Case 003Option pricing and arbitrage checksCore

A vendor option chain on a stock at 1,000 shows the 950 call at 45, the 1,000 call at 30 against the 1,000 put at 22, and the 1,050 call at 32. Find every error and the arbitrage each allows.

The situation

The risk team at Sudhagad Capital runs a sanity check on a vendor's option chain for Ratangad Motors before the desk uses it for marks. The stock is at Rs 1,000, the one-month rate is 6% a year with continuous compounding, and the stock pays no dividend before expiry, which is one month away.

The chain shows the 950 call at Rs 45, the 1,000 call at Rs 30, the 1,000 put at Rs 22 and the 1,050 call at Rs 32. All options are European and the team is told to ignore transaction costs.

Your task

Which quotes are impossible, what rule does each break, and what trade would a desk put on if the quotes were real?

Quick check

Before any arithmetic: is the 1,050 call at 32 above the 1,000 call at 30 a possible pair of prices?

Worked solution

Try it on paper, then open one step at a time.

30-second answerThe answer to give first

All three rows are wrong. The 950 call at 45 is below its lower bound of about 54.7, the stock less the discounted strike. The 1,000 call less the 1,000 put is 8, but put-call parity says it must be about 4.99, so the pair is about 3 too wide. And the 1,050 call at 32 costs more than the 1,000 call at 30, which no option can. Each error hands a desk a trade that cannot lose: buy the cheap call against short stock, run a conversion, and sell the 1,050 call against the 1,000 call.

Step 1What is the first rule to check, and why is 45 too cheap for the 950 call?

Start with floors, because they need no model. A coupon worth Rs 50 off any purchase cannot sell for less than Rs 50 to someone who is going to buy anyway. A call must be worth at least the stock price less the discounted strike, because holding the call plus the discounted strike in cash can never do worse than holding the stock. Here the one-month discount factorWhat one rupee due at a future date is worth today: e to the power of minus the rate times the time. is e to the minus 0.06 over 12, 0.9950, so the floor is 1,000 less 950 times 0.9950, which is 54.74. A quote of 45 is 9.74 below it.

The trade: buy the call for 45, sell the stock short for 1,000, and lend the 955 difference for a month at 6%. At expiry the loan returns 959.79. If the stock is above 950 you exercise, pay 950, and keep 9.79; if it is below 950 you buy the stock in the market for less than 950 and keep even more. The worst case locks in about Rs 9.8 a share with no capital at risk, which is why the quote cannot survive in a liquid market.

Step 2How does parity expose the 1,000 call and put?
The relationship
C−P=S−Ke−rT=1,000−1,000×0.9950=4.99quoted: 30−22=8C - P = S - K e^{-rT} = 1{,}000 - 1{,}000 \times 0.9950 = 4.99 \qquad \text{quoted: } 30 - 22 = 8
C, Pthe call and put prices at the same strike and expiry
Sthe stock price, 1,000
K e^{-rT}the strike discounted at 6% for one month
What it says in wordsA call less a put at the same strike must equal the stock less the discounted strike, because the two sides have the same payoff at expiry. The quoted gap of 8 is about 3 too wide.

The call and put together replicate a forward on the stock, so their difference is pinned by the stock and the rate, not by any view. The quoted 8 against the correct 4.99 means either the call is too dear or the put too cheap, and the trade does not care which. Sell the call at 30, buy the put at 22, and buy the stock at 1,000, a net outlay of 992; at expiry you deliver the stock for 1,000 whatever happens, and 992 financed at 6% for a month costs 996.97, so about Rs 3.0 a share is locked in. This is a conversion, and a desk would run it in size until the quotes moved.

Three cells in the chain break three rules, and each one is a tradeRatangad Motors, spot 1,000, 1 month, rate 6%StrikeCallPut950451,00030221,05032Flag 1: row 950 callFlag 2: row 1,000 call against putFlag 3: row 1,050 call against 1,000 call1950 call below its floorFloor = S - K e^(-rT) = 1,000 - 950 x 0.9950 = 54.74Buy call 45, short stock, lend: locks about Rs 9.82Call minus put too wideParity: C - P = S - K e^(-rT) = 4.99; quoted 30 - 22 = 8Sell call, buy put, buy stock: locks about Rs 3.03Higher strike call dearerA 1,050 call can never be worth more than a 1,000 callSell 1,050 call 32, buy 1,000 call 30: Rs 2 now, never pays out
The chain breaks three rules: the 950 call at 45 sits below its floor of 54.74, the 1,000 call less the 1,000 put is 8 where parity says 4.99, and the 1,050 call at 32 costs more than the 1,000 call at 30, and each error is a trade with no downside.
Step 3Why is the 1,050 call the easiest one to catch, and what does the report say?

Monotonicity needs no rate and no time. The 1,000 call pays everything the 1,050 call pays and Rs 50 more whenever the stock finishes above 1,050. Selling the 1,050 call for 32 and buying the 1,000 call for 30 collects Rs 2 today for a position that can only pay you more later, a vertical spread with negative cost. The risk report should rank the errors by how they would be caught: the dominance error is a pure data fault, the parity error could be a stale put, and the lower-bound breach is most likely a stale call quote from before a move in the stock. Say the limit too: the checks assume no dividend before expiry, European exercise and a single borrowing rate; with a dividend the floor and parity both shift by its present value.

Where candidates lose it

Candidates reach for a pricing model and compute implied volatilities. The chain fails model-free checks first, and an interviewer wants the floors, parity and monotonicity before any volatility. A model fitted to broken quotes gives broken answers with great confidence.

The second loss is stating that the 950 call is too cheap because it should be at least its intrinsic value of 50. The right floor is against the discounted strike, 54.74, and the difference is exactly the point of asking for the rate.

What the interviewer asks next

  • The stock pays a Rs 10 dividend in two weeks. Which of the three checks change, and by how much?
  • How would these checks differ for American options?
  • The bid-ask spread on each quote is Rs 2. Which of the three trades still works?
  • Design the daily check you would run on the whole chain, in three lines.

Company names and figures are illustrative.