Notation
Every symbol used on this site, in the order it first appears, with the section that explains it. Use your browser's find (Ctrl+F or ⌘F) to look something up.
| Symbol | Meaning | Explained in |
|---|---|---|
| Chapter 1 · Brownian Motion | ||
| $t$ | Time elapsed since we started watching. | A Jittering Particle |
| $\Delta x$ | How far the particle has moved sideways (left or right) from where it started. Positive means right, negative means left. The Greek letter Δ ("delta") just means "change in". | A Jittering Particle |
| $\Delta x^2$ | That distance multiplied by itself. Squaring makes left and right moves both count as positive, so they can't cancel each other out. | A Jittering Particle |
| $\langle \;\cdot\; \rangle$ | The average over many particles, or over many repeats of the same experiment. | A Jittering Particle |
| $D$ | The diffusion coefficient: one number that says how fast the particle spreads out. A hotter fluid (the Energy slider) makes it larger. | A Jittering Particle |
| $n$ | Number of coin flips per unit of time. The slider sets it. | 1.1 From Random Walk to Brownian Motion |
| $S_k$ | The walk's position after $k$ flips: the running total of +1s and −1s. For example, heads, heads, tails gives $S_3 = 1 + 1 - 1 = 1$. | 1.1 From Random Walk to Brownian Motion |
| $\lfloor nt \rfloor$ | How many whole flips have happened by time $t$. The brackets mean "round down". With $n = 10$ and $t = 0.37$ that is $\lfloor 3.7 \rfloor = 3$ flips. | 1.1 From Random Walk to Brownian Motion |
| $\frac{1}{\sqrt n}$ | The shrink factor applied to every step, explained above. | 1.1 From Random Walk to Brownian Motion |
| $W^{(n)}_t$ | The shrunken walk's height at time $t$: one point on the orange path. | 1.1 From Random Walk to Brownian Motion |
| $\longrightarrow$ | "Gets closer and closer to, as $n$ grows without limit." | 1.1 From Random Walk to Brownian Motion |
| $W_t$ | Brownian motion: the random path the walks settle into. The letter W honours Norbert Wiener, who first put it on a rigorous footing. | 1.1 From Random Walk to Brownian Motion |
| $W_1 \sim N(0,1)$ | "Where the path ends at time 1 follows the standard bell curve." The $\sim$ reads "is distributed as". $N(0,1)$ is the normal (bell-shaped) distribution centred at 0, with average squared distance from 0 equal to 1. | 1.1 From Random Walk to Brownian Motion |
| $s,\ t$ | Two moments in time, with $s$ earlier than $t$. On the chart they are the left and right edges of a band. | 1.2 Defining Properties |
| $W_t - W_s$ | The increment: where the path is at time $t$ minus where it was at time $s$. It is positive if the path rose over that stretch, negative if it fell. | 1.2 Defining Properties |
| $\sim N(0,\; t-s)$ | "Follows a bell curve centred at 0, with variance $t - s$." Variance is the average squared size of the rise, so a band twice as long gives rises whose squares are twice as large on average. | 1.2 Defining Properties |
| independent | Knowing one of the two rises tells you nothing about the other. In the scatter plot, the cloud has no tilt. | 1.2 Defining Properties |
| correlation | A number from −1 to 1 measuring how much two quantities tend to move together. 0 means no tilt, 1 means a perfect upward line. Printed below the chart. | 1.2 Defining Properties |
| dashed ellipse | Where theory says about 95% of the dots should land. Its width and height come from the variances, its tilt from the correlation. | 1.2 Defining Properties |
| $h$ | The width of the time window you are looking at. Zooming in makes $h$ smaller. At ×1,000, $h = 0.001$. | 1.3 Rough Paths |
| $W_{t+h} - W_t$ | How much the path rises across the window: the increment from section 1.2. | 1.3 Rough Paths |
| $\lvert \;\cdot\; \rvert$ | The size of a number, ignoring its sign. So $\lvert -3 \rvert = 3$. | 1.3 Rough Paths |
