Upload
rohanharsh
View
223
Download
0
Embed Size (px)
Citation preview
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 1/76
Option Strategies &
Exotics
1
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 2/76
Note on Notation
• Here, T denotes time to expiry as well as time
of expiry, i.e. we use T to denote indifferentlyT and δ = T – t
• Less accurate but handier this way, I think
2
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 3/76
3
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)
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 4/76
4
Positions in an Option & the
Underlying
Profit
S T K
Profit
S T
K
Profit
S T
K
Profit
S T K
(a) (b)
(c) (d)
Basis of Put-Call Parity: P + S = C + Cash ( Ke-rT )
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 5/76
5
Bull Spread Using Calls
K 1 K 2
Profit
S T
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 6/76
Bull Spread Using Calls
Example
• Create a bull spread on IBM using the following 3-
month call options on IBM:
Option 1:
Strike: K 1 = 102
Price: C1 = 5
Option 2:
Strike: K 1 = 110
Price: C2 = 2
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 7/76
Long Call (at K 1)
plus
Short Call (at K 2 > K 1)
equals
Call Bull Spread
+10
+1
Profit
Share Price
K 1
5
-3
K 1=102
K 2=110
S BE=105
00
-1
K 2
+1
0
0
Gamble on stock price rise and offset cost
with sale of call
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 8/76
Payoff:
Long call (K 1) + short call (K
2) = Bull Spread:
{ 0, +1, +1} + {0, 0, -1} = {0, +1, 0 }
= Max(0, ST-K 1) – C1 – Max(0, ST-K 2) + C2
= C2 - C1 if ST K 1 K 2
= ST - K 1 + (C2 - C1) if K 1 < ST K 2
= (ST - K 1 - C1) + (K 2 - ST + C2) =
= K 2 - K 1 + (C2 - C1) if ST > K 1 > K 2
„Break -even‟:
SBE = K 1 + (C1 – C2) = 102 + 3 = 105
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 9/769
Bear Spread Using Puts
K 1 K 2
Profit
S T
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 10/7610
Bull Spreads with puts
& Bear Spreads with Cal ls
• Of course can do bull spreads with puts and bear
spreads with calls (put-call parity)
• Figured out how?
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 11/7611
Bull Spread Using Puts
K 1 K 2
Profit
S T
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 12/7612
Bear Spread Using Calls
K 1 K 2
Profit
S T
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 13/76
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 (K 1) + short call (K 2)
{0,+1,0} = {+1,+1,+1} + {-1,0,0} + {0,0,-1}
Equi ty Collar
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 14/76
+1+1
+1
-10 0
Long Stock
Long Put
Short Call
0 0-1
0
0
+1Equity Collar
plus
plus
equals
Equi ty Collar : Payoff Profi le
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 15/76
ST < K 1 K 1 ST K 2 ST > K 2
Long Shares ST ST ST
Long Put (K 1) K 1 – ST 0 0
Short Call (K 2) 0 0 – (ST – K 2)
Gross Payoff K 1 ST K 2
Net Profit K 1 – (P – C) ST – (P – C) K 2 – (P – C)
Net Profit = Gross Payoff – (P – C)
Equity Collar Payoffs
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 16/76
16
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
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 17/76
Short Put
plus
Long Call
equals
Long Futures
+1
+1
0
0+1
+1
A Basic Combination: A Synthetic
Forward/Futures
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 18/76
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 K 1 and buying a call with strike K 2 (with K 2 > K 1)
• When currency is to be received it involves buying a put with
strike K 1 and selling a call with strike K 2
• Normally the price of the put equals the price of the call
18
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 19/76
Range Forward Contract
19
Payoff
Asset
Price K 1 K 2
Payoff
Asset
Price
K 1 K 2
Short
Position
Long
Position
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 20/76
Volati l i ty Combinations • Mainly
• Straddle
• Strangles
• These are strategies that show the true „character‟ of
options
• But also
• Strip
• Straps
• Etc.
