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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.

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    $\begingroup$ When are the premiums paid? Also when is the 19000 paid? Thx $\endgroup$
    – dm63
    Mar 22, 2019 at 3:16
  • $\begingroup$ All at the exact same moment. Theoretically no friction and perfectly liquid. $\endgroup$
    – MonteCarloSims
    Mar 22, 2019 at 3:33
  • $\begingroup$ Now, or in one month? $\endgroup$
    – dm63
    Mar 22, 2019 at 3:34
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    $\begingroup$ You are indeed somehow extracting the equity funding cost (which on top of risk-free funding rate could include dividend yield, repo margins) from call put parity. $\endgroup$
    – Quantuple
    Mar 22, 2019 at 11:18
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    $\begingroup$ You are welcome just note that the CP parity actually writes $C_0 - P_0 = DF(0,T) ( F(0,T) - S_0 )$ where $F(0,T)$ is the equity forward price (related to equity funding cost) and $DF(0,T)$ the discount factor (related to cash or collateral funding cost) $\endgroup$
    – Quantuple
    Mar 22, 2019 at 18:22

1 Answer 1

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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.

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  • $\begingroup$ I agree that there is no arbitrage, as the formula holds perfectly: 3.32 + 189.62 = 2.94 + 190 || 3.32 = 2.94 + .38 I believe you are correct because by paying 19,000 I am essentially lending at the risk free rate. (Easier for me to understand as an arbitrageur would need to pay 38 to borrow the 19,000 if the trade were reversed - Selling Put, Buying Call, Selling Stock.) Thank you for taking the time to answer this question. $\endgroup$
    – MonteCarloSims
    Mar 22, 2019 at 14:07

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