| $\approx \sqrt h$ | "Is typically about $\sqrt h$". Section 1.2 showed the squared rise averages $h$, so the rise itself is typically about $\sqrt h$. For a small window that is much bigger than $h$: with $h = 0.01$ the rise is about $0.1$, ten times the window's width. | 1.3 Rough Paths |
| $\Longrightarrow$ | "Therefore". | 1.3 Rough Paths |
| $\frac{W_{t+h} - W_t}{h}$ | The slope across the window: rise divided by run, the steepness of the straight line joining the window's two ends. For a smooth curve it settles down to the derivative as $h$ shrinks. The panels show the same idea as "height ÷ width". | 1.3 Rough Paths |
| $\to \infty$ | "Grows without limit." | 1.3 Rough Paths |
| Chapter 2 · Variation & Martingales | ||
| $n$ | The number of equal pieces the time interval is cut into. The slider sets it. | 2.1 Quadratic Variation |
| $\Delta W_i$ | The rise of the path across piece number $i$: its value at the piece's right end minus its value at the left end. It is negative if the path fell. | 2.1 Quadratic Variation |
| $\sum_{i=1}^{n}$ | "Add up, for $i = 1, 2, \dots, n$": one term for every piece. The Greek letter Σ ("sigma") stands for sum. | 2.1 Quadratic Variation |
| $\lvert \Delta W_i \rvert$ | The size of the rise, ignoring whether it went up or down. Adding these up measures how much ground the path covers. For a curve, that is its total up-and-down travel, called its total variation. | 2.1 Quadratic Variation |
| $(\Delta W_i)^2$ | The rise multiplied by itself. Adding these up gives the quadratic variation ("quadratic" means "squared"). | 2.1 Quadratic Variation |
| $t$ | The length of the time interval, here $t = 1$. | 2.1 Quadratic Variation |
| $\longrightarrow \infty$, $\longrightarrow t$ | "Grows without limit" and "settles down to exactly $t$" as the pieces get finer. | 2.1 Quadratic Variation |
| $s,\ t$ | "Now" ($s$, the freeze time) and a later time $t$. The chart runs to $t = 1$. | 2.2 Martingales |
| $\mathbb{E}[\;\cdot\;]$ | The expected value: the average over all possible futures. The blue line estimates it with 400 simulated futures. | 2.2 Martingales |
| $\big|\ \mathcal{F}_s$ | "Given everything that has happened up to time $s$". The bar means "given", and $\mathcal{F}_s$ stands for the whole history of the path so far (the orange part). | 2.2 Martingales |
| $W_s$ | Where the path is at the freeze time: the orange dot. | 2.2 Martingales |
| $W_t^2 - t$ | The path's value squared, minus the time. Squares are never negative, so $W^2$ tends to drift upward. It drifts at exactly rate 1 (section 1.2: the average squared rise equals elapsed time), and subtracting $t$ cancels that drift. | 2.2 Martingales |
| martingale | A random process whose best forecast of any future value is simply its current value: $\mathbb{E}[\text{later} \mid \text{history}] = \text{now}$. | 2.2 Martingales |
| $a$ | The level, above the starting point 0. The slider sets it. | 2.3 Reflection Principle |
| $\max_{s \le t} W_s$ | The highest point the path reaches at any time $s$ up to $t$ (here $t = 1$). | 2.3 Reflection Principle |
| $P(\;\cdot\;)$ | The probability: the fraction of all possible paths for which the statement in brackets is true. The stats line estimates it by counting 3,000 simulated paths. | 2.3 Reflection Principle |
| $P(\max \ge a)$ | The fraction of paths that reach the level at some point: the blue ones. | 2.3 Reflection Principle |
| $P(W_t \ge a)$ | The fraction of paths that end at or above the level. This is easy to compute, because $W_t$ follows a bell curve (section 1.1). It is the area under the bell curve to the right of $a$. | 2.3 Reflection Principle |
| Chapter 3 · The Itô Integral | ||
| $\theta$ | Where inside each piece the path is read: 0 for the left end, ½ for the middle, 1 for the right end. | 3.1 Where You Evaluate Matters |
| $t_i^{\theta}$ | The reading point in piece $i$: a fraction $\theta$ of the way from its left end to its right end. The orange dots mark them. | 3.1 Where You Evaluate Matters |
