Capital Analytics

Mathematical Principles & Theorems

Quantitative mathematical finance formulas: Compound interest \(A = P(1 + r/n)^{nt}\) and continuous compounding \(A = P e^{rt}\); Annuity amortization periodic payment \(M = P \frac{r(1+r)^N}{(1+r)^N - 1}\); Black-Scholes European option pricing model \(C = S_0 N(d_1) - K e^{-rT} N(d_2)\) where \(d_1 = \frac{\ln(S_0/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}\) and \(d_2 = d_1 - \sigma\sqrt{T}\); Option Greeks: Delta \(\Delta = N(d_1)\), Gamma \(\Gamma = \frac{N'(d_1)}{S_0 \sigma \sqrt{T}}\).

Operating Instructions

• Select financial tool: Compound Interest, Loan Amortization, or Black-Scholes Option Pricing.
• Enter principal \(P\), annual interest rate \(r\), term length, strike price \(K\), and volatility \(\sigma\).
• Click Calculate to generate financial schedules, amortization breakdowns, and risk curves.
• Inspect payment composition charts and option pricing sensitivity surfaces.