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CFA® Level I · Derivatives · free lesson
Valuing a Derivative Using a One-Period Binomial Model
The one-period binomial model is the simplest setting in which an option can be priced exactly by no arbitrage. This module builds a riskless portfolio from the underlying and the option using the hedge ratio, derives risk-neutral probabilities, values calls and puts with them, and explains why real-world probabilities and risk preferences drop out of the price while volatility does not.
What you will learn
1. Describe the one-period binomial model and value an option by constructing a riskless hedge portfolio using the hedge ratio.
- Two end-of-period states: S_0 u and S_0 d, with d < 1 + r < u.
- Hedge ratio h = (c_u - c_d) / (S_u - S_d): units of underlying per option.
- Long h units of the underlying and short one call is riskless and must earn r.
- Call value: c_0 = h S_0 - (h S_d - c_d) / (1 + r).
- A market price different from the model value creates an arbitrage.
2. Calculate risk-neutral probabilities and use them to value calls and puts in a one-period binomial model.
- Risk-neutral probability of an up move: pi = (1 + r - d) / (u - d).
- Option value = risk-neutral expected payoff discounted at the risk-free rate.
- The same pi values calls, puts and any other derivative on the same underlying and period.
- No arbitrage requires d < 1 + r < u, so that 0 < pi < 1.
- Binomial call and put values satisfy put-call parity.
3. Explain why risk-neutral valuation does not depend on real-world probabilities or risk preferences, and how the volatility of the underlying affects option values in the binomial model.
- Real-world probabilities, expected returns and risk aversion do not enter the option price.
- They are already embedded in the underlying's price, which the replication uses.
- Price as if risk-neutral: risk-neutral probabilities and discounting at the risk-free rate.
- Never combine real-world probabilities with risk-free discounting, or risk-neutral probabilities with a risky rate.
- A wider spread between u and d (higher volatility) raises both call and put values.
Module at a glance
- One period, two outcomes: S_0 u or S_0 d, with d < 1 + r < u.
- Hedge ratio h = (c_u - c_d) / (S_u - S_d); h units of the underlying per written call is riskless.
- Risk-neutral probability pi = (1 + r - d) / (u - d).
- Option value = [pi x up payoff + (1 - pi) x down payoff] / (1 + r), for calls and puts alike.
- Real-world probabilities, expected returns and risk aversion do not enter the price.
- A wider u-d spread (higher volatility) raises both call and put values.
- If a market price differs from the model value, trade the option against the hedge portfolio for a riskless profit.
This free module has 3 lessons, 14 practice questions with full explanations.