| $W_{t_i^{\theta}}$ | The path's value at that reading point: the "height" of the rectangle. | 3.1 Where You Evaluate Matters |
| $\Delta W_i$ | The rise of the path across piece $i$ (section 2.1): the "width" of the rectangle, measured in rises of $W$ instead of in time. | 3.1 Where You Evaluate Matters |
| $\tfrac12 W_t^2$ | The answer ordinary calculus would give: the rule $\int x\,dx = \tfrac12 x^2$, with $W$ in place of $x$. | 3.1 Where You Evaluate Matters |
| $(\theta - \tfrac12)\,t$ | An extra amount that depends on the reading point. For a smooth curve it would be 0. Here it isn't. | 3.1 Where You Evaluate Matters |
| $\int_0^1 \cdots\, dW_t$ | The Itô integral from time 0 to time 1: the total of "holding × rise", added up over all the time. | 3.2 Building the Integral |
| $H_t$ | Your holding at time $t$: the height of the bar for the period that contains $t$. It is chosen at the start of the period, before the price moves (Itô's left-end rule from section 3.1). | 3.2 Building the Integral |
| $H_j$ | The holding during period $j$. | 3.2 Building the Integral |
| $t_{j-1},\ t_j$ | The start and end of period $j$. Here, $t_j = j/8$. | 3.2 Building the Integral |
| $W_{t_j} - W_{t_{j-1}}$ | How much the price rises during period $j$. | 3.2 Building the Integral |
| $\int_0^t H_s\,dW_s$ | Your total trading gains up to time $t$ (section 3.2). Here $t = 1$. | 3.3 Itô Isometry |
| $\mathbb{E}[\;\cdot\;]$ | The average over all possible price paths (section 2.2). The chart estimates it with up to 10,000 paths. | 3.3 Itô Isometry |
| $\big(\int H\,dW\big)^2$ | Your gains squared: a measure of how big your gain or loss is, whichever way it went. | 3.3 Itô Isometry |
| $\int_0^t H_s^2\,ds$ | An ordinary integral over time: the size of your holding, squared, added up over time. It measures how much risk you carried. | 3.3 Itô Isometry |
| $\operatorname{sign}(W)$ | +1 if $W$ is above 0, −1 if below. | 3.3 Itô Isometry |
| isometry | "Same size". The size of the random gains (left side) equals the size of the holdings (right side). | 3.3 Itô Isometry |
| Chapter 4 · Itô's Lemma | ||
| $f$ | Any smooth function, chosen below: $e^x$, $\sin x$ or $x^3$. | 4.1 The Surviving Second-Order Term |
| $f',\ f'',\ f'''$ | Its first, second and third derivatives. $f'$ is the slope, $f''$ is how fast the slope changes (curvature), and $f'''$ is how fast the curvature changes. For $x^3$: $f' = 3x^2$, $f'' = 6x$, $f''' = 6$. | 4.1 The Surviving Second-Order Term |
| $\Delta t,\ \Delta W$ | The length of one small piece of time, and the path's rise across it. | 4.1 The Surviving Second-Order Term |
| $\cdots$ | Higher-order terms, which use fourth and higher derivatives. They shrink even faster than the third-order term. | 4.1 The Surviving Second-Order Term |
| Taylor expansion | The name of this recipe: a change in $f$ is the slope × the step, plus a curvature correction × the step squared, and so on. | 4.1 The Surviving Second-Order Term |
| $d$ | "A tiny change in". $df(W_t)$ is the tiny change in $f(W_t)$ over a tiny time step $dt$, and $dW_t$ is the tiny rise of the path over it. Adding up these tiny changes means taking an integral. | 4.2 Itô's Lemma in Action |
| $f'(W_t)\,dW_t$ | The ordinary chain-rule term: slope × change. Added up, it is an Itô integral (Chapter 3). | 4.2 Itô's Lemma in Action |
| $\tfrac12 f''(W_t)\,dt$ | The extra Itô term: half the curvature × the time step. Added up, it is an ordinary integral over time. | 4.2 Itô's Lemma in Action |
| chain rule | Ordinary calculus's rule $df(x) = f'(x)\,dx$: the change in $f$ is its slope × the change in $x$. It is correct for smooth paths, and it misses the second term for Brownian ones. | 4.2 Itô's Lemma in Action |
| $X_t$ | A quantity that grows or shrinks in proportion to itself, starting at $X_0 = 1$: a price moving in percentage terms, for example. | 4.3 Itô vs Stratonovich |
