The project explores derivative pricing using two core numerical methods:
- Monte Carlo Simulation (MC)
- Finite Difference Methods (PDE Solvers)
The goal is to compare the efficiency, accuracy, and usability of the approaches in European options pricing under the Black–Scholes model.
- Perform Monte Carlo simulations for European option pricing.
- Numerically solve the Black–Scholes PDE with finite difference methods.
- Compare two approaches based on accuracy, efficiency, and convergence.
- Present a modular and extensible computational finance codebase.
The derivative price ( V(S, t) ) of a derivative under the Black–Scholes PDE is described by:
[ \frac{\partial V}{\partial t} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + r S \frac{\partial V}{\partial S} - rV = 0, ]
where:
- ( S ) = underlying asset price
- ( t ) = time
- ( r ) = risk-free interest rate
- ( \sigma ) = volatility
For a European call option with maturity ( T ) and strike ( K ):
[ V(S, T) = \max(S - K, 0). ]
We model the underlying asset dynamics by using Geometric Brownian Motion (GBM):
[ dS_t = r S_t dt + \sigma S_t dW_t, ]
which has the solution: [ S_T = S_0 \exp\left[\left(r - \tfrac{1}{2}\sigma^2\right)T + \sigma W_T\right], ] where '''math ( W_T \sim \mathcal{N}(0, T) ).
The Monte Carlo price of a European call is:
[ C_0 = e^{-rT} \cdot \mathbb{E}[\max(S_T - K, 0)]. ]
We discretize the Black–Scholes PDE on a spatial and temporal grid.
- Explicit Scheme: Forward in time, central in space (conditionally stable).
- Implicit Scheme: Backward in time, unconditionally stable.
- Crank–Nicolson Scheme: Averages explicit and implicit, second-order accurate.
The basic update formula for grid point ( (i, j) ) is based on the coefficients of the PDE:
[ V^{n+1}i = a_i V^n{i-1} + b_i V^n_i + c_i V^n_{i+1}, ]
with coefficients depending on ( r, \sigma, S, \Delta t, \Delta S ).
mc_simulation.py→ Monte Carlo pricing engine.pde_solver.py→ Finite difference solvers (Explicit, Implicit, Crank–Nicolson).utils.py→ Common functions (payoffs, error metrics, plotting).main.py→ Entry point for running experiments.
We tested with parameters:
-
( S_0 = 100, K = 100, r = 0.05, \sigma = 0.2, T = 1 ).
-
Monte Carlo Price (1M paths): ≈
10.45 -
PDE (Crank–Nicolson): ≈
10.43 -
Black–Scholes Closed Form: ≈
10.45
This shows excellent agreement between methods.
- Created a general framework for option pricing.
- Highlighted significant differences between stochastic simulation and deterministic PDE methods.
- Verified correctness against closed-form Black–Scholes solution.