Stratégies Options - Call-put Parity: A Typically European Relation

I - A very simple reason, the following one

Let us imagine that there is no interest rate.
If we buy a 1 year call struck at 100 for example, and if we sell a 1 year put struck at 100 on the same underlying, in one year we would benefit from an increase from the underlying if it would be above the strike 100, or we suffer from a decline that is the underlying is below that strike 100.

We thus have C - P = S - K
That we can write K = S + P - C, at expiry.

II - Present Value

If the rates are not zero, what one would have today is worth:
( K*exp( - r*t ) ) where r is the riskless rate over the period t .

This relation is called the call - put parity. It is fundamental because based on a principle of arbitrage rather from a model. Indeed, no hypothesis is made on the behavior of the underlying S to derive this equation.

The main interest of this relation lies on the fact that it shows the value of a call according to a put and vice versa.

We have: C = P + S - K * (exp ( -r*t) )

The value of a call struck at K is identical to that of a portfolio of a put struck at K and long position on spot S funded by borrowing an amount K during a period t at the rate r.
This is called, "Synthetic Call"

And:P = C + K * (exp (-r*t) )

The value of a put struck at K is identical to a portfolio of a call struck at K and a short sell of S that funded an investment of an amount K in a cash account during a period t at the rate r
This is called, "Synthetic Put"

But also:S = C - P + K*(exp (- r*t ))

We can find the value of spot S by placing an amount that equalizes the present value of the strike K during t year(s) at the rate r, buying 1 year call struck at K and by selling short a 1 year put struck at K, call and put having a maturity that is t.
This is called, "Synthetic Spot"

And finally:(K*exp ( - r*t ) = S - C + P

We can find the value of a riskless investment which will be worth K at maturity by buying the underlying S, by selling a call C struck at K and by buying a put P struck at K, call and put having a maturity t.
This is called, "Synthetic CaSH ACCOUNT"

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Relationships Between Option Sensitivities - INDEX Relationships Between Option Sensitivities - CHAPTER I
Relationships Between Option Sensitivities - CHAPTER II
Relationships Between Option Sensitivities - CHAPTER III

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