## Put-Call Parity

Put-Call parity is a simple result connecting the prices of puts and calls in a model-independent way via the forward price.

Consider the three graphs below, showing independently the payoff at expiry of a vanilla call, a vanilla put, and a forward contract. We can see that the payoff of a long call plus the payoff of a short put will precisely overlap the forward contract payoff (assuming they have the same strike and expiry…).

This can be seen from the algebra as well:

\begin{align*}&space;C_{\rm&space;call}(T)&space;-&space;C_{\rm&space;put}(T)\&space;&&space;=&space;\big(&space;S(T)&space;-&space;K&space;\big)^+&space;-&space;\big(&space;K&space;-&space;S(T)&space;\big)^+&space;\\&space;&&space;=&space;S(T)&space;-&space;K&space;\\&space;&&space;=&space;F(T,T)&space;\end{align*}

If two portfolios have the same payoff, it’s a fundamental rule of derivatives pricing that they must have the same price, which gives the fundamental put-call parity relationship

$C_{\rm&space;call}(t)&space;-&space;C_{\rm&space;put}(t)\&space;=&space;Z(t,T)\cdot\Big(&space;F(t,T)&space;-&space;K\Big)$

We can take the BS expressions for put and call prices given in a previous post and observe that they obey the relationship [since $\inline&space;\Phi(x)&space;+&space;\Phi(-x)&space;=&space;1$], but importantly this result is model independent. As long as there is a forward contract available with the strike and expiry of the options, then this result MUST hold in ANY model.

One implication of this is that if we use a model to determine the price of a call option, put-call parity fixes the price of the put option to be

$C_{\rm&space;put}(t)&space;=&space;C_{\rm&space;call}(t)&space;-&space;Z(t,T)\cdot\Big(&space;F(t,T)&space;-&space;K\Big)$

We can actually use this to improve our code. Out-of-the-money options typically require more paths to converge in Monte Carlo simulations because they rely on the detailed behaviour of a few paths that travel a long way. However, we can calculate in-the-money prices quickly and due to put-call parity these will lock the price of the matching out-of-the-money options.

Put-call parity is also an important test of the implementation of a model – if the prices that we are getting out don’t obey this relationship then we’ve done something seriously wrong [as long as both prices have converged!]. An equivalent statement is that the implied BS volatility of matching puts and calls should be exactly the same in any model, since the implied vol is the BS vol that would produce the given price for a put/call, and this is locked by put-call parity.

We can get some other similar relationships between put and call prices that I’ll look at in the future. Also note that put-call parity holds for european-style options, but not for many more complicated path-dependent options or those with early exercise features like American options.