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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

0–5

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

5–10

0

5–10

5–10

 

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, 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
 

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

 





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