Exotic Options Pricing

Monte Carlo pricing for path-dependent options that have no simple closed-form solution (or where the path matters, not just the terminal price).

Functions

Function	Description
barrier_option(S0, K, H, r, sigma, T, ...)	Down/up knock-in/knock-out options
asian_option(S0, K, r, sigma, T, ...)	Arithmetic or geometric average price
lookback_option(S0, r, sigma, T, ...)	Floating or fixed strike lookback

Option Types

Barrier Options

Price path must cross (or stay away from) a barrier level H. - Down-and-Out: Expires worthless if S drops below H - Down-and-In: Only activates if S drops below H - Up-and-Out / Up-and-In: Symmetric for upside barriers - Key identity: Down-Out + Down-In = Vanilla (in/out parity)

Asian Options

Payoff based on average price, reducing volatility exposure and manipulation risk. - Arithmetic average: max(avg(S) - K, 0) — no closed form - Geometric average: Has a closed-form approximation (Kemna-Vorst)

Lookback Options

Benefit from hindsight — payoff uses the best price during the option's life. - Floating strike call: S_T - min(S) — "buy at the lowest price" - Fixed strike call: max(max(S) - K, 0) — "use the highest price"

Example

from exotic_options import barrier_option, asian_option

# Down-and-out call: knocked out if price drops below 90
price = barrier_option(S0=100, K=100, H=90, r=0.05, sigma=0.20, T=1.0,
                       barrier_type="down-out")

# Asian call (arithmetic average)
asian_price = asian_option(S0=100, K=100, r=0.05, sigma=0.20, T=1.0)