QuantModels — Binomial and Black-Scholes Option Pricing

A C++ project implementing both the binomial tree and Black-Scholes models for option pricing.

Overview

This repository provides implementations of two widely used option pricing models:

1. Binomial Tree Model: A discrete-time model for pricing European and American options.
2. Black-Scholes Model: A continuous-time model for pricing European options.

Features

Binomial Tree Model

• Implements a Depth-First Search (DFS) approach for traversing the binomial tree.
• Supports European and American options.
• Handles call and put options.
• Configurable for discrete or continuous interest rates.
• Outputs the binomial tree structure, option values, and delta values.

Black-Scholes Model

• Supports European options only.
• Handles call and put options.
• Outputs the option price and Greeks.

Quick Build

Requirements

• CMake
• C++17-compatible compiler

Build Instructions

# Create a build directory and navigate into it
mkdir -p build
cd build

# Generate build files using CMake
# Note: You need to write your own CMakeLists.txt file to configure the build process.
cmake ..

# Compile the project
make -j

Run

Provide a configuration file as the single command-line argument. An example configuration file is included at config/setting.cfg.

./financeModel config/setting.cfg

Runtime Flow

1. Configuration Loading: cfgManager::loadCommonInfo reads the configuration file and initializes the models.
2. Model Execution: cfgManager::process runs the binomial tree and Black-Scholes models.
3. Binomial Tree Logic: binomialTree::mainProcess calculates option values and deltas.
4. Black-Scholes Logic: blackScholesModel::mainProcess calculates the option price and delta.

Configuration

The configuration file specifies the following parameters:

• spot: Initial stock price
• strike: Strike price of the option
• rate: Risk-free interest rate
• sigma: Volatility of the underlying asset
• time: Time to maturity
• steps: Number of steps in the binomial tree (only for the binomial model)
• regionType: Option type (EU for European, US for American; only for the binomial model)
• interestType: Interest rate type (Discrete or Continuous; only for the binomial model)
• optionType: Option style (call or put)

Theory

Binomial Tree Model

The binomial tree model calculates option prices using the following steps:

1. Time Step:
   $$\Delta t = \frac{T}{N}$$

2. Up/Down Factors:
   $$u = 1 + \sigma \cdot \sqrt{\Delta t}, \quad d = 1 - \sigma \cdot \sqrt{\Delta t}$$

3. Risk-Neutral Probability:

   • Discrete: $$p = 0.5 + \frac{r \cdot \sqrt{\Delta t}}{2 \cdot \sigma}$$
   • Continuous: $$p = 0.5 + \frac{\exp(r \cdot \Delta t) - 1}{2 \cdot \sigma \cdot \sqrt{\Delta t}}$$

4. Discount Factor:

   • Discrete: $$df = \frac{1}{1 + r \cdot \Delta t}$$
   • Continuous: $$df = \exp(-r \cdot \Delta t)$$

5. Backward Induction:
   $$V = df \cdot (p \cdot V_{\text{up}} + (1 - p) \cdot V_{\text{down}})$$

   For American options, the immediate exercise value is compared with the continuation value:
   $$V = \max(\text{exercise value}, \text{continuation value})$$

Black-Scholes Model

The Black-Scholes formula calculates the price of European options as follows:

1. Key Terms:
   $$d_1 = \frac{\ln(S / K) + (r - q + \sigma^2 / 2) \cdot T}{\sigma \cdot \sqrt{T}}, \quad d_2 = d_1 - \sigma \cdot \sqrt{T}$$ where $q$ is the continuous dividend yield.

2. Call Option Price:
   $$C = S \cdot e^{-q \cdot T} \cdot \Phi(d_1) - K \cdot e^{-r \cdot T} \cdot \Phi(d_2)$$

3. Put Option Price:
   $$P = K \cdot e^{-r \cdot T} \cdot \Phi(-d_2) - S \cdot e^{-q \cdot T} \cdot \Phi(-d_1)$$

4. Greeks:

   • Delta:
     • Call: $\Delta = e^{-q \cdot T} \cdot \Phi(d_1)$;
     • Put: $\Delta = e^{-q \cdot T} \cdot (\Phi(d_1) - 1)$
   • Gamma: $\Gamma = \frac{e^{-q \cdot T} \cdot \phi(d_1)}{S \cdot \sigma \cdot \sqrt{T}}$
   • Theta:
     • Call: $\Theta = -S \cdot e^{-q \cdot T} \cdot \phi(d_1) \cdot \sigma / (2\sqrt{T}) - r \cdot K \cdot e^{-r \cdot T} \cdot \Phi(d_2) + q \cdot S \cdot e^{-q \cdot T} \cdot \Phi(d_1)$
     • Put: $\Theta = -S \cdot e^{-q \cdot T} \cdot \phi(d_1) \cdot \sigma / (2\sqrt{T}) + r \cdot K \cdot e^{-r \cdot T} \cdot \Phi(-d_2) - q \cdot S \cdot e^{-q \cdot T} \cdot \Phi(-d_1)$
   • Vega: $\nu = S \cdot e^{-q \cdot T} \cdot \phi(d_1) \cdot \sqrt{T}$
   • Rho:
     • Call: $\rho = K \cdot T \cdot e^{-r \cdot T} \cdot \Phi(d_2)$;
     • Put: $\rho = -K \cdot T \cdot e^{-r \cdot T} \cdot \Phi(-d_2)$

License

Check the headers of third-party libraries (e.g., rapidcsv/rapidcsv.h) for licensing details.