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 21/76
21
A Straddle Combination
Profit
S T K
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 22/76
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
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 23/76
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
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 24/76
24
A Strangle Combination
K 1 K 2
Profit
S T
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 25/76
25
K S T K S T
Strip Strap
Strip & Strap
ProfitProfit
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 26/76
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 ashort 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
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 27/76
27
Calendar Spread Using Calls
S T
K
Profit
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 28/76
28
Calendar Spread Using Puts
S T
K
Profit
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 29/76
„Quasi - Elementary‟ Securities • Arrow(-Debrew) introduces so called Arrow-
Debrew elementary securities,
i.e. contingent claims with $1 payoff in one state and $0in all other states
• These can be seen as “bet” options
• Butterflies look a lot like them
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 30/76
30
Butterf ly Spread Using Calls
K 1 K 3 S T K 2
Profit
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 31/76
31
Butterf ly Spread Using Puts
K 1 K 3
Profit
S T K 2
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 32/76
Butter f l ies 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 theunderlying 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
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 33/76
Short Butterf l ies 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 theunderlying in either direction (e.g. result of reference tothe 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
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 34/76
34
Short Butterf ly Spread Using Calls
K 1 K 3
Profit
S T K 2
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 35/76
Var iations Using I nterest Rate
Options
35
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 36/76
I nterest Rate Options
• Interest rate option
gives holder the right but not the obligation to receiveone interest rate (e.g. floating\LIBOR) and pay
another (e.g. the fixed strike rate LK )
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 37/76
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 t k +1 on each caplet is N dk max( Lk - L K , 0) where
N is the notional amount, dk = t k +1 - t k , L K is the cap rate, and
Lk is the rate at time t k for the period between t k and t k +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
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 38/76
Caplet Payoff
38
t0 = 0 t1 = 30 t2 = 120 days
Expiry \ Valuation
of option, (LIBOR 1 - LK )
Strike rate LK
fixed in
the contract
δ = 90 days
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 39/76
Planned Borrowing + Caplet (Call
on Bond )
4
6
8
10
12
14
16
18
5 7 9 11 13 15
LIBOR at expiry
A n n u a l i s e d C
o s t o f
B o r r o w i n g
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 40/76
Loan + I nterest Rate F loor let (Put
on Bond )
0
5
10
15
20
4 6 8 10 12 14 16
LIBOR at expiry
A n n u a l i z e d r e
t u r n o n
l o a n
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 41/76
41
Funding cost
iT K
Return rate
iT
K
iT
K
(c)
(a) (b)
Return rate
Long
caplet
Short
caplet
Long
floorlet
iT
K
(d)
Funding cost
Short
floorlet
Positions in an Option & the Under lying
(notice variables on vertical axis )
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 42/76
Collar
42
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%.
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 43/76
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 intoa swap
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 44/76
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 toenter into a swap where they pay the fixed leg andreceive the floating leg (long IRS).
A receiver swaption gives the buyer the right toenter into a swap where they will receive the fixedleg, and pay the floating leg (short IRS).
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 45/76
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 fundthis 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 riseover the 3 years and hence the cost of the fixed rate payments in theswap will be higher than at inception.
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 46/76
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
s K = 10%.
• Payoff is the annuity value of N δmax{ sT – s K , 0}. So, value of swaption atT is:
• f = $10m[ sT – s K ] [(1 + L2,3)-1 + (1 + L2,4)
-2 + (1 + L2,5)-3]
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 47/76
Exotics
47
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 48/76
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 oneasset for another
• Options involving several
assets
• Volatility and Varianceswaps
• etc., etc., etc.
48
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 49/76
Packages
• Portfolios of standard options
• Classical spreads and combinations: bullspreads, bear spreads, straddles, etc
• Often structured to have zero cost
• One popular package is a range forwardcontract
49
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 50/76
Non-Standard American Options • Exercisable only on specific dates
(Bermudans)
• Early exercise allowed during only partof life (initial “lock out” period)
• Strike price changes over the life
(warrants, convertibles)
50
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 51/76
Forward Start Options
• Option starts at a future time, T 1
• Implicit in employee stock option plans
• Often structured so that strike price equals asset
price at time T 1
51
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 52/76
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
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 53/76
Chooser Option “As You Like It”
• Option starts at time 0, matures at T 2
• At T 1 (0 < T 1 < T 2) buyer chooses whether it is a
put or call
• This is a package!
