Question

Consider a European-style call option with an exercise price of 105 euros expiring exactly in a year and a European-style put option with an exercise price of 105 euros expiring exactly in a year. The current stock price is 110 euros, the risk-free interest rate is 10% per annum, the price of this call option is 14.50 euros and the price of put option is 2.50 euros. Suppose that you can borrow and lend the money at risk-free rate. For simplicity, suppose also that the transaction costs are zero. Assume also that there are no counter-party risks. Identify the mispricing and demonstrate (step-by-step) how to apply a risk-free arbitrage strategy that allows you to earn positive net profits? Identify clearly what instruments you will buy and what instruments you will sell short, etc. Calculate also the net profit from arbitrage. (Recall that the value of e is approximately 2.7183).

Answer 1

In the given case,

Strike price=105 euro

Current market price=110 euros

Risk free interest=10%

Call option=14.50

Put option=2.50

In a european style of option, we calculate the mis pricing using put-call parity theory.

Put call parity is:

Call price+strike price/(1+risk free interest)time to maturity=Put price+spot price

14.50+105/(1+10%)1=2.50+110

109.9595=112.50

The put call parity says that the value of left hand side should be equal to right hand side.

But, in the given case, the value is not equal

This leads to mispricing between left hand side and right hand side of:

112.50-109.9595=2.5454 Euros

This mis pricing gives us an opportunity of risk free profit by buying the cheaper asset and selling the expensive one.

This means, we have to buy the cheaper asset of 109.9595 and sell the expensive asset of 112.50.

This can be done using the following strategy.

Sell right hand side for 112.50= Sell the put option for 2.50 euros and receive the payment and also short sell the share the underlying stock and receive the payment of 110 euros. Total money recieved from the following shall be 112.50 euros.

Buy right hand side for 109.9595= Buy the call option for 14.50 euros and pay the payment and also buy the risk free bond for 95.4595 and pay the sum of 109.9595.

The profit from the strategy is

=Money recieved-Money Paid

=112.50-109.9595

=2.5454 euros

The net profit of the strategy is 2.5454 euros no matter the stock moves in any direction. Even if the stock goes above the market price of 105 euros or below the stock price, the strategy would fetch a profit of 2.5454 euros in arbitrage in a risk free manner.