put-call parity equation

Question

I'm reading this book and I'm looking at page 4, and we are considering the case where $C_t - P_t - S_t$ is negative, which means that selling the call did not offset the cost of the stock and the put together. So, case 1), $S_T > K$, which means that the buyer of the call will exercise their option, so we will also have to give the buyer $K$ for the stock at time of maturity.

So, $C_t - P_t - S_t$ is money that we had to borrow in order to buy the cost of the stock and the put that was not offset by the money that we received by selling the call. So the interest at maturity of that money that we owe is $(C_t - P_t - S_t)e^{r(T-t)}$. To that money that we owe, we add the money that we owe to the contract buyer, since we are in case 1) where the strike is larger than the call price. So

$$(C_t - P_t - S_t)e^{r(T-t)} + K < 0$$

Is money that we owe. But on the reference, they put the opposite sign; $$(C_t - P_t - S_t)e^{r(T-t)} + K > 0$$ like it was profit!

Is this a mistake, or am I misunderstanding something?

Answer 1

I think this is where your logic goes wrong:

$(C_t − P_t − S_t)e^{r(T−t)} + K$

With reference to the above equation, you are saying that "...To that money that we owe, we add the money that we owe to the contract buyer.."

Yes, $(C_t − P_t − S_t)e^{ r(T−t)}$ is the money that we owe, but $K$ is not referring to money that we also owe the contract buyer. $K$ is the strike price, so it is the money we receive from the contract buyer at maturity when he exercises the option. Yes, the call option is in a losing position, but $K$ is not referring to the actual loss.

So, the equation is really saying that the money we borrowed to finance this strategy (plus the accrued interest) is less than the amount we ended up receiving when the contract buyer exercised his option and bought the stock from us at the strike price. A net gain will also result if $S_t < K$ as well, which demonstrates that an arbitrage opportunity exists when $C_t - P_t > S_t - Ke^{-r(T-t)}$.