Question

Explain how you would exploit any arbitrage.

A stock which is currently trading at $14 has a 20% chance of going up to $18 tomorrow and an
80% chance of dropping to $12 tomorrow. P is a binary put option on the stock with an exercise
price of $14. In the market, the binary puts are selling for $0.50. Call the portfolio set up to
eliminate risk. The risk-free rate of return is 0.8.

Answer 1

Stock can go up to $18 with 20% chance and down to $12 with 80% chance.

If we were to calculate payoff of put option on this stock using probabilities given above, then payoff will be $0 with 20% chance and $2 with 80% chance.

Therefore, the expected value of put option will be $1.6

Further, it is given that risk free rate of return is $0.8, hence, the risk premium in the above put option will be $0.8 (1.6-0.8).

Binary Put option is trading at $0.5 (which is essentially risk-free since call portfolio is set up).

Since Binary Put option is cheap therefore, arbitrage opportunity exists.

At time zero, we will purchase the Binary Put option and simulatenously sell Put option on the stock with strike $14. Then next day both will lead to same return thus achieving risk-free profit.