Exotic Options

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: ST - Smin — buy at the minimum, sell at current
• Floating-strike lookback put: Smax - ST — 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: ST N(a1) - Smin 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 ST if ST > 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}/St - 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 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

## Source & license

This open-source skill is cataloged on AgentStack and links to its original source — we do not rehost the code.

- **Author:** [brainbytes-dev](https://github.com/brainbytes-dev)
- **Source:** [brainbytes-dev/everything-claude-trading](https://github.com/brainbytes-dev/everything-claude-trading)
- **License:** MIT

Install and usage instructions live in the source repository linked above.