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  1. 060A one-year European call and put on a non-dividend stock, both struck at 100, trade at 12 and 7. The stock is at 100. What interest rate does put-call parity imply?Options and GreeksCoreBank market risk

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    Which relationship do you use?

    Show the worked solution

    About 5.26% a year with annual compounding, or 5.13% continuously compounded. Parity says call minus put equals stock minus the present value of the strike. Here 12 minus 7 is 5, so the present value of 100 must be 95. A one-year discount factor of 0.95 means 100 over 95 minus 1, about 5.26%.

    Why must call minus put equal the stock minus the discounted strike?

    Agreeing today to buy a house next year at a fixed price is the same as buying it now with money borrowed until then: either way you own the house next year and pay the fixed price. A long call plus a short put at the same strike is that agreement. At expiry the pair pays the stock minus the strike in every state, so today it must cost the same as owning the stock and owing the strike in a year, which is S minus the present value of K.

    Long call plus short put is a forward: its price reveals the interest ratelong callshort puttogether: stock - 100strike 1000Payoff at expiryCall - put12 - 7 = 5= Stock - PV(strike)100 - PV = 5So PV(strike)95Implied 1-year rate100 / 95 - 1 = 5.26%continuous compounding: ln(100/95) = 5.13%
    A long call and a short put struck at 100 together pay the stock price minus 100 in every outcome. Call 12 minus put 7 is 5, which must equal the stock at 100 minus the present value of the strike, so that present value is 95 and the implied rate is 5.26% a year.

    Why would a risk manager care about the rate hidden in option prices?

    Because it is a check that comes free. If the rate implied by parity sits far from the funding rate the desk actually pays, either the marks are stale, a dividend has been missed, or the options are American and parity no longer holds exactly. Model validation teams run exactly this test on option books to catch mispriced marks before they show up as a loss.

    The relationship
    C−P=S−K1+r  ⇒  5=100−1001+r  ⇒  r=10095−1C - P = S - \frac{K}{1+r} \;\Rightarrow\; 5 = 100 - \frac{100}{1+r} \;\Rightarrow\; r = \frac{100}{95} - 1
    C, Pthe call and put prices, 12 and 7
    Sthe stock price, 100
    Kthe common strike, 100
    rthe one-year interest rate implied by the prices
    What it says in wordsThe gap between an at-the-money call and put is the interest on the strike, so the prices reveal the rate.

    The limitation: parity holds exactly only for European options on a stock paying no dividend before expiry. A dividend would lower the stock's forward and push the implied rate the other way, so state the assumption before you give the number.

    Where candidates lose it

    The common error is to write 5 over 100 and answer 5%. That treats 5 as the interest on 100, but 5 is what you save today, and the rate is measured on the 95 you actually pay. 100 over 95 minus 1 is 5.26%.

    The other trap is forgetting the conditions. Say European, no dividends, and one year, then give the number; an interviewer will often follow up with a dividend to see if you adjust.

    What the interviewer asks next

    • The stock pays a dividend of 2 in six months. What rate is implied now?
    • The call trades at 13 with the put unchanged. What trade locks in a profit?
    • Why does parity not hold exactly for American options?