Boundary Conditions: What They Are and How They Work

Boundary conditions define the theoretical minimum and maximum prices an option can have based on the current price of the underlying asset, the strike (exercise) price, time to expiration, and whether the option is American or European. They provide intuitive limits and simple sanity checks for option pricing models.

Key takeaways

• An option’s price can never be negative — the absolute minimum is zero.
• Boundary values differ for calls and puts and for American versus European options because American options can be exercised early.
• For calls the price is capped by the underlying asset’s current price; for puts the upper bound is tied to the strike price (discounted for European puts).
• Boundary conditions remain useful as intuition and checks even after modern pricing models (binomial trees, Black‑Scholes) became standard.

Intuition behind boundary conditions

Boundary conditions express limits implied by no‑arbitrage reasoning and basic payoff properties:
* A call gives the right to buy the underlying. It cannot be worth more than owning the underlying itself (you’d never pay more for the option than for the asset).
* A put gives the right to sell the underlying for the strike price. If the underlying became worthless, the put’s maximum payoff equals the strike; the present value of that payoff limits the option’s price today.
* Early exercise possibility for American options typically raises their value relative to equivalent European options, so their bounds reflect that feature.

Common boundary inequalities (where S = current asset price, K = strike, r = risk‑free rate, T = time to expiration)

Lower bounds (no arbitrage):
* European call: C_E ≥ max(0, S − K e^(−rT))
* European put: P_E ≥ max(0, K e^(−rT) − S)
* American call: C_A ≥ max(0, S − K)
* American put: P_A ≥ max(0, K − S)

Upper bounds:
* Calls (European or American): C ≤ S
* European put: P_E ≤ K e^(−rT) (present value of K)
* American put: P_A ≤ K

Notes:
* K e^(−rT) is the present value of the strike using continuous discounting; with discrete compounding use K/(1+r)^T.
* For non‑dividend paying stocks, early exercise of an American call is generally suboptimal, so American and European call values often coincide; dividends change that conclusion.

Practical uses

• Quick sanity checks: If a computed option price violates these bounds, the model or inputs are wrong.
• Intuition for pricing behavior: how underlying price moves, time decay, interest rates, and early exercise rights affect limits on option value.
• Historical role: Before widespread use of binomial and Black‑Scholes models, traders used boundary reasoning to bracket reasonable option prices.

Limitations

• Boundary conditions are bounds, not exact prices. They do not replace full pricing models that account for volatility, dividends, and stochastic factors.
• Underlyings are theoretically unbounded upward, so practical upper limits for modeling are set by statistical or market considerations rather than absolute mathematical ceilings.

Boundary conditions are simple, robust constraints that improve understanding and help validate option valuations. Use them as quick checks and for intuition alongside a complete pricing model.