Question

10 Answer the following a. Suppose data are collected for a certain stock: Stock price Call price (1-year expiration, E $105) Put price (1-year expiration, E 105) $110 $17 $5 5% per year Risk-free interest rate Is there a mispricing of the call and put? If yes, can you exploit this mispricing to create arbitrage proft? b. Design a portfolio using only call options and the underlying stock with the following payoff at expiration: 0 10 20 30 40 S0 6 70 0 90 100 110 c. Show that at-the-money call option must cost more that an at-the-money put option. Both options are written on the same underlying stock with the same maturity and the same exercise price and the stock pays no

Answer 1

A) As per put-call parity theorem, Put price+Stk.price=Call price+P.v of E.P
To finf misprice in option price, back calculate with one unknown parameter and compare it with market price.
Put price+110=17+105*e^-0.05*1
Put price=6.88 (theoritical price)
But market price of put option = $5 < $6.88(put price is undervalued compared to theoritical price)
Hence, there is a mispricing in option prices. To exploit it, but the undervalued option and sell the overvalued option.
C) There will be a unlimited proft/(loss) for option holder/(writer). But this doesn't hold true in case of put option because of lower bound limit(i.e., stock will fall maximum to zero).
Example: Payoff for call option:
Stock price=$100, E.P=$100, Payoff for call holder= In-the money if current stock price is more than 100 and there will be no upper limit(unlimited profit)

Payoff for Put option:
Stock price=$100, E.P=$100, Payoff for Put holder=In-the money if current stock price is less than 100 and there will be a limit on profit i.e., max. profit for put holder will be $100.

Hence, because of limit on profitability/ loss for put option, at-the call option will trade richer than at-the money put option for same E.P.