Verifying the Black-Scholes PDE with Call Option Greeks

Summary

The document shows how to check that a Black-Scholes call price satisfies the Black-Scholes partial differential equation by substituting expressions for delta, gamma, and theta. The PDE relates the option value to its time sensitivity, underlying-price sensitivity, curvature, and the risk-free rate. In the displayed algebra, the terms involving gamma and theta cancel, leaving the expression stated to match the left-hand side, the risk-free rate multiplied by the call value.

This is a direct substitution check rather than a derivation of the Greeks from the call pricing formula. The result depends on using consistent definitions for time to expiry, discounting, and the normal distribution terms. The document gives no broader discussion of assumptions, such as the standard Black-Scholes market model, and its displayed notation should be checked carefully before reuse.

Key ideas

• The Black-Scholes PDE can be checked by substituting the call’s delta, gamma, and theta.
• The gamma contribution cancels the volatility-related term in theta in the displayed calculation.
• The remaining terms are equated with the risk-free rate multiplied by the call value.
• The note checks the identity but does not derive the Greeks or review the model assumptions.

Tags

# Can one use the Greeks (delta,gamma,theta) to show that the Black-Scholes call formula satisfies the Black-Scholes PDE?


# Can one use the Greeks (delta,gamma,theta) to show that the Black-Scholes call formula satisfies the Black-Scholes PDE?












If so, is there a derivation anywhere that shows this? I was told that this could be done in a class but I don't see how it's possible.

## Answer by ajc3 (score 2, accepted)

https://quant.stackexchange.com/a/11120

It's quite straightforward when you plug in the formulae for the greeks into the PDE.

Preliminaries:

$\Delta =\frac{\partial c_t}{\partial S_t}=\Phi(d_1)$

$\Gamma=\frac{\partial^2 c_t}{\partial S_t^2}=\frac{\phi(d1)}{S_t\sigma\sqrt{u}}$

$\Theta=\frac{\partial c_t}{\partial t}=-rKe^{ru}\Phi(d_2)-S_t\phi(d_1)\frac{\sigma}{2\sqrt{u}}$

The Black Scholes PDE:

\begin{eqnarray} rc_t&=&\Theta+rS_t\Delta + \frac{1}{2}S_t^2\sigma^2\Gamma\\ RHS&=&-rKe^{ru}\Phi(d_2)-S_t\phi(d_1)\frac{\sigma}{2\sqrt{u}}+rS_t\Phi(d_1)+\frac{1}{2}S_t^2\sigma^2 \frac{\phi(d1)}{S_t\sigma\sqrt{u}}\\ &=&-rKe^{ru}\Phi(d_2)-\frac{S_t\phi(d_1)\sigma}{2\sqrt{u}}+rS_t\Phi(d_1)+ \frac{S_t\sigma\phi(d1)}{2\sqrt{u}}\\ &=&rS_t\Phi(d_1)-rKe^{ru}\Phi(d_2)\\ &=&rc_t\\ &=&LHS \end{eqnarray}

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.