| $\circ\, dW_t$ | The circle marks the Stratonovich reading: $X$ is read at the midpoint of each small step, like $\theta = \tfrac12$ in section 3.1. | 4.3 Itô vs Stratonovich |
| $dW_t$ (no circle) | The Itô reading: $X$ is read at the start of each step. | 4.3 Itô vs Stratonovich |
| $\Longleftrightarrow$ | "Describes exactly the same process as". | 4.3 Itô vs Stratonovich |
| $\tfrac12 X_t\,dt$ | The conversion drift: the amount you must add to an Itô equation to make it match the Stratonovich one. | 4.3 Itô vs Stratonovich |
| $W^{(1)},\ W^{(2)}$ | Two Brownian motions, blue and purple. Each one on its own is an ordinary Brownian motion (Chapter 1). | 4.4 Two Brownian Motions |
| $\rho$ | Their correlation, from −1 to 1 (section 1.2): 1 means they move in lockstep, 0 means independently, −1 means they are mirror images. | 4.4 Two Brownian Motions |
| $dW^{(1)}\,dW^{(2)} = \rho\,dt$ | Multiplying their tiny rises and adding up gives $\rho$ × elapsed time. With $\rho = 1$ this is section 2.1's $(dW)^2 = dt$, and with $\rho = 0$ the products cancel out to 0. | 4.4 Two Brownian Motions |
| $X,\ Y$ | Any two processes built from these Brownian motions. Here simply $X = W^{(1)}$ and $Y = W^{(2)}$. | 4.4 Two Brownian Motions |
| $d(XY)$ | The tiny change in the product $X \times Y$. | 4.4 Two Brownian Motions |
| $dX\,dY$ | The extra term of the Itô product rule. In ordinary calculus it vanishes. Here it equals $\rho\,dt$. | 4.4 Two Brownian Motions |
| Chapter 5 · Stochastic Differential Equations | ||
| $X_k$ | The computed value after $k$ steps (the orange dots). It starts at $X_0 = 1$. | 5.1 Euler–Maruyama |
| $\Delta t$ | The size of one time step: $1/N$ for $N$ steps. | 5.1 Euler–Maruyama |
| $a(x)$ | The drift: the predictable part of the change per unit of time. Here $a(x) = \mu x$ with $\mu = 0.2$: growth of 20% per unit of time, like interest. | 5.1 Euler–Maruyama |
| $b(x)$ | The volatility: how strongly the random kicks act. Here $b(x) = \sigma x$ with $\sigma = 0.8$: kicks proportional to the current value, so the price moves in percentage terms. | 5.1 Euler–Maruyama |
| $\Delta W_k$ | The Brownian path's rise during step $k$: a bell-curve random number with variance $\Delta t$ (section 1.2). | 5.1 Euler–Maruyama |
| exact solution | For this model Itô's lemma gives a formula: $X_t = e^{(\mu - \sigma^2/2)t + \sigma W_t}$. The $-\sigma^2/2$ is the Itô correction from Chapter 4. | 5.1 Euler–Maruyama |
| $dX_t$ | The tiny change in $X$ over a tiny time step $dt$ (section 4.2). | 5.2 A Zoo of Diffusions |
| $a(X_t)\,dt$ | Drift: the predictable push, which depends on where $X$ is now. | 5.2 A Zoo of Diffusions |
| $b(X_t)\,dW_t$ | Noise: a random kick whose size depends on where $X$ is now. | 5.2 A Zoo of Diffusions |
| GBM | Geometric Brownian motion, $a = \mu x$, $b = \sigma x$. Grows and wobbles in percentage terms, never below 0. The standard model of a stock price. | 5.2 A Zoo of Diffusions |
| OU | Ornstein–Uhlenbeck, $a = \theta(0 - x)$, $b = \sigma$. A pull back toward 0 that gets stronger the further away $X$ is, like a spring. Models velocities, interest rates and anything "mean-reverting". | 5.2 A Zoo of Diffusions |
| CIR | Cox–Ingersoll–Ross, $a = \kappa(1 - x)$, $b = \sigma\sqrt{x}$. Pulled toward 1, and the noise fades near 0. Models interest rates and volatility, which should not go negative. | 5.2 A Zoo of Diffusions |
| $p(t, x)$ | The density: how crowded position $x$ is at time $t$. The area under the curve between two positions is the fraction of particles between them. | 5.3 Paths ↔ Densities (Fokker–Planck) |
| $\frac{\partial p}{\partial t}$ | How fast the density at one fixed position is changing over time. The curly ∂ ("partial") means only one variable changes while the others are held fixed. | 5.3 Paths ↔ Densities (Fokker–Planck) |
| $\frac{\partial}{\partial x}$, $\frac{\partial^2}{\partial x^2}$ | The slope, and the curvature, of a curve drawn across positions at a fixed time. | 5.3 Paths ↔ Densities (Fokker–Planck) |
