Does Stock Options' Put-Call Parity Make Sense?

Put-call parity formula is for European options, which can be only exercised at expiration.

Put-cal parity is based on the no-arbitrage pricing theory, which says whenever there is an arbitrage opportunity, there will be enough of that arbitrage trading to make the underlying assets' prices move to where there is no arbitrage opportunity.

No-arbitrage pricing theory says two equal things must have the same values. Otherwise, one can sell the expensive and buy the cheap to make a risk-free profit.

Formula

Options can be priced based on the put-call parity formula which is the following:

C + PV(x) = P + S

Where C is the price of call, PV(x) is the present value of strike price, P is the price of put, and S is the spot price of

For example, XYZ stock spot price is $100, 1-year risk-free rate is 1%, for 1-year call and put options with the same strike price of 101, the following formula holds.

C + PV(101) = P + S

C - P = 100 - 101/(1 + 0.01) = 0 

Derivation of the Put-Call Parity Formula

Two equal things have equal price. We need to see C + PV(x) equal P + S to prove C + PV(x) = P + S.

Using the previous example, we can see that at the options' expiration date, their values are always the same regardless.

XYZ Spot Price at expiration	C + PV(x)	P + S
96	0 + 101	5 + 96
97	0 + 101	4 + 97
98	0 + 101	3 + 98
99	0 + 101	2 + 99
100	0 + 101	1 + 100
101	0 + 101	0 + 101
102	1 +101	0 + 102
103	2 + 101	0 + 103
104	3 + 101	0 + 104

In Reality

On August 9, 2021, SPY closed at 442.13.

2021/9/10 call for SPY at 442 was $6.71, and the put was 6.35$.

1-month treasury rate is 0.04%. 0.04%/12 is the 1-month risk-free rate, which is 0.000033333.

C + PV(x) = P + S

C + PV(x) = 6.71 + 442/(1 + 0.000033333) = 450.70

P + S = 6.35 + 442.13 = 448.48

450.70 - 448.48 = 2.22

I guess this $2.22 is the extra fee for the right to exercise the option earlier rather than at the expiration date. SPY options are American-style options, which can be exercised prior to expiration dates.