FMP 15: Exotic Options
Listed contracts come in two shapes. A European option is exercisable only at maturity, an American option at any moment up to it, and each settles against one fixed strike price. Those are the plain vanilla contracts. Anything carrying a feature outside that description, in the trigger, the payoff, the exercise dates or the underlying, is exotic.
Dealers assemble exotics to fit one client, and they change hands in the over-the-counter markets rather than on an exchange. The commercial appeal is straightforward: bid-offer spreads here are wide, so writing them pays far better than quoting a listed contract against a crowd of competitors.
Why exotic derivative products get developed
Three motives account for most issuance. Hedging comes first, because an exposure with an awkward shape, one turning on an average rather than a closing price, is often covered more efficiently by an instrument built around that shape. Expressing a view comes second: a firm with a definite opinion on interest rates, exchange rates or commodity prices may find an ordinary call a blunt way to act. Tax and regulatory treatment supply a third motive.
Packages: plain vanilla options assembled into one position
A package is a portfolio of plain vanilla options on one asset. Bull spreads, bear spreads, butterfly spreads, calendar spreads, straddles and strangles all qualify, and they are sometimes counted among the exotics because the assembled position expresses one market view at a chosen risk level.
Buying a butterfly spread bets that an asset finishes near a particular level, at limited risk. Selling a straddle or a strangle expresses much the same opinion and earns premium, but the risk is far larger.
Every derivative can be turned into a zero-cost product by moving the payment to the end. Take a contract maturing at time T with premium f. Rather than collect f today, the seller lets the buyer settle at maturity.
For a European call this reshapes the payoff. Write A for the accumulated premium, so A equals c(1 + R) to the power T, with c the ordinary premium.
That is a forward contract: the buyer purchases the option payoff at maturity for the fixed sum A. Settle the same obligation as a futures contract and the result is a futures-style option, marked to market daily and settling at max(S_T – K, 0). The CME Group and Eurex list such products, unlike the usual equity-style option, where a premium is paid at the start.
Turning standard American options into nonstandard American options
An exchange-traded American option can be exercised whenever the holder likes, at one fixed strike price. Over-the-counter variants relax each part of that. Confining exercise to specified dates gives a Bermudan option, common in interest rate products where an American-style bond option is exercisable only on interest payment dates. An initial lock-out period is normal for employee stock options, which become vested once it has run. A moving strike is normal for a callable corporate bond, whose call price steps down as maturity nears. Warrants sometimes carry all three.
Binomial trees cope with every variant after one adjustment. The standard construction tests each node for whether early exercise beats holding on; here the test runs only where exercise is permitted, at whatever strike applies.
A client wants a two-year European call struck at USD 55 but will not fund a premium today. The dealer quotes USD 3.20 and applies 4 percent per year to the deferred payment.
In a gap option two prices do separate jobs. One decides whether any payment is made, the other sets the size of it. Write K2 for the trigger and K1 for the price used in the calculation. Both versions are European.
Separating the prices allows what no ordinary option permits: a negative payoff. With K1 at 15 and K2 at 10, the call triggers above 10 but is measured against 15, so any finish between costs the holder money. The put mirrors that: with K1 at 15 and K2 at 20, a finish above 15 and up to 20 pays negatively. Where that region carries enough probability, the cost of the gap option itself turns negative.
Why an insurance policy with a deductible is a gap put option
An asset is insured for USD 50,000 under a policy carrying a deductible of USD 500, so the insured party absorbs the first USD 500 of any fall. Ignore administration and the insurer has written a put struck at USD 49,500. Now add the USD 1,000 it costs to assess a claim. The level at which cost first arises has not moved, but the amount lost now comes off a higher figure, so the prices have parted.
Take the policy above and write S_T for the value of the asset at the end of the year.
A forward start option is agreed today and comes alive on a stated future date, struck at the money by convention on that date, so neither party knows the strike when they deal. Employee stock options behave this way whenever an employer commits to granting them later.
Chain several together with a rule for setting successive strikes and a cliquet option results. One version bundles five call options: one running from today for a year, then one-year options starting in one, two, three and four years. Four of the five legs are forward start options, and the rule can simply require each to start at the money.
How a chooser option decomposes into a call and a put
A chooser option gives the buyer a window in which to declare the contract a call or a put, perhaps after one year on a two-year deal. Call that date T1, with both candidates European options struck at K and maturing at T2.
If the asset pays no income, put-call parity pins down that difference and the maximum becomes recognisable.
Read the right-hand side as a package: a call struck at K maturing at T2, plus the payoff of a put struck at PV(K) maturing at T1. So a chooser is not a new instrument but two ordinary options with different strikes and maturities.
