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CFA® Level II · Derivatives · free lesson
Valuation of Contingent Claims
Options are valued by replication: a combination of the underlying and borrowing or lending that reproduces the option's payoff must cost the same as the option. This module builds that idea from one- and two-period binomial trees, extends it to interest rate options, and then moves to the continuous-time Black-Scholes-Merton model and its variants for dividends, currencies, futures, caps and swaptions. It finishes with the Greeks, which are what traders actually use to hedge, and with implied volatility, the market's way of quoting option prices.
What you will learn
1. Value European and American options with one- and two-period binomial models, and identify and exploit option mispricing.
- Hedge ratio h = (c+ - c-) / (S+ - S-); the option equals h shares plus borrowing (call) or minus shares plus lending (put).
- Risk-neutral probability pi = (1 + r - d) / (u - d); real-world probabilities do not affect the price.
- Two-period trees are solved backward; American options compare exercise and continuation value at each node.
- American calls on non-dividend-paying shares are never exercised early; American puts may be.
- Overpriced call: sell the call, buy h shares, borrow; underpriced call: buy the call, short h shares, lend.
2. Value interest rate options such as caplets and floorlets using a binomial interest rate tree.
- Use a calibrated binomial interest rate tree with risk-neutral probabilities of 0.5.
- Caplet (interest rate call) payoff = NA x max(0, r - X) x accrual; floorlet = NA x max(0, X - r) x accrual.
- Payoffs paid in arrears are discounted one period at the rate set at the expiry node.
- Roll back node by node, discounting at each node's own one-period rate.
- A cap is a portfolio of caplets; long cap + short floor at the same strike = pay-fixed swap.
3. State the assumptions of the Black-Scholes-Merton model and interpret it as a dynamically rebalanced leveraged position in the underlying.
- BSM assumes lognormal prices, constant volatility and interest rates, frictionless continuous trading, no arbitrage and European exercise.
- Call = N(d1) units of the underlying financed by borrowing X exp(-rT) N(d2): a leveraged long position.
- Put = short N(-d1) units of the underlying plus lending X exp(-rT) N(-d2).
- N(d1) is the delta; N(d2) is the risk-neutral probability that the call expires in the money.
- Replication needs continuous rebalancing, which fails when prices jump or liquidity disappears.
4. Apply BSM to options on dividend-paying equities and currencies, and the Black model to options on futures, caplets and swaptions.
- Dividend yield: replace S with S exp(-delta T); higher dividends lower calls and raise puts.
- Garman-Kohlhagen: the foreign rate acts as the yield; S is domestic per foreign currency.
- Black model: c = exp(-rT)[F N(d1) - X N(d2)], with no r in d1.
- Caplet = NA x accrual x discount factor to payment date x [FRA N(d1) - X N(d2)].
- Payer swaption = call on the forward swap rate, valued with the swap's annuity factor; payer - receiver = forward pay-fixed swap.
5. Interpret delta, gamma, vega, theta and rho, construct a delta hedge, and explain gamma risk and implied volatility.
- Call delta is between 0 and 1, put delta between -1 and 0; hedge with -(portfolio delta) / (option delta) options.
- Gamma is highest for at-the-money options near expiry; delta-gamma approximation: change = delta x dS + 0.5 x gamma x dS^2.
- Short gamma loses on large moves either way and must buy high and sell low to rebalance; long gamma pays theta.
- Vega is positive for long options and largest at the money; rho is positive for calls, negative for puts.
- Implied volatility equates model and market prices; the equity index skew prices fat left tails.
Module at a glance
- Options are valued by replication: hedge ratio h = (c+ - c-) / (S+ - S-), risk-neutral probability pi = (1 + r - d) / (u - d).
- Solve two-period trees backward and test early exercise of American options at every node; American calls on non-dividend shares are never exercised early.
- Interest rate options use a calibrated rate tree with 0.5 probabilities, node-specific discount rates and payoffs usually paid in arrears.
- BSM assumes lognormal prices, constant volatility and rates, and frictionless continuous trading; a call is a leveraged long position in the underlying.
- Adjust BSM for yields (dividends, foreign rates); use Black for futures options, caplets (discount to payment date) and swaptions (annuity factor).
- Delta hedges need options = -(portfolio delta) / (option delta); gamma measures how quickly the hedge goes stale.
- Short options are short gamma and short vega; implied volatility skews show the market pricing fat tails.
This free module has 5 lessons, 17 practice questions with full explanations.