 on each caplet is Ndk max(Lk - LK, 0) where
N is the notional amount, dk = tk+1 - tk , LK is the cap rate, and
Lk is the rate at time tk for the period between tk and tk+1
• It has the effect of guaranteeing that the interest rate in each of
a number of future periods will not rise above a certain level
Caplet Payoff
Strike rate LK
fixed in
the contract
t0 = 0
Expiry \ Valuation
of option, (LIBOR1 - LK)
t1 = 30
t2 = 120 days
δ = 90 days
38
Annualised Cost of
Borrowing
Planned Borrowing + Caplet (Call
on Bond)
18
16
14
12
10
8
6
4
5
7
9
11
LIBOR at expiry
13
15
Annualized return on
loan
Loan + Interest Rate Floorlet (Put
on Bond)
20
15
10
5
0
4
6
8
10
12
LIBOR at expiry
14
16
Positions in an Option & the Underlying
(notice variables on vertical axis)
Long
floorlet
Return rate
Return rate
Short
caplet
K
K
iT
iT
(b)
(a)
Long
caplet
Short
floorlet
Funding cost
Funding cost
K
K
(c)
iT
iT
(d)
41
Collar
Comprises a long cap and short floor.
It establishes both a floor and a ceiling on a corporate or bank’s (floating
rate) borrowing costs.
Effective Borrowing Cost with Collar (at T  tk+1 = tk + 90) =
= [Lk – max[{0, Lk – LK} + max {0, LK – Lk}]N(90/360)
= Lk,CAP N(90/360)
if Lk > Lk,CAP
= Lk,FL N(90/360)
if Lk < Lk,FL
= Lk (90/360)
if Lk,FL < Lk < Lk,CAP
Collar involves borrowing cost at each payment date of either Lk,CAP = 10%
or Lk,FL = 8% or Lk = LIBOR if the latter is between 8% and 10%.
42
Combining options with swaps
• Cancelable swaps - can be
cancelled by the firm entering into
the swap if interest rates move a
certain way
• Swaptions - options to enter into
a swap
Swaptions
• OTC option for the buyer to enter into a swap
at a future date and a predetermined swap rate
 A payer swaption gives the buyer the right to
enter into a swap where they pay the fixed leg and
receive the floating leg (long IRS).
 A receiver swaption gives the buyer the right to
enter into a swap where they will receive the fixed
leg, and pay the floating leg (short IRS).
Swaptions Example
• A US bank has made a commitment to lend at fixed rate $10m
over 3 years beginning in 2 years time and may need to fund
this loan at a floating rate.
• In 2 years time, the bank may wish to swap the floating rate
payments for a fixed rate,
• Perhaps at that time, the bank may think that interest rates may rise
over the 3 years and hence the cost of the fixed rate payments in the
swap will be higher than at inception.
Example
• Bank might need a $10m swap, to pay fixed and receive floating
beginning in 2 years time and an agreement that swap will last for further
3 years
• The bank can hedge by purchasing a 2-year European payer swaption,
with expiry in T = 2, on a 3 year “pay fixed-receive floating” swap, at say
sK = 10%.
• Payoff is the annuity value of Nδmax{sT – sK, 0}. So, value of swaption at
T is:
• f = $10m[sT – sK] [(1 + L2,3)-1 + (1 + L2,4)-2 + (1 + L2,5)-3]
Exotics
47
Types of Exotics
• Package
• Nonstandard American
options
• Forward start 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
• Volatility and Variance
swaps
• etc., etc., etc.
48
Packages
• Portfolios of standard options
• Classical spreads and combinations: bull
spreads, bear spreads, straddles, etc
• Often structured to have zero cost
• One popular package is a range forward
contract
49
Non-Standard American Options
• Exercisable only on specific dates
(Bermudans)
• Early exercise allowed during only part
of life (initial “lock out” period)
• Strike price changes over the life
(warrants, convertibles)
50
Forward Start Options
• Option starts at a future time, T1
• Implicit in employee stock option plans
• Often structured so that strike price equals asset
price at time T1
51
Compound Option
• Option to buy or sell an option

Call on call
 Put on call
 Call on put
 Put on put
• Can be valued analytically
• Price is quite low compared with a regular option
52
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
• This is a package!
53
Chooser Option as a Package
At time T1 the value is max(c, p)
From put - call parity
p  c  e r (T2 T1 ) K  S1e q (T2 T1 )
The value at time T1 is therefore
max(c, p)  c  e q (T2 T1 ) max(0, Ke ( r q )( T2 T1 )  S1 )


max 0,e r ( T2 T1 ) K  S1e q ( T2 T1 )

This is a call maturing at time T2 plus a put maturing at time T1
with strike Ke ( r q )( T2 T1 )
54
Barrier Options
• Option comes into existence only if stock price
hits barrier before option maturity

