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  1. 083A stock trades at Rs 100 and pays no dividend. A one-year European call struck at Rs 100 costs Rs 10, and the one-year interest rate is 5%. What should a put with the same strike and expiry cost?Rates, risk and optionsHardDebt capital marketsSales and trading

    Try it first

    Pick the put price before you work it.

    Show the worked solution

    About Rs 5.24. A call plus cash worth the strike's present value pays the larger of the share price and 100 at expiry, and so does a put plus the share. Identical payoffs must cost the same today: 10 + 100/1.05 = P + 100. The present value of 100 is 95.24, so the put is 5.24. No pricing model or volatility estimate is needed.

    Why can you price the put without knowing volatility?

    If two combo meals contain exactly the same food, a restaurant cannot charge different prices for long: everyone would buy the cheaper one. Options work the same way. Two portfolios that pay the same in every outcome must cost the same today, so if you know the price of one, the other follows without any view on where the share goes. Volatility matters for what a call is worth, but the call's price already contains it. The put inherits it through the parity relationship.

    Build the two portfolios. A: buy the call and put aside 95.24 at 5%, which grows to exactly 100. B: buy the put and buy one share. If the share ends at 130, A exercises the call for 30 and adds its 100 cash, B holds a share worth 130 and lets the put expire: both have 130. If the share ends at 70, A lets the call expire and holds 100 cash, B exercises the put to sell the share for 100: both have 100.

    Two portfolios, one payoff line, so one priceA: call + cash of 95.24601001401000callcash, grows to 100total: larger of S and 100Share price at expiryB: put + one share601001401000putsharetotal: larger of S and 100Share price at expirySame payoff, same price today: C + PV(K) = P + S10 + 95.24 = P + 100, so P = 5.24
    A call plus cash of 95.24 and a put plus one share both pay the larger of the share price and 100 at expiry, so their prices today must match, which pins the put at 5.24.

    What does the formula say, and what if the put traded at Rs 7?

    The relationship
    C+K1+r=P+S⇒P=10+1001.05−100≈5.24C + \frac{K}{1+r} = P + S \quad\Rightarrow\quad P = 10 + \frac{100}{1.05} - 100 \approx 5.24
    Ccall price, Rs 10
    Pput price, the unknown
    Sshare price today, Rs 100
    K/(1+r)the strike discounted one year at 5%, 95.24
    What it says in wordsA call plus the discounted strike is worth the same as a put plus the share.

    If the put traded at 7, portfolio B would cost 107 against 105.24 for A, so you would sell B and buy A, pocketing 1.76 today with payoffs that cancel at expiry. Selling B means writing the put and short selling the share. Traders arbitraging that gap is what holds parity in place, which is why quoted put and call prices sit close to it in practice.

    Say the assumptions: European options, so no early exercise; no dividend before expiry; and annual compounding. With continuous compounding the strike's present value is 100 x e to the minus 0.05, about 95.12, and the put comes out near 5.12. Give the convention, not just the number.

    Where candidates lose it

    The instinctive answer is Rs 10, on the idea that at-the-money calls and puts must cost the same. They do not, because the call holder defers paying the strike for a year, and that deferral is worth something when rates are positive.

    The second loss is remembering parity but flipping a sign and answering 4.76. Build the two portfolios out loud before writing the formula and the signs take care of themselves.

    What the interviewer asks next

    • If the stock paid a Rs 2 dividend before expiry, how would the put price change?
    • Why does parity hold exactly for European options but only as a bound for American ones?
    • What position replicates a long put using only the call, the share and borrowing?