Black–Scholes Pricing Methods

A small quantitative finance project comparing several numerical approaches to pricing European options under the Black–Scholes model.

The project implements and compares three fundamental approaches to derivative pricing:

• Analytical solution (Black–Scholes closed form)
• Monte Carlo simulation of the underlying stochastic process
• Finite-difference PDE solvers (explicit scheme and Crank–Nicolson)

The goal is to illustrate how different mathematical formulations of the same pricing problem lead to consistent results while exhibiting different numerical properties.

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Mathematical Background

Under the risk-neutral measure, the stock price follows geometric Brownian motion:

$$ dS_t = r S_t , dt + \sigma S_t , dW_t $$

where

• $r$ is the risk-free interest rate
• $\sigma$ is volatility
• $W_t$ is standard Brownian motion

Under these dynamics, the price of a European derivative with payoff $f(S_T)$ is given by the risk-neutral pricing formula:

$$ V_0 = e^{-rT}\mathbb{E}[f(S_T)] $$

For a European call option:

$$ f(S_T) = \max(S_T - K, 0) $$

This expectation can be computed in several ways.

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1. Analytical Solution

Black and Scholes derived a closed-form solution:

$$ C = S_0 N(d_1) - K e^{-rT} N(d_2) $$

where

$$ d_1 = \frac{\ln(S_0/K) + (r + \tfrac12\sigma^2)T}{\sigma\sqrt{T}}, \quad d_2 = d_1 - \sigma\sqrt{T} $$

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2. Monte Carlo Simulation

Using the exact solution of the GBM SDE:

$$ S_T = S_0 \exp\left((r - \tfrac12\sigma^2)T + \sigma \sqrt{T} Z\right) $$

with $Z \sim N(0,1)$.

Simulating many such paths allows the expectation to be approximated numerically.

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3. PDE Formulation

The option price also satisfies the Black–Scholes PDE:

$$ \frac{\partial V}{\partial t} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + r S \frac{\partial V}{\partial S} - rV = 0 $$

with terminal condition:

$$ V(S,T) = \max(S-K,0) $$

This PDE can be solved numerically using finite-difference methods.

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Implemented Methods

Method	Description
Analytical	Closed-form Black–Scholes formula
Monte Carlo	Risk-neutral simulation of GBM paths
Explicit finite difference	Direct discretization of the Black–Scholes PDE
Crank–Nicolson	Stable semi-implicit finite-difference scheme

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Example Results

For the following parameters:

• $S_0 = 100$
• $K = 100$
• $T = 1$
• $r = 0.05$
• $\sigma = 0.20$

The computed option prices are:

Method	Price
Analytical	10.4506
Monte Carlo	10.4876 (sampling noise)
Crank–Nicolson	10.4481
Explicit FD	10.4482

This illustrates that:

• Monte Carlo converges slowly but is flexible
• Crank–Nicolson provides stable and accurate PDE solutions
• Explicit finite differences are conditionally stable

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Runtime Comparison

Approximate runtimes for the above configuration:

Method	Runtime
Analytical	0.000007 s
Monte Carlo	0.010647 s
Crank–Nicolson	0.084510 s
Explicit FD	0.738353 s

This highlights the tradeoffs between pricing methods:

• Analytical solutions are fastest when available.
• Monte Carlo is flexible but converges slowly.
• PDE solvers are deterministic and accurate, though computationally heavier.

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Monte Carlo Convergence

Monte Carlo pricing error decreases with the expected rate

$$ \text{error} \propto N^{-1/2} $$

where $N$ is the number of simulated paths.

Example convergence plot:

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Simulated GBM Paths

Example sample paths under the risk-neutral measure:

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Price vs Strike Comparison

Pricing methods remain consistent across a range of strike prices.

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Running the Experiments

Install the project in editable mode:

pip install -e .

Generate all figures:

Generate all figures:

python experiments/generate_all.py

Project Structure

black_scholes_methods/
    analytic_black_scholes.py
    monte_carlo_pricing.py
    gbm_simulation.py
    finite_difference_pricing.py
    crank_nicolson_pricing.py

experiments/
    convergence_tests.py
    runtime_comparison.py
    strike_sweep.py
    gbm_paths.py

Key Insights

This project highlights the relationship between three perspectives on derivative pricing:

Stochastic calculus  → Monte Carlo simulation
PDE methods          → finite difference solvers
Closed-form analysis → Black–Scholes formula

These approaches are mathematically connected through the Feynman–Kac theorem, which links stochastic expectations to solutions of parabolic PDEs.

Possible Extensions

Future work could include:

• American option pricing
• stochastic volatility models (Heston)
• implied volatility surface construction
• variance reduction techniques for Monte Carlo