Proving the put call parity

In my course notes on the put-call parity, the proof is presented by going over two inequalities, namely $$\text{RHS} > \text{LHS}$$ implies arbtirage and $$\text{RHS} < \text{LHS}$$ implies arbitrage. Therefore, they conclude, $$\text{RHS} = \text{LHS}$$.

This strategy is legit, but I have the feeling the following proof is more straightforward.

$$\textbf{Lemma 1 (law of one price):}$$ If two portfolios have the same profit at maturity time $$T$$, then for all prior times $$t the price of the portfolio's must be equal.

$$\textbf{Proof:}$$ The proof can easily be done by deriving arbitrage by contradiction.

$$\textbf{Theorem (put-call parity)}:$$ Let $$P_0$$ be the price of a European put with strike $$K$$ and maturation date $$T$$. Let $$C_0$$ be the price of a European call with same parameters as the put, and $$r$$ be a risk-free rate. Let $$S_0$$ be the price of a stock at $$t=0$$. Then $$S_0 + P_0 = D(r)K + C_0,$$ where $$D(r)$$ is the discount of the risk-free bank account.

$$\textbf{Proof:}$$ Work out that the portfolios $$\{\text{own a put, stock}\}$$ and $$\{D(r)K\text{ in risk-free bank, own a call}\}$$ make the same profit at time $$T$$. Then by lemma 1 at all times $$t they must be worth the same, so too for $$t=0$$.

Is there something wrong with this proof?

• Looks fine to me. It's basically a summarized version of what you said there was in your notes – Slade Dec 8 '19 at 20:35
• Side note that may help you: Don't be too much clouded by the formalism. Put/call parity is a very easy concept. It's basically saying that if you have option to buy something (and money in the bank to cover the cost) you are in the same situation as if you already had the thing, and had an option to "return it" for the same price. The put call parity just states this obvious thing in a form of equation. – airguru Dec 8 '19 at 22:34

There are usually two ways to write proofs of equalities (like put-call parity) in quantitative finance.

1. By replication,

2. by constructing arbitrage.

Both of these are actually the same, since the first one is done by making, say, two portfolios, $$A$$ and $$B$$, and showing that they have the same outcome at time $$t=T$$. Then, by argument of LOOP (law of one price), one can argue that the two portfolios must be priced the same for all $$t\leq T$$.

But note that the LOOP is actually just a corollary of the no-arbitrage assumption. So the two methods of proof are just arguments by no-arbitrage assumption.

So, the proof posted in the question is correct, yet you might come across a proof in a textbook that argues directly from the no-arbitrage assumption in the following way:

1. suppose $$S_0 + P_0 > D(r)K + C_0$$, and derive an arbitrage position and,

2. suppose $$S_0 + P_0 < D(r)K + C_0$$, and derive an arbitrage position.

Conclude $$S_0 + P_0 = D(r)K + C_0$$.

Although the argument by LOOP is often shorter, sometimes it is preferable to argue directly from a no-arbitrage assumption.