Financial Engineering
David Oakes
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IC302
7. Exotic Options
Autumn Term 2020/1
Exotic Options
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Exotic Options
• Standard European and American puts and calls are plain vanilla products that in many cases trade in liquid markets.
• The risk properties of these products are well understood, although practical hedging can pose significant difficulties, especially for positions that involve multiple options.
• Exotic options are non-standard products that have features not contained in plain vanilla puts and calls.
• Their risk behaviour can be difficult to understand and practical hedging can prove extremely difficult.
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Uses of Exotic Options
• Features of exotic options are often negotiable and may be designed to meet the objectives of specific clients.
• They may express specific views about potential future market movements or serve as customized hedges.
• Exotic options are often used as components of structured investment products and risk management solutions.
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Profit and Risk in Exotics
• Derivatives dealers can demand higher margins from clients for managing the complex risk of exotic options.
• This can make exotics a significant source of profit for derivatives dealers and structuring banks.
• Exotics can also, however, be a significant source of risk.
• In this lecture, we describe some of the more important exotic options, outline the methods used to value them, and identify key concerns in risk management for exotics.
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Packages and Non-Standard Options
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Packages
• Packages are portfolios that contain multiple derivatives with contrasting features.
• They be marketed and sold as individual products.
• Simple examples include bull and bear spreads based on
vanilla puts or calls, straddles, and risk reversals.
• Strictly speaking, these simple packages are not exotics, but they may well exhibit complex risk behaviour.
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Range Forwards (Collars)
• Some packages may be classified as structured products, even though they contain only vanilla instruments.
• An example of this is a range forward contract, which consists of a long call and short put (or vice versa) with the call strike higher than the put strike.
• Typically, the strikes are selected so that they bracket the forward price and result in zero up-front cost.
• The zero-cost interest rate collar we discussed in the last lecture is an interest-rate version of this structure.
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Non-Standard Options
• Some options have non-standard terms related to the exercise price, exercise date or payment date.
• These terms affect the value of the option but do not usually result in significantly altered risk behaviour.
• Examples include deferred payment, non-standard American options, forward-start options, and gap options.
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Deferred Payment
• Any derivative can be transformed into a zero-cost product by deferring payment until maturity.
• Consider a European call with payment made at the option maturity date rather than upfront.
• If 𝑐 is the upfront cost then 𝐴 = 𝑐𝑒!”is the cost if payment is deferred to the option maturity date 𝑇. The payoff is:
max 𝑆! −𝐾,0 −𝐴=max 𝑆! −𝐾−𝐴,−𝐴
• This is distinct from a contingent option (sometimes called a pay-later option) in which payment is deferred to expiry but is only made if the option is exercised.
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Bermudan Options
• Standard American options can be exercised at any time during their life. As we have seen, this affects their value.
• Bermudan options can be exercised on a discrete set of pre-specified dates during the option’s life.
• This places them between European and American options in terms of flexibility and value.
• Bermudan swaptions can be used to create contingent swap hedges and to hedge callable bonds.
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Lockouts and Multiple Strikes
• Other non-standard American options may not be exercised during a lock-out period early in their life or may have multiple strike prices over their life.
• These features are common in warrants (i.e. securities issued by companies that give the holder the right to buy newly issued stock of the company at a specified price).
• Most non-standard American options can be valued using the binomial tree methods we discussed in lecture 5.
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Gap Options
• A gap call option is a European call option that pays 𝑆” −
𝐾- when 𝑆” > 𝐾.. 𝐾. is often called the trigger price.
• This increases the payoff by 𝐾. − 𝐾- when 𝑆” > 𝐾..
• This amount may be positive or negative, depending on whether𝐾. >𝐾- or𝐾. <𝐾-. If𝐾. <𝐾-,thepayoffto the gap call may be negative in some circumstances.
• A gap put option is defined analogously.
• Gap options can be valued by a small modification to the Black-Scholes-Merton formula (see Hull, ch. 26).
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Forward Start Options
• Forward start options start at some time in the future.
• Consider a forward-start at-the-money (ATM) European call option that starts at time 𝑇- and matures at time 𝑇..
• In the Black-Scholes-Merton setting, the value of an ATM European call is proportional to the underlying asset price.