| $a(x)$, $b(x)$ | The drift and volatility from section 5.2. Here (OU): $a(x) = -x$, $b(x) = 1$. | 5.3 Paths ↔ Densities (Fokker–Planck) |
| Chapter 6 · Applications | ||
| $S_t$ | The stock price, modelled as geometric Brownian motion (section 5.2), starting at 100 with 20% volatility. | 6.1 Black–Scholes by Hedging |
| $K$ | The strike: the agreed price, here 100. | 6.1 Black–Scholes by Hedging |
| $C(t, S)$ | The Black–Scholes value of the option at time $t$ when the stock is at $S$ (thick grey line). | 6.1 Black–Scholes by Hedging |
| $V_t$ | Your hedge portfolio: the money received for the option, plus gains from trading the stock (orange line). | 6.1 Black–Scholes by Hedging |
| $\Delta_t$ | "Delta": how many shares to hold, equal to how much the option's value changes when the stock moves by 1. $N(d_1)$ is its formula, a bell-curve area like $P(W_t \ge a)$ in section 2.3. | 6.1 Black–Scholes by Hedging |
| $dV_t = \Delta_t\,dS_t$ | Your portfolio changes by (shares held) × (change in stock price): an Itô integral (section 3.2), with the stock as the "price". | 6.1 Black–Scholes by Hedging |
| $P$ | The original probabilities: every simulated path equally likely. | 6.2 Change of Measure (Girsanov) |
| $Q$ | New probabilities for the same set of paths: a re-weighting. | 6.2 Change of Measure (Girsanov) |
| $\frac{dQ}{dP}$ | The weight each path gets: how much more (or less) likely it is under $Q$ than under $P$. It is called the Radon–Nikodym derivative. Weights average to 1, so they only shift likelihood between paths. | 6.2 Change of Measure (Girsanov) |
| $W_1$ | Where the underlying Brownian path ends at $t = 1$. The weight depends only on that. | 6.2 Change of Measure (Girsanov) |
| $\lambda$ | The drift rate, set by the slider. | 6.2 Change of Measure (Girsanov) |
| $u(t, x)$ | The unknown function, with one value for each time $t$ (horizontal) and position $x$ (vertical): the colour map. | 6.3 Feynman–Kac |
| $g(x)$ | The known final values at $t = 1$: $g(x) = \max(0,\,1 - \lvert x\rvert)$, a triangle of height 1 over $-1 \le x \le 1$. | 6.3 Feynman–Kac |
| $x + W_{1-t}$ | Where a Brownian path started at position $x$ at time $t$ ends up at time 1, after running for the remaining time $1 - t$. | 6.3 Feynman–Kac |
| $\mathbb{E}[\;\cdot\;]$ | The average over all such random paths (section 2.2). | 6.3 Feynman–Kac |
| $\frac{\partial u}{\partial t} + \frac12 \frac{\partial^2 u}{\partial x^2} = 0$ | A partial differential equation (PDE): the rate of change of $u$ in time is tied to its curvature in space (section 5.3's notation). This one is the heat equation run backward from $t = 1$. | 6.3 Feynman–Kac |
| Chapter 7 · Diffusion Models | ||
| $X_0$ | One data sample: a number from one of the two bumps. | 7.1 Forward: Data to Noise |
| $X_t$ | The same sample after being noised for time $t$. | 7.1 Forward: Data to Noise |
| $-X_t\,dt$ | A drift pulling every sample toward 0. This is the Ornstein–Uhlenbeck process of section 5.2. | 7.1 Forward: Data to Noise |
| $\sqrt2\,dW_t$ | Random kicks. The $\sqrt2$ is chosen so the pull and the kicks balance at a long-run spread of exactly 1. | 7.1 Forward: Data to Noise |
| $Z$ | A fresh standard bell-curve number, drawn from $N(0, 1)$. | 7.1 Forward: Data to Noise |
| $e^{-t}$ | How much of the original sample survives: 37% at $t = 1$, 5% at $t = 3$. | 7.1 Forward: Data to Noise |
| $p(t, x)$ | The density of the noised data at time $t$ (section 5.3): the black curve of section 7.1, and the blue curve here. | 7.2 The Score |
| $\log p$ | The natural logarithm of the density. Taking the log makes the score equally informative in crowded and sparse regions. | 7.2 The Score |
| $\frac{\partial}{\partial x}$ | The slope as $x$ changes, with $t$ held fixed (section 5.3). | 7.2 The Score |
| $s(t, x)$ | The score. Positive means "more likely to the right", negative means "more likely to the left", and its size says how steeply. | 7.2 The Score |
