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.

1. Stochastic differential equations

Fix \((\Omega,\mathcal F,\mathbb F,\mathbb P)\) with a Brownian motion \(W\) and \(\mathbb F=\mathbb F^W\). An SDE is an equation of the form

where \(\mu:[0,\infty)\times\mathbb R\to\mathbb R\) is the drift function and \(\sigma:[0,\infty)\times\mathbb R\to\mathbb R\) the diffusion (dispersion) function. Unlike an Itô process, the coefficients depend on the unknown \(X\) itself.

A solution is a process \(X\) such that for all \(t\ge0\)

with all integrals well defined.

GBM is an SDE. With \(\mu(t,x)=\mu x\) and \(\sigma(t,x)=\sigma x\): \(dX_t=\mu X_tdt+\sigma X_tdW_t\), solved by \(X_t=x\,e^{(\mu-\frac12\sigma^2)t+\sigma W_t}\) — the Black–Scholes stock.

Existence and uniqueness. Not every SDE has a solution, and few have closed-form ones. Under mild conditions one can prove existence and uniqueness; the key property is that \(\mu\) and \(\sigma\) are Lipschitz in space,

just as for ordinary differential equations. In this course we always assume solutions exist.

2. The Ornstein–Uhlenbeck process

\(\theta\): long-run mean; \(\kappa\): speed of mean reversion; \(\sigma\): volatility. When \(X\) is above \(\theta\) the drift pulls it down, and vice versa — a spring with noise.

Solving it. (1) Assume a solution exists and find its form; (2) check it. Multiply by the integrating factor \(e^{\kappa t}\) and apply Itô (product rule with a deterministic factor):

Integrate: \(e^{\kappa t}X_t=x+\theta(e^{\kappa t}-1)+\int_0^te^{\kappa u}\sigma\,dW_u\), so

Check: differentiating, \(dX_t=-\kappa e^{-\kappa t}\big((x-\theta)+\int_0^te^{\kappa u}\sigma dW_u\big)dt+\sigma\,dW_t=\kappa(\theta-X_t)dt+\sigma dW_t\). ✓

Distribution. The integrand \(e^{\kappa u}\sigma\) is deterministic, so by Part 04 the stochastic integral is Gaussian, with variance from the Itô isometry:

Hence

The starting point is forgotten exponentially fast; \(\theta\) is the long-run mean and \(\sigma^2/(2\kappa)\) the long-run variance.

Time-dependent coefficients. With non-random \(\kappa(t),\theta(t),\sigma(t)\) (they may vary with \(t\) but not with \(\omega\)), \(dX_t=\kappa(t)\big(\theta(t)-X_t\big)dt+\sigma(t)dW_t\) is solved by

3. The Cox–Ingersoll–Ross process

Same mean reversion as OU, but the instantaneous conditional variance is \(\xi^2X_t\,dt\): noise shrinks as \(X\) approaches \(0\). The solution must stay positive for \(\sqrt{X_t}\) to make sense; strict positivity holds if \(\kappa\theta\ge\frac12\xi^2\) (the Feller condition). There is no closed-form solution, but a solution exists and the distribution of \(X_t\) can be computed. It is a standard interest-rate and variance model.

4. The Markov property

Solutions of \(dX_t=\mu(t,X_t)dt+\sigma(t,X_t)dW_t\) are Markov: to evaluate the future of the process at time \(t\), knowing \(X_t\) is enough — all other information in \(\mathcal F_t\) is irrelevant. In practice:

Why: at time \(t\), the next step \(dX_t\) is determined by \(X_t\) (through \(\mu\) and \(\sigma\)) and by \(dW_t\), which is independent of \(\mathcal F_t\). Expanding \(E[g(X_t+dX_t)\mid\mathcal F_t]\) and \(E[g(X_t+dX_t)\mid X_t]\) gives the same answer, step after step.

