The one-year European 90/110 box spread is bid at 19.20 while the market lends and borrows at 5% continuously compounded. What rate does the box imply, which side do you take, and what is the present value of the edge?

The one-year European 90/110 box spread is bid at 19.20 while the market lends and borrows at 5% continuously compounded. What rate does the box imply, which side do you take, and what is the present value of the edge?

Approach: A box pays the strike difference with certainty at expiry, so treat its price as a zero coupon bond and back out the yield, then compare that yield to the market rate.

4.08%. A European box spread pays 110 - 90 = 20 at expiry whatever the stock does, so paying 19.20 today implies a rate of ln(20/19.20), or ln(1.041667), which is 0.0408 or 4.08%. That is below the 5% market rate, so you sell the box: receive 19.20 now against paying 20 in a year, which is borrowing at 4.08% when the market charges 5%. Fair value is 20*e^{-0.05}, or 19.025, so the arbitrage is worth 0.175 per box in present value terms. The construction is put-call parity applied twice, long the 90 call and short the 90 put against short the 110 call and long the 110 put, and the whole trade is a funding trade with no market exposure. It only works with European exercise, since an American short leg can be assigned early and turn a certain payoff into an open position.

Follow-up: The options are American and the stock pays no dividend. Which leg is the assignment risk and how does that change the price you would pay?

Key concepts: box spread, implied funding rate, put-call parity, arbitrage.