Partial Derivatives and Total Differentials

Motivation: why this matters in quant finance

An option price is not a function of one variable — it depends on the spot

S

, time

t

, volatility

\sigma

, interest rate

r

, and strike

K

simultaneously. The Greeks are nothing more than the partial derivatives of this multivariate function:

\Delta = \partial v / \partial S

\Theta = \partial v / \partial t

\mathcal{V} = \partial v / \partial \sigma

\rho = \partial v / \partial r

. You cannot define, compute, or interpret the Greeks without understanding partial differentiation.

More critically, the distinction between a partial derivative and a total derivative is exactly what makes the Black-Scholes derivation work. The partial derivative

\partial v / \partial t

measures time decay with

S

held fixed. The total derivative

dv/dt

includes the effect of

S

changing with

t

. The total differential

dv = v_t\,dt + v_S\,dS + \frac{1}{2}v_{SS}(dS)^2

is the Itô expansion — and it is the total differential, not any single partial derivative, that describes how the option value actually evolves.

Definition

Partial derivative

Let

f(x_1, x_2, \ldots, x_n)

be a function of

n

variables. The partial derivative of

f

with respect to

x_i

is the ordinary derivative taken with all other variables held constant:

\frac{\partial f}{\partial x_i} = \lim_{h \to 0} \frac{f(x_1, \ldots, x_i + h, \ldots, x_n) - f(x_1, \ldots, x_i, \ldots, x_n)}{h}

Notation: $\partial f / \partial x_i = f_{x_i} = \partial_i f = D_i f$ . The $\partial$ symbol (as opposed to $d$ ) signals that other variables are held fixed.

Example: Let

v(S, t) = S\,\Phi(d_1) - Ke^{-r(T-t)}\Phi(d_2)

(the Black-Scholes call price). Then:

\Delta = \frac{\partial v}{\partial S} = \Phi(d_1) + S\phi(d_1)\frac{\partial d_1}{\partial S} - Ke^{-r(T-t)}\phi(d_2)\frac{\partial d_2}{\partial S}

After cancellation (using $S\phi(d_1) = Ke^{-r(T-t)}\phi(d_2)$ and $\partial d_1/\partial S = \partial d_2 / \partial S$ ), this simplifies to $\Delta = \Phi(d_1)$ .

Higher-order partial derivatives

Second-order partial derivatives:

f_{xx} = \frac{\partial^2 f}{\partial x^2}, \qquad f_{xy} = \frac{\partial^2 f}{\partial y\,\partial x}, \qquad f_{yy} = \frac{\partial^2 f}{\partial y^2}

Clairaut's theorem (symmetry of mixed partials): If

f_{xy}

and

f_{yx}

are both continuous, then

f_{xy} = f_{yx}

— the order of differentiation does not matter. This holds for all standard pricing functions in quant finance.

In the Greeks language:

Second-order Greek	Symbol	Definition	Interpretation
Gamma	$\Gamma$	$v_{SS}$	Curvature w.r.t. spot; hedging cost
Charm		$v_{St} = v_{tS}$	Rate of change of delta w.r.t. time
Vanna		$v_{S\sigma}$	Sensitivity of delta to vol
Volga / Vomma		$v_{\sigma\sigma}$	Curvature w.r.t. vol
Speed		$v_{SSS}$	Third-order: rate of change of gamma

All of these are partial derivatives of increasing order, and Clairaut's theorem guarantees that $v_{St} = v_{tS}$ , $v_{S\sigma} = v_{\sigma S}$ , etc.

The total differential

The total differential of

f(x_1, \ldots, x_n)

is:

df = \sum_{i=1}^{n} \frac{\partial f}{\partial x_i}\,dx_i = f_{x_1}\,dx_1 + f_{x_2}\,dx_2 + \cdots + f_{x_n}\,dx_n

This is the first-order approximation to the change in

f

when all variables change simultaneously. It is the multivariable chain rule written in differential notation.

