Black–Scholes Pricing of European Calls and Puts in Python
Summary
This article explains how to implement closed-form Black–Scholes prices for European vanilla calls and puts in Python. It defines the inputs—spot price, strike, risk-free rate, volatility, and time to maturity—and describes the standard normal cumulative distribution and density functions needed by the formulas. Put pricing is also related to call pricing through put–call parity. The implementation is organized into statistical helper functions and option-pricing functions, with an approximation used for the normal cumulative distribution.
The article emphasizes clarity before optimization and assumes readers already know basic statistics and the Black–Scholes derivation. It frames the implementation as a starting point for later finite-difference and Monte Carlo work on more complex derivatives. Validation against known price bounds and put–call parity is identified as a necessary next step, but is deferred to a later article. The model uses constant volatility and the European exercise convention, so its formulas do not directly cover early exercise or changing volatility.
Key ideas
- Black–Scholes gives closed-form prices for European vanilla calls and puts under its model assumptions.
- The call formula uses the standard normal cumulative distribution, and the put price can be related to the call through put–call parity.
- The implementation separates normal distribution calculations from option pricing calculations.
- A normal cumulative distribution approximation makes the formula practical to implement numerically.
- Pricing code should be checked against known bounds and put–call parity before use.
Tags
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.