Why Put Price Bounds Matter
An option price does not have to be predicted exactly to be checked for consistency. A European put’s expiration payoff is constrained: it cannot be negative and cannot exceed its exercise price. Comparing those payoff limits with the value of a risk-free bond gives useful bounds on the put’s current price. A quoted price outside the bounds permits an arbitrage under the assumptions below.
This tutorial uses the European exercise payoff introduced in the earlier material on option payoffs. The focus is on price bounds, not on estimating a particular option’s fair value. A price inside the bounds is not necessarily the market’s correct price; the bounds only rule out prices that contradict the assumed no-arbitrage market.
The bounds in this tutorial assume frictionless trading: investors can buy and short-sell the stock and options, borrow and lend at the same deterministic risk-free rate, and trade in any required quantities without transaction costs, taxes, or margin constraints. The stock price at expiration is nonnegative, there are no dividends or other distributions before expiration, and the option is European, so it is exercised only at expiration. The risk-free bond used in the arguments pays a fixed amount at \(T\).
The European Put Bounds
A zero-coupon risk-free bond paying \(K\) at time \(T\) costs \(Kv^T\) today. Under the assumptions above, the European put price must satisfy:
The lower bound is the larger of zero and the present value of the exercise price minus the current stock price. The upper bound is the present value of the exercise price. If \(T\) is measured in years and \(i\) is an effective annual rate, then \(v^T=(1+i)^{-T}\). For an integer number of years \(T\), this is the usual annual compounding calculation.
Why the Bounds Hold
The inequalities follow from comparing expiration payoffs. The put payoff is never negative. Also, for every possible \(S_T\), the put payoff is at least \(K-S_T\), because \(\max(K-S_T,0)\geq K-S_T\). Thus its payoff is at least the payoff of a portfolio that holds a bond paying \(K\) and shorts one share of stock.
For every \(S_T\geq 0\), \(0\leq\max(K-S_T,0)\leq K\), and \(\max(K-S_T,0)\geq K-S_T\).
With no arbitrage, a portfolio that always pays at least as much as another cannot cost less today. The bond paying \(K\) costs \(Kv^T\); the stock costs \(S_0\).
The put price is at least \(Kv^T-S_0\), and it is at least zero. It is at most the bond price \(Kv^T\). Therefore \(\max(Kv^T-S_0,0)\leq p_0\leq Kv^T\).
The payoff comparison also explains the arbitrage strategies when a bound is violated. If \(p_0<Kv^T-S_0\), buy the put and one share, and short the bond that pays \(K\). The initial proceeds are positive, and the expiration payoff is never negative. If \(p_0>Kv^T\), sell the put and buy the bond. Again, the initial proceeds are positive, and the expiration payoff is never negative.
Worked Example: Calculating Both Bounds
A non-dividend-paying stock is priced at $42. A European put has an exercise price of $50 and expires in two years. The effective annual risk-free rate is 4%. Find the lower and upper price bounds. Carry full precision through the calculations, then report the bounds to the nearest cent.
Method. Calculate the two-year present value of the exercise price, then apply the bound formula. The relevant risk-free discount factor is \(v^T=(1.04)^{-2}\).
First calculate the accumulation factor:
The present value of the $50 exercise price is:
The lower bound is:
The upper bound is the present value of the exercise price, \(46.2278106509\ldots\). Therefore, rounded to the nearest cent, the put price must lie between $4.23 and $46.23, inclusive. The lower bound is not the stock price minus the discounted exercise price; the order is discounted exercise price minus stock price.
Using the Bounds to Test a Quoted Price
A price below the lower bound or above the upper bound is inconsistent with the model assumptions. It is important to show why, rather than merely label the price “too low” or “too high.” The relevant positions establish a positive initial cash flow and a nonnegative expiration payoff.
Worked Example: A Put Quoted Below Its Lower Bound
A non-dividend-paying stock is priced at $49. A European put has an exercise price of $60, expires in two years, and is quoted at $4 per share. The effective annual risk-free rate is 5%. Determine whether the quote violates a bound, and if so, describe the arbitrage per share. Report money to the nearest cent.
Method. Compare the quote with the lower bound. The two-year accumulation factor is:
The present value of the exercise price is:
Thus the lower bound is:
The $4 quote is below the lower bound of about $5.42. To exhibit the arbitrage, buy one share for $49 and buy the put for $4, for a total cost of $53. At the same time, short a risk-free bond that pays $60 at expiration, receiving \(60/1.1025=54.4217687075\ldots\) today. The initial cash received is:
or $1.42 per share to the nearest cent. At expiration, the stock and put together are worth \(S_T+\max(60-S_T,0)\), and repayment of the bond requires $60. The net expiration payoff is:
The trade therefore has positive initial proceeds and no negative expiration payoff under the stated assumptions. This avoids the common mistake of comparing the put quote with the undiscounted $60 exercise price; the bond comparison uses the discounted amount.
