Discount factor and Put-call parity theory: Difference between pages

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'''1.'''
Put-call parity theory links put and call option values via ‘no arbitrage’ market pricing assumptions and the related:
#underlying asset price
#option strike price
#time to maturity and
#theoretically risk-free rate of return.


(DF).


The number less than one which we multiply a single future cash flow by, to work out its present value as:
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:


PV = DF x future cashflow.
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


The periodic discount factor is calculated from the periodic [[yield]] as:
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.


DF = (1 + periodic yield)<SUP>-n</SUP>


== 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).


Commonly abbreviated as DF(n,r) ''or'' DF<SUB>n,r</SUB>
For example both the left side and the right side of the put-call parity formula represent portfolios with the same terminal value:


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


n = number of periods.


r = periodic yield (or periodic cost of capital).
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.


<span style="color:#4B0082">'''Example 1: Discount factor calculation'''</span>


Periodic yield or cost of capital (r) = 6%.  
The right side portfolio is a deposit of cash.


Number of periods in the total time under review (n) = 1.  
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.


Discount factor = (1 + r)<sup>-n</sup>
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.


= 1.06<sup>-1</sup>
Therefore market supply and demand pressures will act to quickly re-establish the no arbitrage pricing relationship, following any temporary pricing mis-alignments.
 
= 0.9434.
 
 
The greater the time delay, the smaller the Discount Factor.
 
 
<span style="color:#4B0082">'''Example 2: Increasing number of periods delay'''</span>
 
Periodic yield or cost of capital = 6%.
 
The number of periods delay increases to 2.
 
Discount factor = (1 + r)<sup>-n</sup>
 
= 1.06<sup>-2</sup>
 
= 0.8890.
 
''(A smaller figure than the 0.9434 we calculated previously for just one period's delay.)''
 
 
 
'''2.'''
 
The yield or cost of capital used for the purpose of calculating Discount Factors, as defined above. 
 
For example the 6% rate applied in definition 1. above.




== See also ==
== See also ==
* [[Annuity factor]]
* [[Arbitrage]]
* [[Certificate in Treasury Fundamentals]]
* [[Interest rate parity]]
* [[Certificate in Treasury]]
* [[Option]]
* [[Compounding effect]]
* [[Parity]]
* [[Compounding factor]]
* [[Put option]]
* [[Cumulative Discount Factor]]
* [[Risk-free rate of return]]
* [[Day count conventions]]
* [[Factors]]
* [[Present value]]

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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