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Financial Instruments, Markets and Conventions / A First Look at Options: Payoffs, Bounds, and the Binomial Model

No-Arbitrage Bounds and Put-Call Parity

Before a model is specified — before Black-Scholes, before the binomial tree, before any assumption about how stock prices move — no-arbitrage alone constrains option prices to lie within certain bounds. These constraints are valuable because they are model-free: they hold under any well-posed model and can be used to detect mispricing in the market without taking a view on volatility.
The most important constraint is put-call parity: for European options with the same strike and maturity on a non-dividend-paying stock, c+Ke−rT=p+S0c + Ke^{-rT} = p + S_0. This single equation links the entire European options market together. A market maker who prices a call must price the corresponding put consistently, or an arbitrageur will exploit the gap.
The second key result is that American calls on non-dividend-paying stocks are never optimally exercised early — they are therefore worth exactly as much as European calls. This surprising fact follows directly from the lower-bound inequality.
Learning objectives. After this note you should be able to:
  • state the six factors that affect option prices and the direction of their effect;
  • derive the upper and lower bounds for European call and put prices using portfolio replication arguments;
  • apply put-call parity to infer one option price from the other, or to identify an arbitrage;
  • prove that American calls on non-dividend-paying stocks should never be exercised early;
  • explain why American puts may be exercised early, and identify the conditions that make early exercise attractive;
  • extend the bounds and put-call parity to dividend-paying stocks.

In this lesson

  1. Six factors affecting option prices
  2. Notation and no-arbitrage assumptions
  3. Upper bounds
  4. Lower bounds for European options
  5. Put-call parity
  6. American calls on non-dividend-paying stocks
  7. American puts on non-dividend-paying stocks
  8. Effect of dividends

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