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# Put-call parity

In financial mathematics, put-call parity defines a relationship between the price of a call option and a put option - both with the identical strike price and expiry. No assumptions other than a lack of arbitrage in the market are made in order to derive this relationship.

An example, using stock options follows, though this may be generalised. Specifically consider a call option and a put option with the same strike K for expiry at the same date T on some share. Suppose the share has value S on expiration.

First consider a portfolio that consists of one put option and one share. This portfolio has value:

``` [itex] \{^{K\ if\ S<=K\ (the\ put\ has\ value\ (K-S)\ and\ the\ share\ has\ value\ S)}_{S\ if\ S>=K\ (the\ put\ has\ value\ 0\ and\ the\ share\ has\ value\ S)}[/itex]
```

Now consider a portfolio that consists of one call option and K bonds that each pay (with certainty) at time T. This portfolio has value:

``` [itex] \{^{K\ if\ S<=K\ (the\ call\ has\ value\ 0\ and\ the\ bonds\ have\ value\ K)}_{S\ if\ S>=K\ (the\ call\ has\ value\ S-K\ and\ the\ bonds\ have\ value\ K)}[/itex]
```

Notice that, whatever the final share price S is at time T, each portfolio is worth the same as the other. This implies that these two portfolios must have the same value at any time t before T. To prove this suppose that, at some time t, one portfolio were cheaper than the other. Then one could purchase (go long) the cheaper portfolio and sell (go short) the more expensive. Our overall portfolio would, for any value of the share price, have zero value at T. We would be left with the profit we made at time t. This is known as a risk-less profit and represents an arbitrage opportunity.

Thus the following relationship exists between the value of the various instruments at a general time t:

[itex] C(t) + K.B(t,T) = P(t)+S(t) [/itex]

where

C(t) is the time-t value of the call
P(t) is the time-t value of the put
S(t) is the time-t value of the share
and B(t,T) is the time-t value of a bond that pays at T.

If the bond interest rate is assumed to be constant, with value r, B(t,T) is equal to [itex] e^{-r(T-t)}[/itex].

Using the above, and given the (fair) value of any three of the call, put, bond and stock prices one can compute the (implied) fair value of the fourth.

Other arbitrage relationships Note that there are several other (theoretical) properties of option prices which may be derived via arbitrage considerations. These properties define price limits, the relationship between price, dividends and the risk free rate, the appropriateness of early exercise, and the relationship between the prices of various types of options. See, for example, Thayer Watkins Arbitrage Relationships for Options (http://www.sjsu.edu/faculty/watkins/arb.htm).

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