5. Feynman–Kac

Let \(dX_t=\mu(t,X_t)dt+\sigma(t,X_t)dW_t\), let \(g\) be a payoff and \(r(t,x)\) a discount rate. Then

corresponds one-to-one with the PDE

Without discounting, \(u(t,x)=E[g(X_T)\mid X_t=x]\) solves \(0=u_t+\mu u_x+\frac12\sigma^2u_{xx}\) with \(u(T,x)=g(x)\).

Why it is true (one line). By Markov, \(e^{-\int_0^tr\,du}f(t,X_t)=E\big[e^{-\int_0^Tr\,du}g(X_T)\mid\mathcal F_t\big]\) is a martingale. Apply Itô to it: the \(dt\) coefficient is \(e^{-\int_0^tr}\big(f_t+\mu f_x+\frac12\sigma^2f_{xx}-rf\big)\), and a martingale has no \(dt\) term.

Black–Scholes is a special case. Take \(X=S\) under \(\mathbb Q\), i.e. \(\mu(t,s)=rs\), \(\sigma(t,s)=\sigma s\), constant \(r\), and \(g(s)=(s-K)^+\). The PDE becomes exactly the Black–Scholes PDE of Part 05, and the expectation becomes the risk-neutral formula of Part 06. The two halves of the course — PDEs and expectations — are the same thing.

6. Dividend-paying stocks

If the stock pays a continuous dividend at rate \(a_t\), its price drops by the dividend:

Wealth. Holding \(\Delta_t\) shares over \([t,t+dt]\), we also receive dividends \(\Delta_tS_ta_t\,dt\):

The wealth dynamics are the same as without dividends (the dividend you receive offsets the price drop), but the stock dynamics change. So European option prices do change, because the stock price is different. Under \(\mathbb Q\),

and under \(\mathbb Q\) the stock now drifts at \(r_t-a_t\). In practice, write the payoff as a function of \(S_T=S_t\cdot\frac{S_T}{S_t}\), freeze \(S_t=s\), and compute — exactly as in the Black–Scholes derivation.

7. Stopping times

A random time \(\tau\) is a stopping time if

i.e. at every time you can tell from current information whether you have already stopped. Example: a non-random time \(\tau\equiv t_0\), for which \(\{\tau\le t\}\) is \(\Omega\) if \(t_0\le t\) and \(\varnothing\) otherwise.

Hitting times. \(\tau(\omega)=\inf\{t:X_t(\omega)=\lambda\}\), the first time \(X\) hits level \(\lambda\). For adapted \(X\) with continuous paths, hitting times are stopping times. (By contrast, “the last time \(X\) hits \(\lambda\) before \(T\)” is not: you need the future to know it.)

Stopped values and stopped processes. \(X_\tau\) is the random variable \(\omega\mapsto X_{\tau(\omega)}(\omega)\) — e.g. \(X_1(\omega)\) if \(\tau(\omega)=1\), \(X_3(\omega)\) if \(\tau(\omega)=3\). The stopped process \(X^\tau\) is

it follows \(X\) until \(\tau\) and then stays flat.

8. The optional sampling theorem

Theorem (OST). Let \(M\) be a martingale and \(\alpha\le\tau\) two bounded stopping times. Then

Corollaries for a bounded stopping time \(\tau\):

1. the stopped process \(t\mapsto M_{t\wedge\tau}\) is a martingale;
2. \(E[M_\tau]=M_0\);
3. if \(\tau\le T\), the price of a claim at the random time \(\tau\) is \(V_\tau=E^{\mathbb Q}\big[e^{-r(T-\tau)}V_T\mid\mathcal F_\tau\big]\) — the pricing formula of Part 06 with \(t\) replaced by \(\tau\).

Intuition: a fair game stays fair even if you decide when to look using a rule that does not peek into the future. Boundedness matters: “play until you are ahead” is a stopping time, but an unbounded one, and with it \(E[M_\tau]>M_0\) is possible.

