Derivatives & Volatility

The Wiener process, the continuous-time random walk that drives every diffusion model in finance, defined by its four axioms, with its quadratic variation, martingale and scaling properties, and the nowhere-differentiability that forces us into Itô calculus.

The chain rule of stochastic calculus, how a smooth function of a diffusion evolves, why the second-order (dW)²=dt term survives, derived from a Taylor expansion, and applied to geometric Brownian motion to get the lognormal solution.

The fundamental theorem of asset pricing, why no-arbitrage is equivalent to the existence of an equivalent martingale measure, why under it every asset drifts at the risk-free rate, and why prices are discounted expectations of payoffs.

The founding model of option pricing, the PDE derived two ways (delta-hedging and risk-neutral expectation), the closed-form call price with N(d₁) and N(d₂), the meaning of those two terms, and the assumptions that the volatility smile later broke.

The model-free arbitrage relationship linking a European call, a put, the stock, and a bond, derived purely from replication with no distributional assumptions, plus its consequences for synthetic positions and implied-vol consistency.

The sensitivities of an option's value, delta, gamma, vega, theta, rho and the key second-order Greeks, with their Black-Scholes formulas, sign intuition, and the gamma-theta tradeoff that the pricing PDE encodes.

The fundamental P&L equation of a delta-hedged option, why a hedged position earns gamma times the difference between realized and implied variance, derived step by step, with the discrete-hedging error and its variance.

The volatility that makes Black-Scholes match a market price, why the inversion is unique, how it is computed, the ATM approximation, and the crucial distinction between implied, realized, and the variance risk premium between them.

Why implied volatility varies by strike, the direct contradiction of Black-Scholes, the equity skew versus the FX smile, the risk-reversal and butterfly that parametrize it, and the fat-tail and leverage stories that generate it.

The unique deterministic volatility function that reprices the entire option surface, derived via Breeden-Litzenberger and Dupire's forward equation, why it fits every vanilla exactly, and why its wrong smile dynamics make it misprice exotics.

Making volatility itself random, the Heston SDEs with mean-reverting CIR variance, the roles of vol-of-vol and correlation in shaping the smile, the semi-closed-form characteristic-function solution, and why stochastic vol beats local vol on dynamics.

The stochastic-alpha-beta-rho model that dominates rates and FX smile-fitting, its CEV-plus-lognormal-vol SDEs, the meaning of β, ρ and ν, Hagan's implied-vol expansion, and why its closed-form smile made it the market standard for interpolation.

The clean instrument for trading realized variance, its payoff, the log-contract derivation that shows realized variance replicates via a static strip of options weighted 1/K², and the model-free fair strike that underlies the VIX.

The market's fear gauge as model-free implied variance, the CBOE replication formula that is a discretized variance swap, why it is a 30-day risk-neutral vol expectation, and the futures term structure and roll that make VIX products behave.

Implied volatility as a function of maturity, contango versus backwardation, the variance-additivity that defines forward volatility, the no-calendar-arbitrage constraint, and how calendar spreads trade the slope.

The change-of-measure engine behind risk-neutral pricing, how the Radon-Nikodym derivative reweights probabilities to remove a drift, why volatility is invariant, the Novikov condition, and how choosing the market price of risk turns P into Q.

The bridge between PDEs and expectations, why the solution of a parabolic PDE equals a discounted expectation of a diffusion's terminal payoff, derived by making a discounted process a martingale, and why it makes the Black-Scholes PDE and the pricing integral the same thing.

Options beyond the vanilla payoff, barriers, Asians, digitals, and lookbacks, the closed forms that exist (reflection principle, digital = call spread) and the path dependence that forces Monte Carlo or PDE methods, plus the discontinuous Greeks that make them hard to hedge.