Editing Put–call parity (section)

==Derivation==
We will suppose that the put and call options are on traded stocks, but the [[underlying]] can be any other tradeable asset.  The ability to buy and sell the underlying is crucial to the "no arbitrage" argument below.

First, note that under the assumption that there are no [[arbitrage]] opportunities (the prices are [[arbitrage-free]]), two portfolios that always have the same payoff at time T must have the same value at any prior time.  To prove this suppose that, at some time ''t'' before ''T'', one portfolio were cheaper than the other. Then one could purchase (go long) the cheaper portfolio and sell (go short) the more expensive one.  At time ''T'', this long/short portfolio excluding cash would, for any value of the share price, have zero value (all the assets and liabilities have canceled out).  The cash profit we made at time ''t'' is thus a riskless profit, but this violates our assumption of no arbitrage.

We will derive the put-call parity relation by creating two portfolios with the same payoffs ([[static replication]]) and invoking the above principle ([[rational pricing]]).

Consider a call option and a put option with the same strike ''K'' for expiry at the same date ''T'' on some stock ''S'', which pays no dividend.  We assume the existence of a [[Bond (finance)|bond]] that pays 1 dollar at maturity time ''T''.  The bond price may be random (like the stock) but must equal 1 at maturity.

Let the price of ''S'' be S(t) at time t.  Now assemble a portfolio by buying a call option ''C'' and selling a put option ''P'' of the same maturity ''T'' and strike ''K''.   The payoff for this portfolio is ''S(T) - K''.  Now assemble a second portfolio by buying one share and borrowing ''K'' bonds.  Note the payoff of the latter portfolio is also ''S(T) - K'' at time ''T'', since our share bought for ''S(t)'' will be worth ''S(T)'' and the borrowed bonds will be worth ''K''.

By our preliminary observation that identical payoffs imply that both portfolios must have the same price at a general time <math>t</math>, the following relationship exists between the value of the various instruments:

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

Thus given no arbitrage opportunities, the above relationship, which is known as '''put-call parity''', holds, and for any three prices of the call, put, bond and stock one can compute the implied price of the fourth.

In the case of dividends, the modified formula can be derived in similar manner to above, but with the modification that one portfolio consists of going long a call, going short a put, and going long ''D(T)'' bonds that each pay 1 dollar at maturity ''T'' (the bonds will be worth ''D(t)'' at time ''t''); the other portfolio is the same as before - long one share of stock, short ''K'' bonds that each pay 1 dollar at ''T''.  The difference is that at time ''T'', the stock is not only worth ''S(T)'' but has paid out ''D(T)'' in dividends.