‘In’ options
• Option dies if stock price hits barrier before option
maturity

‘Out’ options
55
Barrier Options (continued)
• Stock price must hit barrier from below

‘Up’ options
• Stock price must hit barrier from above

‘Down’ options
• Option may be a put or a call
• Eight possible combinations
56
Parity Relations
c = cui + cuo
c = cdi + cdo
p = pui + puo
p = pdi + pdo
57
Binary Options
• Cash-or-nothing: pays Q if ST > K, otherwise pays
nothing.
Value according to B&S = e–rT Q N(d2)
• Asset-or-nothing: pays ST if ST > K, otherwise
pays nothing.
Value according to B&S = S0e-qT N(d1)
58
Decomposition of a Call Option
Long Asset-or-Nothing option
Short Cash-or-Nothing option where payoff is K
Value according to B&S = S0e-qT N(d1) – e–rT KN(d2)
59
Asian Options
• Payoff related to average stock price
• Average Price options pay:
Call: max(Save – K, 0)
 Put: max(K – Save , 0)

• Average Strike options pay:


Call: max(ST – Save , 0)
Put: max(Save – ST , 0)
60
Asian Options
• No exact analytic valuation
• Can be approximately valued by assuming that
the average stock price is lognormally distributed
61
Lookback Options
• Floating lookback call pays ST – Smin at time T (Allows buyer to
buy stock at lowest observed price in some interval of time)
• Floating lookback put pays Smax– ST at time T
(Allows buyer to sell stock at highest observed price in some
interval of time)
• Fixed lookback call pays max(Smax−K, 0)
• Fixed lookback put pays max(K −Smin, 0)
• Analytic valuation for all types
62
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
63
Exchange Options
• Option to exchange one asset for another
• For example, an option to exchange one
unit of U for one unit of V
• Payoff is max(VT – UT, 0)
64
Basket Options
• A basket option is an option to buy or sell a
portfolio of assets
• This can be valued by calculating the first two
moments of the value of the basket and then
assuming it is lognormal
65
Volatility and Variance Swaps
• Agreement to exchange the realized volatility between
time 0 and time T for a pre-specified fixed volatility with
both being multiplied by a pre-specified principal
• Variance swap is agreement to exchange the realized
variance rate between time 0 and time T for a pre-specified
fixed variance rate with both being multiplied by a
prespecified principal
• Daily expected return is assumed to be zero in calculating
the volatility or variance rate
66
Variance Swaps
• The (risk-neutral) expected variance rate between times 0 and
T can be calculated from the prices of European call and put
options with different strikes and maturity T
• Variance swaps can therefore be valued analytically if enough
options trade
• For a volatility swap it is necessary to use the approximate
relation
 1  var(V )  
ˆ
ˆ
E ( )  E V 1  
2 
ˆ
 8  E (V )  
67
VIX Index
• The expected value of the variance of the S&P
500 over 30 days is calculated from the CBOE
market prices of European put and call options on
the S&P 500
• This is then multiplied by 365/30 and the VIX
index is set equal to the square root of the result
68
How Difficult is it to
Hedge Exotic Options?
• In some cases exotic options are easier to
hedge than the corresponding vanilla options
(e.g., Asian options)
• In other cases they are more difficult to hedge
(e.g., barrier options)
69
Static Options Replication
(Hard Topic)
• This involves approximately replicating an exotic
option with a portfolio of vanilla options
• Underlying principle: if we match the value of an exotic
option on some boundary , we have matched it at all
interior points of the boundary
• Static options replication can be contrasted with
dynamic options replication where we have to trade
continuously to match the option
70
Example
• A 9-month up-and-out call option an a non-dividend
paying stock where S0 = 50, K = 50, the barrier is
60, r = 10%, and  = 30%
• Any boundary can be chosen but the natural one is
c (S, 0.75) = MAX(S – 50, 0) when S < 60
c (60, t ) = 0 when 0  t  0.75
71
Example (continued)
We might try to match the following points on
the boundary
c(S , 0.75) = MAX(S – 50, 0) for S < 60
c(60, 0.50) = 0
c(60, 0.25) = 0
c(60, 0.00) = 0
72
Example continued
We can do this as follows:
+1.00 call with maturity 0.75 & strike 50
–2.66 call with maturity 0.75 & strike 60
+0.97 call with maturity 0.50 & strike 60
+0.28 call with maturity 0.25 & strike 60
73
Example (continued)
• This portfolio is worth 0.73 at time zero compared
with 0.31 for the up-and out option
• As we use more options the value of the replicating
portfolio converges to the value of the exotic option
• For example, with 18 points matched on the
horizontal boundary the value of the replicating
portfolio reduces to 0.38; with 100 points being
matched it reduces to 0.32
74
Using Static Options
Replication
• To hedge an exotic option we short the
portfolio that replicates the boundary
conditions
• The portfolio must be unwound when any
part of the boundary is reached
75
Exercises
• 8.1
• 10.1
76