Exotic Options
Barriers, Asians, lookbacks, digitals, cliquets, autocallables, and quanto options. Pricing, hedging, and risk management.
When to Activate
- User pricing or hedging barrier options (knock-in, knock-out)
- Pricing Asian options (arithmetic or geometric average)
- Analyzing lookback, digital/binary, or cliquet options
- Understanding autocallable structures and their risks
- Quanto options and cross-currency derivative pricing
- Hedging exotic payoffs and understanding Greeks behavior
Core Concepts
Barrier Options
Classification
- Knock-out: option ceases to exist if barrier is hit (down-and-out, up-and-out)
- Knock-in: option comes into existence only if barrier is hit (down-and-in, up-and-in)
- In-out parity: knock-in + knock-out = vanilla (same strike and expiry)
- Rebate: fixed payment made when the barrier is hit (for knock-outs) or at expiry if barrier is never hit (for knock-ins)
Pricing
- Continuous monitoring: closed-form solutions exist under BSM (reflection principle)
- Discrete monitoring (daily, weekly): no closed-form; use MC with Brownian bridge correction or lattice methods
- Broadie-Glasserman-Kou continuity correction: shift barrier by beta * sigma * sqrt(dt) where beta = 0.5826
- Near-barrier behavior: Greeks become extreme (delta can flip sign, gamma spikes)
Key Risks
- Pin risk near the barrier: small moves determine whether the option exists or not
- Barrier shift risk: if the barrier is monitored at fixing times, the effective barrier differs from contractual
- Hedging difficulty: near the barrier, delta changes rapidly and gamma is very large
- Gap risk: underlying can gap through the barrier (especially over weekends, overnight)
Asian Options
Types
- Fixed-strike Asian call: max(avg(S) - K, 0) — compare average price to fixed strike
- Floating-strike Asian call: max(S_T - avg(S), 0) — compare terminal price to average
- Average can be arithmetic (standard) or geometric (tractable)
- Averaging period: full life or partial (e.g., last 3 months of a 1-year option)
Pricing
- Geometric average Asian: closed-form under GBM (geometric average of log-normals is log-normal)
- Geometric Asian call: BSM formula with adjusted vol (sigma/sqrt(3)) and drift
- Arithmetic average Asian: no closed-form; use MC with geometric Asian as control variate
- Turnbull-Wakeman approximation: match first two moments of arithmetic average to log-normal
- As averaging dates increase, Asian option value decreases (averaging reduces effective volatility)
Properties
- Cheaper than vanilla options (averaging reduces payoff volatility)
- Popular for commodity hedging (averaging reflects actual purchase/sale prices)
- Less sensitive to manipulation near expiry (one closing price cannot dominate)
- Vega is lower than vanilla; delta behavior depends on how much averaging has occurred
Lookback Options
Types
- Fixed-strike lookback call: max(S_max - K, 0) — payoff based on maximum price observed
- Floating-strike lookback call: S_T - S_min — buy at the minimum, sell at current
- Floating-strike lookback put: S_max - S_T — sell at the maximum, buy at current
- Partial lookback: lookback feature applies to only part of the option's life
Pricing
- Closed-form under continuous monitoring (Goldman, Sosin, Gatto 1979)
- Floating-strike lookback call: S_T * N(a1) - S_min * exp(-rT) * N(a2) - S_T * sigma^2/(2r) * [...]
