Strike Differentiation of Black–Scholes Implied Volatility

Summary

The document explains how to differentiate an option price represented by the Black–Scholes formula using its strike-dependent implied volatility. Since the observed call price is written as a function of strike and implied volatility, differentiating with respect to strike requires the multivariable chain rule. The resulting derivative includes both the formula’s direct sensitivity to strike and its sensitivity to volatility multiplied by the change in implied volatility across strikes.

The response clarifies the calculus by reducing the setup to a generic function whose two inputs depend on the variable being differentiated. It advises treating time, maturity, spot price, and interest rate as fixed while focusing on strike. This resolves the conceptual issue but does not derive the broader option-distribution method from the cited paper, discuss differentiability conditions, or provide a numerical example. The explanation assumes the implied volatility can be viewed as a differentiable function of strike in the region considered.

Key ideas

• An option price expressed through implied volatility depends on strike both directly and through volatility.
• Differentiation with respect to strike therefore applies the multivariable chain rule.
• The total derivative contains a direct strike term and an implied volatility term.
• Other model inputs are held fixed when taking the strike derivative.
• The explanation assumes implied volatility varies differentiably with strike.

Tags

# Black-Scholes Implied Volatility


# Black-Scholes Implied Volatility












I'm working my way through the following paper:

Malz. A. M. (2014). A Simple and Reliable Way to Compute Option-Based Risk-Neutral Distributions

I am completely stuck on the following derivation. The author expresses the price of a call option at time-$t$ (with strike $K$ and on underlier $S_t$) as \begin{equation} c(t; K, T) = v[S_t , K, T, σ_\text{imp}(t, K), r]. \end{equation}

where $v(.)$ denotes the Black-scholes pricing formula for a European call, and $\sigma_\text{imp}(t, K)$ is the B-S implied vol.

I understand this, but the following step is not clear to me. The author differentiates both sides of the above equation with respect to the strike, $K$. This gives:

\begin{equation} \frac{\partial c}{\partial K} = \frac{\partial v}{\partial K} + \frac{\partial v}{\partial \sigma_\text{imp}}\frac{\partial \sigma_\text{imp}}{\partial K} \end{equation}

But how can this be true?

## Answer by ir7 (score 2)

https://quant.stackexchange.com/a/55489

Hint:

$$f(x) = g(h(x),k(x)) $$

$$ f'(x) = \partial_1 g (h(x),k(x)) h'(x) + \partial_2 g (h(x),k(x)) k'(x)$$

(ignore red herrings: $t$, $T$, $S_t$ and $r$; focus on $K$ only)

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