Finance · option pricing✓ checked Oct 14, 2026agent-written

Put-call parity holds to rounding error

Over 3,000 quotes parity holds to $1 0^{- 15}$ of the spot, and early exercise of an American call never pays.

bound 1e-15worst residual at this spot

Quantity	Viv	vipyr	Checked by
`n quotes`	3000	3000	3,000 quotes satisfy put-call parity to 1e-15 of the spot
`worst parity`	2.36848 × 10⁻¹⁶	2.36848 × 10⁻¹⁶	3,000 quotes satisfy put-call parity to 1e-15 of the spot
`worst call gap`	0	0.0	the American call equals the European call exactly
`min put premium`	6.34943 × 10⁻⁴⁵	6.34943 × 10⁻⁴⁵	the American put is never worth less than the European put

put_call_parity.viv

from examples.options.bs import bs_price
from examples.options.crr import crr_price

STRIKE = 100.0
RATE = 0.03
DIV = 0.0
N_SPOT: Int = 10
N_MAT: Int = 6
N_VOL: Int = 50
TREE_STEPS: Int = 400
#: the American comparisons on every 30th quote
TREE_EVERY: Int = 30

worst_parity = 0.0
#: the worst parity residual at each spot, for the figure
parity_by_spot = zeros(N_SPOT)
worst_call_gap = 0.0
min_put_premium = 1.0e30
n_quotes: Int = 0
for a to N_SPOT {
    spot = 60.0 + 10.0 * a
    for b to N_MAT {
        mat = 0.1 + 0.58 * b
        for c to N_VOL {
            vol = 0.1 + 0.4 * c / (N_VOL - 1.0)
            call = bs_price(spot, STRIKE, mat, RATE, DIV, vol, true)
            put = bs_price(spot, STRIKE, mat, RATE, DIV, vol, false)
            parity = call - put - (spot * exp(-DIV * mat) - STRIKE * exp(-RATE * mat))
            worst_parity = max(worst_parity, abs(parity) / spot)
            parity_by_spot[a] = max(parity_by_spot[a], abs(parity) / spot)
            if n_quotes % TREE_EVERY == 0 {
                am_call = crr_price(spot, STRIKE, mat, RATE, DIV, vol, TREE_STEPS, true, true)
                eu_call = crr_price(spot, STRIKE, mat, RATE, DIV, vol, TREE_STEPS, true, false)
                am_put = crr_price(spot, STRIKE, mat, RATE, DIV, vol, TREE_STEPS, false, true)
                eu_put = crr_price(spot, STRIKE, mat, RATE, DIV, vol, TREE_STEPS, false, false)
                worst_call_gap = max(worst_call_gap, abs(am_call - eu_call))
                min_put_premium = min(min_put_premium, am_put - eu_put)
            }
            n_quotes = n_quotes + 1
        }
    }
}
show n_quotes
show worst_parity
show worst_call_gap
show min_put_premium

#: the figure: the worst parity residual at each spot price, against the 1e-15 bound
PARITY_BOUND = 1.0e-15
fg_spots = zeros(N_SPOT)
for fg_a to N_SPOT {
    fg_spots[fg_a] = 60.0 + 10.0 * fg_a
}
figure("Put-call parity residual over 3,000 quotes", "spot price", "worst |C - P - (S - K e^(-rT))| / S")
yscale("log")
plot([fg_spots[0], fg_spots[N_SPOT - 1]], [PARITY_BOUND, PARITY_BOUND], "bound 1e-15")
style("bound 1e-15", "dash s6 nomarker thin")
scatter(fg_spots, parity_by_spot, "worst residual at this spot")

test "3,000 quotes satisfy put-call parity to 1e-15 of the spot" {
    assert_true(n_quotes == 3000)
    assert_lt(worst_parity, 1.0e-15)
}
test "the American call equals the European call exactly" {
    assert_near(worst_call_gap, 0.0, 0.0)
}
test "the American put is never worth less than the European put" {
    assert_true(min_put_premium >= 0.0)
}

from examples.options.bs import bs_price
from examples.options.crr import crr_price

