12.3 Exotic Options
Key Takeaways
- Exotics differ from vanillas in path dependence, barriers, averages, multiple underlyings, or nonstandard exercise—often to match a hedging need at lower premium
- Zero-cost collars/conversions and structured packages can set net premium near zero by selling optionality against purchased optionality
- Barrier, binary, Asian, lookback, compound, chooser, forward-start, gap, exchange, and basket options each change payoff timing or measurement of the underlying
- Variance swaps pay realized variance versus a strike; volatility swaps target realized vol—replication and quoting conventions differ
- Static replication prices many exotics as portfolios of vanillas (or bonds) held without dynamic delta rebalancing under the replicating measure’s assumptions
Exotic Options
FMP–15 surveys exotic derivatives: contracts whose payoffs depend on the path of the underlying, on hitting barriers, on averages, on choices during the life, or on more than one asset. Vanillas (standard calls/puts) are European or American with payoff based only on ST versus K. Exotics exist because end users want cheaper hedges, tailored risk transfer, or exposures that vanillas cannot replicate without a dynamic book.
Exotic Versus Vanilla
| Feature | Vanilla | Exotic examples |
|---|---|---|
| Payoff driver | Usually ST only | Path, average, barrier, min/max, another asset |
| Liquidity / quoting | Deep listed markets | Often OTC, model- and credit-sensitive |
| Premium | Higher for full optionality | Often lower when barriers/averages reduce expected payout |
| Risk management | Standard Greeks | Gaps, discontinuities, sticky barriers, smile risk |
Why exotics exist: (1) Cost — a knock-out call can be much cheaper than a vanilla if the buyer will abandon the hedge if the market rallies through a barrier. (2) Fit — an Asian average-rate option matches a corporation’s average FX rate over a billing quarter. (3) Structuring — notes embed exotics to shape coupons. (4) Views — binaries express pure event bets; choosers delay the call/put decision.
Zero-Cost Conversion and Related Structures
A zero-cost (or nearly zero-cost) structure sets net premium ≈ 0 by financing long options with short options.
Classic collar: long put (floor) financed by short call (cap). If put premium equals call premium, net debit is zero (zero-cost collar). The holder of stock plus collar has downside protected and upside capped—economically similar to a bull spread on the stock.
Conversion (options market): long stock + long put + short call with the same K locks a forward-like payoff; with European options this relates directly to put-call parity. Dealers use conversions/reversals to extract carry when parity is off. “Zero-cost conversion” language in structured products often means packaging so the investor pays no upfront premium—financing comes from sold optionality or reduced participation.
Worked zero-cost collar
Stock at 100. Buy 95-put for 3.20; sell 108-call for 3.20. Net premium 0. Between 95 and 108 the collar expires worthless; below 95 the put pays; above 108 upside is called away. Effective sale price if called: 108; effective floor if put exercised: 95.
Nonstandard American Options
Not all “American” contracts allow exercise any day for cash intrinsic value:
- Bermudan: exercise only on specified dates (common in callable bonds / swaptions).
- Shout options: holder “shouts” once to lock intrinsic; final payoff is at least the shouted intrinsic.
- Compound early-exercise features: options on options with American-style decisions on the mother or daughter contract.
- Reset / cliquet-style Americans: strikes reset on schedules; early exercise rules become path-dependent.
FRM point: early-exercise analysis from FMP–13 does not transplant unchanged—exercise rights and locked values differ by contract terms.
Catalogue of Common Exotics
Gap options
Payoff resembles a call/put but uses two strikes: one determines whether the option pays (trigger), another sizes the cash payoff. Example: gap call pays ST − K2 if ST > K1. If K2 ≠ K1, payoff can be discontinuous (a “gap”) at K1—hence binary-like jump risk for the seller.
Forward-start options
Strike is set in the future as a function of the then spot (for example, ATM at t1). Useful for employee options or cliquet building blocks. Value today depends on forward volatility between start and expiry.
Compound options
Options on options: call on a call, call on a put, put on a call, put on a put. Two strikes and two expiries. Used when a firm may need an option only if a project is approved. At the first expiry the holder pays K1 to receive the underlying vanilla.
Chooser options
After a choice date, the holder selects whether the option is a call or a put (usually same K and final expiry). Value relates to a put-call package; roughly, a chooser can be replicated with a call plus a put with adjusted maturity under Black–Scholes assumptions.
