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Determine the upper and lower bounds for option prices, whether the underlying stock pays dividends or not.

FRM® Part I · Financial Markets and Products · Properties of Options · FRM1.FM.13.b

The idea

Bounds follow from no-arbitrage. Let S be the stock price, K the strike, r the continuously compounded rate, T the maturity and D the present value of dividends before expiry. A call can never be worth more than S, and a put never more than K for American or K × exp(−rT) for European. A European call is worth at least max(S − D − K × exp(−rT), 0). A European put is worth at least max(K × exp(−rT) + D − S, 0). American options must also be worth at least intrinsic value, so an American put is at least K − S, which is higher than the European bound. A common slip is forgetting to discount the strike, or applying the European bound where the American one is tighter. Exams ask for the tightest bound.

Formulas

Lower bound on a European call on a dividend-paying stock

lower bound on c = max(S − D − K × e^(−r × T), 0)

S
the current stock price
D
the present value of dividends paid before expiry
K
the strike price
r
the continuously compounded risk-free rate
T
the time to expiry, in years

Set D to zero for a stock with no dividends. The usual slip is leaving the strike undiscounted.

Lower bound on a European put on a dividend-paying stock

lower bound on p = max(K × e^(−r × T) + D − S, 0)

K
the strike price
r
the continuously compounded risk-free rate
T
the time to expiry, in years
D
the present value of dividends paid before expiry
S
the current stock price

European puts only. An American put must be worth at least K − S, which is the tighter bound because it can be exercised at once.

A practice question

From the question bank, with the reason each wrong option is wrong.

Which of the following is the upper bound on the price of a call option, European or American, on a stock currently trading at S?

  1. S minus the strike price, because that is the payoff if the call is exercised nowWhy not: Uses intrinsic value, which is a lower bound for an American call, not an upper bound.
  2. The strike price, because it is the most the holder will payWhy not: Applies the put's upper bound; a call's value is capped by the stock price, not the strike.
  3. S, because the call can never be worth more than the stock itself - correct
  4. The present value of the strike price, because it is discounted from expiryWhy not: Uses the upper bound for a European put; a call's cap is S.

A call gives the right to buy the stock. If it ever cost more than the stock, buying the stock directly would be cheaper. So c is at most S and C is at most S.

A harder one

A non-dividend stock trades at USD 50. A European call has a strike of USD 48 and expires in six months. The continuously compounded risk-free rate is 6%. What is the lower bound on the call price, closest to?

  1. USD 3.42 - correct
  2. USD 2.00Why not: Uses S - K without discounting the strike, which understates the bound.
  3. USD 0.54Why not: Compounds the strike forward, 50 - 48 x exp(0.03), instead of discounting it.
  4. USD 46.58Why not: Reports the present value of the strike, 48 x exp(-0.03), instead of the call's lower bound.

Lower bound = max(S - K x exp(-rT), 0) = 50 - 48 x exp(-0.06 x 0.5) = 50 - 46.58 = 3.42. The strike must be discounted at the risk-free rate.

The hardest tier

A non-dividend stock trades at USD 36. A one-year European put has a strike of USD 40 and is quoted at USD 1.50. The continuously compounded rate is 5%. Which statement is correct?

  1. The put violates its lower bound; selling the put and shorting the stock, with the proceeds lent, locks in about USD 0.55Why not: Takes the opposite side of the arbitrage; the put is too cheap, so it should be bought, not sold.
  2. The put violates its lower bound; the arbitrage profit is about USD 2.05, the value of the boundWhy not: Reports the lower bound as the profit; the profit is the bound minus the market price, 2.05 - 1.50.
  3. The put violates its lower bound; the arbitrage profit is USD 2.50, which is K - S - pWhy not: Uses K - S - p without discounting the strike; the European lower bound uses K x exp(-rT) - S.
  4. The put violates its lower bound; buying the put and the stock, financed by borrowing, locks in a present value profit of about USD 0.55 - correct

Lower bound = K x exp(-rT) - S = 40 x exp(-0.05) - 36 = 38.05 - 36 = 2.05, above the 1.50 quote. Borrow 37.50, buy stock and put: at expiry the position is worth at least 40 against a loan of 37.50 x exp(0.05) = 39.42. Profit in present value terms is 2.05 - 1.50 = 0.55.

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