· 7 min
How an option is priced by hedging away its risk. The Black-Scholes partial differential equation derived from Ito's formula and a delta hedge, its risk-neutral solution as a discounted expected payoff, and the closed-form Black-Scholes formula for a European call.
· 5 min
How to trade return against risk optimally. The Markowitz problem of minimising variance at a target return, its closed-form solution by Lagrange multipliers, the efficient frontier it traces, and the tangency portfolio maximising the Sharpe ratio.
· 31 min
The models built on the order book mechanism. The efficient price is a martingale, the bounce and adverse selection set spreads, queue position sets fills, imbalance sets the microprice, Kyle's lambda turns information into impact, and Almgren-Chriss solves execution.
· 6 min
The canonical mean-reverting diffusion. We solve the Ornstein-Uhlenbeck SDE in closed form, derive its Gaussian transition and stationary laws, read off the half-life, and reduce it to an exact AR(1) recursion for estimation.