Option-chain structure

Option Pricing Bounds

Calculate European and American option upper and lower bounds, apply discount and dividend assumptions, and assess executable bound violations.

Mechanics & mathematicsHypothetical examples

What Is It?

Option pricing bounds constrain premiums using simple dominating or replicating portfolios. A European call cannot exceed the prepaid value of the delivered asset; its lower bound compares that value with the discounted strike. A European put uses the corresponding cash-versus-asset comparison.

These inequalities do not need a volatility forecast, but they do need explicit assumptions about dividends, discounting, exercise style and settlement. An observed out-of-bounds midpoint can be a data-quality warning rather than an executable opportunity.

How It Works

A European call has nonnegative terminal payoff and is dominated by receiving the underlying at expiry. Combining it with a bond that pays the strike dominates the prepaid underlying, which gives the lower bound. The analogous put construction compares the strike claim with the underlying claim.

American holders may exercise immediately, so a premium cannot fall below immediate intrinsic value in the frictionless exercisable market. Their upper bounds differ from the European prepaid bounds because exercise may occur before expiry. Negative rates or unusual carry can require different American upper bounds than the familiar nonnegative-rate formulas.

Across two European call strikes K₁ < K₂, a vertical spread pays between zero and K₂ − K₁ at expiry. Its present price must therefore lie between zero and that discounted amount. This slope bound complements absolute bounds and strike convexity.

Mathematical Formula

\max(0,A_0-KD)\le C_E\le A_0

For matched European vanilla contracts with nonnegative underlying, deterministic carry and discount factor D = exp(−rT), use prepaid forward value A₀ = S₀ exp(−qT) for deterministic proportional yield, or S₀ minus PV of known cash dividends.

\max(0,KD-A_0)\le P_E\le KD

European puts are bounded above by the discounted strike payment. These bounds assume feasible replication and compatible settlement; transaction costs create an executable band.

(S_0-K)^+\le C_A\le S_0,\qquad(K-S_0)^+\le P_A\le K

These familiar American bounds assume a nonnegative underlying, nonnegative rates and nonnegative dividends. Immediate exercise gives the lower bounds. Do not use the upper K bound unchanged with negative rates.

0\le C_E(K_1)-C_E(K_2)\le(K_2-K_1)D

Same expiry and other contract terms are required. For puts, reverse the strike-price difference: 0 ≤ P(K₂) − P(K₁) ≤ (K₂ − K₁)D.

Worked Example

Illustrative prices · not market data

Assume European options on a non-dividend-paying asset, S₀ = 100, K = 90, T = 1 and r = 5%. A call is offered at an executable ask of 13.

The discounted strike is 90 exp(−0.05) = 85.6106. The European call lower bound is 100 − 85.6106 = 14.3894.
Buy the call for 13, short the asset at a bid of 100, and invest 85.6106 to receive 90 at expiry. Initial gross proceeds are 1.3894.
At expiry the stock short, call and cash pay max(90 − Sₜ, 0), which cannot be negative. If total present-valued implementation and borrow costs are 0.40, the net initial credit is 0.9894.
If there is a dividend, its compensation on the stock short must be funded, reducing the appropriate lower bound and the available gap.

This is a lower-bound replication, not an instruction to exercise the European call immediately. If stock cannot be borrowed or the prices do not fill together, the apparent violation may remain inaccessible.

Execution Considerations

Check bids and asks in the direction of the proposed hedge. Use the interest rate available on actual collateral, include borrowing costs and confirm all cash-flow dates. Discount known dividend payments individually instead of using an unrelated average yield.

Reject mismatched multipliers, stale quotes and inconsistent currencies before treating a bound failure as an opportunity. A market maker's displayed price may not support the required quantity.

Risks & Limitations

Borrow constraints, interest-rate assumptions, unknown distributions, corporate actions and settlement conventions can invalidate an apparently universal bound. American early exercise adds cash-flow and assignment timing risk.

Even valid theoretical violations can be smaller than commissions, spreads and capital charges. Prices within the bounds can still be expensive or cheap under a model; that is a different claim from arbitrage.

Frequently Asked Questions

Can a European option trade below immediate intrinsic value?

Yes in some carry regimes, because immediate exercise is unavailable. Use discounted, dividend-adjusted European bounds rather than American exercise bounds.

Does a price inside the bounds mean fair value?

No. Bounds exclude some impossible prices; they do not choose a unique premium or validate a volatility forecast.

Are the same bounds valid for inverse crypto options?

Not automatically. Derive the payoff in its actual settlement currency and apply the appropriate numeraire and contract rules.

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References & Further Reading

Background sources for the concepts above. Apply the assumptions stated in this guide to each formula.

NYU Stern, Aswath Damodaran: Options Arbitrage

Educational material only; not investment advice or an offer to trade. Examples are hypothetical. Trading involves execution, financing and capital-loss risks; no return is guaranteed.