Comparing positions
1. The idea
To compare two positions, write each one’s profit, or each plan’s cost, as a function of the price at the end, \(S_T\), and find the price at which the two are equal.
Shares or calls for the same money. With an amount \(M\), an investor can buy \(n_S = M/S_0\) shares or \(n_C = M/c\) calls, where \(S_0\) is the share price today and \(c\) the premium of a call with strike price \(K\). A call costs much less than a share, so \(n_C\) is much larger. If the share ends at or below \(K\), the calls expire worthless and all the money is lost. Above \(K\) the calls gain \(n_C\) dollars for each dollar the share rises, against \(n_S\) for the shares, and they overtake the shares where
\[ n_S\,(S_T - S_0) = n_C\,(S_T - K) - M. \]
An example. With $6,000 and a share at $30, the investor can buy 200 shares, or 5,000 calls with a strike price of $32 at $1.20 each. The calls give the larger profit above \(S_T = \frac{5{,}000 \times 32}{5{,}000 - 200} = \$33.33\), and lose everything if the share ends at $32 or below.
Fixing a purchase price or capping it. A company that will buy something later can fix the price today with a forward at today’s forward price \(F\): it must then pay \(F\) whatever the market price \(S_T\) is when it buys, and it pays nothing today. Or it can buy calls that expire when it buys, with a strike price \(K\) at or above \(F\), for a premium \(c\) per unit paid today. The calls give the right, not the obligation, to buy at \(K\): below \(K\) the company lets them expire and buys at \(S_T\), and above \(K\) it exercises them and pays \(K\). Either way it has paid \(c\), so a unit costs
\[ \min(S_T, K) + c, \]
at most \(K + c\). That is less than \(F\) only where \(S_T + c < F\), so the calls make the purchase cheaper below \(S_T = F - c\).
An example. A jeweller will buy gold in three months. A forward for delivery then fixes the price at $2,410 an ounce, and calls that expire then, with a strike price of $2,450, cost $58 an ounce. With the calls, gold costs at most \(2{,}450 + 58 = \$2{,}508\) an ounce, and less if its price falls; they make the purchase cheaper than the forward only if gold ends below \(2{,}410 - 58 = \$2{,}352\).
2. Try it
If an answer is wrong, we will see what went wrong and how to fix it.
3. The two profits on a chart
First, what happens if the share ends at the strike price.
Now move the share price at expiry. The calls lose everything up to the strike price, then gain much faster than the shares and overtake them.
4. One worked example, then your turn
In the worked example, a coffee roaster compares fixing its purchase price with a forward and capping it with calls.
Now shares against calls, with hints, one step at a time.
Then a purchase price, fixed or capped, with hints: first the cost with the calls at two prices, then where the two plans cost the same.
And one on your own.
5. Practice
Each question has new numbers. You master the skill when you get three right in a row, with at least one right answer for shares against calls and one for fixing a purchase price against capping it. You can always open the solution, but then that question does not count.
6. Coming back to it
When you have mastered a skill, it comes back on the start page for a short review after 1 day. After each right answer the next review comes later: after 3, 7, 14 and then 30 days. After a wrong one, the reviews start again at 1 day.
This skill: