Quant Mechanics Lab

Risk-Neutral Valuation

Categories: Derivatives Pricing
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About Course

Derivative pricing as it actually works on a desk — not a wall of theorems, but something you build.

It starts with one deceptively simple question: what is an option worth? From there you derive risk-neutral valuation from first principles, then turn it into working code — a Monte Carlo pricer with honest error bars and a validation suite that proves the number is right.

WHAT YOU’LL BUILD

By the end you’ll have a working, tested pricer you wrote line by line — not a notebook you downloaded.

  • Replication and no-arbitrage, from the ground up
  • The risk-neutral measure and the martingale property
  • The change of measure that separates the real world from the pricing world
  • Multi-curve discounting, done the modern way — the modelling choice that quietly moves prices
  • A Monte Carlo pricer with confidence intervals and variance reduction
  • A validation suite: put-call parity, martingale tests, convergence checks

WHO IT’S FOR

Aspiring and working quants. Quant developers. Risk and model-validation professionals. Strong engineers and students moving toward a pricing role.

If you can read a derivation and write Python, you’re ready.

WHAT MAKES IT DIFFERENT

Every formula is motivated before it’s used. Every result is checked.

The emphasis is on the skill that actually separates practitioners: not 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.

WHAT YOU KEEP

  • 8 executable, fully documented Jupyter notebooks — the exact code from the videos, ready to run, adapt and reuse
  • A before-you-trust-a-price checklist you can carry onto a desk
  • A clear mental model of the whole subject
  • Lifetime access
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What Will You Learn?

  • Derive risk-neutral valuation from replication and no-arbitrage
  • Understand the change of measure without drowning in measure theory
  • Build a Monte Carlo pricer with confidence intervals and variance reduction
  • Discount correctly in a multi-curve world
  • Validate a price: parity, martingale tests, convergence
  • Defend a number the way a reviewer would expect

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.