| time going backward | Each step moves $t$ down by a small amount, so $dt$ is negative. | 7.3 Noise to Data: the Reverse-Time SDE |
| $-X_t\,dt$ | The forward drift of section 7.1. Run backward it pushes away from 0. | 7.3 Noise to Data: the Reverse-Time SDE |
| $-2\,s(t, X_t)\,dt$ | The score correction. Because $dt$ is negative, it moves each particle along the score arrows of section 7.2, toward more likely places. | 7.3 Noise to Data: the Reverse-Time SDE |
| $d\overline{W}_t$ | Fresh random kicks for the backward run. | 7.3 Noise to Data: the Reverse-Time SDE |
| probability-flow ODE | The same equation with half the score term and no noise: $dX = [-X - s(t, X)]\,dt$. Each noise sample then maps to one data sample, with no randomness along the way. | 7.3 Noise to Data: the Reverse-Time SDE |
| $\pi(x)$ | The target density to sample from: here the two-bump data of this chapter. | 7.4 Langevin Sampling |
| $s(x)$ | Its score: the slope of $\log \pi$, pointing toward more likely places (section 7.2 at noise level 0). | 7.4 Langevin Sampling |
| $\sqrt2\,dW_t$ | Random kicks. The $\sqrt2$ balances the drift so that the resting distribution is exactly $\pi$. | 7.4 Langevin Sampling |
| $t \to \infty$ | "In the long run". The theorem says nothing about how long, and that turns out to matter. | 7.4 Langevin Sampling |
| Chapter 8 · Beyond Brownian Motion | ||
| $N_t$ | A Poisson process: the number of jumps that have happened by time $t$. Jumps arrive at random moments, on average $\lambda$ per unit of time, independently of each other. | 8.1 Jumps |
| $\lambda$ | The jump rate ("lambda"): how many jumps per unit of time on average. | 8.1 Jumps |
| $J_k$ | The size of jump number $k$: here a bell-curve number with mean 0, so jumps go up or down. | 8.1 Jumps |
| $\sigma W_t$ | The ordinary Brownian part, scaled by $\sigma$. | 8.1 Jumps |
| $\operatorname{Var}(X_t)$ | The variance: the Brownian part contributes $\sigma^2 t$ and the jumps contribute $\lambda\,\mathbb E[J^2]\,t$. | 8.1 Jumps |
| $[X]_t$ | The quadratic variation (section 2.1). Now it is random: $\sigma^2 t$ plus the sum of the squared jumps. | 8.1 Jumps |
| log scale | Each step up the vertical axis multiplies by 10, so a bar ten times shorter is still easy to see. | 8.1 Jumps |
| $X_t$ | The hidden signal, following $dX = -\theta X\,dt + \sigma\,dW$ with $\theta = \sigma = 1$. | 8.2 The Kalman–Bucy Filter |
| $dY_t$ | What is measured in a tiny time step: $X_t\,dt + r\,dV_t$, the signal plus independent noise $dV$. | 8.2 The Kalman–Bucy Filter |
| $r$ | How noisy the measurements are (the slider). | 8.2 The Kalman–Bucy Filter |
| $m_t$ | The filter's estimate: the best guess of $X_t$ given every measurement up to time $t$. | 8.2 The Kalman–Bucy Filter |
| $P_t$ | The variance of the estimate's error: how unsure the filter is. The band is $m_t \pm 2\sqrt{P_t}$. | 8.2 The Kalman–Bucy Filter |
| $dY_t - m_t\,dt$ | The innovation: the part of the new measurement that the estimate did not predict, the "surprise". | 8.2 The Kalman–Bucy Filter |
| $P_t / r^2$ | The Kalman gain: how much of the surprise to believe. It is large when the filter is unsure ($P$ big) or the data are clean ($r$ small). | 8.2 The Kalman–Bucy Filter |
| Chapter 9 · Pricing Lab | ||
| $S_T^{(i)}$ | The stock price at maturity on simulated path $i$, drawn from the risk-neutral GBM (growth rate $r = 3\%$, volatility 20%). | 9.1 Monte Carlo & Variance Reduction |
| $(S_T - K)^+$ | The call's payoff: $S_T - K$ if positive, otherwise 0. | 9.1 Monte Carlo & Variance Reduction |
| $e^{-rT}$ | Discounting: a payoff in one year is worth less today. | 9.1 Monte Carlo & Variance Reduction |
| sd, standard error | sd is the spread of individual payoffs. The standard error, $\text{sd}/\sqrt n$, is the typical error of their average. | 9.1 Monte Carlo & Variance Reduction |
