Capital Markets Arbitrage question?
Consider the following market quotes:
• The current price of a stock is S0=$100;
• there is a European call option struck at X= 100 maturing in 3 months which is trading for $0.7 ; and
• there is a European put option struck at X= 100 maturing in 3 months which is trading for $2
• the risk free rate is 8% [quarterly compounding].
You do not know what values the stock can take at expiration.
Assume that you are not allowed to sell short [to write ] call options but you can short any other traded asset.
Detect an arbitrage opportunity and construct the trading strategy to exploit this opportunity.
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Verified answer
This one is easy.
Buying the ATM call and shorting the ATM put is creating a synthetic futures contract which ought to cost 0 so the call ought to cost the same as the put. Here the put is much more expensive than the call which means either a) the call is too cheap or b) the put is too expensive. So we buy the thing that might be too cheap and sell the thing that might be too expensive. That means we are long the ATM call and short the ATM put, i.e., we are long the synthetic futures contract. To make it an arbitrage that means we short the stock.
That means we make $1.30/share + interest on our 1.30 for 3 months + interest in the short stock position for 3 months.