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How to derive Δ for a Non-Dividend-Paying European Call Option in Simple Terms

To derive \( \Delta \) for a non-dividend-paying European call option, we look at the Black-Scholes formula for a call option's price:

\[ C = S \cdot N(d_1) - K \cdot e^{-rT} \cdot N(d_2) \]

In this formula:

  • \( C \): Call option price
  • \( S \): Current stock price
  • \( K \): Option's strike price
  • \( T \): Time until the option expires
  • \( r \): Risk-free interest rate
  • \( N() \): Cumulative distribution function for the standard normal distribution

The values \( d_1 \) and \( d_2 \) are calculated by:

\[ d_1 = \frac{\ln(S/K) + \left(r + \sigma^2 / 2\right) \cdot T}{\sigma \cdot \sqrt{T}}, \quad d_2 = d_1 - \sigma \cdot \sqrt{T} \]

\( \Delta \), the delta for a call option, measures how much the price of the option changes if the stock price changes by one unit. To find \( \Delta \), we take the derivative of \( C \) with respect to \( S \):

\[ \Delta = \frac{dC}{dS} \]

Differentiating the Black-Scholes Formula

Differentiate the Black-Scholes formula with respect to \( S \):

\[ \Delta = \frac{d(S \cdot N(d_1))}{dS} - \frac{d(K \cdot e^{-rT} \cdot N(d_2))}{dS} \]

The derivative of \( S \cdot N(d_1) \) is:

\[ \frac{d}{dS} \left[S \cdot N(d_1)\right] = N(d_1) + S \cdot N'(d_1) \cdot \frac{d(d_1)}{dS} \]

For \( K \cdot e^{-rT} \cdot N(d_2) \), the derivative with respect to \( S \) is:

\[ \frac{d}{dS} \left[K \cdot e^{-rT} \cdot N(d_2)\right] = K \cdot e^{-rT} \cdot N'(d_2) \cdot \frac{d(d_2)}{dS} \]

Since \( d_2 = d_1 - \sigma \cdot \sqrt{T} \), we have \( \frac{d(d_1)}{dS} = \frac{d(d_2)}{dS} \), and \( N'(d_1) = N'(d_2) \). Substituting these into the equation:

\[ \Delta = N(d_1) + S \cdot N'(d_1) \cdot \frac{d(d_1)}{dS} - K \cdot e^{-rT} \cdot N'(d_2) \cdot \frac{d(d_2)}{dS} \]

Simplify further:

\[ \Delta = N(d_1) + S \cdot N'(d_1) \cdot \frac{d(d_1)}{dS} - S \cdot e^{-rT} \cdot N'(d_2) \cdot \frac{d(d_2)}{dS} \]

The \( S \cdot N'(d_1) \cdot \frac{d(d_1)}{dS} \) and \( S \cdot e^{-rT} \cdot N'(d_2) \cdot \frac{d(d_2)}{dS} \) terms cancel, leaving:

\[ \Delta = N(d_1) \]

\( N'(d_1) \) is the probability density function for the standard normal distribution, which is the derivative of \( N(d_1) \). When multiplied by \( \frac{d(d_1)}{dS} \), the \( S \) terms cancel out, simplifying to \( N(d_1) \). This reflects the risk-neutral probability of the option ending in the money, adjusted for the sensitivity of the option's price to the stock price.


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About the Author

 

 Florian Campuzan is a graduate of Sciences Po Paris (Economic and Financial section) with a degree in Economics (Money and Finance). A CFA charterholder, he began his career in private equity and venture capital as an investment manager at Natixis before transitioning to market finance as a proprietary trader.

 

In the early 2010s, Florian founded Finance Tutoring, a specialized firm offering training and consulting in market and corporate finance. With over 12 years of experience, he has led finance training programs, advised financial institutions and industrial groups on risk management, and prepared candidates for the CFA exams.

 

Passionate about quantitative finance and the application of mathematics, Florian is dedicated to making complex concepts intuitive and accessible. He believes that mastering any topic begins with understanding its core intuition, enabling professionals and students alike to build a strong foundation for success.