| antithetic | Use each random number $Z$ twice, as $Z$ and $-Z$, and average the pair. Their errors partly cancel. | 9.1 Monte Carlo & Variance Reduction |
| $\hat C_{\text{cv}}$, $\beta$ | Control variate: subtract the known error of something correlated. $S_T$ has known mean $S_0e^{rT}$, so if the sample's $\bar S_T$ came out high, the call estimate probably did too. $\beta$ sets how much to correct. | 9.1 Monte Carlo & Variance Reduction |
| importance sampling | Draw $Z$ from a shifted bell curve so paths land near the strike, then multiply each payoff by a likelihood-ratio weight so the average stays unbiased. It is Girsanov's theorem (section 6.2) used as a tool. | 9.1 Monte Carlo & Variance Reduction |
| $\Delta$ | Delta: how much the option's value changes when the stock moves up by one dollar. | 9.2 Greeks by Simulation |
| $h$ | The bump size in the finite-difference estimate $\big(V(S_0 + h) - V(S_0 - h)\big)/2h$. | 9.2 Greeks by Simulation |
| common random numbers | Reprice both bumped values with the same random draws, so their noise cancels in the difference. | 9.2 Greeks by Simulation |
| $\mathbf 1_{\{S_T > K\}}$ | 1 if the option finishes in the money, 0 otherwise. | 9.2 Greeks by Simulation |
| pathwise | Differentiate each path's payoff with respect to $S_0$ and average. Here $\partial S_T/\partial S_0 = S_T/S_0$. | 9.2 Greeks by Simulation |
| $Z$ | The standard normal number that generated the path: $S_T = S_0 e^{(r - \sigma^2/2)T + \sigma\sqrt T Z}$. | 9.2 Greeks by Simulation |
| likelihood ratio (LR) | Leave the payoff alone and differentiate the probability of the path instead. The weight $Z/(S_0\sigma\sqrt T)$ is the derivative of the log density. | 9.2 Greeks by Simulation |
| $B$ | The barrier: 90. If the stock touches it at any moment, the option is worthless. | 9.3 Barrier Options |
| $x_i,\ x_{i+1}$ | The log stock price at two consecutive checking dates, both above $\ln B$. | 9.3 Barrier Options |
| $\Delta t$ | The time between checks. | 9.3 Barrier Options |
| Brownian bridge | A Brownian path pinned at both ends (section 1.1's bridge construction). Its chance of dipping below a level has the exact formula above. | 9.3 Barrier Options |
| $\tau$ | A stopping time: an exercise rule that decides using only the prices seen so far. | 9.4 American Options |
| $\sup_{\tau}$ | The best possible value over all such rules. | 9.4 American Options |
| $t_j$ | The exercise dates in the simulation: 50 dates over the year. | 9.4 American Options |
| $\widehat{\mathbb E}[\,\cdot \mid S_{t_j}]$ | The continuation value: what holding on is worth, estimated by a regression of the discounted future cash flows on $1, S, S^2, \dots$ across the simulated paths. | 9.4 American Options |
| exercise boundary | The critical stock price below which exercising immediately is optimal. It rises toward $K$ as maturity nears. | 9.4 American Options |
| Chapter 10 · Volatility | ||
| $C_{\text{market}}(K)$ | The observed call price at strike $K$: here computed from the jump model. | 10.1 Implied Volatility & the Smile |
| $\sigma_{\text{imp}}(K)$ | Implied volatility: the unique $\sigma$ that makes the Black–Scholes price match. It is found numerically, because Black–Scholes can't be solved for $\sigma$ in closed form. | 10.1 Implied Volatility & the Smile |
| smile, skew | The shape of $\sigma_{\text{imp}}$ against $K$. Equity markets show a downward skew: low strikes (crash protection) have higher implied volatility. | 10.1 Implied Volatility & the Smile |
| $q(K)$ | The risk-neutral density of $S_T$: how likely each final price is, as priced by the market. The second derivative of call prices in the strike recovers it (Breeden–Litzenberger). | 10.1 Implied Volatility & the Smile |
| $v_t$ | The instantaneous variance. Its square root is the volatility. Here it starts at $v_0 = 0.04$ (20% volatility). | 10.2 Stochastic Volatility (Heston) |
