An option has no cash flows to discount and no risk-adjusted rate you can observe. So the entire valuation apparatus is built on a different question: what portfolio of the underlying and borrowing produces exactly the option's payoff?
Every model here follows from one idea. If two positions produce identical future cash flows in every state of the world, they must cost the same today. That is the law of one price. The arbitrageur enforces it under two constraints: do not use your own money, and do not take any price risk.
Five assumptions make this work: replicating instruments are identifiable and investable, there are no market frictions (no transaction costs or taxes), short selling is allowed with full use of proceeds, the underlying follows a known statistical distribution, and borrowing and lending occur at a known risk-free rate.
At expiration, a call is worth and a put is worth , where X is the exercise price. Before expiration, the option also carries time value, the market's valuation of upside potential relative to limited...
Common mistakes
- Discounting one period on a two-period tree. With r = 5%, dividing a $10.705 expected payoff by 1.05 gives $10.20 instead of the correct $9.71. The discount factor is .
- Using real-world probabilities. A stem that supplies a subjective probability of an up move is baiting you. Recompute from u, d, and the risk-free rate every time.
- Applying the expectations formula to American puts. The closed-form two-period expression is only valid when early exercise is never optimal. For American puts you must roll backward node by node and compare against exercise value at each one.
Bottom line
- Hedge ratio: h = (c⁺ − c⁻) / (S⁺ − S⁻); non-negative for calls, negative for puts
- Risk-neutral probability: π = [FV(1) − d] / (u − d); never a real-world forecast
- European value = present value of the risk-neutral expected payoff, discounted at the risk-free rate over the full number of periods
- American options require backward induction with an exercise-versus-continuation check at every node; calls on non-dividend stock are never exercised early
Exam shortcut
Compute first, always, before touching payoffs. If the vignette hands you a probability, ignore it. Count periods before discounting; the single most common trap answer on a two-period tree is the value discounted only once. Whenever the word "American" and the word "put" appear together, expect the answer to require a node-by-node exercise check and expect the distractor to be the European value.
The full lesson (about 3,772 words, 25 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.
Learning objectives
- contingent claims
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