Arbitrage, Put-Call Parity, and the No-Free-Lunch Principle
Summary
The discussion addresses whether a portfolio can earn more than the risk-free rate in every market state. It distinguishes that goal from simply earning a positive return: a risk-free bond already does so, while arbitrage means a superior return without risk under the stated assumptions. The central lesson is that a consistent no-arbitrage market does not permit a riskless excess return.
Put-call parity offers a way to illustrate the concept. If an option is mispriced relative to its theoretical value, a portfolio combining options and the underlying asset can be constructed to exploit the discrepancy, with payoffs designed to remain positive under the assumed conditions. The example depends on the pricing relationship being violated and on assumptions that make the trades feasible. The material is introductory and does not specify transaction costs, short-sale constraints, funding, or execution frictions, all of which can undermine an apparent opportunity in practice.
Key ideas
- Arbitrage means earning more than the risk-free return without exposure to market risk.
- Under no-arbitrage assumptions, a riskless portfolio cannot systematically outperform the risk-free rate.
- Put-call parity can identify option prices inconsistent with the underlying asset and related options.
- A theoretical arbitrage construction relies on mispricing and assumptions about feasible trading.
Tags
Full text
# Portfolio Strategies Project # Portfolio Strategies Project My first assignment for my Quantitative Finance Masters is to design a portfolio that theoretically makes money under any market movement. I am also asked to state all necessary assumptions. What I'm investigating: I wrote a C++ application that generates a payoff diagrams at maturity with any combination of financial derivatives. I have been researching various well known option strategies (such as straddle, bear-spread, strangle etc). The problem I am facing: All of these strategies only work under certain predictions about the market. None of them work for all market movement. Question: Under suitable assumptions, is it possible to design a portfolio that theoretically makes money under any market movement? I suppose what I am really asking: Does there exist theoretical arbitrage opportunities under certain market assumptions? I am not looking for anything overly complex here. I have very little finance knowledge. If this topic is deemed too broad to answer, I would appreciate some direction toward particular readings. Edit: I am still looking for more readings on possible models. ## Answer by SmallChess (score 2, accepted) https://quant.stackexchange.com/a/19550 If I understand the question correctly, you're asked to invent a strategy that if under some unrealistic assumptions, your strategy will always earn at a rate higher than the risk-free rate without any risk. This is important because simply buying a risk-free zero-coupon-bond will make money for you, but this is not arbitrage. One of the simplest assumptions is the Put-call parity. If you can assume your call option is always cheaper than it's theoretical value, you can take a long position on the cheaper call option and a short position on the put option. You can cover up your investment by short selling your stock. If you draw a payoff diagram, your portfolio will always generate positive profits. There is a PDF document that may be helpful. ## Answer by SRKX (score 0) https://quant.stackexchange.com/a/19551 Reading your comment on Student T's answer, it could be possible that your professor is expecting you to come up with the simple fact that, under assumptions of no arbitrage, there is now way to generate a return superior to the risk-free rate without taking any risk. Note that making money is probably not the right terminology, making on average more money than the risk-free bond is probably what he meant, and in all market conditions would mean without risk. This is a typical question you ask to beginners in finance in general, to generate the concept that there is no free lunch.
Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.