For $v(S, t)$ :

dv = \frac{\partial v}{\partial S}\,dS + \frac{\partial v}{\partial t}\,dt = \Delta\,dS + \Theta\,dt

This is the deterministic total differential. Itô's Lemma adds the second-order correction:

dv = \Delta\,dS + \Theta\,dt + \frac{1}{2}\Gamma\,\sigma^2 S^2\,dt

The total differential is the object that the Black-Scholes derivation manipulates: it describes how $v$ changes instant by instant, combining contributions from all variables.

Total derivative vs partial derivative

When

v = v(S, t)

and

S = S(t)

is itself a function of

t

, the total derivative of

v

with respect to

t

is:

\frac{dv}{dt} = \frac{\partial v}{\partial t} + \frac{\partial v}{\partial S}\frac{dS}{dt}

The partial derivative $\partial v / \partial t = \Theta$ isolates the direct time dependence (with $S$ held fixed). The total derivative $dv/dt$ includes the indirect effect through $S$ changing. In the Black-Scholes world, the P&L of an option position is determined by the total differential, not by any single Greek.

The gradient and the Jacobian

Gradient

The gradient of

f(x_1, \ldots, x_n)

is the vector of all first-order partial derivatives:

\nabla f = \left(\frac{\partial f}{\partial x_1}, \frac{\partial f}{\partial x_2}, \ldots, \frac{\partial f}{\partial x_n}\right)

In quant finance, the gradient of the pricing function is the vector of all first-order Greeks: $\nabla v = (\Delta, \Theta, \mathcal{V}, \rho, \ldots)$ . The total differential can be written compactly as:

dv = \nabla v \cdot d\mathbf{x}

where $d\mathbf{x} = (dS, dt, d\sigma, dr, \ldots)$ is the vector of input changes.

Jacobian

For a vector-valued function

\mathbf{F}: \mathbb{R}^n \to \mathbb{R}^m

, the Jacobian matrix is:

J = \begin{pmatrix} \partial F_1/\partial x_1 & \cdots & \partial F_1/\partial x_n \\ \vdots & \ddots & \vdots \\ \partial F_m/\partial x_1 & \cdots & \partial F_m/\partial x_n \end{pmatrix}

In quant finance, if you have a portfolio of $m$ options depending on $n$ underlying risk factors, the Jacobian is the matrix of all first-order sensitivities — the "Greeks matrix." This is the object used in multi-factor risk decomposition and in computing portfolio-level VaR via the delta-normal method.

The Hessian

The Hessian matrix of

f(x_1, \ldots, x_n)

is the matrix of all second-order partial derivatives:

H = \begin{pmatrix} f_{x_1 x_1} & f_{x_1 x_2} & \cdots \\ f_{x_2 x_1} & f_{x_2 x_2} & \cdots \\ \vdots & & \ddots \end{pmatrix}

By Clairaut's theorem,

H

is symmetric. For the option pricing function, the Hessian contains gamma, charm, vanna, volga, and all other second-order cross-Greeks. The second-order Taylor expansion is:

\Delta v \approx \nabla v \cdot \Delta\mathbf{x} + \frac{1}{2}\Delta\mathbf{x}^T H\,\Delta\mathbf{x}

This is the multi-factor version of the delta-gamma approximation. The quadratic form $\frac{1}{2}\Delta\mathbf{x}^T H\,\Delta\mathbf{x}$ captures all second-order risk — not just gamma, but all cross-Greeks.

Examples and applications

Example 1: the theta-delta-gamma identity

For a European call in the Black-Scholes model, the option value satisfies the Black-Scholes PDE:

\Theta + rS\Delta + \frac{1}{2}\sigma^2 S^2 \Gamma = rv

This is a relationship between the partial derivatives $v_t$ , $v_S$ , $v_{SS}$ and the function value $v$ . It shows that the Greeks are not independent — if you know any two, the PDE constrains the third (given the model). Traders use this identity to sanity-check Greek calculations and to understand the theta-gamma trade-off: in a delta-hedged portfolio ( $\Delta = 0$ ), $\Theta \approx -\frac{1}{2}\sigma^2 S^2 \Gamma + rv$ , so theta and gamma have opposing signs.