| $\kappa,\ \theta$ | Variance is pulled back toward its long-run level $\theta = 0.04$ at speed $\kappa = 2$ (half-life about 4 months). | 10.2 Stochastic Volatility (Heston) |
| $\xi$ | The volatility of volatility: how wildly $v$ itself moves. It curves the smile. | 10.2 Stochastic Volatility (Heston) |
| $\rho$ | The correlation between price and variance moves (section 4.4). It tilts the smile. | 10.2 Stochastic Volatility (Heston) |
| Fourier price | Heston's model has a known characteristic function for $\ln S_T$. An integral of it gives the exact call price, with no simulation needed. | 10.2 Stochastic Volatility (Heston) |
| $\frac1T\int_0^T\sigma_t^2\,dt$ | Realized variance: the average squared volatility over the life of the swap. In practice it is measured as the average squared daily log return, which is section 2.1's quadratic variation. | 10.3 Variance Swaps |
| $\int_0^T dS_t/S_t$ | The gains from continuously holding one dollar's worth of stock: a trading strategy with zero value under risk-neutral pricing (when $r = 0$). | 10.3 Variance Swaps |
| $\ln(S_T/S_0)$ | A "log contract": a payoff that can be built from a strip of options. | 10.3 Variance Swaps |
| $P(K),\ C(K)$ | Put and call prices at strike $K$ today. Out-of-the-money options are used: puts below today's price, calls above. | 10.3 Variance Swaps |
| $K_{\text{var}}$ | The fair variance strike: the value that makes the swap worth zero today. | 10.3 Variance Swaps |
| Chapter 11 · Rates, Control & Data | ||
| $r_t$ | The short rate: the interest rate for borrowing over the next instant. | 11.1 Short-Rate Models (Vasicek) |
| $a,\ b,\ \sigma$ | Mean-reversion speed, long-run level (5%) and volatility of the rate. | 11.1 Short-Rate Models (Vasicek) |
| $P(0, T)$ | Today's price of one dollar paid at time $T$ (a zero-coupon bond): the average, under risk-neutral probabilities, of discounting at the realized path of rates. | 11.1 Short-Rate Models (Vasicek) |
| $A(T),\ B(T)$ | Explicit functions of the parameters. $B$ is the bond's sensitivity to today's rate (its "duration" in this model). | 11.1 Short-Rate Models (Vasicek) |
| yield $y(T)$ | The constant rate that gives the same price: $y(T) = -\ln P(0, T)/T$. Plotted against $T$ it is the yield curve. | 11.1 Short-Rate Models (Vasicek) |
| $X_t$ | Your wealth at time $t$. | 11.2 Merton's Portfolio Problem |
| $\pi$ | The fraction of wealth kept in the stock (above 100% means borrowing to invest more). | 11.2 Merton's Portfolio Problem |
| $\mu,\ \sigma,\ r$ | The stock's expected return (8%) and volatility (20%), and the cash rate (2%). | 11.2 Merton's Portfolio Problem |
| $\gamma$ | Relative risk aversion, in the utility $U(x) = x^{1-\gamma}/(1-\gamma)$. Larger means more afraid of losses. | 11.2 Merton's Portfolio Problem |
| certainty equivalent | The guaranteed growth rate you would accept instead of the risky strategy: $r + \pi(\mu - r) - \tfrac12\gamma\pi^2\sigma^2$. | 11.2 Merton's Portfolio Problem |
| $\pi^*$ | The fraction that maximizes it: the risk premium $\mu - r$ divided by the risk ($\sigma^2$) times the aversion ($\gamma$). | 11.2 Merton's Portfolio Problem |
| $r$ | A daily log return, $\ln(S_{t+1}/S_t)$. | 11.3 Models Meet Data |
| kurtosis | The average fourth power of standardized returns. It measures how heavy the tails are. A bell curve has 3 (section 8.1). | 11.3 Models Meet Data |
| autocorrelation | The correlation between a quantity today and the same quantity some days later (the lag). | 11.3 Models Meet Data |
| volatility clustering | Calm days follow calm days and turbulent days follow turbulent ones: the size of moves is predictable even though their direction is not. | 11.3 Models Meet Data |
| $\alpha,\ \beta,\ \varepsilon$ | A regression of tomorrow's yield on today's, which is the discrete-time version of an OU process. $\beta$ close to 1 means slow mean reversion. | 11.3 Models Meet Data |
| half-life | How long a deviation from the long-run mean takes to shrink by half. | 11.3 Models Meet Data |