Upper and lower bounds on option prices
1. The idea
Put-call parity gives every European option a lowest price. A put is never worth less than zero, so the call in \(c = p + S_0 - Ke^{-rT}\) is worth at least \(S_0 - Ke^{-rT}\). A call is never worth less than zero, so the put in \(p = c + Ke^{-rT} - S_0\) is worth at least \(Ke^{-rT} - S_0\). For European options on a share without dividends:
\[ c \ge \max\big(S_0 - K e^{-rT},\ 0\big), \qquad p \ge \max\big(K e^{-rT} - S_0,\ 0\big), \]
where \(c\) and \(p\) are the call’s and the put’s prices, \(S_0\) the share price today, \(K\) the strike price, \(r\) the risk-free rate with continuous compounding and \(T\) the time to expiry in years.
An example: a share at $51 that pays no dividends, a strike price of $50, six months to expiry and a 6% rate. The strike price’s present value is \(50\,e^{-0.06 \times 0.5} = 48.52\), so a European call costs at least \(51 - 48.52 = \$2.48\). At $2 there would be an arbitrage: buy the call, sell the share short and lend the $49 received. In six months the loan repays \(49\,e^{0.03} = \$50.49\), and with the call, buying the share back costs at most $50. At least $0.49 is gained, whatever the share does.
Each option also has a highest price. A call’s payoff is never more than the share price at expiry, so a call never costs more than the share, \(c \le S_0\). A European put pays at most the strike price, at expiry, so it never costs more than the strike price’s present value, \(p \le Ke^{-rT}\); an American put can pay the strike price at once, so it never costs more than the strike price, \(P \le K\).
Dividends. As in put-call parity, the share counts without the dividends paid before expiry, \(S_0 - D\), where \(D\) is their present value; a stock index with a dividend yield \(q\) counts as \(S_0 e^{-qT}\). So \(c \ge \max(S_0 - D - Ke^{-rT},\ 0)\) and \(p \ge \max(Ke^{-rT} + D - S_0,\ 0)\).
American options. An American call on a share without dividends is worth at least \(S_0 - Ke^{-rT}\), and never less than zero. With a positive rate that is more than \(S_0 - K\), its intrinsic value when it is in the money; out of the money, exercising pays nothing. Selling it always beats exercising it, so it is never exercised early, and it costs the same as a European call. An American put can be worth exercising early, and for American options parity becomes a range:
\[ S_0 - K \le C - P \le S_0 - K e^{-rT}, \]
where \(C\) and \(P\) are the American call’s and put’s prices. Given the call’s price, the put costs at least \(C - S_0 + Ke^{-rT}\), the European put’s price from parity, and at most \(C - S_0 + K\): the right to exercise early adds at most the interest on the strike price, \(K - Ke^{-rT}\).
2. Try it
If an answer is wrong, we will see what went wrong and how to fix it.
3. The bound on a chart
First, the strike price’s present value.
Now move today’s share price. The put’s price, here at a volatility of 30% a year, stays above its lowest price at every share price; far in the money the two come close.
4. One worked example, then your turn
In the worked example, we find the lowest price of a call on a share that pays a dividend.
Now an American put’s lowest price, with hints, one step at a time.
And one on your own, on a stock index.
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 call, one for a put and one for an American put’s range. 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: