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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.

Jose Avilez
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user898975
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    How can we tell if an arbitrage is possible without knowing any prices? Also we need the riskless rate in order to determine the fair forward. There's also a question of dividends, but let's assume there aren't any. – lulu Jan 21 '22 at 00:18
  • Your formulas may benefit from a bit of MathJax – Jose Avilez Jan 21 '22 at 00:25
  • @lulu That's all the information that I was given. I think the question refers to whether or not an arbitrage can be constructed, not to actually construct it but I'm not sure. Does this change things? – user898975 Jan 21 '22 at 00:32
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    No, it doesn't. Arbitrage depends on prices. Perhaps these instruments are being priced correctly, in which case the system would be free of arbitrage. – lulu Jan 21 '22 at 00:37
  • As a side note, the payoff of your portfolio is correct. – Jose Avilez Jan 21 '22 at 00:38
  • @JoseAvilez Thank you! I slightly guessed the LHS (Call(T) - Put(T)). Is it -Put(T) because it is being sold? – user898975 Jan 21 '22 at 00:58
  • @user898975 Yes. To "short" or "write" an option means to sell it. – Jose Avilez Jan 21 '22 at 01:00
  • Ok, thank you!! – user898975 Jan 21 '22 at 01:01

1 Answers1

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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.

Jose Avilez
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