Take a two-year chooser struck at USD 100, the choice falling after one year, at a continuously compounded rate of 4 percent. The discounted strike at the decision date is USD 96.08. With the asset then at USD 90 and the two-year call worth USD 8.20, parity puts the put at 8.20 + 96.08 – 90 = USD 14.28. The decomposition agrees: 8.20 plus an embedded put payoff of 6.08 is USD 14.28, and the holder takes the put.
An alternative reading starts from the put. Writing the value as p + max(c – p, 0) and substituting parity gives p + max(S_1 – PV(K), 0): a put maturing at T2 alongside a call struck at PV(K) maturing at T1. Both readings give identical cash flows.
Binary options pay one of two amounts and nothing between. Four varieties exist, dividing on two questions: is the payment a fixed sum or the asset itself, and is the condition a finish above or below the strike price. The fixed-sum pair are the cash-or-nothing call and the cash-or-nothing put, also known as digital options. Delivering the underlying instead gives the asset-or-nothing call and the asset-or-nothing put.
Building a European option out of binary options
An ordinary call comes apart into two binaries. Buy the asset-or-nothing call and sell the cash-or-nothing call paying the strike price: above K the first leg delivers S_T while the second takes back K, and below K neither pays. The put reverses both legs, leaving the holder short an asset-or-nothing put and long a cash-or-nothing put paying the strike.
Why discontinuous payoffs invite price manipulation
Binaries share the gap option problem of a payoff that jumps. Take a cash-or-nothing call paying USD 100,000 when a stock closes above USD 29. At USD 28.9 the holder gets nothing; at USD 29.1 the full USD 100,000. Two-tenths of a dollar decides a six-figure sum.
The incentive that creates has no counterpart in a continuous payoff. With little time left and the price just under USD 29, the holder has reason to enter buy orders and push it over, while whoever sold the option has equal reason to sell and hold it down. Manipulation of that kind is far more tempting than under a plain vanilla contract, and more so when the underlying is thinly traded.
Cash-or-nothing options also carry a reputation for outright fraud and are banned in several countries. Regulators report a consistent pattern: customers blocked from withdrawing funds, unauthorised use of personal data, and advertised returns inflated on products structured to deliver a negative expected return.
Barrier options carry a second price level besides the strike, and reaching it decides whether the contract exists. Two choices give four varieties: the barrier sits below or above the starting price, and touching it either destroys the option or creates it. A down-and-out contract dies when the asset falls to a level beneath its start; a down-and-in stays dormant until that fall occurs. The up-and-out and up-and-in versions work the same way upward. The first family are knock-out options, the second knock-in options.
Why barrier options cost less, and the discontinuity that comes with it
Price is the attraction. An asset trades at USD 30 and a trader wants a one-year call struck at USD 32, regarding a slide to USD 27 as unlikely. A down-and-out call struck at USD 32 with its barrier at USD 27 pays what the regular call pays whenever the price holds above USD 27, and costs less: at volatility of 20 percent, a risk-free rate of 2 percent and no dividends, USD 1.56 against USD 1.82.
What the trader surrenders is the tail, and payoffs jump at the barrier as they do for gap and binary contracts. Take a call struck at USD 50 with an up-and-out barrier at USD 60. If the barrier is never touched, a finish at USD 59.90 pays USD 9.90. Lift that finish by 0.1 and the payoff is nothing, the barrier having been reached.
These contracts also break the usual link between volatility and value. Prices normally rise with volatility, yet a knock-out option near its barrier can lose value as volatility increases, because the extra movement mainly raises the chance of cancellation. Monitoring frequency works the same way: a down-and-out put is worth less when observed more often, each observation being another chance for the barrier to catch.
In-out parity for barrier options
Hold the knock-in and the knock-out version of one contract, matched on barrier, strike price and maturity, and an ordinary option remains. Exactly one of the pair is alive at maturity. Touch the barrier and the knock-in pays the plain vanilla amount while the knock-out has gone. Miss it and the knock-out pays that amount while the knock-in never existed.
Return to the asset at USD 30, where the one-year regular call struck at USD 32 is worth USD 1.82 and the down-and-out call with a barrier at USD 27 is worth USD 1.56.
A Parisian option tightens the condition, requiring the asset to stay beyond the barrier for a stated number of days rather than reacting the instant it is touched. Where that number is ten, one variety demands ten consecutive days and another any ten days over the contract.
Lookback contracts settle against the best or worst price recorded while they ran, not the final day alone. The floating pair carry no stated strike. A floating lookback call pays the excess of the closing price over the lowest price observed, and a floating lookback put the excess of the highest price observed over the closing price. Neither finishes negative, since the closing price never sits below the running minimum or above the running maximum.
The fixed pair do have a stated strike, and it is the extreme price rather than the closing price that meets it.