Series contents

1. 01 · Probability Spaces, Filtrations, Conditional Expectation and Martingales
2. 02 · Random Walks, Quadratic Variation and Brownian Motion
3. 03 · Stochastic Integration
4. 04 · Itô’s Formula, Itô Processes and Geometric Brownian Motion
5. 05 · Black–Scholes: Replication, the PDE and the Greeks
6. 06 · Risk-Neutral Pricing, Change of Measure and Girsanov
7. 07 · SDEs, Feynman–Kac and Stopping Times (this post)

← Previous: Risk-Neutral Pricing, Change of Measure and Girsanov

This series collects my notes from QST MF 790 Stochastic Calculus at Boston University’s Questrom School of Business, in 7 parts. The content follows the order of the course; wording, intuition and a few corrections are my own. Throughout, \(N(\mu,\sigma^2)\) uses the variance as its second argument. Corrections are welcome. Chinese version of this post: 中文版.

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.

1. From the call to any contingent claim

Recap of the Black–Scholes setup. \(S\sim GBM(\mu,\sigma)\) under \(\mathbb P\); money market at constant \(r\); self-financing strategy \(\Delta\). Discounted wealth:

For the call we found: with \(\widetilde S\sim GBM(r,\sigma)\) under \(\widetilde{\mathbb P}\),

initial capital \(x=c(0,S_0)\), strategy \(\Delta_t=c_s(t,S_t)\), terminal wealth \(X_T=(S_T-K)^+\), so by no-arbitrage \(c(0,S_0)\) is the price. Now generalize.

Contingent claims. A claim pays \(V_T\) at \(T\), where \(V_T\) can be any \(\mathcal F_T\)-measurable random variable with enough integrability:

2. Two assumptions and the pricing formula

1. Risk-neutral measure. There is a measure \(\mathbb Q\) on \((\Omega,\mathcal F)\) under which \(S\sim GBM(r,\sigma)\). Equivalently there is a \(\mathbb Q\)-Brownian motion \(W^{\mathbb Q}\) with

   \[\frac{dS_t}{S_t}=\underbrace{\mu\,dt+\sigma\,dW_t}_{\text{under }\mathbb P}=\underbrace{r\,dt+\sigma\,dW^{\mathbb Q}_t}_{\text{under }\mathbb Q}.\]

2. Replication. There are \(x\) and \(\Delta\) with \(\mathbb P[X^{x,\Delta}_T=V_T]=\mathbb Q[X^{x,\Delta}_T=V_T]=1\).

Discounted wealth is a \(\mathbb Q\)-martingale. Under \(\mathbb Q\) the drift term vanishes:

That is why \(\mathbb Q\) is also called a martingale measure.

Replication over \([t,T]\). The no-arbitrage argument works from any starting time: if you start at \(t\) with \(X_t\) (known at \(t\)), trade self-financingly on \([t,T]\) and end with \(V_T\), then the price at \(t\) must be \(X_t\).

Determining \(V_t\). By replication, \(e^{-rT}X_T=e^{-rT}V_T\). By the martingale property,

And since \(V_T=X_T=X_t+\int_t^TdX_u\), holding \(X_t\) at time \(t\) is exactly what it takes to deliver \(V_T\), so

How to read it. The chain of reasoning is: (i) the replicating portfolio is the claim, so the claim’s price is the portfolio’s value; (ii) under \(\mathbb Q\) every discounted self-financing portfolio is a fair game, so today’s discounted value equals the expected discounted terminal value; (iii) the terminal value is the payoff. Prices are NPVs — not under the real-world measure, but under one where the risky and riskless assets have the same expected return.

The two assumptions raise five questions: how do we get \(\mathbb Q\); how are \(\mathbb P\) and \(\mathbb Q\) related; how do we switch between them; why is \(S\sim GBM(r,\sigma)\) under \(\mathbb Q\); and where does \(W^{\mathbb Q}\) come from. The rest of the post answers them.

