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 at Expiration Call Pay-Off Strike Price Bond Value at Maturity 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 at Expiration 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

Example

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

Solution

B: p + S0 = 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

Example

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?

1. 4.1
2. 5
3. 6
4. 25

Solution

p + S0 =  c + D + Xe-rt