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  1. 034A stock trades at Rs 1,000 and its one-month options are priced at 30% implied volatility. Without a calculator, roughly what does a one-month at-the-money straddle cost, and how far must the stock move for the buyer to break even at expiry?Options and payoffsCoreVolatility and relative value fundsProp and quant trading firms

    Try it first

    Closest estimate for the straddle:

    Show the worked solution

    About Rs 69, so the stock must move about 6.9% either way. Scale the 30% annual volatility to one month by the square root of time: 30% x root(1/12) is about 8.7%. An at-the-money straddle is worth about 0.8 times that move times the price: 0.8 x 8.7% x 1,000 is about Rs 69. At expiry the buyer profits only below about Rs 931 or above Rs 1069.

    Where does the 0.8 x sigma x root T rule come from?

    A straddle pays the size of the move, whichever way it goes, so its value is the expected absolute move. For a normal distribution, the average absolute move is the standard deviation times root(2/pi), which is about 0.8. Over one month the standard deviation of the price is S x sigma x root T = 1,000 x 0.30 x root(1/12), about Rs 87. Times 0.8 gives about Rs 69. Black-Scholes with zero rates gives Rs 69.1, so the shortcut is good to within a rupee for short-dated at-the-money options.

    The straddle loses its premium unless the stock moves about 6.9% either wayRs 931Rs 1069max loss: premium Rs 69payoff before premium8509001,0001,1001,150Stock price at expiry, Rs-500+50+1000.8 x 30% xroot(1/12) x 1,000Rs 69Black-Scholes: 69.1move needed: 6.9%
    A one month straddle struck at Rs 1,000 costs about Rs 69, so at expiry it loses money anywhere between about Rs 931 and Rs 1069 and profits only on a move of more than about 6.9% either way.

    Why is the breakeven move bigger than the typical move traders expect?

    Buying a straddle is like buying insurance against a big move in either direction: you pay the average claim up front. The breakeven at expiry is the premium, about 6.9%, while the average absolute move is the same 6.9% by construction, so a buyer at fair implied volatility breaks even only on average. The buyer makes money when realised volatility turns out higher than the 30% priced in, and the seller makes money when it turns out lower. That is why desks talk about straddles as a bet on volatility, not on direction.

    The relationship
    Straddle≈2π σT S≈0.8×0.30×1/12×1,000≈69\text{Straddle} \approx \sqrt{\tfrac{2}{\pi}}\,\sigma\sqrt{T}\,S \approx 0.8 \times 0.30 \times \sqrt{1/12} \times 1{,}000 \approx 69
    sigmaimplied volatility, 30% a year
    Ttime to expiry in years, 1/12
    Sthe stock price, Rs 1,000
    What it says in wordsAn at-the-money straddle costs about 0.8 times the one standard deviation move over its life.

    Say the limits. The rule assumes the option is at the money and short dated, and rates are small; for long-dated or out-of-the-money options it drifts. A single call or put is half the straddle, about 0.4 x sigma x root T x S, which is a useful second number to have ready.

    Where candidates lose it

    The most common loss is forgetting to scale volatility to one month, which gives an answer near Rs 240 or Rs 300. Implied volatility is quoted per year; divide by root 12 for a month.

    The second loss is forgetting the 0.8 and pricing the straddle at the full one standard deviation move, Rs 87. Say where the 0.8 comes from, the average absolute value of a normal, and the answer sounds reasoned rather than memorised.

    What the interviewer asks next

    • What does a one-week straddle cost on the same stock?
    • Roughly what is the one month at-the-money call worth on its own?
    • The stock moves 5% by expiry. Did the straddle buyer or seller win, and what does that say about realised volatility?
  2. 059A stock trades at Rs 1,000 and pays no dividend. A six-month European call struck at Rs 1,000 costs Rs 60 and the matching put costs Rs 45. Six-month simple interest is 3%. Is there an arbitrage, and how do you lock it in?Options and payoffsCoreVolatility and relative value fundsProp and quant trading firms

    Try it first

    What should the call minus the put be worth?

    Show the worked solution

    Yes: buy the call, sell the put, short the stock and lend Rs 970.87, which collects about Rs 14.13 today with nothing owed at expiry. Put-call parity says the call less the put should equal the stock less the present value of the strike, 1,000 minus 1,000/1.03, about Rs 29.13. The market prices that gap at Rs 15, so the call is cheap against the put by about Rs 14.

    What does a call minus a put actually pay?

    Imagine agreeing today to buy a used car in six months at a fixed price. If car prices rise you gain the difference; if they fall you lose it. A long call with a short put at the same strike does exactly that: at expiry it pays the stock price minus the strike, whichever way the stock moved. So it must cost the same as buying the stock today with money borrowed against the strike, 1,000 minus 970.87, which is Rs 29.13. That identity is put-call parityFor European options on the same stock, strike and expiry, call minus put equals the stock price minus the present value of the strike, less the value of any dividends..

    The relationship
    C−P=S−K1+r=1,000−1,0001.03≈29.13C - P = S - \frac{K}{1 + r} = 1{,}000 - \frac{1{,}000}{1.03} \approx 29.13
    C, Pthe call and put prices, Rs 60 and Rs 45
    Sthe stock price today, Rs 1,000
    Kthe strike, Rs 1,000
    rsix-month simple interest, 3%
    What it says in wordsA call less a put is worth the stock less the strike discounted to today.

    How do you build the trade?

    Buy what is cheap and sell what is dear. The option version of the forward, long call and short put, costs Rs 15; the cash version, long stock bought with borrowed money, is worth Rs 29.13, so buy the option version and sell the cash version. Buy the call for 60, sell the put for 45, short the stock for 1,000 and lend 970.87 at 3%. The cash left over today is Rs 14.13, worth about Rs 14.55 at expiry.

    At expiry the four legs cancel; the profit was collected on day one-200-100+100+20008009001,0001,1001,200Stock price at expiry, Rslong call + short put = stock - 1,000short stock + loan repaid = 1,000 - stocknet: 0 at every priceCash today, RsBuy the call-60.00Sell the put+45.00Short the stock+1,000.00Lend PV of strike-970.87Collected today+14.13Fair call - put = 29.13Market: 15. Gap 14.13
    At expiry the long call and short put pay the stock price minus Rs 1,000 while the short stock and the maturing Rs 1,000 loan pay Rs 1,000 minus the stock price, so the package nets to zero at every price and the Rs 14.13 collected at the start is kept.

    What could stop the arbitrage working?

    Say what you assumed. The lock holds only if the options are European, the stock pays no dividend before expiry, and you can short the stock without a borrowing fee that eats the Rs 14. A dividend lowers the forward and so the fair gap; an American put can be exercised against you early; a hard-to-borrow stock charges a fee. A gap this size in a liquid name usually means one of those is present, so your first question back is which one.

    Where candidates lose it

    The common loss is comparing the call and the put directly, seeing the call is dearer and deciding it must be the one to sell. Parity says the call should be dearer, by about Rs 29; the trade depends on whether the actual gap is too wide or too narrow.

    The second is forgetting to discount the strike and calling zero fair, or using 3% of the stock, Rs 30, which is close but wrong. Say the formula before the numbers, then build the four legs.

    What the interviewer asks next

    • The stock will pay a Rs 20 dividend before expiry. Is there still an arbitrage, and which way?
    • The call is Rs 80 and the put Rs 45. What trade do you put on now?
    • Why does parity hold exactly for European options but only as a band for American ones?