3. What a change of measure changes

Not affected by the measure (they only involve sets and maps):

1. \(\Omega\), the set of outcomes;
2. \(\mathcal F\), the information (collection of subsets of \(\Omega\));
3. \(\mathbb F\), the flow of information;
4. a random variable \(X:\Omega\to\mathbb R\) — it is just a map;
5. a process \(\{X_t\}\);
6. adaptedness: \(\{\omega:X_t(\omega)\in E\}\in\mathcal F_t\).

Affected by the measure (they involve likelihood):

1. probabilities \(P(A)=E^P[\mathbf 1_A]\);
2. expectations \(E\big[\sum_i\alpha_i\mathbf 1_{A_i}\big]=\sum_i\alpha_iP(A_i)\);
3. moment generating functions \(\alpha\mapsto E^P[e^{\alpha X}]\);
4. distributions \(P(X\le t)\);
5. the martingale property \(E[X_t\mid\mathcal F_s]=X_s\);
6. independence \(P(A\cap B)=P(A)P(B)\).

So the stock price paths are the same; only the weights we put on them change — and with them, the drift.

4. Change of measure: the construction

Given \((\Omega,\mathcal F,\mathbb P)\), take any random variable \(Z\) with \(\mathbb P[Z>0]=1\) (write \(Z>0\)) and \(E^{\mathbb P}[Z]=1\). Define

\(\mathbb Q\) is a probability measure: \(\mathbb Q(A)\ge0\); \(\mathbb Q(\Omega)=E^{\mathbb P}[Z]=1\); and countable additivity holds, e.g. for disjoint \(A,B\), \(\mathbb Q(A\cup B)=E^{\mathbb P}[(\mathbf 1_A+\mathbf 1_B)Z]=\mathbb Q(A)+\mathbb Q(B)\).

Example: fair coin to unfair coin. Two tosses, fair and independent under \(\mathbb P\). Set

Then \(\mathbb Q(HH)=\frac14\cdot\frac49=\frac19\), \(\mathbb Q(HT)=\mathbb Q(TH)=\frac29\), \(\mathbb Q(TT)=\frac49\). Under \(\mathbb Q\) the tosses are still independent (\(\frac19=\frac13\cdot\frac13\), etc.) but the coin is unfair: \(\mathbb Q(H)=\frac13\).

Equivalence. Because \(Z>0\): \(\mathbb P(A)>0\Rightarrow\mathbb Q(A)=E^{\mathbb P}[Z\mathbf 1_A]>0\), and conversely. So \(\mathbb P(A)>0\iff\mathbb Q(A)>0\) and we call \(\mathbb P,\mathbb Q\) equivalent. Equivalent is not equal: they agree on what can and cannot happen, not on how likely it is (\(\mathbb P(A)=0.01\), \(\mathbb Q(A)=0.99\) is fine).

Radon–Nikodym derivative. \(\dfrac{d\mathbb Q}{d\mathbb P}:=Z\), also called the likelihood ratio. On a finite space,

5. Shifting the mean of a normal

Let \(X\sim N(0,\sigma^2)\) under \(\mathbb P\), \(\theta\in\mathbb R\), and \(Z=e^{-\theta X-\frac12\sigma^2\theta^2}\). Then \(Z>0\) and \(E[Z]=e^{-\frac12\sigma^2\theta^2}E[e^{-\theta X}]=1\). Under \(\mathbb Q(A)=E^{\mathbb P}[Z\mathbf 1_A]\):

so \(X\overset{\mathbb Q}{\sim}N(-\sigma^2\theta,\ \sigma^2)\), i.e. \(X+\sigma^2\theta\overset{\mathbb Q}{\sim}N(0,\sigma^2)\). Tilting by an exponential shifts the mean and leaves the variance alone.

