# 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:

$$(3.32-2.94) \cdot 100 = \38$$

$$\dfrac{\38}{\190\cdot100} = 0.2\%$$

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.

• When are the premiums paid? Also when is the 19000 paid? Thx
– dm63
Mar 22, 2019 at 3:16
• All at the exact same moment. Theoretically no friction and perfectly liquid. Mar 22, 2019 at 3:33
• Now, or in one month?
– dm63
Mar 22, 2019 at 3:34
• 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. Mar 22, 2019 at 11:18
• 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) Mar 22, 2019 at 18:22