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Black-Scholes Greeks from First Principles

This example derives, deploys, and validates exact analytical Greeks for European options using the Black-Scholes model for options pricing.

The Black-Scholes PDE has an exact closed-form solution, which means every Greek (Delta, Gamma, Vega, Theta, Rho, Vanna, Volga, Charm, Speed) comes out as a symbolic partial derivative. That parametric form gives you the full sensitivity surface in one expression, versus a bump-and-revalue approach.

The same workflow applies to any pricing PDE with a known functional form: put-call parity Greeks, digital/barrier options via limits, near-ATM polynomial approximations via Taylor expansion, or multi-asset cross-Greeks via the Jacobian.

Example Outline

1. Geometric Brownian Motion and Ito's Lemma

Symbolic declaration of the Black-Scholes PDE from first principles: GBM dynamics, Ito's lemma on a twice-differentiable option value $V(S,t)$, and the no-arbitrage condition on a delta-hedged portfolio.

2. Analytic Solution Verification

States the known European call formula $C = S,N(d_1) - Ke^{-r\tau}N(d_2)$ and proves it satisfies the PDE — the residual simplifies to exactly zero.

  • diff — partial derivatives forming the PDE operator
  • simplify — verify residual is identically zero

3. Extract All Greeks as Exact Partial Derivatives

Each Greek is a symbolic partial derivative of the option price. First-order (Delta, Vega, Theta, Rho) and second-order (Gamma, Vanna, Volga, Charm, Speed) Greeks are computed in closed form.

  • diff — first and higher partial derivatives
  • simplify — canonical form for each Greek expression

4. Put-Call Parity: Put Greeks by Differentiation

Put-call parity $P = C - S + Ke^{-r\tau}$ is differentiated symbolically to yield all put Greeks without re-derivation. The parity identities (put Delta = call Delta − 1, put Gamma = call Gamma) are verified exactly.

  • diff, simplify — derive and verify parity relations

5. Near-ATM Taylor Expansions

Taylor expansion of Delta and Gamma around $S = K$ gives polynomial approximations for the moneyness regime where most hedging activity occurs. The Brenner-Subrahmanyam ATM call price approximation $C_{ATM} \approx K\sigma\sqrt{\tau}/\sqrt{2\pi}$ is derived by expanding in $\sigma$ under zero rates.

  • taylor — systematic expansion in moneyness and volatility
  • subs — substitute ATM condition before expanding

6. Barrier and Digital Option Limits

Digital (binary) call pricing as the limit of a call spread whose width goes to zero. The result $e^{-r\tau}N(d_2)$ is verified symbolically. Digital Greeks are computed exactly, which is critical near the barrier where finite differences are unreliable.

  • limit — exact evaluation of the call-spread limit
  • subs — construct shifted strikes symbolically

7. Multi-Asset Cross-Greeks via Jacobian

A two-asset portfolio's full cross-Greek matrix (4×4: sensitivities of Delta and Vega for each asset with respect to both spots and both vols) computed in one jacobian call.

  • jacobian — matrix of all partial derivatives simultaneously
  • assume — correlation bounds $-1 < \rho < 1$

8. Code Generation: Deploy to Pricing Engine

Convert symbolic Greek expressions into optimized MATLAB® functions via matlabFunction. Three function files are generated: first-order Greeks, second-order Greeks, and near-ATM Taylor approximations.

  • matlabFunction — generates bsGreeks.m, bsGreeksSecondOrder.m, bsGreeksATM.m

9. Numerical Evaluation: Greek Surface

Evaluate the exact Greeks across the full $(S, \tau)$ surface using realistic equity index parameters: $K = 4500$, $r = 4.5%$, $\sigma = 18%$. Surface plots of Delta, Gamma, Theta, and Vega.

10. Taylor Approximation Accuracy

Compare exact Delta against the near-ATM Taylor approximation across moneyness. The polynomial is extremely accurate within $\pm5%$ of ATM, where most hedging activity occurs.

11. Validate Against Financial Toolbox

All symbolic Greeks validated against blsprice, blsdelta, blsgamma, blsvega, blstheta, blsrho to confirm the derivation is correct end-to-end.

12. Monte Carlo with Exact Greeks

50,000-path GBM simulation with exact Greeks evaluated at every weekly rebalancing point. Delta-hedge P&L distribution shows that discrete rebalancing error (not Greek noise) is the dominant residual. Performance benchmark: exact evaluation vs. bump-and-revalue timing and accuracy.

  • randn — GBM path generation
  • Vectorized Greek evaluation along all paths — no inner bump loop

What Gets Generated

Running the example produces deployable MATLAB functions:

Generated file Signature Use case
bsGreeks.m [Price, Delta, Gamma, Vega, Theta, Rho] = bsGreeks(S, K, r, sigma, tau) First-order Greeks for pricing engines and risk aggregation
bsGreeksSecondOrder.m [Vanna, Volga, Charm, Speed] = bsGreeksSecondOrder(S, K, r, sigma, tau) Second-order Greeks for volatility risk and gamma scalping
bsGreeksATM.m [DeltaATM, GammaATM] = bsGreeksATM(S, K, r, sigma, tau) Near-ATM Taylor approximations for real-time dashboards

All functions accept vectorized inputs. C code equivalents are available via ccode() for integration with low-latency pricing engines.

Quick Start

Requirements

Running the Example

  1. Open MATLAB and navigate to this directory
  2. Open BlackScholesGreeks.m as a Live Script
  3. Run section by section, or run all — the example is self-contained

The generated functions (bsGreeks.m, bsGreeksSecondOrder.m, bsGreeksATM.m) are written to the working directory at runtime.

Files

File Description
BlackScholesGreeks.m Main example (Live Script). Full workflow from PDE derivation through Monte Carlo deployment.
bsGreeks.m Generated at runtime. First-order Greeks (Price, Delta, Gamma, Vega, Theta, Rho).
bsGreeksSecondOrder.m Generated at runtime. Second-order Greeks (Vanna, Volga, Charm, Speed).
bsGreeksATM.m Generated at runtime. Near-ATM Taylor approximations.