Arbitrage with a mispriced forward
1. The idea
Arbitrage is a profit without risk and without using any money of our own. A forward on a share that pays no dividends before delivery has one price at which no arbitrage is possible, the no-arbitrage price: the forward price of the first lesson of this level, the spot price grown at the risk-free rate until delivery, \(S_0 e^{rT}\). (With dividends, the spot price less their present value is grown instead, as in the lesson on dividends.) That is what it costs to borrow the money, buy the share today and hold it until delivery. A forward quoted at any other price lets us lock in the difference:
- Quoted too high: sell the forward, and borrow to buy the share today. This is called cash-and-carry.
- Quoted too low: buy the forward, sell the share short (borrow it and sell it, as the lesson on short selling explains) and invest the money. This is called reverse cash-and-carry.
Either way, the profit at delivery is the gap between the quote and the no-arbitrage price:
\[ \text{profit at delivery} = \left| F_{\text{quote}} - S_0\, e^{rT} \right|, \]
where \(F_{\text{quote}}\) is the quoted forward price, \(S_0\) the spot price, \(r\) the risk-free rate with continuous compounding, as a decimal, and \(T\) the time to delivery in years. This assumes that we can borrow and lend at the risk-free rate, sell a share short and use the money, and trade without costs.
A forward on a coupon bond. A bond’s coupon is an interest payment that the bond makes on set dates. A coupon paid before delivery works like a dividend. The no-arbitrage price is the bond’s spot price less the coupon’s present value, grown at the risk-free rate for delivery:
\[ F_0 = (S_0 - I)\, e^{r_2 T}, \]
where \(F_0\) is the no-arbitrage price, \(S_0\) the bond’s spot price, \(I\) the coupon’s present value, discounted with the risk-free rate for its date, \(r_2\) the risk-free rate for delivery, with continuous compounding, as a decimal, and \(T\) the time to delivery in years. When the forward is quoted too high, we buy the bond with borrowed money, and the coupon repays part of the loan. When it is quoted too low, we sell the bond short, invest the money, and pay the coupon to the bond’s lender.
2. Try it
If an answer is wrong, we will see what went wrong and how to fix it.
3. The profit on a chart
In the question, the nine-month forward is quoted at $67.50, above the no-arbitrage price of $65.95. So we sell the forward, borrow $64 and buy the share today. In nine months we deliver the share for $67.50 and repay the loan with interest, $65.95, which leaves us a profit of $1.55 a share, whatever the share price is by then.
Now move the quoted price. One of the two trades always makes a profit, except at the no-arbitrage price.
4. One worked example, then your turn
In the worked example, the quote is below the no-arbitrage price.
Now one with hints, one step at a time.
Then a forward on a coupon bond, with hints.
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 a forward on a share quoted too high, one for a forward on a share quoted too low, and one for a forward on a coupon bond. 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: