Question

A six-month European call option's underlying stock price is $86, while the strike price is $80 and a dividend of $5 is expected in two months. Assume that the risk-free interest rate is 5% per annum with continuous compounding for all maturities.

1) What should be the lowest bound price for a six-month European call option on a dividend-paying stock for no arbitrage?

2) If the call option is currently selling for $2, what arbitrage strategy should be implemented?

1) theoretical price = $2.85

1) theoretical price = $3.02

1) theoretical price = $3.67

1) theoretical price = 1.98

arbitrage strategy: Buy the call and short the stock.

2) arbitrage strategy: Buy the call and buy the stock.

2) arbitrage strategy: Short the call and buy the stock

2) arbitrage strategy: Short the call and sell the stock

Answer 1

Solution:

Spot = 86, Strike = 80, Time = 6 month = 0.5 years , Dividend = 2 , Time = 2 months = 2/12 years  

Present Value of dividend = 5 * exp ( -0.05* 2/12 ) = 4.958

Future value after month = (S -pot - present value of dividend ) * exp ( interest rate * time)

= (86 -4.958) * exp ( 0.05* 0.5 )

= $83.093

Quest 1 )

Since strike price is 80 so minimum call price so that there is no arbitrage opportunity = 83.093 - 80 = 3.093

Quest 2 )

If call option price is $2

Then there exists an arbitrage opportunity because theoretical call price comes out to be 3.093

So current call option priced at $2 is cheaper so we buy it and short the stock

Buy the call option and short the stock

If theoretical call option price is greater than $2 then buy the call and short the stock

If theoretical call option price is lower than $2 then Short the call and buy the stock