Why lookback options cost more than regular options
Settling against the most favourable point of a path beats settling against an arbitrary point, so these contracts are expensive. Take a non-dividend paying stock at USD 30, a strike price of USD 30, a risk-free rate of 4 percent, volatility of 20 percent and one year to run. Under the Black-Scholes Merton assumptions the European call is worth USD 2.98 and the put USD 1.80, while the four lookbacks cost far more.
| Contract | Price | Plain vanilla | Extra cost | Multiple |
|---|---|---|---|---|
| Floating lookback put | 4.44 | 1.80 | 2.64 | 2.47 |
| Fixed lookback put | 3.85 | 1.80 | 2.05 | 2.14 |
| Fixed lookback call | 5.61 | 2.98 | 2.63 | 1.88 |
| Floating lookback call | 5.03 | 2.98 | 2.05 | 1.69 |
Source: chapter prices in USD for the contract described above. The last two columns are derived and set the ranking.
Observation frequency pushes value the opposite way to the barrier case. Sampling more often produces a more extreme running maximum and minimum, so a lookback gains value as observations grow more frequent. The table assumes continuous observation.
The floating pair make an investor look prescient. The call reproduces buying at the cheapest moment and selling at the end, the put reproduces selling at the dearest. A fixed lookback resembles an American option exercised with perfect hindsight about the best date.
An Asian option settles against an arithmetic average of the asset price over the life of the contract, usually built from regular observations such as each day’s close. Write K for the strike, S_T for the final price and S ave for the average. Four contracts follow, according to whether the average replaces the asset price or the strike.
Why averaging makes an option cheaper and often more useful
An average swings less than the series behind it, so these contracts cost less than the equivalent regular option. Cost is only half the appeal, because for many exposures the average is what matters. A company repatriating foreign earnings weekly ends its year with a profit driven by the average exchange rate, not any single day’s rate. One Asian put caps the damage from a fall in that average, at far less than 52 plain vanilla options.
A one-year Asian contract observes the share price each quarter. The four observations are USD 52.00, USD 48.50, USD 55.00 and USD 56.50, the last also being the closing price. The fixed strike is USD 51.00.
A compound option has another option as its underlying, so two strike prices and two maturity dates are in play. Let the dates be T1 and T2, with T2 the later, and K1 and K2 belong to each.
Four combinations exist. A call on a call lets the holder pay K1 at T1 for a long position in a call buying the asset at K2 on T2. A put on a call pays the holder K1 at T1 and hands over the short side of that call. A call on a put buys a long put selling the asset at K2 on T2, and a put on a put delivers K1 with the short side of that put.
Compound options deliver leverage and sharper volatility sensitivity
Traders reach for these when plain vanilla leverage is not enough. Take a non-dividend paying stock at USD 50, volatility of 20 percent per year and a risk-free rate of 2 percent per year. Under the Black-Scholes Merton assumptions a one-year call struck at USD 55 costs USD 2.47. Structure a compound option instead: after six months the holder may pay USD 3 for that call, which in turn buys the stock at USD 55 after a year. The compound option is worth USD 0.99.
Whether the second option gets bought turns on where the stock stands at the halfway date. Holding volatility and the risk-free rate steady, the model values the call above USD 3 once the stock trades above USD 54.30, the exercise threshold.
Leverage shows up when volatility moves. Lift it from 20 percent to 30 percent per year and the compound option gains 160 percent, climbing to USD 2.57 from USD 0.99, while the regular call gains only 79 percent, reaching USD 4.43 from USD 2.47. Both rise, since higher volatility helps ordinary options, but the compound structure is a claim on a premium rather than an asset price. The arithmetic bears it out: 2.57 divided by 0.99 is 2.60, against 1.79 for the regular call.
A parity relation links two of these contracts. With K1, K2, T1 and T2 held the same, a long call on a call plus the present value of K1 equals a long put on the call plus a European call maturing at T2, an identity behind it.
Everything so far settles against one underlying. Two further families take their payoff from two or more assets at once, so correlation becomes a pricing input.
Asset-exchange options and the better of two assets
An asset-exchange option confers the right to swap one asset for another, and structures of that shape are commoner than the name suggests. For an investor based in the United States, the right to hand over euros for Australian dollars is one asset swapped for another. So is a takeover in which one company offers a set number of its own shares per share of the target.
The family connects to a second construction: the contract delivering whichever of two assets proves worth more. Holding it amounts to holding the first asset plus the right to swap it for the second, and equally to holding the second plus the right to swap back. Either description reduces it to a position plus one exchange option.
Basket options and the role of correlation
A basket option is written on a portfolio, which may hold individual stocks, stock indices, currencies or a mixture. Firms use them to cover aggregate exposure across several assets in one cheaper trade.
