feat: Implement American option pricing model

[joaquinbejar]

Description

Implement pricing support for American options in the Black-Scholes pricing module. American options can be exercised at any time before expiration, which requires different pricing methods than European options.

Current State

• American options return PricingError::UnsupportedOptionType
• No pricing model implemented for early exercise

Target State

• Fully functional American option pricing
• Support for both calls and puts
• Accurate early exercise premium calculation

Tasks

• Research and select appropriate pricing method (Binomial tree, Barone-Adesi-Whaley, or Bjerksund-Stensland)
• Implement the chosen pricing algorithm
• Handle early exercise boundary conditions
• Add dividend handling for American options
• Implement Greeks calculations for American options
• Add comprehensive tests with known values
• Add documentation with examples

Technical Notes

Recommended approaches:

• Binomial Tree (Cox-Ross-Rubinstein): Most accurate
• Barone-Adesi and Whaley: Good balance of speed and accuracy
• Bjerksund-Stensland: Fast analytical approximation

Estimated Effort

High (8-12 hours)

Related Issues

• Issue fix: Resolve 14 TODO/FIXME items in black_scholes_model.rs #214: Resolve TODOs in black_scholes_model.rs (completed)

[jakeheke75]

Hello Joaquin,

I tried to dig a bit into this interesting topic. I think that the optimal choice will mostly depend on the specific use case.
If the ultimate goal is to be extremely fast, then probably the way to go is for an analytical approximation solution like BAW or BS.
If instead it's more important to be right rather than fast, then there are a couple of popular choices.

• Binomial Tree approach with Cox-Ross-Rubinstein discrete time approximation. It seems to be a very common choice and the best compromise for accuracy, flexibility and performance.
• Finite Difference Methods approach (Crank-Nicolson method). It seems to be the most accurate and mathematically rigorous, but less flexible and more computational-heavy.

What do you think?

Thanks,
Gianmario