• The value of the forward start call at the forward start date 𝑇- is therefore 𝑐 𝑆-⁄𝑆/, where 𝑐 is the value at time zero of an ATM European call that lasts for 𝑇. − 𝑇-, 𝑆/ is the asset price at time zero, and 𝑆- is the asset price at 𝑇-.
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Forward Start Options
• Using risk-neutral valuation, the value of the forward start call at time zero is therefore:
𝑒"#!!𝐸3 𝑐𝑆$ 𝑆%
where 𝐸, denotes expectation in the risk-neutral world.
• Since𝑐and𝑆/areknownand𝐸,𝑆- =𝑆/𝑒!01"!,the
value of the forward start option is 𝑐𝑒01"!.
• For a non-dividend-paying asset, 𝑞 = 0, so the value of the forward start call is 𝑐, which is the value of a regular ATM option with the same life as the forward start option.
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Cliquet Options
• A cliquet option is a call or put option that periodically resets its strike to the level of the underlying asset price.
• At each reset date, the cliquet locks in the difference between the new and old strikes and pays that as a profit.
• A cliquet is basically a series of forward start options.
• Some cliquets have special features (e.g. limits on the payoff each period) that complicate valuation, for which numerical methods (e.g. Monte Carlo) may be required.
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Compound and Chooser Options
Compound Options
• Compound options are options on options.
• European-style compound options can be valued analytically in the Black-Scholes-Merton framework in terms of integrals of the bivariate normal distribution.
• In practice, compound options are very sensitive to higher-order derivatives of the underlying asset price, which can make risk management and valuation difficult.
• In corporate finance, compound options have been used to value coupon-paying bonds as contingent claims.
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Chooser Options
• Chooser options are options that the holder can choose to be either a put or a call at some predetermined time.
• A chooser for which the choice date is the same as the expiry date would be identical to a straddle.
• Prior to the choice date, risk characteristics of choosers fall between those of vanilla options and straddles.
• If both underlying options are European and have the same strike price, put-call parity can be used to value the chooser using the Black-Scholes-Merton formula.
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Binary Options
Binary Options
• Binary options make a fixed payout to the option holder if exercised. They may be European or American in exercise.
• A European binary call makes a fixed payoff if the underlying asset terminates above the strike price, while a European binary put makes a fixed payoff if the underlying asset terminates below the strike price.
Payoff Long Binary Call Payoff Long Binary Put
KST KST
K = exercise price (strike price)
ST = asset price at option expiry
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Binary Options
• In effect, the option holder is betting that the underlying asset will trade above (in the case of a call) or below (in the case of a put) the strike price at expiry.
• For this reason, binary options are sometimes called bets.
• The term digital option is also used.
• Strictly speaking, the options we have described are cash- or-nothing binary options. In an asset-or-nothing binary, the option holder receives a long position in the asset (rather than cash) if the option expires in the money.
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Risk Characteristics of Binaries
• The fixed payoff of a binary produces a very different risk profile than the ‘ramped’ payoff of an ordinary option.
• As expiry approaches, the price of a binary (as a function of the underlying asset) will evolve in a manner much like that we observe for the delta of an ordinary option.
Price
near expiry
far from expiry
K ST
K = exercise price (strike price)
ST = asset price at option expiry
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Binary Option Delta
• The delta of a binary behaves like the gamma of an ordinary option, becoming very large near expiry for options that are at or near the money.
Delta
near expiry
far from expiry
• This can make delta hedging of binaries impractical.
K
S
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Binary Option Gamma
• The binary has positive gamma (and pays time decay or theta) when it is out of the money, but it has negative gamma (and earns time decay) when it is in the money.
Gamma
near expiry
far from expiry K
S
• At the money, the binary has little local convexity and will act more like a futures position.
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Binary Options and Skew
• The complex ‘risk reversal’ behaviour of binaries makes them very dependent on skew in implied volatility.
• This dependence on skew can have a significant impact on hedging, replication and valuation of binary options.
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Hedging and Replicating Binaries
• The risk reversal pattern in gamma and vega and the explosive behaviour of delta near the strike price as expiry
approaches suggest that delta hedging in the underlying asset may not be appropriate for a binary option.
• A more promising hedge would be a spread trade in which we go long one option and short another for the same expiry date but for two different strike prices, with the strike prices bracketing the strike price of the binary.
• A bull spread could be used to replicate or hedge a binary call, while a bear spread could be used for a binary put.
