Question

A stock currently sells for $50.00. A 3-month European put option with a strike of $48.00 has a premium of $1.25. The stock has a 3% continuous dividend. The continuously compounded risk-free interest rate is 5%.

Suppose you observe the price of the associated call to be $3.1. Give a portfolio that can be used to take advantage of the arbitrage opportunity, and show that the portfolio that you give is an arbitrage portfolio.

Answer 1

According to put-call parity,

Cash Investment + Call Option premium = (Stock-Dividend) + Put Option Premium

{Note that for put-call parity to hold, Strike price of both call and put options must be same and Cash Investment should be present value of option strike price}

Therefore, Call Option premium must be equal to (Stock-Dividend) + Put Option Premium - Cash Investment

(Stock-Dividend) + Put Option Premium - Cash Investment = 50xe(-3%)(3/12) + 1.25 - 48xe(5%)(3/12)

(Stock-Dividend) + Put Option Premium - Cash Investment = 50 x 0.992528 + 1.25 - 48 x 1.012578

(Stock-Dividend) + Put Option Premium - Cash Investment = 49.63 + 1.25 - 48.60

(Stock-Dividend) + Put Option Premium - Cash Investment = $2.27

Call option premium must be equal to $2.27. However, it is trading at $3.10. Therefore, arbitrage gain is possible.

I will take above position that is buy stock, buy put option and short risk free rate that is borrow at risk free rate to create long synthetic call option. Simanteneously, I will sell call option at premium of 3.10.