Exotic Options

19.1 19.2 Types of Exotics • Package • Binary options Exotic Options • Nonstandard • Lookback options American options • Shout options • Forward start options • Asian options • Compound options • Options to exchange Chapter 19 • Chooser options one asset for another • Barrier options • Options involving several assets

19.3 19.4 Packages Non-Standard American Options

• Portfolios of standard options • Exercisable only on specific dates • Examples from Chapter 9: bull spreads, (Bermudans) bear spreads, straddles, etc • Early exercise allowed during only part of • Often structured to have zero cost life (e.g. there may be an initial “lock out” period) • One popular package is a range forward contract • Strike price changes over the life

19.5 19.6 Forward Start Options Compound Option • Option to buy / sell an option • Option starts at a future time, T 1 – Call on call • Most common in employee stock option – Put on call plans – Call on put • Often structured so that strike price – Put on put

equals asset price at time T1 • 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 Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 19.7 19.8 Chooser Option “As You Like It” Chooser Option as a Package At time Tcp the value is max( , ) • Option starts at time 0, matures at T 1 2 From put-call parity • At T (0 < T < T ) buyer chooses 1 1 2 pce rT()21 T KSe qT () 21 T whether it is a put or call 1 The value at time T is therefore • A few lines of algebra shows that this is 1 qT()21 T ()() r q T 21 T a package ce max(0, Ke S1 )

This is a call maturing at time T2 plus

a put maturing at time T1

19.9 19.10 Barrier Options Barrier Options • Stock price must hit barrier from below • Option comes into existence only if stock price hits barrier before option maturity –‘Up’ options –‘In’ options • Stock price must hit barrier from above • Option dies if stock price hits barrier –‘Down’ options before option maturity • Option may be a put or a call –‘Out’ options • Eight possible combinations

19.11 19.12 Parity Relations Binary Options

c = cui + cuo • Cash-or-nothing: pays Q if S > K at time –rT T, otherwise pays 0. Value = e QN(d2) c = cdi + cdo • Asset-or-nothing: pays S if S > K at time p = pui + puo

T, otherwise pays 0. Value = S0 N(d1) p = pdi + pdo

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 19.13 19.14 Decomposition of a Call Option Lookback Options

Long Asset-or-Nothing option • Lookback call pays ( ST – Smin ) at time T Short Cash-or-Nothing option Allows buyer to buy stock at lowest observed price in some interval of time (where payoff is K)

• Lookback put pays ( Smax– ST ) at time T –rT Value = S0 N(d1) – e KN(d2) Allows buyer to sell stock at highest observed price in some interval of time

19.15 19.16 Shout Options Asian Options • ‘ ’ Buyer can shout once during option life • Payoff related to average stock price • Final payoff is either • Average Price options pay:

– Usual option payoff, max(ST – K, 0), or –max(Save – K, 0) (call), or – Intrinsic value at time of shout, SW – K –max(K – Save , 0) (put) • Payoff: max(ST – SW, 0) + SW – K • Average Strike options pay: • Similar to lookback option but cheaper –max(ST – Save , 0) (call), or • How can a binomial tree be used to value a shout option? –max(Save – ST , 0) (put)

19.17 19.18 Asian Options Exchange Options

• No analytic solution • Option to exchange one asset for another • Can be valued by assuming (as an • For example: approximation) that the average stock an option to exchange U for V price is lognormally distributed • Payoff is max(VT – UT, 0)

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 19.19 19.20 Basket Options How Difficult is it to Hedge Exotic Options ? • A basket option is an option to buy or sell a portfolio of assets • In some cases exotic options are easier to hedge than the corresponding vanilla • This can be valued by calculating the options. (e.g., Asian options) first two moments of the value of the • In other cases they are more difficult to basket and then assuming it is hedge. (e.g., barrier options) lognormal

19.21 19.22 Static Options Replication Example • This involves approximately replicating an • A 9-month up-and-out call option an a non- exotic option with a portfolio of vanilla options dividend paying stock where S0 = 50, K = 50, • Underlying principle: If we match the value of the barrier is 60, r = 10%, and V = 30% an exotic option on some boundary , we have • Any boundary can be chosen but the natural matched it at all interior points of the boundary one is • Static options replication can be contrasted with c (S, 0.75) = MAX(S – 50, 0) when S 60 dynamic options replication where we have to c (60, t ) = 0 when 0 d t d 0.75 trade continuously to match the option

19.23 19.24 Example (continued) Example continued (See Table 19.1, page 449) • We might try to match the following We can do this as follows: points on the boundary +1.00 call with maturity 0.75 & strike 50 c (S , 0.75) = MAX(S – 50, 0) for S 60 c (60, 0.50) = 0 –2.66 call with maturity 0.75 & strike 60 c (60, 0.25) = 0 +0.97 call with maturity 0.50 & strike 60 c (60, 0.00) = 0 +0.28 call with maturity 0.25 & strike 60

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 19.25 19.26 Example (continued) Using Static Options Replication • This portfolio is worth 0.73 at time zero compared with 0.31 for the up-and-out option • To hedge an exotic option we short the • As we use more options the value of the portfolio that replicates the boundary replicating portfolio converges to the value of the exotic option conditions • For example, with 18 points matched on the horizontal boundary the value of the replicating • The portfolio must be unwound when portfolio reduces to 0.38; with 100 points being any part of the boundary is reached matched it reduces to 0.32