Pay off for portfolio consisting of long call and short put both struck at K? Arbitrage construction? I'm very new to financial maths. Please can I get some help on this question, thanks!
Q: Write down the payoff of a put option with a strike of $K$. What is the payoff for a portfolio consisting of a long call and a short put both struck at $K$? Can you construct an arbitrage from this portfolio?
Here's what I have so far:
$$\operatorname{Put}(T) = \max \{ K - S(T), 0 \}$$
$$\operatorname{Call}(T) = \max\{ S(T) - K, 0 \}$$
Payoff of the portfolio is $$\operatorname{Call}(T) - \operatorname{Put}(T) = S(T) - K$$
Is this correct? If not where am I going wrong?
Also can someone explain whether an arbitrage can be constructed? I've so far only had practice in an FX environment.

As discussed in the comments, your portfolio payoff is correct.
However, it's impossible to tell whether there it is possible to construct an arbitrage portfolio or not. Often, when one is given prices for the stock, a call and a put striking at $K$ with the same maturity, and a risk-free rate $r$, then one wishes to use put-call parity to determine whether an arbitrage portfolio is possible. Put-call parity is given by:
$$C-P = S - e^{-rt}K$$
Without knowing the values in the expression above, it's impossible to tell whether one can construct an arbitrage.
