Chapter 2: FAQs 29
portfolio will always be worth−K, a guaranteed amount.
Since this amount is guaranteed we can discount it back
to the present. We must have
C−P−S=−Ke−r(T−t).This is put-call parity.
Another way of interpreting put-call parity is in terms
of implied volatility. Calls and puts with the same strike
and expiration must have the same implied volatility.
The beauty of put-call parity is that it is a model-
independent relationship. To value a call on its own
we need a model for the stock price, in particular its
volatility. The same is true for valuing a put. But to
value a portfolio consisting of a long call and a short
put (or vice versa), no model is needed. Such model-
independent relationships are few and far between in
finance. The relationship between forward and spot
prices is one, and the relationships between bonds and
swaps is another.
In practice options don’t have a single price, they have
two prices, a bid and an offer (or ask). This means
that when looking for violations of put-call parity you
must use bid (offer) if you are going short (long) the
options. This makes the calculations a little bit messier.
If you think in terms of implied volatility then it’s much
easier to spot violations of put-call parity. You must
look for non-overlapping implied volatility ranges. For
example, suppose that the bid/offer on a call is 22%/25%
in implied volatility terms and that on a put (same strike
and expiration) is 21%/23%. There is an overlap between
these two ranges (22–23%) and so no arbitrage opportu-
nity. However, if the put prices were 19%/21% then there
would be a violation of put-call parity and hence an easy
arbitrage opportunity. Don’t expect to find many (or,
indeed, any) of such simple free-money opportunities in