# Put Call Parity

- Consider the Pay-off of a trader who has the following position:
- A Call Option with a Strike Price of 5 and
- A Bond with a maturity value of 5

Share Price |
Call |
Strike Price |
Bond Value |
Bond + Call |

0â€“5 |
0 |
5 |
5 |
5 |

6 |
1 |
5 |
5 |
6 |

7 |
2 |
5 |
5 |
7 |

8 |
3 |
5 |
5 |
8 |

9 |
4 |
5 |
5 |
9 |

10 |
5 |
5 |
5 |
10 |

- Consider, now, the Pay-off of a trader who has:
- A Put Option with a Strike Price of 5 and
- An equivalent unit of the underlying asset

Share Price |
Put Pay-Off (Exercise Price 5) |
Stock Pay-off |
Stock+ Put |

0 |
5 |
0 |
5 |

1 |
4 |
1 |
5 |

2 |
3 |
2 |
5 |

3 |
2 |
3 |
5 |

4 |
1 |
4 |
5 |

5â€“10 |
0 |
5â€“10 |
5â€“10 |

**The Pay-offs are exactly the same**

According to Put Call parity for European options, purchasing a put option on ABC stock will be equivalent to

Buying a call, buying ABC stock and buying a Zero Coupon bond

Buying a call, selling ABC stock and buying a Zero Coupon bond

Selling a call, selling ABC stock and buying a Zero Coupon bond

Buying a call, selling ABC stock and selling a Zero Coupon bond

B: p + S_{0} = c + Ke^{-rT}

- Put Call parity provides an equivalence relationship between the Put and Call options of a common underlying and carrying the same strike price:
- It can be expressed as:
- Value of call + Present value of strike price = value of put + share price
- If value of put is not available, it can be derived as:
- Value of put = Value of call + present value of strike price - share price
- Put-call parity relationship, assumes that the options are not exercised before expiration day, i.e. it follows European options
- This holds true for American options only if they are not exercised early
- In case of dividend-paying stocks, either the amount of dividend paid should be known in advance or it is assumed that the strike price factors the future dividend payment
- The mathematical representation of Put Call Parity is:

= Initial stock price (S) + Put premium (P)

Put Call Parity is valid only for European options, for American Options this relationship turns into an inequality

Consider a 1-year European call option with a strike price of $27.50 that is currently valued at $4.10 on a $25 stock. The 1-year risk-free rate is 6%.What is the value of the corresponding put option?

- 4.1
- 5
- 6
- 25

p + S_{0} = c + D + Xe^{-rt}