General version. To move \(X\) from \(N(\mu_P,\sigma^2)\) under \(\mathbb P\) to \(N(\mu_Q,\sigma^2)\) under \(\mathbb Q\), use \(\frac{d\mathbb Q}{d\mathbb P}=\frac{e^{-\theta X}}{E^{\mathbb P}[e^{-\theta X}]}=e^{-\theta X-\frac12\sigma^2\theta^2+\theta\mu_P}\). Then

6. Application to Brownian motion and the stock

Fix \(T\); \(W_T\sim N(0,T)\) under \(\mathbb P\). Take \(\theta=\frac{\mu-r}{\sigma}\) and

By Section 5 (with \(\sigma^2=T\), \(\mu_P=0\)), \(W^{\mathbb Q}_T:=W_T+\theta T\sim N(0,T)\) under \(\mathbb Q\). Since \(\mu-\sigma\theta=r\),

So at the fixed time \(T\), \(S_T\) looks like a \(GBM(r,\sigma)\) under \(\mathbb Q\). To have \(S\sim GBM(r,\sigma)\) under \(\mathbb Q\) for all \(t\le T\) we need more: is \(t\mapsto W^{\mathbb Q}_t=W_t+\theta t\) a \(\mathbb Q\)-Brownian motion on \([0,T]\)? Equivalently, is

The core is to verify that \(W^{\mathbb Q}\) is a Brownian motion. (Note \(Z_t=E^{\mathbb P}[Z_T\mid\mathcal F_t]=e^{-\theta W_t-\frac12\theta^2t}\) is a \(\mathbb P\)-martingale — the exponential martingale of Part 04.)

7. Girsanov’s theorem

Simple version. If \(W\) is a \(\mathbb P\)-Brownian motion, \(Z_T=e^{-\theta W_T-\frac12\theta^2T}\) and \(\mathbb Q(A)=E^{\mathbb P}[\mathbf 1_AZ_T]\), then

General version. Fix \((\widetilde\Omega,\widetilde{\mathcal F},\widetilde{\mathbb F},\widetilde{\mathbb P})\) with a \(\widetilde{\mathbb P}\)-Brownian motion \(\widetilde W\), and an adapted \(\widetilde\theta\). Let

Then \(\widetilde W^{\widetilde{\mathbb Q}}_t=\widetilde W_t+\int_0^t\widetilde\theta_u\,du\), \(t\le T\), is a \(\widetilde{\mathbb Q}\)-Brownian motion.

Intuition: reweighting the paths by \(Z\) makes paths that drift down more likely and paths that drift up less likely, by exactly the amount that turns “Brownian motion plus drift \(\theta\)” back into a driftless Brownian motion.

Lévy’s characterization

Proving something is a Brownian motion straight from the definition is painful. Lévy’s characterization is the shortcut: if \(\widetilde M\) is a \(\widetilde{\mathbb Q}\)-martingale with continuous paths, \(\widetilde M_0=0\) and \([\widetilde M,\widetilde M]_t=t\), then \(\widetilde M\) is a \(\widetilde{\mathbb Q}\)-Brownian motion.

So to prove Girsanov we only need: (1) \([\widetilde W^{\widetilde{\mathbb Q}},\widetilde W^{\widetilde{\mathbb Q}}]_t=t\) and (2) \(\widetilde W^{\widetilde{\mathbb Q}}\) is a \(\widetilde{\mathbb Q}\)-martingale.

Price dynamics under \(\mathbb Q\)

By Girsanov, \(W^{\mathbb Q}_t=W_t+\theta t\) is a \(\mathbb Q\)-Brownian motion on \([0,T]\), and since \(\mu-\sigma\theta=r\),

\(\theta=\dfrac{\mu-r}{\sigma}\) is the market price of risk: excess return per unit of volatility, i.e. the Sharpe ratio. This answers questions 1–5 of Section 2 for assumption 1.

8. Assumption 2: finding the hedge

Good news first: \(\frac{d\mathbb Q}{d\mathbb P}\big\vert_{\mathcal F_T}=Z_T>0\), so \(\mathbb P\) and \(\mathbb Q\) are equivalent and it suffices to show \(\mathbb Q[X_T=V_T]=1\).

