Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
19.1
Exotic Options
Chapter 19
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
19.2
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
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
19.3
Packages (page 435)
? Portfolios of standard options
? Examples from Chapter 9,bull spreads,
bear spreads,straddles,etc
? Often structured to have zero cost
? One popular package is a range forward
contract
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
19.4
Non-Standard American
Options (page 436)
? 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
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
19.5
Forward Start Options (page 437)
? Option starts at a future time,T1
? Most common in employee stock option
plans
? Often structured so that strike price
equals asset price at time T1
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
19.6
Compound Option (page 437)
? Option to buy / 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
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
19.7
Chooser Option,As You Like It”
(page 438)
? Option starts at time 0,matures at T2
? At T1 (0 < T1 < T2) buyer chooses
whether it is a put or call
? A few lines of algebra shows that this is
a package
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
19.8
Chooser Option as a Package
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Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
19.9
Barrier Options (page 439)
? 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
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
19.10
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
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
19.11
Parity Relations
c = cui + cuo
c = cdi + cdo
p = pui + puo
p = pdi + pdo
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
19.12
Binary Options (page 441)
? Cash-or-nothing,pays Q if S > K at time
T,otherwise pays 0,Value = e–rT Q N(d2)
? Asset-or-nothing,pays S if S > K at time
T,otherwise pays 0,Value = S0 N(d1)
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
19.13
Decomposition of a Call Option
Long Asset-or-Nothing option
Short Cash-or-Nothing option where
payoff is K
Value = S0 N(d1) – e–rT KN(d2)
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
19.14
Lookback Options (page 441)
? 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
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
19.15
Shout Options (page 443)
? 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
? How can a binomial tree be used to
value a shout option?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
19.16
Asian Options (page 443)
? 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)
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
19.17
Asian Options
? No analytic solution
? Can be valued by assuming (as an
approximation) that the average stock
price is lognormally distributed
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
19.18
Exchange Options (page 445)
? Option to exchange one asset for
another
? For example,an option to exchange
U for V
? Payoff is max(VT – UT,0)
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
19.19
Basket Options (page 446)
? 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
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
19.20How 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)
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
19.21
Static Options Replication
(Section 19.14,page 447)
? 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
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
19.22
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 s = 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
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
19.23
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
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
19.24
Example continued
(See Table 19.1,page 449)
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
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
19.25
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
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
19.26
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