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Replication with Spreads
•
Bull and bear spreads can be constructed using calls or puts. Here we use a call spread to replicate a binary call:
Payoff Payoff Bull spread with calls
•
KST KST K = exercise price (strike price)
ST = asset price at option expiry
For the hedge or replication to be effective, the spread must be quite narrow around the binary strike. But this means that we must trade in large quantities in order to match the payoff to the binary option.
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Replication and Transactions Costs
• In practice, binary options are seldom replicated with narrow spreads, especially if there is some time to expiry.
• One reason for this is that transactions costs will undermine the performance of the hedge.
• In order to create the spread we must buy one option (at the offer) and sell another (at the bid), both in large underlying amounts. This will be expensive.
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Replication and Skew
• Another reason is that the volatility skew in options can have a significant effect on the value of the binary.
• Suppose that we replicate the binary using a narrow spread on calls, as described earlier. Any volatility skew in options prices will affect the cost of the spread.
• For a narrow spread, the effect will be small (since the strikes are close together), but in cash terms it may be significant, since we will be trading in large amounts.
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Practical Replication
• For this reason, it may be more practical to hedge a European binary that has some time to expiry using a
wider spread on a smaller nominal amount, thereby incorporating the volatility skew into the binary price.
• As the option comes toward expiry, the spread on which the hedge is based can be narrowed.
• For options away from the money, the hedging will become simpler as we near expiry. For options at or near the money, the risk in the binary will become a sort of pin risk, which cannot really be hedged.
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Black-Scholes-Merton Valuation
• A cash-or-nothing call pays 𝑄 if the asset price is above
the strike price at 𝑇 and zero if it is below the strike.
• In the BSM setting, the risk-neutral probability that the
assetpriceendsupabovethestrikepriceis𝑁𝑑. .
• The risk-neutral value of the cash-or-nothing call is:
𝑄𝑒"#!𝑁 𝑑&
• The risk-neutral value of the cash-or-nothing put is: 𝑄𝑒"#!𝑁 −𝑑&
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American Binary Options
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American Binary Options
• American binary options make a fixed payoff to the option holder the first time that the underlying asset ‘touches’ the strike price during the option’s life.
• If the option expires without the underlying asset touching the strike, there is no payoff.
• The expected life of an American binary is uncertain.
• In a sense, it is a bet on the amount of time that will pass before the underlying asset price touches the strike.
First Exit or Stopping Time
• This time is known formally as a first exit time or stopping
time.
• It is random, and its distribution will govern the value of an American binary option.
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American Binaries as Barriers
• Because the payoff to a binary option is fixed, once the strike price has been touched there is no benefit to the American binary option holder in continuing to wait.
• This allows us to model the American cash-or-nothing binary as a barrier option which knocks out at a barrier level equal to the option strike price, with a cash payoff or rebate equal to the binary option payoff.
• This rebate is paid if the barrier is hit (i.e. if the binary strike is touched).
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American Binary Value
• The probability of an American binary making a payoff is at least as great as the probability for an otherwise equivalent European binary option.
• Typically, American binary options cost about twice as much as otherwise equivalent European binary options.
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American Binary Risks
• American binary options do not usually exhibit the ‘risk reversal’ behaviour typical of European binary options.
• Higher volatility increases the probability of hitting the barrier, making the option more valuable, so an American binary is long volatility (i.e. has positive vega).
• Since the option cannot cross the barrier, its gamma is also typically positive and does not switch in sign, although this may not be true when there is positive carry on the underlying asset that is used as a hedge.
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American Binary Risks
• As expiry approaches, the price of the binary will fall towards zero below the strike and move towards the level of the cash payoff near the strike.
Price
far from expiry
near expiry
K ST
• This will make the delta at levels well below the strike near zero and the delta at levels near the strike very high.
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American Binary Risks
• The gamma will also fall towards zero at price levels well below the strike and rise to a peak just below the strike.
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Barrier Options
Barrier Options
• Barrier options are options that either come into existence (‘knock in’) or are terminated (‘knock out’) when the underlying asset trades at a specific price (the barrier).
• Barrier options may be calls or puts.
• The barrier may be set either above or below the strike.
• The underlying options are usually European in exercise
Regular and Reverse Barriers
• Regular barrier options knock in or out when they are out of the money. For a call, this means that the barrier is below the strike; for a put, the barrier is above the strike.
