Overview
The last part zooms out. Stock prices, interest rates and volatilities are all modelled as solutions of stochastic differential equations. We solve the Ornstein–Uhlenbeck SDE explicitly, meet the CIR process, and see why every SDE solution is Markov. Then Feynman–Kac closes the loop of the whole course: a conditional expectation of a payoff and a PDE are the same object — Black–Scholes is just one instance. We finish with dividend-paying stocks, stopping times and the optional sampling theorem.
Overview
Part 05 priced a call. This post prices any payoff with one formula,
\(V_t=E^{\mathbb Q}\big[e^{-r(T-t)}V_T\mid\mathcal F_t\big]\):
the price is a net present value — but under a different probability measure \(\mathbb Q\), one in which the stock earns the risk-free rate. Then we answer the obvious questions: what is a change of measure, why do \(\mathbb P\) and \(\mathbb Q\) agree on what can happen but not on how likely it is, and how does Girsanov’s theorem turn \(\mu\) into \(r\) by shifting the Brownian motion.
Overview
The price of an option is the capital needed to replicate it. This post sets up the Black–Scholes market, writes down the wealth of a self-financing trader, and matches it term by term (\(dW\) terms and \(dt\) terms) against the option’s price process. Matching the \(dW\) terms gives the delta hedge; matching the \(dt\) terms gives the Black–Scholes PDE. Solving the PDE gives the Black–Scholes formula, and differentiating it gives the Greeks.
A surprise worth noticing early: the stock’s expected return \(\mu\) disappears from the price.
Overview
Itô’s formula is the chain rule of stochastic calculus. Compared with the ordinary chain rule it has one extra term, \(\frac12 f''\,d[W,W]_t=\frac12f''\,dt\), and the reason is exactly \([W,W]_t=t>0\) from Part 02.
In practice you only need a second-order Taylor expansion plus the multiplication table
\(dt^2=0,\quad dt\,dW_t=0,\quad (dW_t)^2=dt.\)
This post derives the formula, extends it to Itô processes, and uses it to build the standard stock-price model, geometric Brownian motion.
Overview
A stochastic integral is just cumulative trading P/L: hold \(\Delta\) shares while the price moves by \(dW\), and add it up. This post builds \(\int_0^t\Delta_s\,dW_s\) in three steps — discrete time, “buy-and-hold” (simple) strategies, and general strategies by approximation — and shows the same five properties survive every step:
linear · adapted · martingale · Itô isometry \(\mathrm{Var}[I_t]=E\int_0^t\Delta_s^2ds\) · \([I,I]_t=\int_0^t\Delta_s^2ds\).