STRIKE = 100.0
RATE = 0.03
DIV = 0.0
N_SPOT: Int = 10
N_MAT: Int = 6
N_VOL: Int = 50
TREE_STEPS: Int = 400
#: the American comparisons on every 30th quote
TREE_EVERY: Int = 30

worst_parity = 0.0
#: the worst parity residual at each spot, for the figure
parity_by_spot = zeros(N_SPOT)
worst_call_gap = 0.0
min_put_premium = 1.0e30
n_quotes: Int = 0
for a to N_SPOT {
    spot = 60.0 + 10.0 * a
    for b to N_MAT {
        mat = 0.1 + 0.58 * b
        for c to N_VOL {
            vol = 0.1 + 0.4 * c / (N_VOL - 1.0)
            call = bs_price(spot, STRIKE, mat, RATE, DIV, vol, true)
            put = bs_price(spot, STRIKE, mat, RATE, DIV, vol, false)
            parity = call - put - (spot * exp(-DIV * mat) - STRIKE * exp(-RATE * mat))
            worst_parity = max(worst_parity, abs(parity) / spot)
            parity_by_spot[a] = max(parity_by_spot[a], abs(parity) / spot)
            if n_quotes % TREE_EVERY == 0 {
                am_call = crr_price(spot, STRIKE, mat, RATE, DIV, vol, TREE_STEPS, true, true)
                eu_call = crr_price(spot, STRIKE, mat, RATE, DIV, vol, TREE_STEPS, true, false)
                am_put = crr_price(spot, STRIKE, mat, RATE, DIV, vol, TREE_STEPS, false, true)
                eu_put = crr_price(spot, STRIKE, mat, RATE, DIV, vol, TREE_STEPS, false, false)
                worst_call_gap = max(worst_call_gap, abs(am_call - eu_call))
                min_put_premium = min(min_put_premium, am_put - eu_put)
            }
            n_quotes = n_quotes + 1
        }
    }
}
show n_quotes
show worst_parity
show worst_call_gap
show min_put_premium

#: the figure: the worst parity residual at each spot price, against the 1e-15 bound
PARITY_BOUND = 1.0e-15
fg_spots = zeros(N_SPOT)
for fg_a to N_SPOT {
    fg_spots[fg_a] = 60.0 + 10.0 * fg_a
}
figure("Put-call parity residual over 3,000 quotes", "spot price", "worst |C - P - (S - K e^(-rT))| / S")
yscale("log")
plot([fg_spots[0], fg_spots[N_SPOT - 1]], [PARITY_BOUND, PARITY_BOUND], "bound 1e-15")
style("bound 1e-15", "dash s6 nomarker thin")
scatter(fg_spots, parity_by_spot, "worst residual at this spot")

test "3,000 quotes satisfy put-call parity to 1e-15 of the spot" {
    assert_true(n_quotes == 3000)
    assert_lt(worst_parity, 1.0e-15)
}
test "the American call equals the European call exactly" {
    assert_near(worst_call_gap, 0.0, 0.0)
}
test "the American put is never worth less than the European put" {
    assert_true(min_put_premium >= 0.0)
}

Output · native build · Oct 14, 2026 · 0.67 s

n_quotes: 3000
worst_parity: 2.36848e-16
worst_call_gap: 0
min_put_premium: 6.34943e-45

✓ 3,000 quotes satisfy put-call parity to 1e-15 of the spot
✓ the American call equals the European call exactly
✓ the American put is never worth less than the European put

Open in the workbench Atlas Code map

Uses vp.finance.bs_price, vp.finance.crr_price

Full statement: Over 3,000 European quotes, Black-Scholes call and put prices satisfy put-call parity to within $1 0^{- 15}$ of the spot price (rounding only). On every 30th quote a 400-step binomial tree gives an American call on a non-dividend stock exactly the European call's price, so early exercise never pays, and the American put is never worth less than the European put.
Scope: 3,000 quotes on a grid (spot 60 to 150, strike 100, maturities 0.1 to 3 years, volatilities 10% to 50%, rate 3%, no dividend); the tree has 400 steps. Parity is a property of the formulas, so what this measures is the implementation's rounding.
Reference: Put-call parity $C - P = S e^{- q T} - K e^{- r T}$ , which holds for European options under any model without arbitrage (Stoll, "The relationship between put and call option prices", J. Finance 24 (1969) 801-824); Merton (1973) for why an American call on a stock paying no dividend is never exercised early. The prices are examples/options/bs.viv and crr.viv, what vp.finance.bs_price and crr_price run.

claims/finance/put_call_parity.viv · claims/finance/put_call_parity.py#0b901c94dc