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CFA Level I · CFA Level I Exam

Pricing and Valuation of Options for CFA Level I

Option pricing finds the fair value of a call or put today. You start from the payoff at expiry, then use no-arbitrage tools: put-call parity, bounds, the binomial tree with risk-neutral probabilities, and Black-Scholes-Merton. The Greeks measure how that value changes when inputs move.

What this chapter covers

This chapter explains what an option is worth and why. You begin with payoffs at expiry and moneyness. Then you add no-arbitrage links, such as put-call parity and price bounds. Finally you move to models that give a value today: the one-period and two-period binomial tree and the Black-Scholes-Merton model.

The second half is about risk and use. The Greeks (delta, gamma, vega, theta, rho) tell you how option value reacts to changes in the underlying price, volatility, time and rates. Delta hedging uses delta to cut price risk. Implied volatility turns the pricing idea around: you take the market price and solve for the volatility it implies. Option strategies then combine these ideas.

The chapter links to several other areas. It builds on the basics of forwards, futures and swaps in Derivatives. It uses the time value of money and the normal distribution from Quantitative Methods. It also supports Portfolio Construction and Fixed Income, where embedded options and risk management appear. Learn the logic here and you will find derivative questions elsewhere easier.

Derivatives and Risk Management is a smaller topic area than some others, but option questions are very testable because they are mostly rule-based and numerical. A single parity or binomial calculation can be solved in under 90 seconds once the method is automatic. With three-option MCQs and no penalty for wrong answers, direction questions on the Greeks and on how inputs move value let you eliminate two options by logic alone. The chapter rewards short, focused practice more than long reading.

Pricing and Valuation of Options: topics in the order to study them

  1. 1Option Payoffs and MoneynessEverything else rests on the payoff at expiry, so you must be fluent in profit and loss for buyers and sellers of calls and puts first.
  2. 2Put-Call ParityIt is your first no-arbitrage relationship and links calls, puts, the underlying and a risk-free bond in one equation.
  3. 3Factors Affecting Option Value and BoundsOnce parity is clear, you learn which inputs raise or lower value and the minimum and maximum prices, which sets up the models.
  4. 4Binomial Option Pricing ModelIt prices an option step by step with risk-neutral probabilities and shows the logic of replication before any formula with the normal distribution.
  5. 5Black-Scholes-Merton ModelIt is the continuous-time version of the same idea, so it is easier to follow after the binomial tree.
  6. 6Option Greeks and Delta HedgingThe Greeks come from the pricing model inputs, so you study them once you know what drives value.
  7. 7Implied Volatility and Option StrategiesIt closes the chapter by reversing the model to find volatility and by combining options into strategies using all earlier tools.

How to prepare Pricing and Valuation of Options

Treat this chapter as a set of small methods you can repeat quickly, backed by a few direction rules you simply know.

  1. Draw payoff and profit diagrams for long and short calls and puts until you can sketch each in seconds, and write the expiry payoff formulas: call = max(0, S − X), put = max(0, X − S).
  2. Learn put-call parity in one form, c + X ÷ (1 + r)^T = p + S, and practise rearranging it to find any one of the four items, then spot which side is cheap.
  3. Make a table of the factors (underlying price, strike, time, volatility, risk-free rate, dividends) against the effect on calls and puts, and test yourself until it needs no thought.
  4. Practise one-period binomial trees: find u and d, the risk-neutral probability, the payoffs at the up and down nodes, then discount the expected value at the risk-free rate. Then do a two-period tree working backward.
  5. Use the Black-Scholes-Merton model as a set of inputs and a reading of N(d1) and N(d2); do not memorise long derivations, and focus on what each input does.
  6. Work through Greeks and delta hedging with small numbers: compute hedge ratios, and know that gamma and vega are positive for long options, while theta is usually negative.
  7. Finish with mixed timed sets of three-option MCQs, and review each wrong answer for the rule you missed. Use your BA II Plus or HP 12C for powers and discounting, for example 1.05 y^x 0.5 on the BA II Plus.

Common mistakes in Pricing and Valuation of Options

  • Mixing up payoff and profit

    Fix: Compute the payoff first, then subtract the premium paid for a buyer or add the premium received for a seller.

  • Using the real-world probability in the binomial model

    Fix: Price with the risk-neutral probability from u, d and r, and use the real-world probability only if the question asks for something other than the option value.

  • Forgetting to discount in parity and trees

    Fix: Always write X ÷ (1 + r)^T or discount at (1 + r) per period before comparing values.

  • Getting the direction of input effects wrong for puts

    Fix: Reason through the payoff: a higher strike helps a put and hurts a call, and a higher rate lowers a put's value.

  • Mixing delta signs and hedge direction

    Fix: Write the position delta with its sign first, then choose the offsetting position in the underlying.

  • Treating implied volatility as a forecast from history

    Fix: Remember it is backed out of the current option price and reflects the market's expectation, not a historical measure.

Last-day revision: Pricing and Valuation of Options

  • Long call payoff at expiry is max(0, S − X); long put payoff is max(0, X − S).
  • A call is in the money when S > X; a put is in the money when S < X.
  • Put-call parity: c + X ÷ (1 + r)^T = p + S, for European options on an asset with no cash flows.
  • Higher underlying price raises call value and lowers put value.
  • Higher volatility raises both call and put values.
  • Longer time to expiry generally raises option value; this is not guaranteed for deep in-the-money European puts.
  • Higher risk-free rate raises call value and lowers put value.
  • Binomial risk-neutral probability of an up move: π = (1 + r − d) ÷ (u − d).
  • Option value today is the discounted risk-neutral expected payoff, found by working backward through the tree.
  • Delta of a call is between 0 and 1; delta of a put is between −1 and 0.
  • Gamma measures the change in delta; it is highest for at-the-money options near expiry.
  • Implied volatility is the volatility that makes the model price equal the market price.

Pricing and Valuation of Options practice questions

Pricing and Valuation of Options in other exams

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

Pricing and Valuation of Options: frequently asked questions

Do I need to memorise the Black-Scholes-Merton formula for CFA Level I?

Focus on understanding the inputs and how each affects value rather than on long derivations. You should know what N(d1) and N(d2) represent and the direction of every input effect. Check the current curriculum for what is expected.

Is put-call parity always true?

It holds for European options with the same strike and expiry on the same underlying, when there are no arbitrage opportunities and no cash flows on the asset. Adjust for dividends or other cash flows if the question includes them.

How do I find the risk-neutral probability in a binomial tree?

Use π = (1 + r − d) ÷ (u − d), where u and d are the up and down factors. The down probability is 1 − π. Then discount the expected option payoff at the risk-free rate.

How much time should I spend on the Greeks?

Spend enough to know the sign and meaning of each Greek and to do a simple delta hedge. Questions are usually direct, so quick, repeated practice works better than a long study session.