Practitioner course in derivatives
The Black-Scholes Model
Option pricing from the ground up: payoffs and no-arbitrage bounds, the binomial model, the replication argument, the Black-Scholes equation and formula, risk-neutral valuation, and numerical pricing in Python.
Fischer Black and Myron Scholes published the Black-Scholes model of option pricing in 1973, with Robert Merton developing the theory in parallel. Its key insight is that an option can be replicated by continuously trading the underlying stock, so its price is fixed by the cost of the hedge, not by anyone’s view of where the stock is going.
How is the Black-Scholes formula derived?
By building a portfolio of the option and the stock whose risk cancels over each instant. A riskless portfolio must earn the risk-free rate, and that condition gives a partial differential equation for the option price. Solving it with the option’s payoff as the boundary condition gives the formula. You will work through each step, then reach the same answer by risk-neutral valuation and with a binomial tree.
Why does the model matter if its assumptions are wrong?
Because it gives a common language and a hedging method. Departures from its assumptions show up as the volatility smile, which the later courses in the derivatives track study directly. Understanding exactly where the model fails is the starting point for every model that improves on it.
Who is this course for?
Quant analysts, traders and developers entering derivatives, and students preparing for quant interviews. It counts toward the Certificate in Derivatives Pricing.
What you will be able to do
- Price options with a binomial tree and show its convergence.
- Derive the Black-Scholes equation from a hedging argument.
- Apply the Black-Scholes formula and explain each of its terms.
- Price options by risk-neutral Monte Carlo and finite differences.
- State the model's assumptions and where real markets depart from them.
Syllabus, module by module
Options and arbitrage
Calls, puts, payoffs, put-call parity and no-arbitrage bounds.
The binomial model
One-step replication, risk-neutral probabilities and Cox-Ross-Rubinstein trees.
Continuous time
Geometric Brownian motion, Ito's lemma and the delta-hedged portfolio.
The Black-Scholes equation
Deriving the PDE, boundary conditions and the closed-form solution.
Risk-neutral valuation
Change of measure in outline, and pricing as a discounted expectation.
Numerical methods
Monte Carlo pricing, finite differences and American exercise.
Assumptions and reality
Constant volatility, continuous hedging, costs and the volatility smile.
Project lab
Supervised work on the assessed pricing project.
How the certificate is earned
An assessed project: you implement three pricing methods for European and American options, verify them against each other and the closed form, and submit the code with a report. On a pass you receive the Course certificate in the Black-Scholes Model, issued by the Institute with a verification code. Read how certification works.
Before you start
Calculus, probability and working Python. The course teaches Ito's lemma in its second module, so no stochastic calculus is assumed.
This course counts toward
Questions applicants ask
What does the Black-Scholes model assume?
That the stock price follows geometric Brownian motion with constant volatility, that interest rates are constant, that trading is continuous and free of costs, and that there is no arbitrage. Real markets break each of these to some degree.
Is the Black-Scholes model still used?
Yes, mostly as a quoting convention. Traders quote options by their implied volatility in the Black-Scholes formula, even though they know volatility is not constant. The course explains why that works.