Two facts:

1. \(e^{-rt}X_t\) is a \(\mathbb Q\)-martingale: \(d(e^{-rt}X_t)=e^{-rt}\Delta_tS_t\sigma\,dW^{\mathbb Q}_t\).
2. \(M^V_t:=E^{\mathbb Q}[e^{-rT}V_T\mid\mathcal F_t]\) is a \(\mathbb Q\)-martingale on \([0,T]\), by Tower: \(E^{\mathbb Q}[M^V_t\mid\mathcal F_s]=E^{\mathbb Q}[e^{-rT}V_T\mid\mathcal F_s]=M^V_s\).

We want to make these two martingales the same process. Suppose \(M^V\) can be written as a stochastic integral, \(dM^V_t=\Gamma_t\,dW^{\mathbb Q}_t\) for some adapted \(\Gamma\). Then choose

so that \(d(e^{-rt}X_t)=e^{-rt}\Delta_tS_t\sigma\,dW^{\mathbb Q}_t=\Gamma_t\,dW^{\mathbb Q}_t=dM^V_t\). Starting both at \(x=M^V_0=E^{\mathbb Q}[e^{-rT}V_T]\), “same start + same dynamics” gives \(e^{-rt}X_t=M^V_t\) for all \(t\), and at \(T\), \(X_T=V_T\).

The remaining question — is every \(\mathbb Q\)-martingale of the form \(\int\Gamma\,dW^{\mathbb Q}\)? — is answered (yes, when the filtration is generated by the Brownian motion) by the martingale representation theorem. In words: in a Brownian world, every fair game is a trading strategy in the stock, so every claim can be hedged — the market is complete.

9. Quadratic covariation and integration by parts

For continuous \(X,Y\) and a partition \(\Pi\) of \([0,t]\),

For Itô processes driven by the same \(W\), \(dX_t=\Theta_tdt+\Delta_tdW_t\) and \(dY_t=\phi_tdt+\psi_tdW_t\), the heuristic \(d[X,Y]_t=dX_t\,dY_t\) gives

and \([X,Y]\equiv0\) if either \(\Delta\equiv0\) or \(\psi\equiv0\).

Stochastic integration by parts.

again an Itô process. For smooth (no-\(dW\)) \(f,g\) the bracket vanishes and we recover the ordinary product rule \(d(f_tg_t)=f_t\,dg_t+g_t\,df_t\).

Series contents

1. 01 · Probability Spaces, Filtrations, Conditional Expectation and Martingales
2. 02 · Random Walks, Quadratic Variation and Brownian Motion
3. 03 · Stochastic Integration
4. 04 · Itô’s Formula, Itô Processes and Geometric Brownian Motion
5. 05 · Black–Scholes: Replication, the PDE and the Greeks
6. 06 · Risk-Neutral Pricing, Change of Measure and Girsanov (this post)
7. 07 · SDEs, Feynman–Kac and Stopping Times

← Previous: Black–Scholes: Replication, the PDE and the Greeks　·　Next: SDEs, Feynman–Kac and Stopping Times →

This series collects my notes from QST MF 790 Stochastic Calculus at Boston University’s Questrom School of Business, in 7 parts. The content follows the order of the course; wording, intuition and a few corrections are my own. Throughout, \(N(\mu,\sigma^2)\) uses the variance as its second argument. Corrections are welcome. Chinese version of this post: 中文版.

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.

1. The market

Fix \((\Omega,\mathcal F,\mathbb F,\mathbb P)\) and a Brownian motion \(W\). Two assets trade:

1. Stock (risky), \(S\sim GBM(\mu,\sigma)\): \(\dfrac{dS_t}{S_t}=\mu\,dt+\sigma\,dW_t\), \(S_0>0\). \(S_t(\omega)\) is the price of one share at time \(t\) in scenario \(\omega\).
2. Money market (riskless), \(\dfrac{dB_t}{B_t}=r\,dt\), \(B_0=1\): one dollar at time \(0\) grows to \(e^{rt}\) dollars at time \(t\), with constant \(r>0\).

2. Self-financing strategies and wealth

A trading strategy \(\Delta\) is an adapted process; \(\Delta_t(\omega)\) is the number of shares held at \((t,\omega)\). Start with capital \(x\). Self-financing: all money not in the stock sits in the money market, and no money enters or leaves after time \(0\).

