Replicate an option before assigning probabilities

Learning goals

• Calculate the one-period European call value.
• Explain why 0.50 applies to this case.
• Identify the case-specific assumption or failure mode before taking action.

Draft · approximately 75 minutes, excluding further revision.

Shared case and decision context

A stock is 100 and can be 120 or 90 in one period. A European call strike is 100 and the period risk-free gross return 1.05. No dividends, costs or short-sale constraints apply.

Economic or professional mechanism

A binomial replication combines delta shares with borrowing or lending to match option payoffs in both states. The risk-neutral probability is chosen to reproduce the current stock price under discounting, not to estimate the actual likelihood of an up move. No arbitrage requires the risk-free growth factor to lie between the down and up factors.

Apply, challenge and communicate

The up-state probability is 0.5, delta 2/3 and call value 9.5238. In richer models delta changes as price, time and volatility change, so a hedge needs rebalancing and retains model and transaction-cost risk. European and American exercise rights cannot be treated identically.

Use put-call parity to test an option package

For European options on a non-dividend stock with the same strike and expiry, C+PV(K)=P+S under standard no-arbitrage assumptions. If S100, PV(K)95, put4 and call12, the left side107 exceeds the right104 by3. The overpriced package could be sold and the matching underpriced package bought if financing, shorting and execution are feasible. American exercise, dividends or different settlement terms require appropriate modifications. A displayed theoretical gap is not a guaranteed retail profit after bid-ask and borrowing costs.

Work backward through two option periods

A two-period binomial option tree evaluates terminal exercise payoffs first, then discounts risk-neutral expected values one node at a time. An American option also compares continuation value with immediate exercise value at each eligible node. In a European call example S100,u1.2,d.8,K100,R1.05 per period, q=.625. Terminal call payoffs are44,0,0; price is44×.625²/1.05²=15.5896. Do not discount an already discounted node value twice or use the physical up probability merely because it seems plausible.

Value an interest-rate option with local discounting

A rate option payoff can depend on the rate observed at a reset date and be paid at a later date. A caplet protecting a borrower has a payoff proportional to notional×accrual fraction×max(reference rate−strike,0), subject to its contract. With notional1million, delta.5, rate6% and strike4%, the end-period payoff is10,000. In a short-rate tree use each node’s relevant discount factor when rolling values backward; a single average rate can misprice path-dependent claims. A floorlet reverses the rate difference to protect against low rates.

Know what Black–Scholes–Merton assumes

The classical Black–Scholes–Merton setup uses a diffusion with continuous price paths, a specified constant volatility and risk-free rate, frictionless continuous trading and European exercise, with dividends handled through the selected extension. Real prices can jump, volatility can vary and discrete rehedging incurs costs. The model can be a useful benchmark without those assumptions being literally true. Do not interpret its precision as proof of no model risk or apply the non-dividend formula unchanged to a dividend-paying stock.

Interpret the option as a state-dependent financed position

For a non-dividend European call the BSM decomposition is S×N(d1)−PV(K)×N(d2). The first term resembles a stock position and the second a financed strike-related claim; the weights depend on model parameters. N(d1) is the call delta in this setup, while N(d2) has a risk-neutral exercise-probability interpretation, not the same role. Neither should be called a verified physical forecast. As the stock price changes, both the replicating holdings and financing needs change.

Match equity and currency option carry

A continuous dividend yield reduces the prepaid stock value used in BSM from S to Sexp(−qT). For a currency call quoted domestic per foreign, the foreign risk-free rate plays the dividend-yield role and the domestic rate discounts the strike. Thus currency pricing needs two interest rates and a clearly stated quote orientation. Reversing the quote also reverses which currency is the underlying. Check expiry, strike units and rate compounding before using a model output in a hedge decision.

Use a futures-based option model consistently

Black’s European futures-option formula discounts the expected option payoff using a futures price as the underlying model input. In the standard setup call=e^(−rT)[F×N(d1)−K×N(d2)]. It does not require replacing F with spot while retaining all other terms. Futures options still need contract-specific exercise and settlement checks, and American exercise may require a different method. The futures position has its own mark-to-market and margin mechanics, which are distinct from the option premium.

Scale rate-option formulas by annuity or accrual

A Black caplet valuation uses a forward rate, strike, volatility and the appropriate discounted notional-accrual scale. A European swaption commonly uses the forward swap rate as underlying and the fixed-leg annuity as the scaling factor under its quotation convention. A payer swaption gives the right to pay fixed and receive floating, helping protect against higher swap rates; a receiver gives the opposite right. Formula units matter: a rate quoted in percent cannot be inserted as if it were a decimal. Alternative normal or shifted models may be needed for market conventions and negative-rate settings.

Invert price to implied volatility without forecasting certainty

Implied volatility is the volatility input that makes a selected option model reproduce an observed market option price, with the other inputs fixed. It is model- and contract-dependent. Different strikes and maturities can produce a smile or surface rather than one constant value. It is neither a guaranteed future realized volatility nor a model-free forecast. Compare volatility positions using sensitivity, maturity and strike exposures; a high implied value can reflect tail demand, supply constraints or model assumptions as well as expected variation.

Formula and assumptions

q=(R−d)/(u−d); C=(q*Cu+(1−q)*Cd)/R; delta=(Cu−Cd)/(Su−Sd)

Case analysis with explicit assumptions

1. A stock is 100 and can be 120 or 90 in one period. A European call strike is 100 and the period risk-free gross return 1.05. No dividends, costs or short-sale constraints apply.
2. Method: q=(R−d)/(u−d); C=(q*Cu+(1−q)*Cd)/R; delta=(Cu−Cd)/(Su−Sd)
3. Case result: 9.52. The up-state probability is 0.5, delta 2/3 and call value 9.5238. In richer models delta changes as price, time and volatility change, so a hedge needs rebalancing and retains model and transaction-cost risk. European and American exercise rights cannot be treated identically.

Errors to catch

• Do not replace the stated assumptions with an unstated shortcut. Reconcile units, timing and the claim being valued or duty being assessed.
• The verified real-world up frequency is not a defensible conclusion for these case facts.

Original knowledge check

Answer every question before checking. Review explanations after submitting; retry starts a new attempt.

Sources and coverage status

These original lessons are drafts. Public outlines are used to check scope; the official learning-outcome text and proprietary question banks are not reproduced here. Follow the source for exam-year requirements and updates.

• Reference · www.cfainstitute.org
• Reference · www.cfainstitute.org