Q1. Put-call parity Does put-call parity mean the put and the call option (of th

Question

Q1. Put-call parity

Does put-call parity mean the put and the call option (of the same stock, with same expiration, with same strike price) have the same value/price?

If not, for put and call to have the same prices, what must be the relationship between the strike price and the current stock price?

Hint: Find the Apple option page from Yahoo Finance and look for the strike price where the call and put price are similar.

Q2.A European call option and a European put option on a stock both have the same strike price of $45 and expire in 6 months. Currently, the market call price is $10 and the put price is $6. The risk-free rate is 2% per annum, and the current stock price is $49. Identify the arbitrage opportunity open to the trader. All the interest rates are with continuous compounding.

Explanation / Answer

Q1) No, Put -Call Parity does not mean the put and call options have the same price. It means that the value of protective put and the value of fiduicary call should have the same value.

Protective Put is buying a stock (S) and a put option(P) on the stock with strike price X and expiration T

Fiduciary Call us Buying a call option (C) on the stock with strike price X and expiration T and a Risk free bond that gives X at maturity.

Thus, the relation looks like

C + Xe-rT = P + S

where, r is the risk free rate with continuous compunding.

Thus, for Put and Call options to have the same price,

Xe-rT = S

That is, the stock price should be equal to the present value of the strike price discounted at the continuously compounded risk free rate.

Q2) Stock Price = S = $49

Strike price =X = $45

Expiration = T = 6 months = 0.5 years

Continuously compounded risk free rate = r = 2% per annum

Market call price = $10

Market put price = $6

For no arbitrage opportunity to exist, put - call parity should exist, that is,

C + Xe-rT = P + S (since they will have similar cash flows in the future)

C + Xe-rT = 10 + 45e-0.5*2% = 10 + 44.55 = $54.55

P + S = 6 + 49 = $55

Since C + Xe-rT is not equal to P + S, an arbitrage opportunity exists as C + Xe-rT and P + S will have similar cash flows in the future.

Thus, the arbitrage opportunity open to the trader is:

Buy a Fiduciary Call that is buy a call option on the stock for $10 and a continuously compounded risk free bond that gives $45 (strike price) in 6 months.

Cost = C + Xe-rT = 10 + 45e-0.5*2% = 10 + 44.55 = $54.55

Sell a Protective put that is sell a call option on the stock for $6 and sell (or short sell) a stock for $49.

Proceeds = P + S = 6 + 49 = $55

Arbitrage Profit = 55 - 54.55 = $0.45

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