Black-Scholes Option Pricing Model in C++

This project provides a C++ implementation of the Black-Scholes model. Additionally, it computes the option Greeks.

Features

• Option Pricing (European call and put)
• Greeks Calculation:
  • Delta
  • Gamma
  • Theta
  • Vega
  • Rho

Prerequisites

C++

Compilation

g++ -std=c++11 -o option_pricing option_pricing.cpp  -lm

Usage

./option_pricing

Mathematical Formulas

Call Option Price (C):

$$C = S_0 \times N(d_1) - K \times e^{-r \times T} \times N(d_2)$$

Put Option Price (P):

$$P = K \times e^{-r \times T} \times N(-d_2) - S_0 \times N(-d_1)$$

Where:

• $S_0$: Current price of the underlying asset
• $K$: Strike price of the option
• $r$: Risk-free interest rate
• $T$: Time to expiration (in years)
• $N(\cdot)$: Cumulative distribution function of the standard normal distribution
• $d_1$ and $d_2$ are calculated as:

$$d_1 = \frac{\ln(\frac{S_0}{K}) + (r + \frac{\sigma^2}{2}) \times T}{\sigma \times \sqrt{T}}$$

$$d_2 = d_1 - \sigma \times \sqrt{T}$$

• $\sigma$: Volatility of the underlying asset's returns

Option Greeks:

• Delta (Δ): Measures the sensitivity of the option's price to changes in the price of the underlying asset.

  • Call Option: $\Delta_{\text{call}} = N(d_1)$
  • Put Option: $\Delta_{\text{put}} = N(d_1) - 1$

• Gamma (Γ): Measures the rate of change of Delta with respect to changes in the underlying asset's price. $$\Gamma = \frac{N'(d_1)}{S_0 \times \sigma \times \sqrt{T}}$$

• Theta (Θ): Measures the sensitivity of the option's price to the passage of time.

  • Call Option: $$\Theta_{\text{call}} = -\frac{S_0 \times N'(d_1) \times \sigma}{2 \times \sqrt{T}} - r \times K \times e^{-r \times T} \times N(d_2)$$
  • Put Option: $$\Theta_{\text{put}} = -\frac{S_0 \times N'(d_1) \times \sigma}{2 \times \sqrt{T}} + r \times K \times e^{-r \times T} \times N(-d_2)$$

• Vega (ν): Measures the sensitivity of the option's price to changes in the volatility of the underlying asset. $$\nu = S_0 \times N'(d_1) \times \sqrt{T}$$

• Rho (ρ): Measures the sensitivity of the option's price to changes in the risk-free interest rate.

  • Call Option: $$\rho_{\text{call}} = K \times T \times e^{-r \times T} \times N(d_2)$$
  • Put Option: $$\rho_{\text{put}} = -K \times T \times e^{-r \times T} \times N(-d_2)$$

References

• Black–Scholes model - Wikipedia
• European vanilla option pricing with C++ and analytic formulae
• Black-Scholes Model - GeeksforGeeks