Editing Put–call parity (section)

==Statement==
'''Put–call parity''' can be stated in a number of equivalent ways, most tersely as:
:{{center|<math>C - P = D\cdot(F - K)</math>}}
where <math>C</math> is the (current) value of a call, <math>P</math> is the (current) value of a put, <math>D</math> is the [[discount factor]], <math>F</math> is the [[forward price]] of the underlying asset, and <math>K</math> is the strike price. The left side corresponds to a portfolio of a long call and a short put; the right side corresponds to a forward contract. The assets <math>C</math> and <math>P</math> on the left side are given in present values, while the assets <math>F</math> and <math>K</math> are given in future values (forward price of asset, and strike price paid at expiry), which the discount factor <math>D</math> converts to present values.

Now the spot price <math>S = D\cdot F</math> can be obtained by discounting the forward price <math>F</math> by the factor <math>D</math>. Using spot price <math>S</math> instead of forward price <math>F</math> gives us:
:{{center|<math>C - P = S - D\cdot K</math>.}}

Rearranging the terms gives a first interpretation:
:{{center|<math>C + D \cdot K = P + S</math>.}}
Here the left-hand side is a [[fiduciary call]], which is a long call and enough cash (or bonds) to exercise it by paying the strike price. The right-hand side is a [[Married put]], which is a long put paired with the asset, so that the asset can be sold at the strike price on exercise. At expiry, the intrinsic value of options vanish so both sides have payoff <math>\max(K, S)</math> equal to at least the strike price <math>K</math> or the value <math>S</math> of the asset if higher.

That a long call with cash is equivalent to a long put with asset is one meaning of put-call parity.

Rearranging the terms another way gives us a second interpretation:
:{{center|<math>D \cdot K - P = S - C</math>.}}
Now the left-hand side is a cash-secured put, that is, a short put and enough cash to give the put owner should they exercise it. The right-hand side is a [[Covered option|covered call]], which is a short call paired with the asset, where the asset stands ready to be called away by the call owner should they exercise it. At expiry, the previous scenario is flipped. Both sides now have payoff <math>\min(K, S)</math> equal to either the strike price <math>K</math> or the value <math>S</math> of the asset, whichever is ''lower''.

So we see that put-call parity can also be understood as the equivalence of a cash-secured (short) put and a covered (short) call. This may be surprising as selling a cash-secured put is typically seen as riskier than selling a covered call.<ref>{{cite web |last1=Noël |first1=Martin |title=Call Put Parity: How to Transform Your Positions |url=https://www.optionmatters.ca/call-put-parity-transform-positions/ |website=OptionMatters.ca |date=17 May 2017 |publisher=Bourse de Montréal Inc}}</ref>

To make explicit the time-value of cash and the time-dependence of financial variables, the original put-call parity equation can be stated as:
:{{center|<math> C(t) - P(t) = S(t)- K \cdot B(t,T)</math>}}
where
:<math>C(t)</math> is the value of the call at time <math>t</math>,
:<math>P(t)</math> is the value of the put of the same expiration date,
:<math>S(t)</math> is the [[spot price]] of the underlying asset,
:<math>K</math> is the strike price, and
:<math>B(t,T)</math> is the present value of a [[zero-coupon bond]] that matures to $1 at time <math>T</math>, that is, the discount factor for <math>K.</math>

Note that the right-hand side of the equation is also the price of buying a [[forward contract]] on the stock with delivery price <math>K</math>.  Thus one way to read the equation is that a portfolio that is long a call and short a put is the same as being long a forward.  In particular, if the underlying is not tradable but there exists forwards on it, we can replace the right-hand-side expression by the price of a forward.

If the bond [[interest rate]], <math>r</math>, is assumed to be constant then
:{{center|<math> B(t,T) = e^{-r(T-t)}</math>}}

Note: <math>r</math> refers to the [[force of interest]], which is approximately equal to the effective annual rate for small interest rates. However, one should take care with the approximation, especially with larger rates and larger time periods. To find <math>r</math> exactly, use <math>r = \ln (1+i)</math>, where <math>i</math> is the effective annual interest rate.

When valuing European options written on stocks with known dividends that will be paid out during the life of the option, the formula becomes:

:{{center|<math> C(t) - P(t) + D(t) = S(t) - K \cdot B(t,T)</math>}}

where <math>D(t)</math> represents the total value of the dividends from one stock share to be paid out over the remaining life of the options, discounted to [[present value]].
 
We can rewrite the equation as:

:{{center|<math> C(t) - P(t) = S(t) - K \cdot B(t,T)\ - D(t)</math>}}

and note that the right-hand side is the price of a forward contract on the stock with delivery price <math>K</math>, as before.