Correlation sets the price. Assets whose returns move together behave almost like one asset, so the portfolio swings widely and options on it are dear. Lower the correlations and diversification damps those swings, cutting portfolio volatility and with it the premium. So a call on ten highly correlated assets costs more than the same call on ten unrelated ones.
That logic settles a common question. A one-year at-the-money call on a basket of three stocks is worth less than three separate one-year at-the-money calls, one per stock. Separate calls collect on every stock that rises and ignore those that fall, while the basket call nets gains against losses first. Only perfect correlation would equalise them.
Some exotic instruments pay according to how much an asset moved rather than where it ended. A plain vanilla option gives exposure to volatility, but bundled with exposure to the asset price, and a trader with a pure view on volatility does not want the second part.
Volatility per day is the standard deviation of daily returns, annualised by multiplying by the square root of 252, the trading days in a year, since volatility accumulates over trading days rather than calendar days.
A variance swap is built the same way, except that the quantity exchanged is the variance rate, the square of volatility. Where a contract is written with a volatility principal, calculus links the conventions: a small change in the square of a quantity is roughly twice that quantity times the change, so matching the two means scaling the variance principal down.
A trader takes the pay-fixed side of a three-month volatility swap on the S&P 500. The principal is USD 1 million, the pre-specified volatility 15 percent per year, and realised volatility works out at 1 percent per day.
Exotic options do not all present the same hedging problem. An average price option is easier to manage than a regular one: as maturity nears, more of the prices feeding its average are already fixed, so the payoff grows more certain and late movements barely register. A plain vanilla option behaves the opposite way.
Barrier options sit at the difficult end. Near the barrier, uncertainty about whether the contract will exist becomes acute and the sensitivities delta hedging relies on turn unstable. Traders answered with static options replication, set out by Derman, Ergener and Kani in a 1995 paper (Journal of Derivatives, volume 2, issue 4, pages 78-95).
The principle behind static options replication
One result carries the method. If two portfolios are worth the same at every point along a boundary drawn in asset price and time, they are worth the same at every interior point, meaning every point reachable before the boundary is met. So a portfolio of plain vanilla options matching the exotic contract along that boundary can be sold short as a hedge, needing no rebalancing while the price stays inside.
Constructing the hedge step by step
Take an up-and-out call running one year, strike price USD 30, barrier USD 50, on a stock at USD 25 today. The natural boundary has two edges. Along the horizontal edge, with the asset at the barrier, the contract is worth zero. Along the vertical edge, at the one-year date, it is worth max(S – 30, 0) if the barrier was never reached.
Start with a one-year European call struck at USD 30, matching the vertical edge exactly. That leaves the horizontal edge. Pick N equally spaced points along it and add N European options maturing inside the year, so they contribute nothing on the vertical edge: options struck at 50 maturing at 1 divided by (N + 1) years, 2 divided by (N + 1) years and so on to N divided by (N + 1) years. Value each at each point, then solve for the position sizes driving the total to zero there.
Raising N matches the horizontal edge more finely and improves the hedge. The portfolio is left alone until the asset reaches the boundary, at which point the trader unwinds it and builds a fresh one.
Each contract here departs from the plain vanilla shape in one place, and locating it is enough to rebuild the payoff from memory.
| Contract | Defining feature | Typical use |
|---|---|---|
| Gap option | Trigger and payoff strike are separate prices | Insurance with a deductible and an assessment cost |
| Forward start option | Begins on a future date, struck at the money then | Employee stock options promised for later grant |
| Compound option | The underlying is another option | Highly levered speculation on a rise |
| Chooser option | Call or put decided after purchase | A large move of unknown direction |
| Barrier, knock-in | Created by touching a level | Cheap cover that starts once a level breaks |
| Barrier, knock-out | Destroyed by touching a level | Cutting the cost of a call when a big adverse move looks unlikely |
| Binary, cash-or-nothing | Payoff size does not vary with the asset | A view on one threshold being crossed |
| Lookback option | The extreme price of the path settles it | An expected short-term spike |
| Shout option | The holder locks in a level once, by shouting | Protecting a gain while keeping the upside |
| Asian option | An average replaces the price or the strike | Hedging an exposure that is itself an average |
| Asset-exchange option | Two assets and no cash strike price | Share-for-share takeovers, cross-currency choices |
| Basket option | The underlying is a portfolio | Covering aggregate exposure in one trade |
Source: compiled from the payoff definitions above, in order of first appearance.
The list is far from exhaustive, and dealers have been inventive with names as much as structures. Beside the shout option sit Madonna options, Himalaya options and pyramid options.
One caution belongs at the end. The saying that you get what you pay for travels badly here, because exotic options are priced far less competitively than listed contracts, and a buyer may get less than the price paid.