- Very expensive: the ability to buy at the low / sell at the high has significant value
- Typically 2-3x the price of an ATM vanilla option
- Discrete monitoring: use MC with Brownian bridge correction for continuous max/min estimation
Digital (Binary) Options
Types
- Cash-or-nothing call: pays fixed amount Q if S_T > K, else 0
- Asset-or-nothing call: pays S_T if S_T > K, else 0
- Standard call = asset-or-nothing call - K * cash-or-nothing call (decomposition)
Pricing under BSM
- Cash-or-nothing call: Q * exp(-rT) * N(d2)
- Asset-or-nothing call: S * exp(-qT) * N(d1)
- Delta of digital: Q * exp(-rT) * phi(d2) / (S * sigma * sqrt(T)) — can be very large near ATM at expiry
Hedging Challenges
- Digital options have discontinuous payoff — delta approaches infinity near ATM at expiry
- In practice: replicate with tight call spread (buy K call, sell K+epsilon call, scale by Q/epsilon)
- The call spread replication has bounded delta and gamma
- Skew sensitivity: digital price is very sensitive to the vol skew (slope of IV around the strike)
- Overhedge: use call spread width that accounts for realistic hedging frequency
Cliquet (Ratchet) Options
Structure
- Series of forward-starting options, each resetting at the end of the previous period
- Each period: captures return = max(0, S_{t+1}/S_t - 1), typically with local cap and floor
- Total payoff: sum of capped/floored periodic returns
- Embedded in equity-linked insurance products and structured notes
Pricing Considerations
- Cliquet value depends critically on forward volatility and forward skew
- Cannot be hedged with vanilla options alone — requires forward-starting option hedges
- Very sensitive to vol-of-vol and correlation structure
- Stochastic volatility models (Heston, SABR) are essential for accurate pricing
- Local vol models can misprice cliquets because they underestimate forward smile dynamics
Autocallable Notes
Structure
- Periodic observation dates (e.g., quarterly, semi-annually)
- If underlying > autocall barrier (e.g., 100% of initial) on observation date: note is called, investor receives principal + coupon
- If not called and underlying > coupon barrier: investor receives coupon for that period
- At maturity, if underlying < put strike (e.g., 60% of initial): investor bears downside loss
- Most popular structured product globally (especially in Asia and Europe)
Risk Profile
- Short a deep OTM put (downside exposure)
- Short a series of digital options at the autocall barrier
- Correlation with dividends and rates (forward price matters for autocall probability)
- Path-dependent: early autocall eliminates remaining coupons and downside risk
- Issuer is typically net long vol and long correlation for worst-of autocallables
Quanto Options
Definition
- Option on a foreign asset with payoff converted at a fixed FX rate (not the prevailing rate)
- Example: USD investor buys a call on Nikkei 225 with payoff in USD at a fixed JPY/USD rate
- Eliminates FX risk from the investor's perspective
Pricing Adjustment
- Under BSM, quanto adjustment modifies the drift of the foreign asset:
- r_quanto = r_foreign - rho * sigma_S * sigma_FX
- rho = correlation between foreign asset and FX rate
- sigma_FX = volatility of the FX rate
- If equity and FX are negatively correlated (typical for developed markets: currency weakens when stocks fall), the quanto drift adjustment is positive — quanto option is worth more
- Volatility of the quanto option = sigma_S (no FX vol component)
Examples
Down-and-Out Call Pricing
S=100, K=100, Barrier B=85, r=5%, sigma=25%, T=0.5
Vanilla call (BSM): $7.62
Down-and-out call (closed-form): $5.84
Knock-out discount: $1.78 (23% cheaper due to barrier risk)
If S drops to 86 (1 point above barrier):
Vanilla call value: $2.10
D&O call value: $0.15 (nearly worthless — about to knock out)
Delta of D&O: -2.5 (negative — further drop kills the option)
Gamma: extremely large (discontinuity at barrier)
Arithmetic Asian Option
S=100, K=100, r=5%, sigma=30%, T=1.0, monthly averaging (12 dates)
Vanilla call: $14.23
Geometric Asian call (closed-form): $8.42
Arithmetic Asian call (MC with control variate): $8.91
Asian discount: 37% cheaper than vanilla.
Effective vol of average: approximately 30% / sqrt(3) = 17.3% (for geometric).
Worst-of Autocallable
Underlying: worst of SPX, EuroStoxx 50, Nikkei 225
Autocall barrier: 100%, observed quarterly
Coupon: 8% p.a. if worst-of > 70%
Put strike: 60% of initial (investor bears loss below 60%)
Maturity: 3 years
Pricing requires:
- 3-asset correlated MC simulation
- Calibrated local-stochastic vol model for each underlying
- Quanto adjustment for cross-currency underlyings
- Dividend assumptions for forward price computation
Key sensitivities:
- Correlation: lower correlation = lower autocall probability = higher coupon value but more downside risk
- Vol: higher vol = wider range of outcomes = more expensive embedded put
- Dividends: higher dividends = lower forward = less likely to autocall
Quality Gate
- Barrier options: verify with in-out parity (knock-in + knock-out = vanilla within tolerance)
- Barrier monitoring: specify continuous vs. discrete and apply appropriate corrections
- Asian options: arithmetic Asian price must be bounded by geometric Asian (lower) and vanilla (upper)
- Digital options: replicate with call/put spread in practice; never hedge a pure digital
- Cliquet pricing must use a stochastic vol model — local vol alone systematically misprices forward skew
- Autocallables: Monte Carlo must have sufficient paths (>100,000) and correct early termination logic
- Quanto: verify the correlation sign and magnitude between asset and FX are reasonable
- All exotic prices must be bounded by appropriate upper and lower limits (vanilla, intrinsic value)
- Greeks must be computed via bump-and-revalue with appropriately small bumps (especially near barriers)
- Model risk assessment: compare prices from at least two models (BSM, local vol, stochastic vol)
- Hedging strategy must be defined before trading: specify instruments, rebalancing frequency, Greeks targets