M.4 Stochastic Calculus

Basic Prerequisites: Probability & Measure Theory. (these are the folder-level prerequisites for pages 02–06; page 01 states its own, smaller, entry requirements)

1. Intuition & Practical Objective

Stochastic calculus is the calculus whose “clock” ticks randomly. Ordinary calculus assumes $df=f^{\prime }(x)dx$ because $dx$ carries no randomness and $(dx{)}^2=0$. But the driving process of financial prices - Brownian motion $W$ - has nonzero quadratic variation: over every interval it accumulates squared increments at rate 1, i.e. $(dW{)}^2=dt$. That single fact forces the correction term with the $\frac{1}{2}$ in every formula, changes how you integrate, and is the deep reason the drift $\mu$ of a stock never enters the option price.

This folder is the mathematical engine room of the whole Atlas. It is a hub: the Theorem Lookup below (job #1) and six sub-pages taking you from zero intuition through Brownian motion, the Itô integral & Itô–Doeblin lemma, SDEs and simulation, Girsanov / risk-neutral measure, and the advanced bridges (Feynman–Kac, martingale representation, Fundamental Theorems). Everything the Black–Scholes–Merton pillar needs to derive rather than quote lives here.

The one-sentence essence. “A calculus where the second-order term survives - because Brownian motion moves fast enough that its squares add up deterministically ($(dW{)}^2=dt$); every Itô result is the consequence of that bookkeeping.”

2. Mathematical Ground Truth & Theorem Lookup

All formulas below are transcribed from Shreve Vol II (Ch 3–5), Shreve Vol I (Ch 13–18), Björk (Ch 4–5, 10–14) and Glasserman (Ch 1–3), all cross-checked in the verified corpus; the numeric checks in §3 reproduce them exactly.

Brownian Motion - $W(t),t\ge 0$, $W(0)=0$, continuous, independent increments $W(t)-W(s)\sim N(0,t-s)$, $W(\cdot )$ is an $\mathcal{F}(t)$-martingale (Shreve II Def 3.3.1, Thm 3.3.4). Covariance $\mathbb{E}[W(s)W(t)]=\min (s,t)$; finite-dimensional laws jointly normal with $C_{ij}=t_i\wedge t_j$ (Shreve I §13.6–13.7). Quadratic variation $[W,W](T)=T$, so $(dW{)}^2=dt$ (Shreve II §3.4; Def 3.4.1, Thm 3.4.3).

Exponential martingale - $Z(t)=\exp \{\sigma W(t)-\frac{1}{2}{\sigma }^{2}t\}$ is a martingale (Shreve II Thm 3.6.1). First-passage Laplace transform: $\mathbb{E}[e^{-\alpha {\tau }_m}]=e^{-|m|\sqrt{2\alpha }}$, and $\mathbb{E}{\tau }_m=\infty$ (Shreve II Thm 3.6.2, Remark 3.6.3).

Itô integral - $I(t)={\int }_0^t\Delta (u)dW(u)$, built for simple adapted $\Delta$ then extended to $\mathbb{E}{\int }_0^T{\Delta }^{2}du<\infty$. It is a martingale, mean $0$, with Itô isometry $\mathbb{E}[I^2]={\int }_0^t\mathbb{E}[{\Delta }^2]du$ and quadratic variation $[I,I](t)={\int }_0^t{\Delta }^{2}du$ (Shreve II Thm 4.2.1–4.2.3).

Itô–Doeblin - for $dX=\Theta dt+\Delta dW$ and $f\in C^{1,2}$ (Shreve II Thm 4.4.6):

GBM - $dS=\mu Sdt+\sigma SdW$ has solution $S(t)=S(0)e^{\sigma W(t)+(\mu -\frac{1}{2}{\sigma }^2)t}$; $\mathbb{E}[S(t)]=S(0)e^{\mu t}$ (Shreve I 15.1, Thm 15.3; Björk 5.2). The $\frac{1}{2}{\sigma }^2$ is the Itô correction.

