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: \[ \max(a,b) = \begin{cases} a, & \text{when } a > b \\ b, & \text{when } a \leq b \end{cases} \] The graph below shows the plot of a function \(N(S)\) versus \(S\). \(N(S)\) can be expressed as ______.
Show Hint
A ramp that starts rising at one point and flattens out at another can always be built as the difference of two shifted \(\max(S-K,0)\) functions.
Step 1: Look at where the slope of \(N(S)\) changes.
From the graph, the slope of $N(S)$ is $0$ for $S$ between $0$ and $10$, then jumps to $1$ for $S$ between $10$ and $20$, then drops back to $0$ for $S$ beyond $20$. So the slope changes by $+1$ at $S=10$ and by $-1$ at $S=20$.
Step 2: Recall the slope behavior of a single hinge function.
The function $H_K(S)=\max(S-K,0)$ has slope $0$ before $S=K$ and slope $1$ after $S=K$. So adding a term $H_K(S)$ to a sum bumps the slope up by $1$ right at $S=K$, and does nothing before that point.
Step 3: Build $N(S)$ as a sum of slope changes.
We need a $+1$ change in slope at $S=10$ and a $-1$ change in slope at $S=20$. A $+1$ change comes from adding $H_{10}(S)$, and a $-1$ change comes from subtracting $H_{20}(S)$. So \[ N(S)=H_{10}(S)-H_{20}(S) \]
Step 4: Check this against the value at $S=20$.
At $S=20$: $H_{10}(20)=10$ and $H_{20}(20)=0$, giving $N(20)=10$, which matches the graph's flat height of $10$ after the ramp. For $S>20$ both terms grow at the same rate of $1$ per unit, so their difference stays fixed at $10$, matching the flat part of the graph.
Step 5: Eliminate the wrong options by their slope pattern.
$H_{10}(S)-2H_{20}(S)$ has a slope change of $-2$ at $S=20$, which sends the graph sloping downward for large $S$ instead of flattening out, so this is wrong. Reversing the signs to $-H_{10}(S)+H_{20}(S)$ gives a slope change of $-1$ at $S=10$ and $+1$ at $S=20$, the mirror image of what is needed, so this is wrong too. Using $H_{15}(S)-H_{20}(S)$ shifts the first hinge to $S=15$ instead of $S=10$, which does not match where the graph starts to rise.
Step 6: Conclude.
Only $H_{10}(S)-H_{20}(S)$ has the right hinge points and the right flat height.
\[ \boxed{H_{10}(S)-H_{20}(S)} \]
