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

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

C = S * N(d1) - K * e^(-rT) * N(d2)

In this formula, C is the call option price, S is the current stock price, K is the option's strike price, T is the time until the option expires, r is the risk-free interest rate, and N() is the cumulative distribution function for the standard normal distribution. The values d1 and d2 are calculated by:

d1 = (ln(S/K) + (r + σ^2 / 2) * T) / (σ * √(T))
d2 = d1 - σ * √(T)

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

Δ = dC/dS

Differentiate the Black-Scholes formula with respect to S:
Δ = d(S * N(d1))/dS - d(K * e^(-rT) * N(d2))/dS

d/dS [S * N(d1)] = N(d1) + S * N'(d1) * d(d1)/dS

For K * e^(-rT) * N(d2), differentiate with respect to S:

d(K * e^(-rT) * N(d2))/dS = K * e^(-rT) * N'(d2) * d(d2)/dS

Since d2 = d1 - σ * √T, we have d(d1)/dS = d(d2)/dS and N'(d1) = N'(d2) and we get:

Δ = d(S * N(d1))/dS - d(K * e^(-rT) * N(d2))/dS

=> Δ = N(d1) + S * N'(d1) * d(d1)/dS-K * e^(-rT) * N'(d2) * d(d2)/dS

=> Δ = N(d1) + S * N'(d1) * d(d1)/dS-e^(-rT) * S * N'(d2) * d(d2)/dS

=> Δ = N(d1) + S * N'(d1) * d(d1)/dS-(1) * S * N'(d2) * d(d2)/dS

=> Δ = N(d1)

N(d1) is the probability density function for the standard normal distribution, which is the derivative of N(d1). Multiplying N'(d1) by d(d1)/dS, the S terms cancel out, simplifying to N(d1).
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