Barrier options
Payoff activates or extinguishes when a barrier H is hit:
| Type | Behavior |
|---|---|
| Down-and-out call | Dies if S hits H < S0 |
| Up-and-out call | Dies if S hits H > S0 |
| Down-and-in / up-and-in | Comes to life only if barrier hit |
| With rebate | Pays fixed amount if knocked out |
Cheaper than vanillas when knock-out probability is material. Risk: barrier discontinuity—delta can spike near H; hedging is notoriously difficult in jumps.
Binary (digital) options
Pay a fixed cash amount (cash-or-nothing) or the asset (asset-or-nothing) if a condition holds at expiry (or on hitting a barrier for one-touch). Highly sensitive to the density at the strike—smile and digital risk matter.
Lookback options
Payoff uses the max or min of S over the life (for example, floating-strike lookback call pays ST − min(S)). Expensive: holder gets hindsight. Fixed-strike lookbacks replace ST with max(S) in a call payoff.
Asian options
Payoff uses an average of S (arithmetic or geometric; discrete or continuous). Average-rate calls pay max(A − K, 0). Asians reduce manipulation risk and match average-price exposures; they are typically cheaper than vanillas because averaging dampens volatility.
Exchange options
Margrabe-style: right to exchange asset U for asset V—payoff max(UT − VT, 0). Volatility input is the volatility of the ratio (or difference of vols minus correlation term). Used in M&A contingency and quanto-like structures.
Basket options
Payoff on a portfolio (basket) of underlyings. Correlation is first-order: higher correlation among components raises basket vol and usually raises option value for a given weighted spot.
Volatility Swaps Versus Variance Swaps
| Contract | Floating leg | Typical quoting |
|---|---|---|
| Variance swap | Realized variance (often 252 × sum of squared returns) | Variance strike K_var; payoff ∼ notional × (σ_real^2 − K_var) |
| Volatility swap | Realized volatility (square root of variance) | Vol strike; payoff ∼ notional × (σ_real − K_vol) |
Variance swaps admit a famous static replication with a continuum of vanillas (log contract / portfolio of puts and calls across strikes). Volatility swaps are harder to replicate statically because of the square-root; dealers often risk-manage them via variance swaps plus dynamic adjustments. Convexity: E[σ] ≠ sqrt(E[σ^2]), so vol-swap fair strikes differ from variance-swap implied vols.
Worked variance-swap idea
Vega notional = $100,000 per volatility point. Realized vol = 18%, strike vol = 20% on a variance swap quoted in vol terms means K_var = 0.20^2 = 0.04 and realized variance = 0.18^2 = 0.0324. Under the standard vega-notional convention the payoff to the long is N_vega × (σ²_real − K_var) / (2 × K_vol) = 100,000 × (0.0324 − 0.04) / (2 × 0.20) ≈ −$1,900. Watch the convention: if the contract instead states a variance notional of $100,000, the payoff is simply 100,000 × (0.0324 − 0.04) = −$760. The two quoting bases differ by the factor 2 × K_vol, so always read which notional the question specifies. Directionally either way: realized below strike → long variance loses.
Static Replication Premise
Static replication means holding a fixed portfolio of liquid instruments (vanillas, bonds, shares) that matches the exotic’s payoff in all states, without continuous rebalancing. If payoff f(ST) is a function of terminal spot only, Breeden–Litzenberger / Carr–Madan results say f can be written as bonds + forwards + continuum of puts and calls. Path-dependent exotics generally need dynamic hedging or extra state variables—unless a clever identity reduces them to vanillas (as with continuously monitored barriers in Black–Scholes, or the log contract for variance).
Exam use: when asked why variance swaps are popular among dealers, cite robust static replication with vanillas; when asked why barrier hedging fails in practice, cite jumps and discrete hedging near discontinuous payoffs.
Synthesis
Choose exotics when the alternative vanilla hedge is too expensive or poorly matched. Price and risk-manage with clear eyes: barriers and binaries create cliff Greeks; Asians and baskets embed averaging and correlation; variance swaps are the cleanest OTC vol transfer when static replication is available.
A down-and-out call is typically cheaper than an otherwise identical vanilla call because:
An Asian average-rate call payoff is based on:
Relative to variance swaps, volatility swaps are generally:
Static replication of a European-style payoff that depends only on ST means: