A stock trades at 100 with no dividends. The 3-month 100-strike call is 4.20 and the 100-strike put is 3.00. The continuously compounded rate is 4%. How much is the arbitrage per share and what do you trade?

A stock trades at 100 with no dividends. The 3-month 100-strike call is 4.20 and the 100-strike put is 3.00. The continuously compounded rate is 4%. How much is the arbitrage per share and what do you trade?

Approach: Compare the market value of C - P against S - K discounted at the funding rate over the option life, then trade the side that is rich.

0.205. Put-call parity with no dividends gives C - P = S - K*e^{-rT} = 100 - 100*e^{-0.04*0.25} = 100 - 99.005 = 0.995, using the discount factor e^{-0.01} = 0.99005. The market prints C - P = 4.20 - 3.00 = 1.20, so the call side is rich by 1.20 - 0.995 = 0.205. The arbitrage is a conversion: sell the call, buy the put, buy the stock, and borrow 99.005 to fund it. At expiry the stock is delivered against whichever option is in the money, the loan repays exactly 100, and the 0.205 collected today is kept whatever the stock does.

Follow-up: How large a stock borrow fee would be needed to make that 0.205 gap fair rather than an arbitrage?

Key concepts: put-call parity, conversion, discount factor, arbitrage.