Question

The price of a European call that expires in six months and has a strike price of $49 is $4.5. The underlying stock price is $50, and a dividend of $1.00 is expected in three months. The term structure is flat, with all risk-free interest rates being 10%.

a. What is the price of a European put option that expires in six months and has a strike price of $49? [1 mark]

b. Explain in detail the arbitrage opportunities if the European put price is $1.60. How much will be the arbitrage profit? [4 marks]

Answer 1

Put call parity with known dividend : C + PV (Strike Price) = S - PV (Dividend) + P

where C is the price of call option, P is the price of Put option and S is the current stock price.

PV strike price = 49 * e^-10%*0.5 = 46.61

PV dividend = 1 * e^-10%*0.25 = 0.98

Plugging in the values we get:

4.5 + 46.61 = 50 - 0.98 + P ; or P = 2.09

b. However if the Put price is 1.60, then the RHS of the put call parity will be less than the LHS side of the equation. Hence a trader can profit by arbitraging as below:

• Sell 1 call for $4.5 and borrow PV of the strike price which will be $46.61 for 6 months and PV of dividend which will be 0.98 for 3 months - total inflow will be $52.09
• Purchase 1 unit of stock for $50 and purchase 1 put for $1.60 - total outflow = 51.6
• Hence the trader has residual left of (52.09 - 51.6) = $0.49
• After 3 months, the dividend of $1 is received and is used to repay the 3 month borrowing of 0.98 which in 3 months will grow to $1.
• At the end of six months : the repayment on the borrowings will be = $49. We can have two possibel pay off scenarios - either expiry stock price (EP) is less than 49 or equal to / more than 49:
  • EP is less than $49:
    • Short Call pay off = 0
    • Repayment of loan = 49
    • Long stock pay off = EP
    • Long put pay off = 49 - EP
    • Net pay off = 0 - 49 + EP + 49 - EP = 0
  • EP is equal to or more than $49:
    • Short Call pay off = 49 - EP
    • Repayment of loan = 49
    • Long stock pay off = EP
    • Long put pay off = 0
    • Net pay off = 49 - EP - 49 + EP + 0 = 0
• ​Thus we see that in either case on the expiry, the pay off will be 0 and the residual left at initiation will be the arbitrage profit which is exactly equal to the difference between the price of Put option as it should be as per put call parity and the market price.