Explain put-call parity, then apply it when valuing American and European stock options (paying dividends or not), including its expression through forward prices.

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

The idea

Put-call parity links European options with the same strike K and maturity T. For a stock paying no dividends, c + K × exp(−rT) = p + S. When D is the present value of dividends, c + D + K × exp(−rT) = p + S. Equivalently, using the forward price F, c − p = (F − K) × exp(−rT), so a call and put struck at the forward price have equal value. A violation lets you buy the cheap side and sell the expensive side for a riskless profit. American options only obey inequalities. For a non-dividend stock, S − K is at most C − P, which is at most S − K × exp(−rT). Do not use the exact equality for American options, and remember to discount both strike and dividends.

Formulas

European put price from put-call parity with dividends

p = c + D + K × e^(−r × T) − S

c: the European call price, same strike and expiry
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
S: the current stock price

European options only, and D is zero when no dividend falls before expiry. Forgetting to discount the strike or the dividends is the usual error.

Put-call parity in forward form

c − p = (F − K) × e^(−r × T)

c: the European call price
p: the European put price, same strike and expiry
F: the forward price of the underlying for the option's maturity
K: the strike price
r: the continuously compounded risk-free rate
T: the time to expiry, in years

European options only. When the strike equals the forward price, the call and put are worth the same.

Bounds on American call minus put, non-dividend stock

S − K ≤ C − P ≤ S − K × e^(−r × T)

S: the current stock price
K: the strike price
C: the American call price
P: the American put price, same strike and expiry
r: the continuously compounded risk-free rate
T: the time to expiry, in years

Early exercise means American options obey only this range, not an equality. Using the exact European parity for American options is the trap.

A practice question

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

A European call and put have the same strike and maturity. The strike equals the forward price of the underlying for that maturity. What does put-call parity imply about the two prices?

They are equal, because c - p = (F - K) x exp(-rT) equals zero - correct
The call is worth more, because calls have unlimited upsideWhy not: Ignores that put-call parity ties the difference to F - K, which is zero here; upside and downside are balanced at the forward.
The put is worth more, because the strike is received in cashWhy not: Misreads the parity; the cash received on a put is already built into the discounted forward difference.
The call exceeds the put by the discounted strikeWhy not: Mixes up c - p = S - K x exp(-rT) with the value of a call alone.

In forward form, c - p = (F - K) x exp(-rT). When K = F, the difference is zero, so call and put have the same price.

A harder one

A European call on a non-dividend stock trades at USD 4.20. The stock price is USD 50, the strike is USD 52, the maturity is six months and the continuously compounded rate is 5%. What is the European put price implied by put-call parity, closest to?

USD 6.20Why not: Adds the strike without discounting it: 4.20 + 52 - 50.
USD 7.52Why not: Compounds the strike forward instead of discounting: 4.20 + 52 x exp(0.025) - 50.
USD 4.92 - correct
USD 3.48Why not: Has the sign wrong on the strike term: 4.20 - 52 x exp(-0.025) + 50.

p = c + K x exp(-rT) - S = 4.20 + 52 x exp(-0.05 x 0.5) - 50 = 4.20 + 50.72 - 50 = 4.92.

The hardest tier

A non-dividend stock trades at USD 42. An American call with strike USD 40 and six months to expiry costs USD 5.00. The continuously compounded rate is 6%. Using put-call parity relationships for American options, the price of the American put with the same strike and maturity must lie:

at exactly USD 1.82Why not: Applies the European equality to an American option; for American options parity gives only an inequality.
between USD 1.82 and USD 3.00 - correct
between about USD 1.82 and about USD 4.22Why not: Compounds the strike forward for the upper limit, 5 - 42 + 40 x exp(0.03), instead of using K itself.
between about USD 3.00 and about USD 4.22Why not: Inverts the interval, taking the limit with K as the lower bound and a compounded strike as the upper one.

For a non-dividend stock, S - K <= C - P <= S - K x exp(-rT), so 2.00 <= 5.00 - P <= 42 - 38.82 = 3.18. Hence P lies between 5.00 - 3.18 = 1.82 and 5.00 - 2.00 = 3.00.

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