Put-Call Parity and Forward Delivery Prices

# Finding circumstances for price of call = price of put


# Finding circumstances for price of call = price of put












Here is a problem in Hull's book and the given solution:

- My approach was to compute the profit $\pi = \pi_{SP} + \pi_{LC}$ (short put, long call).

One can show that $\pi = \pi_{SP} + \pi_{LC} = S_T - K + p_{SP} - p_{LC}$, where $p_{SP}, p_{LC}$ are the prices of the options.

So if we want $p_{SP} = p_{LC}$, then we must have $\pi = S_T - K$

Is that right?

- I'm not quite sure I understand the last two sentences of the answer in the solutions manual. What is the difference between forward price and delivery price? What I think the answer means:

Because the payoff from LC and SP (long call and short put) is $S_T - K$, because the payoff from a fwd contract is $S_T - F$ and because a fwd contract is LC and SP combined, $p_{SP} = p_{LC}$ when $K = F$?

Am I understanding right? How exactly does the conclusion follow if so? If not, what exactly does the solutions manual mean to say?

## Answer by Louis. B (score 1, accepted)

https://quant.stackexchange.com/a/21557

You are mostly right, I don't really get what you don't understand. The answer in the book is quite clear, but let me put it that way :

Selling a put and buying a call on the same underlying $S$ with same maturity and same stike $K$ is always equivalent to a long position in a forward contract on $S$ with delivery price $K$. The easiest way to see that is to draw the payoff of such a strategy in a simple graph, you should get a 45° line crossing the $x$-axis at $K$. Then we know that a forward contract with delivery price $F$ (the forward rate) costs nothing. Then if you want your strategy to cost nothing, you should set $K=F$.

It seems that it's the beginning of the book, but you may want to look at the Put-Call parity. Without details, it is a relation that links the price of a call with the price of a put on the same underlying, same maturity and same strike. The Put-Call parity can be expressed as follows (with $r$ the risk-free rate) : \begin{equation} C-P=S-Ke^{-rt} \end{equation}

With $F = Se^{rt}$ you can see that : \begin{equation} C-P=e^{-rt}(F-K) \end{equation}

So we have $F=K$ $\iff$ $C=P$. The demonstration should be in the Hull's book.