Example 2: multi-factor risk decomposition

Suppose a portfolio $P$ depends on three risk factors: equity level $S$ , interest rate $r$ , and volatility $\sigma$ . The first-order P&L approximation is:

\Delta P \approx \frac{\partial P}{\partial S}\,\Delta S + \frac{\partial P}{\partial r}\,\Delta r + \frac{\partial P}{\partial \sigma}\,\Delta\sigma = \Delta_S\,\Delta S + \rho\,\Delta r + \mathcal{V}\,\Delta\sigma

This is the total differential applied to the portfolio. Risk managers use this decomposition to attribute P&L to individual risk factors and to determine which hedges are needed.

Example 3: computing vega from the Black-Scholes formula

The Black-Scholes call price $C = S\Phi(d_1) - Ke^{-rT}\Phi(d_2)$ depends on $\sigma$ through $d_1$ and $d_2$ . Vega is:

\mathcal{V} = \frac{\partial C}{\partial \sigma} = S\phi(d_1)\frac{\partial d_1}{\partial \sigma} - Ke^{-rT}\phi(d_2)\frac{\partial d_2}{\partial \sigma}

Using $S\phi(d_1) = Ke^{-rT}\phi(d_2)$ and $\partial d_1/\partial \sigma - \partial d_2/\partial \sigma = \sqrt{T}$ ... wait, the calculation is cleaner: since $d_2 = d_1 - \sigma\sqrt{T}$ , $\partial d_2/\partial \sigma = \partial d_1/\partial \sigma - \sqrt{T}$ . The $\partial d_1/\partial \sigma$ terms cancel, leaving:

\mathcal{V} = Ke^{-rT}\phi(d_2)\sqrt{T} = S\phi(d_1)\sqrt{T}

Vega is always positive (higher vol means higher option value for vanilla options), proportional to $\sqrt{T}$ (longer-dated options are more sensitive to vol), and peaks near the money (where $\phi(d_1)$ is largest). All derived from partial differentiation.

Common confusions and pitfalls

Confusing $\partial v / \partial t$ with the "total" time derivative. The partial

\partial v / \partial t = \Theta

measures time decay with the stock price frozen. The actual change in option value over a time step also includes the stock's movement (

\Delta\,dS

) and the Itô correction (

\frac{1}{2}\Gamma(dS)^2

). Theta alone does not determine P&L — the total differential does.

Forgetting cross-partials exist. First-order Greeks (

\Delta

\Theta

\mathcal{V}

\rho

) are not the whole story. Cross-partials like vanna (

v_{S\sigma}

) and charm (

v_{St}

) capture how one Greek changes with respect to another variable. For large portfolios or exotic options, these cross-terms can dominate risk.

Assuming partial derivatives commute for any function. Clairaut's theorem requires continuity of the mixed partials. For functions with kinks or discontinuities (e.g., payoff functions at expiration), mixed partials may not exist or may not commute at isolated points.

Where this goes next

Partial derivatives are the language of the Greeks. The total differential is the language of Itô's Lemma. Together, they feed directly into:

Itô's Lemma: The stochastic total differential $dv = v_t\,dt + v_S\,dS + \frac{1}{2}v_{SS}(dS)^2$ .
The Black-Scholes PDE: A second-order PDE in the partial derivatives $v_t$ , $v_S$ , $v_{SS}$ .
Implicit Differentiation: When you cannot solve for a variable explicitly (e.g., implied volatility), implicit differentiation uses partial derivatives to compute sensitivities indirectly.
Taylor Series: The multivariable Taylor expansion $\Delta v \approx \nabla v \cdot \Delta\mathbf{x} + \frac{1}{2}\Delta\mathbf{x}^T H\,\Delta\mathbf{x}$ is the Greeks expansion carried to second order.

References

Stewart, J. (2008). Single Variable Calculus: Early Transcendentals (6th ed.). Thomson Brooks/Cole. Ch. 2.8 and Ch. 11.11 for derivative-as-function and higher-order approximation background; multivariable notation follows the vault calculus sequence.