# Put-call parity for European options

The put-call parity is the equation of parity between the put options and call options. It is valid for European options only. According to the put-call parity, two different portfolios (one consisting of European call option and the other consisting of European put option) will have the same payoff at the expiry regardless of the movement of underlying.

One portfolio consists of a European call option at an exercise price X and a bond that will pay X at the expiration. This portfolio strategy is also known as a **fiduciary call**. The other portfolio consists of a European put option at the same exercise price X and the underlying. This portfolio strategy is also known as protective put. So, we can say that the put-call parity is the equivalence of fiduciary call and protective put for the same exercise price.

Let's look at the payoff of both the portfolios at expiry with different underlying spot price.

Payoffs |
S |
S |
S |

Fiduciary Call |
|||

European call |
S |
0 |
0 |

Bond |
X |
X |
X |

Total |
S |
X |
X |

Protective Put |
|||

European put |
0 |
0 |
X - S |

Underlying |
S |
S |
S |

Total |
S |
S |
X |

We can see from the table that the payoff of both the portfolios will be the same at the expiry regardless of the movement of underlying. That's why both portfolios should always trade at the same price. If the portfolios trade at a different price, then an arbitrage profit can be made by shorting the overvalued portfolio and buying the undervalued portfolio.

**Graphical interpretation from put-call parity:** The graph of the different options or a combination of options with other assets are drawn with the payoff on the Y-axis and the underlying price of the X-axis. The payoff of the bond will always be equal to X regardless of the underlying price. So, it is like a constant in the equation. So, we can say that the payoff graph of c_{0} is equivalent to the payoff graph of p_{0} plus S_{0}.

Payoff graph of c_{0} = Payoff of (p_{0} + S_{0}) or payoff of protective put

Payoff of (S_{0} - c_{0}) or payoff of covered call = Payoff of - p_{0} or payoff short put option

Similarly, we can say that the payoff graph of a European put option will be equal to the payoff graph of portfolio combining a European call long position and short underlying position. The payoff of the underlying position will be equivalent to the payoff of long European call option plus the payoff of short European put option.

__Check your concepts:__

(58.22) If there is no arbitrage opportunity, then the fiduciary call position will be equivalent to:

(a) Long stock, long put, short bond

(b) Short stock, short put, long bond

(c) Long stock, long put

(58.23) What will be the payoff of the protective put strategy if the stock price is above the exercise price at expiration?

(a) Exercise price

(b) Stock price

(c) Exercise price minus stock price

__Solutions:__

(58.22) Correct Answer is C: The no-arbitrage equation as per put-call parity is that the fiduciary call and protective put should trade at the same price. So, the fiduciary call position is equivalent to a long position in the stock and a long position in the put option.

(58.23) Correct Answer is B: The protective put position is long put plus long stock. At expiration, the payoff of the put option will be equal to zero as the stock price is above the exercise price. Thus, the payoff of the protective put will be equal to the payoff of long stock i.e. stock price.

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