Computational Finance

Derivative pricing implemented from the mathematics in Python — an analytical Black-Scholes model with the full Greeks and a delta-hedging simulation, and a binomial tree pricer checked for convergence against the closed-form solution.

Most models in quantitative finance have no tractable closed form, so the practical question is always how a numerical method behaves: whether it converges, how fast, and to what. These two notebooks are a paired answer — one model that has an exact solution, and one numerical method measured against it.

Black-Scholes with Greeks

Black-Scholes Option Pricing Model.ipynb implements the closed-form European option price and all five first- and second-order sensitivities, derived directly rather than differenced numerically:

Greek	Measures
Delta	sensitivity to the underlying price
Gamma	rate of change of delta — the convexity that makes hedging non-trivial
Theta	time decay
Rho	sensitivity to the risk-free rate
Vega	sensitivity to volatility

Gamma is the one that matters operationally: because delta itself moves as the underlying moves, a delta-hedged position stops being hedged the moment the price does anything. The notebook includes a hedging simulation that rebalances over the life of the option to show that drift in practice.

Worked example — S₀ = 100, K = 110, r = 0.04, σ = 0.30, T = 1:

	Value	Delta
Call	9.6254	0.4863
Put	15.3122	−0.5137

The two deltas summing to exactly −1 is put-call parity holding, which is the cheapest available sanity check that the implementation is right.

Binomial tree, and whether it converges

Binomial Tree Option Pricing Model.ipynb builds the Cox-Ross-Rubinstein lattice: model the underlying as discrete up/down moves, then work backwards from expiry through risk-neutral valuation.

The reason to implement it when a closed form exists is that the lattice generalises where Black-Scholes does not — early exercise, American options, path dependence. So the useful measurement isn't the price, it's the convergence: how many steps until the tree agrees with the analytical result, and whether it approaches smoothly or oscillates. The notebook plots tree price against step count with the Black-Scholes value as a horizontal reference.

Worked example — S₀ = 100, K = 99, r = 0.06, σ = 0.20, T = 1, N = 50 steps:

	Value
Call	11.5464
Put	4.7811

Run it

pip install -r requirements.txt
jupyter notebook

Both notebooks are self-contained — no data files, no external feeds. All parameters are set in the first cell of each so they can be varied directly.

Stack

Python, NumPy, SciPy (scipy.stats for the normal CDF and PDF), pandas, matplotlib, Jupyter.