Periodic discount rate and Put-call parity theory: Difference between pages

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Put-call parity theory links put and call option values via ‘no arbitrage’ market pricing assumptions and the related:
A cost of borrowing - or rate of return - expressed as:
#underlying asset price
#option strike price
#time to maturity and
#theoretically risk-free rate of return.


*The excess of the amount at the end over the amount at the start
*Divided by the amount at the end


So for example if the put option value, underlying asset price, strike price, time to maturity, and risk-free rate of return are known, then the call option value can be calculated using the put-call parity relationship:


==Example 1==
Underlying asset price + Put value ''less'' Call value = Present Value of option strike price
GBP 1 million is borrowed.


GBP 1.03 million is repayable at the end of the period.
Call value = Underlying asset price + Put value ''less'' Present Value of option strike price


In the special case where the strike price of the options is equal to the forward price of the underlying asset, the Put value and the Call value are exactly equal.


The periodic discount rate (d) is:


d = (End amount - start amount) / End amount
== Theoretically risk-free portfolios ==
The no-arbitrage pricing relationship is based on the theory that combinations of market assets and liabilities with the same terminal cash flows, must also have the same present values (i.e. the same theoretical current market prices).


= (1.03 - 1) / 1.03
For example both the left side and the right side of the put-call parity formula represent portfolios with the same terminal value:


= 0.029126
Underlying asset + Put ''less'' Call = Present Value of option strike price


= '''2.9126%'''


The left side portfolio is built by buying the underlying asset, buying a put option, and selling a call option with the same strike price.


==Example 2==
The theoretically risk free terminal value of this portfolio is the equal strike price of the two options.
GBP 0.97 million is borrowed or invested


GBP 1.00 million is repayable at the end of the period.  
The present value of this left side portfolio is the present value of the strike price.




The periodic discount rate (d) is:
The right side portfolio is a deposit of cash.


(End amount - start amount) / End amount
This cash portfolio also produces a theoretically risk free terminal value, equal to the strike price of the options.


= (1.00 - 0.97) /  1.00
The current market pricing of these two portfolios must in theory be exactly the same.


= 0.030000
If this relationship did not hold, there would be an arbitrage opportunity to buy the cheaper portfolio and sell the more expensive one, to earn an immediate risk free profit.


= '''3.0000%'''
Therefore market supply and demand pressures will act to quickly re-establish the no arbitrage pricing relationship, following any temporary pricing mis-alignments.




==Example 3==
== See also ==
GBP  0.97 million is borrowed.
* [[Arbitrage]]
 
* [[Interest rate parity]]
The periodic discount rate is 3.0000%.
* [[Option]]
 
* [[Parity]]
Calculate the amount repayable at the end of the period.
* [[Put option]]
 
* [[Risk-free rate of return]]
===Solution===
The periodic discount rate (d) is defined as:
 
d = (End amount - start amount) / End amount
 
d = 1 - (Start amount / End amount)
 
 
''Rearranging this relationship:''
 
(Start amount / End amount) = 1 - d
 
Start amount = End amount x (1 - d)
 
Start amount / (1 - d) = End amount
 
End amount = Start amount / (1 - d)
 
 
''Substituting the given information into this relationship:''
 
End amount = GBP 0.97m / (1 - 0.030000)
 
= GBP 0.97m / 0.97
 
= '''GBP 1.00m'''
 
 
==Example 4==
An investment will pay out a single amount of GBP 1.00m at its final maturity after one period.
 
The periodic discount rate is 3.0000%.
 
Calculate the amount invested at the start of the period.
 
===Solution===
As before, the periodic discount rate (d) is defined as:
 
d = (End amount - start amount) / End amount
 
d = 1 - (Start amount / End amount)
 
 
''Rearranging this relationship:''
 
(Start amount / End amount) = 1 - d
 
Start amount = End amount x (1 - d)
 
 
''Substitute the given data into this relationship:''
 
Start amount = GBP 1.00m x (1 - 0.030000)
 
= '''GBP 0.97m'''
 
 
 
==See also==
 
*[[Effective annual rate]]
*[[Discount rate]]
*[[Nominal annual rate]]
*[[Periodic yield]]
*[[Yield]]

Revision as of 21:06, 5 February 2018

Put-call parity theory links put and call option values via ‘no arbitrage’ market pricing assumptions and the related:

  1. underlying asset price
  2. option strike price
  3. time to maturity and
  4. theoretically risk-free rate of return.


So for example if the put option value, underlying asset price, strike price, time to maturity, and risk-free rate of return are known, then the call option value can be calculated using the put-call parity relationship:

Underlying asset price + Put value less Call value = Present Value of option strike price

Call value = Underlying asset price + Put value less Present Value of option strike price

In the special case where the strike price of the options is equal to the forward price of the underlying asset, the Put value and the Call value are exactly equal.


Theoretically risk-free portfolios

The no-arbitrage pricing relationship is based on the theory that combinations of market assets and liabilities with the same terminal cash flows, must also have the same present values (i.e. the same theoretical current market prices).

For example both the left side and the right side of the put-call parity formula represent portfolios with the same terminal value:

Underlying asset + Put less Call = Present Value of option strike price


The left side portfolio is built by buying the underlying asset, buying a put option, and selling a call option with the same strike price.

The theoretically risk free terminal value of this portfolio is the equal strike price of the two options.

The present value of this left side portfolio is the present value of the strike price.


The right side portfolio is a deposit of cash.

This cash portfolio also produces a theoretically risk free terminal value, equal to the strike price of the options.

The current market pricing of these two portfolios must in theory be exactly the same.

If this relationship did not hold, there would be an arbitrage opportunity to buy the cheaper portfolio and sell the more expensive one, to earn an immediate risk free profit.

Therefore market supply and demand pressures will act to quickly re-establish the no arbitrage pricing relationship, following any temporary pricing mis-alignments.


See also

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