Worked Example: A Put Quoted Above Its Upper Bound
A non-dividend-paying stock is priced at $31. A European put has an exercise price of $40, expires in two years, and is quoted at $38 per share. The effective annual risk-free rate is 5%. Test the quote against the upper bound and construct the corresponding arbitrage if one exists. Report money to the nearest cent.
Method. The upper bound is the present value of the exercise price. Using the two-year accumulation factor \((1.05)^2=1.1025\):
The put quote of $38 exceeds the upper bound of about $36.28 by:
Sell the put for $38 and use $36.2811791383… to buy a risk-free bond paying $40 at expiration. The initial proceeds are about $1.72 per share. At expiration, the bond pays $40 and the short put position costs \(\max(40-S_T,0)\). The net payoff is:
because \(S_T\geq0\). The strategy therefore has positive initial proceeds and a nonnegative expiration payoff. The upper comparison is to the bond’s current cost, not to the undiscounted $40 received at expiration.
Interpreting a Price That Satisfies the Bounds
The bounds are necessary restrictions, not a complete pricing model. If \(p_0\) lies between the bounds, these comparisons alone do not establish that the put is fairly priced or determine its exact value. They also do not say how the put price changes when the stock price, exercise price, volatility, or time to expiration changes; those questions require additional relationships or valuation models.
The lower bound can equal zero when the discounted exercise price is no greater than the current stock price. That does not mean the put must be worthless: the bound only says that the payoff comparison does not require a strictly positive minimum price. Similarly, a put price may equal either bound without violating the inequality. A strict violation, rather than equality, is what creates the positive initial cash flow in the arbitrage constructions.
| Price test | Comparison | Position indicated by the bound |
|---|---|---|
| Below the lower bound | \(p_0<Kv^T-S_0\), when this lower bound is positive | Buy the put and stock; short the bond paying \(K\) |
| Within the bounds | \(\max(Kv^T-S_0,0)\leq p_0\leq Kv^T\) | No arbitrage follows from these bounds alone |
| Above the upper bound | \(p_0>Kv^T\) | Sell the put; buy the bond paying \(K\) |
Common Exam Traps
- Discounting in the wrong direction. The exercise price is paid at expiration, so its current value is \(Kv^T=K/(1+i)^T\), not \(K(1+i)^T\).
- Reversing the lower-bound subtraction. The positive candidate lower bound is discounted exercise price minus current stock price: \(Kv^T-S_0\).
- Forgetting the zero floor. A negative value of \(Kv^T-S_0\) does not imply a negative put price. Use \(\max(Kv^T-S_0,0)\).
- Claiming an in-range price is the exact value. Bounds identify prices inconsistent with no arbitrage; they do not select one unique price from the permitted range.
- Using the bound without checking its assumptions. The formulas here apply to European puts on non-dividend-paying stock under the stated trading and risk-free borrowing assumptions.
Check Your Understanding
For each question, use the stated effective annual risk-free rate and assume the stock pays no dividends. Round final dollar answers to the nearest cent.
- A stock is priced at $55. A European put has \(K=$60\), \(T=1\) year, and the effective annual risk-free rate is 5%. Find its lower and upper bounds. (Answer: lower bound $2.14; upper bound $57.14.)
- A stock is priced at $70. A two-year European put has \(K=$65\), and the effective annual risk-free rate is 4%. A quote is $61. Does it violate a bound? (Answer: Yes; the upper bound is \(65/(1.04)^2=$60.10\), so the quote exceeds it.)
- A stock is priced at $24. A two-year European put has \(K=$30\), and the effective annual risk-free rate is 5%. Calculate the lower and upper bounds. (Answer: lower bound $3.21; upper bound $27.21.)
- A stock is priced at $100. A one-year European put has \(K=$110\), the effective annual risk-free rate is 10%, and the put is quoted at $102. Identify the violated bound and state the initial arbitrage proceeds per share from the bond strategy. (Answer: The upper bound is \(110/1.10=$100.00\); sell the put and buy the bond for initial proceeds of $2.00 per share.)
- A hypothetical contract covers 100 shares of stock currently priced at $48. A one-year European put has \(K=$52\), the effective annual risk-free rate is 4%, and the quoted premium is $1.75 per share. Find the bounds per share and determine whether the quote is admissible under them. (Answer: The bounds are $2.00 and $50.00 per share, or $200.00 and $5,000.00 per 100-share contract; the $1.75 quote is below the lower bound.)