Quant Core
Stochastic calculus, pricing models, and the Greeks.
Everything a derivatives desk prices flows from a few ideas: returns behave in statistically distinctive ways, prices follow stochastic processes, and no-arbitrage pins down fair value. This track builds that machinery, from time series through Brownian motion, Itô calculus, and risk-neutral pricing, into the pricing engines: Black–Scholes, binomial trees, Monte Carlo, the Greeks, and implied volatility.
Modeling Returns
Stochastic Calculus
Stochastic Processes
Brownian motion with drift and diffusion, exponentiated into geometric Brownian motion, is the first workable model of a stock price.
Itô's Lemma
Itô's lemma adds a second-order correction to the chain rule, and that correction is what turns stochastic dynamics into pricing PDEs.
Risk-Neutral Pricing
No-arbitrage guarantees an equivalent martingale measure, so derivatives price as discounted expected payoffs under it.
Pricing Engines
Black–Scholes
Constant-volatility dynamics plus continuous hedging yield a closed-form option price and the message that options are manufactured, not bet on.
Binomial Trees
Backward induction on a recombining tree makes replication concrete and converges to Black–Scholes as the steps shrink.
Monte Carlo Methods
Simulating risk-neutral paths and averaging discounted payoffs prices the exotics that defeat closed forms, with variance reduction doing the heavy lifting.
The Greeks
Delta, gamma, vega, theta, and rho translate model prices into hedge ratios, hedging costs, and the daily decay that funds them.
Implied Volatility
Inverting Black–Scholes against market prices recovers the volatility the market implies, quoted per strike and expiry as the volatility surface.