Using Put-Call Parity to Identify a Risk-Free Options Arbitrage

Summary

The document presents an options-arbitrage exercise involving European calls and puts with the same strike and expiration. Given a stock price, interest rate, and equal quoted option prices, the question is how to identify a risk-free trade using put-call parity. The accepted response compares the call-minus-put position with the stock-minus-discounted-strike position, which have matching expiration payoffs under the parity relationship.

If one side costs less than the other, the suggested strategy is to buy the cheaper side and sell the more expensive one. The initial price difference is the arbitrage profit, while the matching terminal payoffs offset at expiration. The response describes the decision rule but does not calculate the discounted strike or show which side is mispriced for the stated inputs, since the maturity is not specified. The conclusion also assumes standard European options, consistent contract terms, and the ability to trade the positions at the quoted prices without frictions or constraints.

Key ideas

• Put-call parity equates the call-minus-put position with stock minus the present value of the strike.
• Compare the market prices of the two equivalent payoff packages to locate a possible mispricing.
• Buy the cheaper package and sell the more expensive package when their prices diverge.
• The response does not identify the trade direction for the example because it omits a maturity value.
• The arbitrage argument assumes matching option terms and frictionless execution.

Tags

# Risk-free investment strategy for european call and put option


# Risk-free investment strategy for european call and put option












I have some trouble solving the following question:

We have an european call and put option (with the same maturity date $T$ en strike $E=10$). The stock price now is $S=11$ and we use a continuous compound interest of $r=0.06$. Determine, using the put-call parity, an investment strategy to accomplish a risk-free profit based on the arbitrage principle if both options have value $V=2.5$

I cannot figure out how to approach this problem. The put-call parity alone does not seem to solve this problem. Help is very much appreciated.

## Answer by TLP (score 1, accepted)

https://quant.stackexchange.com/a/9353

The left hand side $(C-P)$ of the put-call partity equation provides the same pay-off as the right hand side $(S-K\times e^{-rT})$. Determine (by filling in the numbers) which part of the equation is relative cheap e.g. $(C-P) < (S-K\times e^{-rT})$. If this is the case, sell the $(S-K\times e^{-rT})$ and use the funds from selling to buy $(C-P)$. The pay-off from $C-P$ can be used to settle $(S-K\times e^{-rT})$ at maturity, the profit from initial selling and buying is the profit.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.