Discount rate 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.


The quoted market rate for traded instruments quoted at a discount.


The market discount rate is quoted based on a percentage of the ''maturity amount''.
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


<span style="color:#4B0082">'''Example 1: Discount rate calculation'''</span>
Call value = Underlying asset price + Put value ''less'' Present Value of option strike price


The maturity amount for an investment is £10m.
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 gain for the single period from the start to the final maturity is £2m.


The periodic discount rate (d) is:
== 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).


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


= 2 / 10
Underlying asset + Put ''less'' Call = Present Value of option strike price


= '''20%'''


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.


In the US the market discount rate is sometimes known as the ''discount yield''.
The theoretically risk free terminal value of this portfolio is the equal strike price of the two options.


This is different from a [[yield]] or interest rate, which is conventionally quoted based on a percentage of the ''starting amount''.
The present value of this left side portfolio is the present value of the strike price.




<span style="color:#4B0082">'''Example 2: Yield calculation'''</span>
The right side portfolio is a deposit of cash.


The starting amount for an investment is £8m.
This cash portfolio also produces a theoretically risk free terminal value, equal to the strike price of the options.


The gain for the single period from the start to the final maturity is £2m.
The current market pricing of these two portfolios must in theory be exactly the same.


The periodic yield (r) is:
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.


(r) = Gain / Start amount
Therefore market supply and demand pressures will act to quickly re-establish the no arbitrage pricing relationship, following any temporary pricing mis-alignments.
 
= 2 / 8
 
= '''25%'''
 
 
Notice that the discount rate and the yield calculated above both relate to exactly the same deal.
 
£8m is invested now, and £10m is repaid at the end of one period.
 
The discount rate of 20% and the yield of 25% both summarise the same deal, using different conventional bases.
 
 
 
2.
 
Cost of capital.
 
The yield used to calculate [[discount factor]]s and present values.
 
 
3.
 
The rate used to discount future liabilities of a Defined benefit pension scheme in order to calculate the present value of the liabilities, often for the purpose of comparing them with the market value of the scheme’s assets. 
 
Historically it was common to use the blended rate of investment return expected on the actual assets in the scheme, but typically now a market rate is used, such as the government bond or AA corporate bond yield for a fixed income security with a similar duration to that of the underlying liabilities.
 
 
4.
 
In the US, the interest rate that member banks pay the Federal Reserve when the banks use securities as collateral.  The discount rate acts as a benchmark for interest rates issued. 
 
Other central banks also have similar discount rates.




== See also ==
== See also ==
* [[CertFMM]]
* [[Arbitrage]]
* [[Cost of capital]]
* [[Interest rate parity]]
* [[Discount]]
* [[Option]]
* [[Discount basis]]
* [[Parity]]
* [[Discount instruments]]
* [[Put option]]
* [[Discounted cash flow]]
* [[Risk-free rate of return]]
* [[Interest rate]]
* [[Monetary policy]]
* [[Nominal annual discount rate]]
* [[Periodic discount rate]]
* [[Periodic rate]]
* [[Yield]]
 
 
===Other links===
[http://www.treasurers.org/node/8837 Students: Triumph with timelines, The Treasurer, March 2013]
 
[[Category:Corporate_finance]]

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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