• Reverse barrier options knock in or out when they are in the money and therefore have some intrinsic value.
• Reverse barriers are more difficult to trade and hedge than regular barriers, since the gain or loss in intrinsic value when the option knocks in or out creates differences in price and behaviour on either side of the barrier.
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Regular and Reverse Barriers
Payoff
Regular Barrier Call Payoff Regular Barrier Put
BKST KBST B = barrier (knock in or knock out)
K = exercise price (strike price) ST = asset price at option expiry
Reverse Barrier Call Payoff Reverse Barrier Put
KBST BKST B = barrier (knock in or knock out)
K = exercise price (strike price) ST = asset price at option expiry
Payoff
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Classification by Spot Level
• Barrier options are also classified according to the level of the barrier relative to the spot price.
• Thus, a ‘down and out’ option will knock out at a barrier level below the current spot, an ‘up and in’ option will knock in at a barrier above the current spot, and so on.
• This results in eight distinct types of barrier option which present varying degrees of trading difficulty and risk.
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Joint Classification and Risk
Option
Barrier Type
Trading Difficulty and Risk
Down and out call
Regular
Low
Down and in call
Regular
Low
Up and out put
Regular
Low
Up and in put
Regular
Low
Option
Barrier Type
Trading Difficulty and Risk
Up and out call
Reverse
High
Up and in call
Reverse
High
Down and out put
Reverse
High
Down and in put
Reverse
High
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Barriers and Vanilla Options
• A combination of a knock-out option and a knock-in option, both on the same underlying asset and with the
same strike price, barrier level and expiry date, is equivalent to an ordinary European option.
• Including a barrier therefore reduces an option’s price, which can help reduce the cost of option-based hedges.
• Some barrier options (usually knock-outs) make a fixed cash payoff or ‘rebate’ to the option holder if the barrier is hit. This feature can be useful in structured products.
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Discontinuous Delta
• The delta of a knock-out option is discontinuous at the barrier, as it will go to zero if the option is knocked out.
• This can make knock-outs difficult to trade and hedge, because unwinding the hedge will result in ‘slippage’.
• Knock-in options also have discontinuous deltas, with the delta changing sign at the barrier level.
• The transaction costs of unwinding the hedge should be taken into account when selling a barrier option.
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Down and Out Call
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Down and In Call
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Barriers and Volatility
• For vanilla options, volatility affects value in a more-or-less linear way (e.g. doubling volatility doubles the price).
• For barrier options, higher volatility brings the barrier closer in a non-linear way.
• Higher volatility reduces the expected life of the option and increases the influence of the barrier on its value.
• As with binary options, skew in implied volatility can have an important impact on pricing and hedging of barriers.
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Reverse Barriers and Exit Time
• As noted earlier, reverse barrier options can be difficult to price and hedge. They also contain complex risks that can be difficult for unsuspecting clients to understand.
• Consider a up-and-out call. As the underlying asset price increases, the delta can become negative because the asset price is moving towards the knock-out barrier level.
• Large promised payouts at expiry may be meaningless if the first exit time or stopping time (i.e. the expected time to when the barrier is hit) is quite short.
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Up and Out Call
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Lookback and Asian Options
Lookback Options
• Lookback options allow the option holder to buy at the low or sell at the high over a specified period.
• This is achieved through a floating strike price.
• A floating lookback call allows the holder to buy the asset
at the minimum price observed during the option’s life.
• A floating lookback put allows the holder to sell the asset at the maximum price observed during the option’s life.
Pricing of Lookback Options
• The payoff to a lookback option is very appealing.
• In practice, however, lookback options seldom trade, simply because they are so expensive (as a rule of thumb, about twice as expensive as conventional options).
• Analytic formulas exist for valuing floating lookbacks in the BSM framework (see Hull, ch. 26).
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Risk in Lookback Options
• The principal risk management problem with lookbacks is related to gamma and implied volatility skew.
• A dealer who sells a lookback call can hedge by buying an ATM ordinary call. If the asset price falls, she will sell that call and replace it with a call at a lower strike.
• If there is negative skew (i.e. higher implied volatility at lower strikes), ‘rolling’ the hedge will cost money, which in turn will increase the price of the lookback call.
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Risk in Lookback Options
• Notice that, if the asset price continues to fall, the gamma of the vanilla option will fall to zero but the gamma of the lookback will not (since its strike follows the spot price).