Short-rate SDEs (Shreve II Ex 4.4.10/11; Glasserman §3.3–3.4):

• Vasicek: $dR=\kappa (\theta -R)dt+\sigma dW$ ⇒ $R(t)=e^{-\kappa t}R(0)+\theta (1-e^{-\kappa t})+\sigma e^{-\kappa t}{\int }_0^t{e}^{\kappa s}dW(s)$, Gaussian, can go negative.
• CIR: $dR=\kappa (\theta -R)dt+\sigma \sqrt{R}dW$ - no closed form, nonnegative, Feller condition $2\kappa \theta \ge {\sigma }^2$ for strict positivity (Glasserman §3.4).

Girsanov (Shreve II Thm 5.2.3) - with $\tilde{W}(t)=W(t)+{\int }_0^t\Theta (u)du$, the RN derivative

makes $\tilde{W}$ a BM under $\tilde{\mathbb{P}}$ ($d\tilde{\mathbb{P}}=Z(T)d\mathbb{P}$). Market price of risk $\Theta =(\mu -r)/\sigma$; it changes the drift ($\mu \to r$) but not the volatility or the paths (Shreve II 5.2.22–23).

Martingale Representation (Shreve II Thm 5.3.1) - w.r.t. the filtration generated by a single $W$, every martingale $M(t)=M(0)+{\int }_0^t\Gamma (u)dW(u)$.

Risk-neutral pricing (Shreve II 5.2.30/31): $V(t)=\tilde{\mathbb{E}}[e^{-{\int }_t^{T}R(s)ds}V(T)\mid \mathcal{F}(t)]$.

Feynman–Kac (Björk 5.5/5.6) - $F_t+\mu F_x+\frac{1}{2}{\sigma }^2{F}_{xx}-rF=0$, $F(T,\cdot )=\Phi$ ⇔ $F(t,x)=e^{-r(T-t)}{\mathbb{E}}_{t,x}[\Phi (X_T)]$ for $dX=\mu dt+\sigma dW$.

Fundamental Theorems (Shreve II 5.4.7/5.4.9; Harrison–Pliska) - FTA-1: no arbitrage ⇔ a risk-neutral (martingale) measure exists. FTA-2: market complete ⇔ the risk-neutral measure is unique.

3. Computational Implementation

A stdlib-only sanity sweep that reproduces the headline numbers of this folder end-to-end (each sub-page has a fuller example):

4. Failure Modes & First-Principles Breakdowns

Hub signposts - the folder’s failure analysis lives in each sub-page. In one line each:

1. $(dW{)}^2=0$ (classical calculus bleed-through). Treating the SDE with ordinary calculus drops the $\frac{1}{2}{\sigma }^2$ Itô term; every pricing formula, every hedge, every simulation is then wrong by the variance term. See 03 · Itô Integral & Doeblin.
2. Drift under the wrong measure. Pricing requires expectation under $\mathbb{Q}$ (drift $r$), not $\mathbb{P}$ (drift $\mu$). Discounting a $\mathbb{P}$-expectation misprices by the unhedgeable drift. See 05 · Girsanov & Risk-Neutral.
3. Discretization error in simulation. Exact lognormal transitions for GBM differ from Euler–Maruyama; coarse $\Delta t$ biases every statistic (Glasserman Ch 6). Simulate the exact transition whenever one exists.
4. Feller condition ignored. Below $2\kappa \theta <{\sigma }^2$ a CIR Euler scheme crosses below zero; Milstein or exact noncentral-${\chi }^2$ sampling is required. See 04 · SDEs & Simulation.
5. MRT needs the BM filtration. Martingale representation and market completeness both require the filtration generated by the Brownian motion - extra randomness (jumps, an untraded factor) breaks them. See 06 · Advanced Extensions.

5. References

• Shreve, Steven E.: Stochastic Calculus for Finance II
• Shreve, Steven E.: Stochastic Calculus for Finance I
• Björk, Tomas: Arbitrage Theory in Continuous Time (3rd ed.)
• Glasserman, Paul: Monte Carlo Methods in Financial Engineering

6. Connected Graph Bridges

• Foundational base: Probability & Measure Theory (martingales, RN derivative, conditional expectation)
• Sub-pages (in-folder): 01 From Zero · 02 Brownian Motion & Martingales · 03 Itô Integral & Doeblin · 04 SDEs & Simulation · 05 Girsanov & Risk-Neutral · 06 Advanced Extensions

Beginner: start at 01 · Practitioner: start at 05