Quant Mechanics Lab

Risk-Neutral Valuation

Categories: Derivatives Pricing
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About Course

Risk-Neutral Valuation — from formula to production

This course teaches how derivative pricing actually works on a desk — not as a wall of theorems, but as something you build. Starting from a single, deceptively simple question — what is an option worth? — you’ll derive risk-neutral valuation from first principles: replication and no-arbitrage, the risk-neutral measure and the martingale property, the change of measure that separates the real world from the pricing world, and discounting done the modern, multi-curve way — the kind of modelling choice that quietly moves prices. Then you’ll turn that theory into working code: a Monte Carlo pricer with a confidence interval and variance reduction, and a validation suite that proves the number is right.

It’s built for people who want quant finance as it’s lived inside a bank — aspiring and working quants, quant developers, and risk or model-validation professionals, as well as strong engineers and students moving toward a pricing role. Every formula is motivated before it’s used, every result is checked, and the emphasis throughout is on the skill that actually separates practitioners: not just computing a price, but being able to defend it — the right measure, the right curve, an honest error bar, and tests a reviewer can re-run in seconds.

And it’s all yours to keep. You’ll walk away with a working, tested pricer you built line by line, a clear mental model of the whole subject, and a practical before-you-trust-a-price checklist you can carry onto a desk — plus eight executable, fully documented Jupyter notebooks: the exact code shown in the videos, ready to run, adapt, and reuse in your own work.

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What Will You Learn?

  • Build option pricing models the way they are implemented on a desk — not just derived on paper
  • Price by replication and no-arbitrage, and see the risk-neutral weight q emerge on its own
  • Make the change of measure from the real world (P) to the risk-neutral world (Q) explicit
  • Discount cash flows under OIS, the way it is actually done in practice
  • Implement a Monte Carlo pricer from scratch, with an honest confidence interval
  • Benchmark and validate a pricer with martingale, convergence and put-call parity checks
  • Learn to defend a price, not just compute it — the skill that separates practitioners
  • Take home ready-to-run Python notebooks for every lesson

Course Content

Foundations
Why a price can't depend on who's holding the position. We start from the failure of intuitive, expectation-based pricing — where every investor gets a different number — and rebuild the logic from replication: construct a portfolio that reproduces the option's payoff, and let no-arbitrage pin down the price. Out of that argument, the risk-neutral weight appears on its own. This is the foundation everything else in the module rests on.

  • Why naive pricing fails
  • Replication & the one-period binomial

From Binomial to Black-Scholes
From a one-step tree to a continuous-time model. We refine the binomial replication argument, then watch the tree converge to Black-Scholes as the steps shrink. Along the way we make the change of measure explicit — from the real world, P, to the risk-neutral world, Q — the step most treatments quietly skip, and the one that finally makes risk-neutral pricing click.

Discounting & Simulation
Turning the theory into numbers. First we discount the way a desk actually does — under OIS, where the choice of curve quietly moves the price. Then we build a Monte Carlo pricer from the ground up: simulating paths, pricing the option, and reading the result against the closed-form benchmark — with an honest confidence interval, never a bare number.

Validation & Wrap-up
Trusting the number before it reaches a book. We put the pricer through the checks a desk would demand — martingale and convergence tests, and benchmark agreement within tolerance — and plant a classic bug to watch the tests catch it. We close by tying the full arc together: from replication, to a price you can defend.