Skip to content
All library documents

Put-Call Parity and the Risk-Free Return on a Synthetic Position

Article Quant Q&A · Author: MonteCarloSims
Read the original

Summary

The document clarifies how to interpret the return from a put-call parity position built from a short call, a long put, and shares of the underlying. The example compares the option price difference with the share purchase cost and finds a return close to the quoted risk-free rate, prompting the question of whether parity should imply zero return or an arbitrage.

The response explains that the position requires substantial capital for the holding period. The small cash difference from the options is earned against the net amount invested, so a return near the risk-free rate is consistent with parity rather than evidence of a free profit. The example uses quoted midpoint prices and a simplified annualization; it does not analyze executable bid-ask prices, financing, dividends, or transaction costs. Those details would matter when assessing an actual trade for arbitrage.

Key ideas

  • Put-call parity links the option combination to the underlying and a risk-free cash position.
  • A parity-consistent synthetic position can earn the risk-free rate over its holding period.
  • The return should be measured against the capital actually invested, not just the option premium difference.
  • A return close to the risk-free rate does not by itself establish an arbitrage opportunity.
  • Midpoint quotes and simplified annualization omit trading and financing frictions.

Tags

OptionsEquitiesArbitrage
Full text
# Should Put/Call Parity result in Zero Return or the Risk-Free Rate?


# Should Put/Call Parity result in Zero Return or the Risk-Free Rate?












Sorry in advance if this is a basic question. I'm examining some potential at-the-money put/call arbitrage. What I found surprised me somewhat:

```
            Bid   Ask    Mid
ATM Call = 3.31 x 3.33 (3.32)
ATM Put  = 2.93 x 2.95 (2.94)

Expiration = ~ 1 Month

Underlying Stock Price = 190.00
```

The resulting Put/Call Parity return is equal to:

> (Sell Call, Buy Put, Buy 100 Shares of Underlying) $(3.32-2.94) \cdot 100 = \$38$ $\dfrac{\$38}{\$190\cdot100} = 0.2\%$ Annualized Return = $0.2\%\cdot12=2.4\%$

This return is very close to the current stated treasury 'risk-free' rate of 2.48%

I would have expected the return on Put/Call Parity to be zero, however, since the combination of assets is risk-free at that point it would make sense that it pays exactly the risk-free rate.

Is it expected that I should see this risk-free rate of return or should I be seeing zero return?

Is this some other component of return - is this an arbitrage opportunity?

Am I merely seeing an algebraically extracted risk free rate from the put call parity formula?

$C_0+X*e^{-r*t} = P_0+S_0$

$3.32+190*e^{-0.024*(1/12)} = 2.94 + 190 = 192.94$

Thanks for any clarification.

## Answer by dm63 (score 4, accepted)

https://quant.stackexchange.com/a/44724

You should see the risk free rate as the return on the strategy. That’s because you actually have to invest money , namely usd 19000 minus usd 38, for the one month period. Hence, there is no arbitrage in the market data you observe.

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.