Put-Call Parity

Formula

C − P = S e^{−qT} − K e^{−rT}

q = 0 (no dividends):   C − P = S − K e^{−rT}
gap = (C − P) − (S e^{−qT} − K e^{−rT})

gap = 0  → parity holds at these inputs
gap > 0  → the call is rich to the put, or the forward is cheap, relative to this identity
The identity is for European options. American options do not obey it as an equality.

What it constrains

• You can back a forward out of the two premiums, which is what the smile tool does before it reads a smile off strikes. A bad forward makes every implied volatility look skewed.
• It forces several Greeks into pairs. Differentiating in S gives Δ_call − Δ_put = e^{−qT} (delta). Differentiating in σ gives equal vegas. Differentiating twice in S gives equal gammas.
• Rho is the sensitivity to the r that discounts K in this identity.

Pitfalls

• European, same strike, same expiry, same underlying. An American put can be worth more than parity allows, because early exercise has value. A gap on American prices is not automatically an arbitrage.
• Dividends have to be in q, or the identity is the wrong one. A discrete cash dividend is not a continuous yield. Using q = 0 on a stock that pays a dividend shows a gap that is just the dividend.
• Costs. A gap smaller than the round-trip spread, fees and borrow is not a trade. The function returns a price difference, not an executable profit.

Python reference implementation

Requires standard library only. Tested before publication; the generator re-executes it on every rebuild. MIT licensed. Download put-call-parity.py · All snippets

from math import exp

def parity_gap(call, put, S, K, T, r, q=0.0) -> float:
    """Call minus put, minus the European forward value S*exp(-qT) - K*exp(-rT).

    Zero when put-call parity holds. Not defined as an equality for American options.
    """
    if S <= 0 or K <= 0 or T < 0:
        raise ValueError("S and K must be positive and T non-negative")
    return float((call - put) - (S * exp(-q * T) - K * exp(-r * T)))

FAQ

Why does parity not mention volatility?

The call-minus-put package pays S_T − K at expiry no matter what path volatility takes. Its present value is the prepaid forward minus the discounted strike. Volatility changes the call and the put by the same amount, so it drops out of the difference.

What does a non-zero gap mean?

At these inputs the two premiums are not consistent with one forward. Causes include a dividend yield you left at zero, American early exercise, stale quotes, or a genuine dislocation larger than the cost of putting the package on. The sign says which side is rich relative to the identity. It does not by itself say there is a trade.