The Black-Scholes Model and the Option Greeks

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.

How does delta hedging work?

An option’s price moves with the underlying, with volatility and with time, and the option Greeks measure each of those sensitivities. For anyone who holds or hedges options, they are the working language of risk. This course explains each one and then puts them to use in hedging simulations.

Delta is how much the option’s price changes for a small move in the underlying. Holding the opposite amount of the underlying cancels that exposure for small moves. Because delta itself changes as the price moves, the hedge must be adjusted, and the cost of those adjustments depends on gamma and on how much the market actually moves. You will simulate it and measure the result.

Who is this course for?

Quant analysts, traders and developers entering derivatives, and students preparing for quant interviews.

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.
• Compute the main Greeks analytically and numerically.

Syllabus, module by module

1. Options and arbitrage

   Calls, puts, payoffs, put-call parity and no-arbitrage bounds.

2. The binomial model

   One-step replication, risk-neutral probabilities and Cox-Ross-Rubinstein trees.

3. Continuous time

   Geometric Brownian motion, Ito's lemma and the delta-hedged portfolio.

4. The Black-Scholes equation

   Deriving the PDE, boundary conditions and the closed-form solution.

5. Delta

   Sensitivity to the underlying, its shape across strikes, and delta hedging.

6. Gamma

   Convexity, gamma near expiry and the cost of rehedging.

What every course contains

Every paid course comes with the first seven and the certificate, whichever level it sits at. The proctored exam and the credential come when the course is taken as part of a Certificate, Diploma or Fellowship. Everything is self-paced: the material is yours to study in any order, at any hour, for the whole of the license.

1. Workbooks. Written to train the thinking patterns and the important concepts of the course, with worked problems and the reasoning behind each answer, so that you learn to think the way the subject is thought.
2. Audio edition. Focused, well-presented recordings with explainers and examples of the material, made to be interesting, relevant and entertaining, so that the general subject can be learned away from the desk.
3. Notes. Pre-produced notes with every fact the course relies on, written out in full, so that nothing has to be transcribed from the films or the audio.
4. Cheat sheet. One sheet with every important learning concept of the course on it, to keep beside you during the work and to revise from before the exam.
5. Named sources. Every fact in the notes names its source, the paper, the book or the dataset, for anyone who wants to dig in further.
6. Explainer films. Focused films, in the style of the sample films on the catalog page, that walk through the most difficult concepts of the course with visual explanations of a high standard.
7. Transcript. A full transcript of every film and every audio recording, searchable and quotable.
8. Proctored exam. Taken as part of a Certificate, Diploma or Fellowship, the course ends in a proctored multiple-choice questionnaire, with a pass mark of 60% and three retakes included. A course taken on its own is assessed by its coursework and has no exam.
9. Verifiable certificate. A certificate of completion that a third party can verify: an employer sends the code on it to the credentials office and receives written confirmation of the course, the date and the result.
10. Credential of the program. The Certificate, Diploma or Fellowship of the program, track or path the course belongs to, awarded when every course in it is passed. Which one depends on the program you choose.

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 and the Option Greeks, issued by the Institute with a verification code. A course taken as part of a Certificate, Diploma or Fellowship ends in a proctored online exam with a pass mark of 60%. The fee includes three retakes; each attempt after that carries a reduced proctored exam fee. A course taken on its own is assessed by its coursework and has no exam. 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.

• Probability for Quantitative Finance, with Statistics
• Python and SQL for Finance