# How to calculate riskless profit out of call options?

I'm having trouble with working out a question that I can't currently ask my lecturer as they're away. Hoping for some help here with why the answer is (a).

A stock price is currently \$40. It is known that at the end of two months it will be either \$36 or \$44. The riskfree interest rate is 6% p.a. with continuous compounding. If a two-month call option on the stock with strike \$42 is trading at \\$1.20, how can you make a riskless profit?

• a) Long 1 stock and short 4 call options
• b) Short 1 stock and long 4 call options
• c) Long 1 stock and short 1 call option
• d) Short 1 stock and long 2 call options

## 1 Answer

The call in the up state has payoff of 2 and it has a payoff of zero in the down state. The stock is worth 44 and 36, in the two states, respectively. So the delta is 0.25=(2-0)/(44-36). So you can hedge one call with 0.25 stock units or 4 calls with one stock.

You would then need to generate the cash flows of the portfolio, funding the stock purchase/sale at the risk free rate, to decide between a) and b). E.g., sell one unit of stock at 40 and invest the proceeds at risk free rate and buy 4 calls(1.2*4=4.8). This would have a net flow of -4.8 at time zero. Work out the cash flows of this portfolio in both up and down states (should be same in both states by definition as portfolio is hedged). Take the present value of this and compare to 4.8. If it is positive then that is arbitrage otherwise reverse the strategy.