SABR Caplet Pricing and Calibration

This project implements and calibrates a SABR model for pricing caplets under the risk-neutral forward measure.
It was developed as part of the ERDOS Institute Fall 2025 Quant Finance Bootcamp.

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Overview

The project:

• Simulates SABR forward and volatility paths via Monte Carlo,
• Benchmarks against the Black-76 and Hagan (2002) analytic approximations,
• Generates synthetic “market” data from a Heston model, and
• Calibrates SABR parameters $(\alpha, \nu, \rho, \beta)$ to fit the observed (or simulated) volatility smile.

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Methodology

1. SABR Simulation: stochastic volatility model for the forward rate
   (df_t = \sigma_t f_t^{\beta} dW_t,\quad d\sigma_t = \nu \sigma_t dZ_t,; \text{Corr}(dW_t,dZ_t)=\rho)

2. Monte Carlo Pricing: simulate forward paths, compute discounted caplet payoffs.

3. Black-76 Benchmark: invert prices to implied volatilities.

4. Hagan (2002) Approximation: analytic link between SABR parameters and Black-style implied vol.

5. Calibration: minimize RMSE between model and “market” implied vols.