Learning Module 10
Valuing a Derivative Using a One-Period Binomial Model
Key Outcomes Summary & Practice Problems
What you must be able to do
Curriculum Year: 2026
Explain how to value a derivative using a one-period binomial model.
Describe the concept of risk neutrality in derivatives pricing.
1 · Why a Model Is Needed for Options
Forwards can be priced via pure no-arbitrage/replication without any assumption about the underlying's future price distribution.
Options have asymmetric payoffs — pricing them requires a model for how the underlying's price might evolve. The one-period binomial model is the simplest such model: the underlying can move to only one of two possible prices over the period.
S1u = RuS0 (up gross return) | S1d = RdS0 (down gross return), with Rd < (1+r) < Ru
2 · Building the Hedge Ratio
Combine a position in the option with a position in the underlying so that the portfolio's value is identical in both the up and down scenarios — eliminating risk.
The hedge ratio (h*) is the number of underlying shares needed per option to construct this risk-free combination.
h* = (c1u − c1d) / (S1u − S1d)
A positive hedge ratio (selling the call, buying the underlying) hedges a short call; a negative hedge ratio for a put means buying both the put and the underlying together.
Once hedged, V1u = V1d = V1 — a certain (risk-free) future value.
3 · Solving for the Option Price
V0 = h×S0 − c0 | V1 = h×S1u − c1u = h×S1d − c1d
c0 = h×S0 − V1(1 + r)−1
The hedged portfolio must earn exactly the risk-free rate — otherwise an arbitrage exists (e.g., constructing a synthetic risk-free asset that out-earns the actual risk-free rate).
This procedure works identically for puts, just substituting put payoffs (p1u, p1d) for call payoffs.
4 · Risk-Neutral Pricing
The hedging approach is mathematically equivalent to computing the discounted expected payoff using risk-neutral probabilities (π) rather than actual/real-world probabilities.
Neither the real-world probability of an up/down move nor the underlying's expected return is needed to price the option — only its volatility (captured by Ru and Rd) and the risk-free rate.
π applies to the underlying's binomial process, not to any specific option — the same π values a call, a put, or any other derivative on that same underlying process.
π = (1 + r − Rd) / (Ru − Rd)
c0 = [π × c1u + (1 − π) × c1d] / (1 + r)T
5 · What Affects (and Doesn't Affect) the Binomial Price
Factor | Affects Option Price? |
|---|---|
Risk-free rate (r) | Yes — used for discounting and embedded in π |
Spread between Ru and Rd (volatility proxy) | Yes — wider spread raises both call and put values |
Actual/real-world probability of an up move (q) | No — irrelevant to the no-arbitrage price |
Investor risk aversion / risk preferences | No — pricing is "risk-neutral" precisely because the perfect hedge removes risk exposure |
A higher risk-free rate raises π (the risk-neutral probability of an up move) and lowers PV of the expected payoff for puts — both effects reduce put values; the reverse logic raises call values.
Increasing Ru and/or decreasing Rd widens the potential price range (higher implied volatility) — raising the value of bothcalls and puts, consistent with the volatility factor from option-pricing fundamentals.