53
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 54/76
Chooser Option as a Package
54
))((
12
,0max
1))(()(
1
)(1
)(
1
12
)12(1
)12(
1212
1212
),0max(),max(
),max(
T T qr
eS K e
T T qr T T q
T T qT T r
Ke
T T
S Keec pc
T
eS K ec p
pcT
T T qT T r
strikewith
timeatmaturingputaplustimeatmaturingcallaisThis
thereforeistimeatvalueThe
paritycall-putFrom
isvaluethetime At
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 55/76
Barr ier 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
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 56/76
Barr ier 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
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 57/76
Parity Relations
c = cui + cuo
c = cdi + cdo
p = pui + puo
p = pdi + pdo
57
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 58/76
Binary Options • Cash-or-nothing: pays Q if S T > K , otherwise pays
nothing.
Value according to B&S = e – rT Q N (d 2)
• Asset-or-nothing: pays S T if S T > K , otherwise
pays nothing.
Value according to B&S = S 0e-qT N (d 1)
58
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 59/76
Decomposition of a Call Option Long Asset-or-Nothing option
Short Cash-or-Nothing option where payoff is K
Value according to B&S = S 0e-qT N (d 1) – e – rT KN (d 2)
59
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 60/76
Asian Options • Payoff related to average stock price
• Average Price options pay:
Call: max(S ave – K , 0)
Put: max( K – S ave , 0)
• Average Strike options pay:
Call: max(S T – S ave , 0) Put: max(S ave – S T , 0)
60
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 61/76
Asian Options • No exact analytic valuation
• Can be approximately valued by assuming that
the average stock price is lognormally distributed
61
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 62/76
Lookback Options • Floating lookback call pays S T – S min at time T (Allows buyer to
buy stock at lowest observed price in some interval of time)
• Floating lookback put pays S max – S T at time T (Allows buyer to sell stock at highest observed price in some
interval of time)
• Fixed lookback call pays max(S max− K , 0)
• Fixed lookback put pays max( K −S min, 0)• Analytic valuation for all types
62
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 63/76
Shout Options • Buyer can „shout‟ once during option life
• Final payoff is either
Usual option payoff, max(S T – K , 0), or Intrinsic value at time of shout, S
t – K
• Payoff: max(S T – S t , 0) + S
t – K
• Similar to lookback option but cheaper
63
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 64/76
Exchange Options
• Option to exchange one asset for another
• For example, an option to exchange oneunit of U for one unit of V
• Payoff is max(V T – U T , 0)
64
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 65/76
Basket Options • A basket option is an option to buy or sell a
portfolio of assets
• This can be valued by calculating the first twomoments of the value of the basket and then
assuming it is lognormal
65
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 66/76
Volati l i ty and Var iance 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
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 67/76
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
67
2)(ˆ
)var(
8
11ˆ)(ˆ
V E
V V E E
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 68/76
VIX I ndex • 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 onthe 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
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 69/76
How Diff icul t is it to
Hedge Exotic Options?
• In some cases exotic options are easier tohedge than the corresponding vanilla options
(e.g., Asian options)
• In other cases they are more difficult to hedge
(e.g., barrier options)
69
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 70/76
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 exoticoption on some boundary , we have matched it at allinterior points of the boundary
• Static options replication can be contrasted withdynamic options replication where we have to trade
continuously to match the option
70
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 71/76
Example
• A 9-month up-and-out call option an a non-dividend
paying stock where S 0 = 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
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 72/76
Example (continued)
We might try to match the following points on
the boundary
c(S , 0.75) = MAX(S – 50, 0) for S < 60c(60, 0.50) = 0
c(60, 0.25) = 0
c(60, 0.00) = 0
72
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 73/76
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
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 74/76
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
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 75/76
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
7/29/2019 L3 - Option Payoffs
http://slidepdf.com/reader/full/l3-option-payoffs 76/76
Exercises
• 8.1
• 10.1