Question

# A six-month European call option's underlying stock price is \$86, while the strike price is \$80...

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: Short the call and buy the stock

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

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

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