Question

Stock that doesn't pay a dividend is trading at $5. A riskless bond that will pay $100 after a year is trading at $94. A European call option on the stock with strike price of $60 and one year to maturity is trading at $6.1. Devise an arbitrage strategy and prove that it works in all scenarios.

Answer 1

Today

Buy the 940 shares today = -$5*940 = -$4700

Sell the 940 call-option = $6.1 * 940= +$5734

Cash-inflow = +$5734-$4700 = $1034

Invest in $1034 riskless bond =1034/94 = 11 bonds

Total net cash-flow today = 0

Case 1: After 1 year stock price S(T) is less than $60

Bond payoff= $100*11 = $1100

Sell the shares for 940*S(T) = $S(T)*940

Short call expires out of the money, hence payoff will be 0

Total cash-inflow = $1100 + $S(T)

Case 2: After 1 year stock price S(T) is greater than $60

Bond payoff= $100*11 = $1100

Sell the shares = $S(T)*940

Close the short call position = +$60*940 - $S(T)*940

Total cash-inflow = $1100 + $S(T)*940 +$60*940 - $S(T)*940 = $1100 +$60*940

Hence, the strategy works irrespective of the stock price ending up at any value

Sell the shares today = 94/5 = 18.8 shares

Total cash in-flow from selling of shres = $18.8

Buy a 1 year risk-less bond for $94

After 1-year, payoff from the bond = $100

If the stock price S(T) is below $60, the call expires worthless. Let's say the stock price after 1-year if $69

Buy the sold stock to close out the position = $69*18.8