For positive real numbers $S$ and $K$, the function $H_K(S)$ is defined as: $H_K(S) = \max(S - K, 0)$. The $\max$ function is defined as: $$\max(a, b) = \begin{cases} a, & \text{when } a > b b, & \text{when } a \le b \end{cases}$$ The graph below shows the plot of a function $N(S)$ versus $S$. $N(S)$ can be expressed as ______.

From the given graph, the function $N(S)$ behaves piecewise as follows:

1. For $0 \le S \le 10$:

   $$N(S) = 0$$

2. For $10 < S \le 20$:

   $$N(S) = S - 10$$

   (The line connects $(10, 0)$ to $(20, 10)$ with slope $m = 1$).

3. For $S > 20$:

   $$N(S) = 10$$

Now, let us evaluate the given options using the definition $H_K(S) = \max(S - K, 0)$:

For Option (A): $N(S) = H_{10}(S) - H_{20}(S)$

• When $S \le 10$:

  $$H_{10}(S) = 0, \quad H_{20}(S) = 0 \implies N(S) = 0 - 0 = 0$$

• When $10 < S \le 20$:

  $$H_{10}(S) = S - 10, \quad H_{20}(S) = 0 \implies N(S) = (S - 10) - 0 = S - 10$$

• When $S > 20$:

  $$H_{10}(S) = S - 10, \quad H_{20}(S) = S - 20 \implies N(S) = (S - 10) - (S - 20) = 10$$

This completely matches the graph of $N(S)$.

Therefore, the correct option is (A).