Valuation of Contingent Claims for CFA Level II

What this chapter covers

Valuation of Contingent Claims: topics in the order to study them

1. 1Binomial Option Pricing ModelIt builds the core idea of replication and risk-neutral pricing that every later topic reuses.
2. 2Put-Call Parity and No-Arbitrage RelationshipsLearn it early because it is short, links calls to puts, and checks your binomial answers.
3. 3Black-Scholes-Merton ModelIt is the continuous-time version of the same idea, so it is easier once binomial pricing is clear.
4. 4Option Greeks and Delta HedgingGreeks come from the BSM model and explain how option values move when inputs change.
5. 5Implied Volatility and Volatility SmileIt reverses the BSM model to find volatility, and shows where the model's assumptions break.
6. 6Black Model for Futures and Interest Rate OptionsIt adapts BSM to futures and forwards, so you need BSM first.
7. 7Binomial Models for Interest Rate OptionsRate trees do not depend on the Black model, so this order is a choice, not a requirement. They need comfort with binomial steps and discounting, and they tie into Fixed Income.
8. 8Valuing Swaptions and Interest Rate Derivative OptionsIt combines swaps, the Black model and rate options, so it comes last.

How to prepare Valuation of Contingent Claims

1. Learn the no-arbitrage idea first. Be able to explain why an option can be priced without forecasting the underlying.
2. Practise the one-period and two-period binomial tree by hand until the steps (up and down values, risk-neutral probability, discount, work backward) are automatic.
3. Do the BSM and Black models with your exam calculator. Write down the inputs from the vignette before you compute, and note which model fits the underlying.
4. Build a one-page table of the Greeks: what each measures, its sign for calls and puts, and how it changes with price and time.
5. Practise interpreting results: delta hedge ratios, what a volatility smile implies, and whether parity is violated and how to exploit it.
6. Work full item sets under timing, about 3 minutes per question (about 12 minutes per item set of 4 questions), and review each wrong answer by naming the model or step you missed.
7. In the last week, redo calculation questions from memory and read your error log.

Common mistakes in Valuation of Contingent Claims

• Using real-world probabilities in a binomial tree instead of risk-neutral probabilities.

  Fix: For pricing, always compute the risk-neutral probability from u, d and the risk-free rate, unless the question clearly asks for an expected value under real-world probabilities.

• Forgetting to check early exercise for American options.

  Fix: At each node, compare the continuation value with the exercise value and take the higher, then continue backward.

• Choosing the wrong model for the underlying.

  Fix: Identify the underlying first. Use the futures or forward price in the Black model, and the asset price in BSM.

• Mixing up the signs and sizes of the Greeks.

  Fix: Tie each Greek to a question: delta for price change, gamma for delta change, vega for volatility, theta for time, rho for rates. Then check signs for long calls and puts.

• Applying put-call parity with the wrong timing or without adjusting for cash flows.

  Fix: Adjust the asset price for the present value of any dividends or carrying benefits and discount the strike for the correct time to expiry.

• Reading the volatility smile as proof that options are mispriced.

  Fix: Explain that the smile reflects market views about return distributions that differ from the model's assumptions, and that implied volatility is a market quote, not an error.

Last-day revision: Valuation of Contingent Claims

• ✓Binomial: u and d give up and down values; risk-neutral probability π = (1 + r − d) ÷ (u − d), with r as the per-period rate.
• ✓Option value today = [π × up payoff + (1 − π) × down payoff] ÷ (1 + r), working backward from the end.
• ✓American options need an early-exercise check at each node of the tree.
• ✓Put-call parity (no dividends): c + X ÷ (1 + r)^T = p + S.
• ✓BSM assumes constant volatility, lognormal prices and a constant risk-free rate; call and put use N(d1) and N(d2).
• ✓Call delta is between 0 and 1; put delta is between −1 and 0.
• ✓Gamma is highest for at-the-money options and is the same for a call and put with the same terms.
• ✓Vega is positive for long calls and puts; theta is usually negative for long options.
• ✓Delta-gamma hedging reduces risk from larger price moves than delta alone.
• ✓Implied volatility is the volatility that makes the model price equal the market price.
• ✓A volatility smile shows that the constant-volatility BSM assumption does not hold in the market.
• ✓Black model uses the futures or forward price as the underlying input; a swaption is an option on a swap rate.

Valuation of Contingent Claims in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

• CFA Level IPricing and Valuation of Options
• CMA FinalOptions
• CA FinalDerivatives Analysis and Valuation
• Actuarial Core Principles (IAI)Black-Scholes derivative-pricing model