• This risk that cannot be hedged with vanilla options.
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Asian Options
• Asian options are options on the average price of the underlying asset during the option’s life.
• An average price call makes a payoff at expiry equal to: max 𝑆'() −𝐾,0
and an average price put makes a payoff equal to: max 𝐾−𝑆'(),0
where 𝑆9:; is the average price of the underlying asset during the option’s life.
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Pricing of Asian Options
• Asian options can be valued using familiar formulas in the
BSM framework, provided 𝑆9:; is lognormal.
• The geometric average price is exactly lognormal in the
BSM framework, but Asian options on it seldom trade.
• The arithmetic average price in BSM is approximately lognormal. This allows Asian options to be valued using Black’s model after fitting 𝑆9:; to the first two moments of the lognormal distribution (see Hull, ch. 26).
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Users of Average Price Options
• Average price options are less expensive than vanilla options and may be better suited to some practical risk management problems faced by option users.
• For example, companies that face FX risk related to foreign currency payments that arrive throughout the year may be more concerned about the average exchange rate over the year than the rate on a specific date.
• An average price option could provide them with a low- cost and effective hedge.
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Multi-Asset Options
Multi-Asset Options
• Multi-asset options make payoffs that depend on more than one underlying asset price.
• This includes structures based on:
– Choice (e.g. best of, worst of, rainbow)
– Linear combinations (e.g. baskets and spreads) – Products or quotients
• A key factor in pricing and hedging multi-asset options is correlation between the assets.
Rainbow Options
• Rainbow options are options with more than one strike price and more than one underlying asset.
• A typical example might be an option that is a call on either of two assets, each with a specific strike price.
• This is sometimes called a best-of call, since the option holder will choose the asset that gives the largest payoff.
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Rainbows and Correlation
• Consider a best-of call on two assets with the same current price and volatility and the same strike price.
• If the correlation between the assets is 1, the fact that there are two assets is irrelevant. The option will trade at the price of a vanilla call on either of the two assets.
• If the correlation is -1, the structure will trade at a price twice that of either vanilla option, because it is guaranteed to expire in the money for one of the assets.
• The option’s price will be inversely related to correlation. 65
Exchange Options
• Exchange options are options to exchange one asset for another.
• Because they depend on the relative price of the two assets, they are sometimes called spread options.
• The payoff to a European option to exchange asset 𝑈 for asset𝑉ismax 𝑉" −𝑈",0 .
• A formula for valuing exchange options in the BSM framework was first developed by Margrabe, so these options are also sometimes called Margrabe options.
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Pricing Exchange Options
• Margrabe’s formula can be derived using the results on martingales and measures that we discussed in lecture 6 (for details, see Hull, ch. 26 and ch. 28).
• As with rainbow options, the value of an exchange option depends critically on the correlation between the assets.
• In this case, higher correlation reduces the option’s value.
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Examples of Exchange Options
• A foreign exchange option is an option to exchange one currency for another at a fixed rate.
• A stock tender offer is an option to exchange shares in one company for shares in another company at a fixed rate.
• Both are examples of exchange options.
• An outperformance option between two assets can also be thought of as one of the assets plus an option on the spread between the two assets.
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Basket Options
• Basket options are options on the weighted sum of two or more assets.
• Typically, they have a strike price on the weighted sum.
• An option on a stock index is a kind of basket option.
• As with other multi-asset options, correlation between the assets is a key factor driving pricing and risk.
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Pricing Basket Options
• In some cases, the basket itself may be treated as a commodity, with options on it priced directly.
• If we assume the value of the basket is lognormal, European options can be priced using the BSM formulas.
• Stock index options are often priced in this way.
• Note that this is not consistent with using the same formulas to price options on individual stocks, since the average (or any linear combination) of lognormally distributed variables is not lognormally distributed.
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Pricing Basket Options
• Ignoring this problem of lognormality can lead to practical problems in hedging basket options in markets with skew.
• Sensitivity to correlation is also an issue for baskets where correlations may change over the option’s life.
• In many cases, this may be better handled by using Monte Carlo methods for valuation. More complex multi-asset structures can often only be valued in this way.
• We explain how this is done in the next lecture.
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References Main reading:
• Hull, ch. 26
• Kosowski and Neftci, ch. 11
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