Option Pricing Explained: From Binomial Trees to Black-Scholes

The Million-Dollar Question: What Is This Option Worth?

Options give you the right to buy or sell at a fixed price. That right has value — but how much? This question launched an entire field of mathematics and won several Nobel Prizes.

The fundamental challenge: an option's payoff depends on the future stock price, which is random. You cannot simply take the expected payoff and call it the price (that would ignore risk). The breakthrough was realising you do not need to estimate expected returns at all. Instead, you can replicate the option's payoff using the stock and cash, and the cost of that replication is the price.

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The One-Step Binomial Model

Start simple. A stock is worth £100 today. In one period, it can go to either £120 (up) or £80 (down). What is a call option with strike £100 worth?

At expiry:

• If the stock goes up: payoff = £120 - £100 = £20
• If the stock goes down: payoff = max(£80 - £100, 0) = £0

The Replicating Portfolio

Can we construct a portfolio of stock and cash that exactly replicates these payoffs?

Hold ( \Delta ) shares and borrow ( B ) at the risk-free rate:

• Up state: ( 120\Delta + B(1+r) = 20 )
• Down state: ( 80\Delta + B(1+r) = 0 )

Solving: ( \Delta = 0.5 ) shares and ( B = \frac{-40}{1+r} ).

The option price equals the cost of this portfolio today:

[ C = 0.5 \times 100 + B = 50 - \frac{40}{1+r} ]

Notice something remarkable: the real-world probability of up vs down never appeared. The price depends only on the possible outcomes, the risk-free rate, and the current stock price.

Risk-Neutral Probabilities

We can rewrite this as an expected value under special "risk-neutral" probabilities:

[ C = \frac{1}{1+r}[p \cdot 20 + (1-p) \cdot 0] ]

where ( p = \frac{(1+r)S_0 - S_d}{S_u - S_d} ).

These are not real-world probabilities — they are mathematical constructs that make the pricing formula work. Under risk-neutral probabilities, all assets earn the risk-free rate on average. This is the foundation of risk-neutral pricing.

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Multi-Step Binomial Trees

Extend to multiple periods and you get a binomial tree. At each node, the stock goes up by factor ( u ) or down by factor ( d ). Work backward from expiry, computing the risk-neutral expected value at each node.

As you increase the number of steps, the binomial model converges to the Black-Scholes formula. This convergence is important — it means the discrete model is consistent with the continuous model.

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The Black-Scholes Formula

In 1973, Fischer Black, Myron Scholes, and Robert Merton derived a closed-form formula for European option prices:

[ C = S_0 N(d_1) - K e^{-rT} N(d_2) ]

where:

[ d_1 = \frac{\ln(S_0/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}, \quad d_2 = d_1 - \sigma\sqrt{T} ]

and ( N(\cdot) ) is the cumulative standard normal distribution.

What Each Term Means

• ( S_0 N(d_1) ): the expected stock cost, weighted by the probability of exercise
• ( K e^{-rT} N(d_2) ): the discounted strike price, weighted by the probability the option is in the money

The Five Inputs

Four inputs are known. Volatility is the tricky one — it is the subject of an entire field of research. See the volatility module for more.

Model Assumptions

Black-Scholes assumes:

• Stock prices follow geometric Brownian motion
• Constant volatility (empirically false — see volatility smile)
• Continuous trading with no transaction costs
• No dividends (extensions exist for dividends)
• Risk-free rate is constant

The model is "wrong" in the sense that these assumptions do not hold perfectly. It is useful in the way that Newton's laws are useful — accurate enough for most practical purposes, with known corrections for when precision matters.

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Implied Volatility

If you plug the market price of an option into Black-Scholes and solve backward for volatility, you get the implied volatility (IV).

[ C_{\text{market}} = \text{BS}(S_0, K, T, r, \sigma_{\text{implied}}) ]

IV is what traders actually quote and trade. It represents the market's expectation of future volatility and varies by strike and expiry (the volatility surface).

Finding implied volatility requires numerical root-finding — Newton's method is the standard approach.

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Beyond Black-Scholes

Real-world pricing models address Black-Scholes's limitations:

• Local volatility models: volatility varies with stock price and time
• Stochastic volatility (Heston, SABR): volatility itself is random
• Jump-diffusion models: stock prices can jump (Merton's model)
• Monte Carlo simulation: simulate thousands of price paths and average the payoffs

Each model trades off accuracy against complexity. The choice depends on the product and the use case.

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Option Pricing in Python

import numpy as np
from scipy.stats import norm

def black_scholes_call(S, K, T, r, sigma):
    d1 = (np.log(S/K) + (r + sigma**2/2)*T) / (sigma*np.sqrt(T))
    d2 = d1 - sigma*np.sqrt(T)
    return S * norm.cdf(d1) - K * np.exp(-r*T) * norm.cdf(d2)

price = black_scholes_call(S=100, K=100, T=1, r=0.05, sigma=0.2)
print(f"Call price: £{price:.2f}")  # About £10.45

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The Full Journey

Option pricing is where calculus, probability, stochastic processes, and financial intuition converge. It is the intellectual heart of quantitative finance.

Quantt builds you up to this point systematically — from exponentials and calculus through to full implementations of binomial trees and Black-Scholes in Python. Every step is connected, and every concept has interactive exercises to make it stick.