Principle:Avhz RustQuant Generalized Black Scholes Merton

Knowledge Sources	Black-Scholes 1973 Merton 1973 Black 1976 Heston 1993 Haug - The Complete Guide to Option Pricing Formulas
Domains	Derivatives, Option_Pricing, Mathematical_Finance
Last Updated	2026-02-07 20:00 GMT

Overview

A unified framework for pricing European options analytically by parameterizing spot, risk-free rate, and cost-of-carry across multiple Black-Scholes-Merton model variants.

Description

The Generalized Black-Scholes-Merton (GBSM) framework unifies several classic option pricing models by abstracting three accessor functions: s() (spot/forward price), r() (risk-free rate), and b() (cost-of-carry). By varying the cost-of-carry parameter, the same pricing formula covers:

Black-Scholes (1973): b = r (no dividends, equity options)
Merton (1973): b = r - q (continuous dividend yield)
Black (1976): b = 0 (futures options)
Asay (1982): b = 0, r = 0 (margined futures options)
Garman-Kohlhagen (1983): b = r_d - r_f (FX options)

Additionally, the framework accommodates models with different pricing mechanics:

Heston (1993): Stochastic volatility via characteristic functions
Bachelier (1900): Normal (arithmetic) Brownian motion pricing

Usage

Use this principle when you need to price European options analytically using closed-form solutions. Select the appropriate model variant based on the underlying asset type (equity, future, FX) and the dividend/carry assumptions.

Theoretical Basis

The generalized BSM call price formula is:

$C = S e^{(b - r) T} N (d_{1}) - K e^{- r T} N (d_{2})$

where:

$d_{1} = \frac{\ln (S / K) + (b + σ^{2} / 2) T}{σ \sqrt{T}}$

$d_{2} = d_{1} - σ \sqrt{T}$

The put price follows from put-call parity:

$P = K e^{- r T} N (- d_{2}) - S e^{(b - r) T} N (- d_{1})$

Model variants by cost-of-carry:

Model	b value	Use case
BlackScholes73	b = r	Equity options (no dividends)
Merton73	b = r - q	Equity options with continuous dividends
Black76	b = 0	Futures options
Asay82	b = 0, r = 0	Margined futures options
GarmanKohlhagen83	b = r_d - r_f	FX options

Related Pages

Implemented By

Implementation:Avhz_RustQuant_BSM_Model_Constructors

Page Connections

Double-click a node to navigate. Hold to expand connections.

Principle

Implementation

Heuristic

Environment