What it means
An option's price depends on several inputs. These are the current price of the underlying asset, the strike price, the time to expiry, the interest rate, any dividends, and the volatility, which measures how much the price is expected to move.
The models combine these into a single value. The central idea is replication.
Because an option's payoff can be copied by holding a mix of the underlying asset and borrowing, its value must equal the cost of that mix, or someone could make a risk-free profit. This argument removes the need to guess investors' appetite for risk.
The binomial model builds the answer step by step. It assumes the price can move up or down by a set amount in each short period, then works backward from the final payoffs to today's value.
The Black-Scholes-Merton model reaches a formula by assuming prices move continuously, and it is widely used for European options, which can only be exercised at expiry. Volatility is the one input that cannot be observed directly.
Traders therefore often work backward from a market price to find the implied volatility, which shows what the market expects. Higher volatility makes both calls and puts more valuable, because a wider range of outcomes increases the chance of a large payoff.
A related result is put-call parity, which links the price of a call and a put with the same strike and expiry to the price of the underlying asset. If the prices drift out of line, traders can lock in a risk-free gain, which pushes them back together.
Analysts use the relationship to check that quoted prices are consistent. The models have limits.
Real markets have sudden jumps, changing volatility and trading costs that the simple models ignore, and the results are only as good as the assumptions. Finance teams use them as a guide for valuing employee share options, hedges and other derivatives, often with adjustments and professional judgement.
In practice
Real-world examples.
Example
A technology company grants employees share options and must record their value as an expense. Its finance team uses a pricing model to estimate the fair value on the grant date and spreads that amount over the vesting period.
Example
A treasurer is quoted $4.20 for a currency option by two banks and $5.10 by a third. She uses a pricing model with the market volatility to check which quote is reasonable before she accepts one.
Example
An analyst notices that a stock's options imply much higher volatility than the stock's recent history. She investigates whether the market is expecting news, because the option prices contain information about future uncertainty. She shares the finding with the portfolio team.
Formula
Calculation
One-period binomial value of a call = [probability of up move x payoff if up + probability of down move x payoff if down] / (1 + interest rate)
Probability of up move (risk-neutral) = (1 + interest rate - down factor) / (up factor - down factor)
A share is worth $100 today and will be either $120 or $80 in one period. To keep the arithmetic clean, assume an interest rate of 0%. The call has a strike of $100, so its payoff is 120 - 100 = $20 if the price rises and $0 if it falls. The up factor is 1.20 and the down factor is 0.80, so the probability of an up move = (1 + 0 - 0.80) / (1.20 - 0.80) = 0.20 / 0.40 = 0.5. Call value = (0.5 x 20 + 0.5 x 0) / 1 = $10. A real model uses many small steps, but the same logic of working backward from the payoffs applies.Case study
Seen in the real world.
Blue Meridian Aerospace is a fictional company that issued share options to its senior managers. The finance director had to value the grant for the accounts and chose a binomial model.
The inputs were the share price, the strike price, the time to expiry, an estimate of volatility and the interest rate. The model gave a value of $8.40 per option, and with 500,000 options granted the total value was 500,000 x 8.40 = $4,200,000, to be expensed over the vesting period.
In this illustrative story the auditors asked the team to justify its volatility estimate, because a small change in that one input moved the answer a lot. The finance director kept a short paper explaining the choice and a table showing the value under higher and lower volatility.
Watch out
Common mistakes.
- Treating a model price as a fact, when it is an estimate that depends on assumptions.
- Using historic volatility without checking whether it fits the future period the option covers.
- Applying a model built for European options to American options without any adjustment.
Questions
People also ask.
What is the Black-Scholes model?
It is a formula for the fair value of a European option based on the price, the strike, time, volatility, interest rate and dividends.
Why is volatility so important?
Because it measures the uncertainty about the future price, and higher uncertainty raises the value of an option.
Do companies use these models in their accounts?
Yes, they are commonly used to value share options and other derivatives for financial reporting.
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