Put-call parity

  To understand put-call parity. We assume. We have two  portfolios. Portfolio A and Portfolio B

One European call option plus an amount of cash equal to  ke^{-rt}   portfolioB: One European put option plus one share.Both are worth.max  (S_{r},k)

At  expiration of the options. Because the option is European, they can not be exercised prior to the expiration date. The portfolio must, therefore, have identical values.
c+ ke^{-rt}=p+S_{o}

This relationship is known as put-call parity. It shows that the value of European call with a certain exercise price and exercise date can be deduced from the value of a European put with the same exercise date and vice-versa. If equation 2nd does not hold.

There is arbitrage opportunity. For example, the stock price is $31, the exercise price is $30, the risk-free interest rate 10% per annum, the price of three-month European call option is $3,and the price of a three-month European put option is 2.25.

 c+Ke^{-rt}=3+30e^{-0.1\times3/12 }=$32.26
p+S_{0}=2.25+31=$33.25

Portfolio B is overpriced relative to portfolio A.The correct arbitrage strategy  to buy the securities in portfolio B.The strategy involves buying the call and shorting both the put and the stock, generating a positive cash flow of
-3+2.25+31=$30.25

upfront,when invested at the risk-free interest rate;this amount grows to 30.25e^{0.1\times 0.25}=$31.02 in three months.

If the stock price at expiration of the option is greater than $30, the call will be exercised.If it is less than $30, the put will be exercised.In either case, the investor ends up buying one share for $30.This can be used to close out the short position. The net profit is therefore
$31.02-$30.00=$1.02
For an alternative situation, suppose that the call price is $3 and the put price is $1.In this case.
c+Ke^{-rT}=3+30e^{-0.1\times 3/12}=$32.26
p+S_{0}=1+31=$32.00

Portfolio A is overpriced relative to portfolio B.An arbitrageur can short the securities in portfolio A and buy the securities in portfolio B to lock in a profit.The strategy involves shorting the call and buying both the put and the stock with an initial investment of
$31+$1-$3=$29

when the investment is financed at the risk-free interest rate, repayment of 29_{e}^{0.1\times 0.25}  = $29.73 is required at the end of the three months. As in the previous case, either the call or the put will be exercised. The short call and long put option position, therefore, leads to the stock sold for $30.00.The net profit is therefore
$30.00-$29.73=$0.27

 

Now Put-call parity for European future option on the assumption that there is no difference between the payoffs from future and forward contracts.

Consider European call and put future options both with strike price k and time to expiration T. We can form two portfolio
Portfolio C: European call future option plus an amount of cash equal to Ke^{-rt}
Portfolio D: European put future options plus a long future contract plus an amount of cash equal to F_{o}e^{-rt}.
In portfolio A the cash be invested at the risk-free rate,r, and will grow to K at time T.Let F_{T} be the futures price at maturity of the option in portfolio C is exercised and portfolio C is worth F_{T}. If  F_{T}>K, the call option in portfolio C is exercised and portfolio C is worth F_{T}. If F_{T}\leq K, the call is not exercised and portfolio C is worth k.The value of portfolio C at a time is therefore
max(F_{T},K)
In portfolio D the cash can be invested at the risk-free rate to grow to F_{0} at time T. The put option provides a payoff of max(K-F_{T},0).The future contract provides a payoff of F_{T}-F_{0}.The value of portfolio D at time T is therefore
F_{0}+(F_{T}-F_{0})+max(K-F_{T},0)=max(F_{T},K)
Because the two portfolio have the same value at time T and there are no early exercise opportunities,it .follows that they are worth the same today.The value of portfolio C today is
c+Ke^{-rT}
where c is the price of the call future option.The marking-to-market process ensures that the future contract in portfolio D is worth zero today.Therefore D is worth
p+F_{0}e^{-rt}
where p is the price of the put future option.Hence
c+Ke^{-rt}=p+F_{0}e^{-rt}

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