The wealth \(X=X^{x,\Delta}\) splits as

and the money-market part buys \((X_t-\Delta_tS_t)/B_t\) “shares” of \(B\).

Wealth dynamics. Fix positions at the start of \([t,t+dt]\):

Substituting the dynamics of \(S\) and \(B\):

so \(X\) is an Itô process. Read it as: earn \(r\) on everything, plus the excess return \(\mu-r\) on the part in stock, plus noise.

3. What “price” means

We want to price a European call: the right to buy \(S\) at strike \(K\) at maturity \(T\), with payoff \((S_T-K)^+=\max(S_T-K,0)\).

Arbitrage-free price = replication cost. If starting from \(x\) and trading self-financingly with some \(\Delta\) we end with \(X^{x,\Delta}_T=(S_T-K)^+\) with probability one, then the call’s price must be \(x\). Any other price would let you sell the expensive one, buy the cheap one, and pocket the difference risk-free.

Same start + same dynamics = same process. If \(dY_t=\Theta^Y_tdt+\Delta^Y_tdW_t\) and \(dZ_t=\Theta^Z_tdt+\Delta^Z_tdW_t\) with \(Y_0=Z_0\), \(\Theta^Y\equiv\Theta^Z\), \(\Delta^Y\equiv\Delta^Z\), then \(Y_t=Z_t\) for all \(t\le T\) with probability one. This is the replication argument in one line: match the starting point and match both the \(dt\) and \(dW\) terms.

4. Matching wealth against the option price

Let \(c(t,s)\) be the call price at time \(t\) when \(S_t=s\). We know \(c(T,s)=(s-K)^+\) and want \(c\) for \(t<T\). Work with discounted processes (it simplifies the algebra): \(e^{-rt}X_t\) and \(e^{-rt}c(t,S_t)\).

Discounted wealth. By the discounting rule from Part 04,

Discounted price. Assume \(c\) is smooth (strictly speaking the payoff is not differentiable at \(s=K\), but \(c\) is smooth for \(t<T\)). With \(d[S,S]_t=\sigma^2S_t^2dt\) and writing \((\cdot)=(t,S_t)\):

Set \(x=c(0,S_0)\) and match:

• \(dW\) terms: \(e^{-rt}\Delta_t\sigma S_t=e^{-rt}c_s(t,S_t)\sigma S_t\), so

  \[\boxed{\ \Delta_t=c_s(t,S_t)\ }\quad\text{— the delta hedge.}\]

• \(dt\) terms: \(\Delta_t(\mu-r)S_t=-rc+c_t+c_s\mu S_t+\frac12c_{ss}\sigma^2S_t^2\). Plug in \(\Delta_t=c_s\); the \(c_s\mu S_t\) terms cancel, leaving

  \[0=-rc(t,S_t)+c_t(t,S_t)+rS_t\,c_s(t,S_t)+\tfrac12\sigma^2S_t^2\,c_{ss}(t,S_t).\]

It suffices to require this for every \(t<T\) and \(s>0\), which gives the Black–Scholes PDE:

\(\mu\) is gone. The hedge cancels the stock’s risk exactly, so how fast the stock is expected to grow is irrelevant to the price.

5. Solving the PDE

Two steps: (1) assume a solution \(c\) exists and find a formula for it; (2) check that the formula solves the PDE.

Step 1. Let \(\widetilde S\sim GBM(r,\sigma)\) under some measure \(\widetilde{\mathbb P}\) with Brownian motion \(\widetilde W\) — note the new measure and the drift \(r\) instead of \(\mu\). By Itô,

so the discounted price is a martingale under \(\widetilde{\mathbb P}\). Together with the terminal condition,

Write \(\widetilde S_T\) in terms of \(\widetilde S_t\):

\(\widetilde S_t\) is known at time \(t\) and \(\widetilde W_T-\widetilde W_t\perp\!\!\!\perp\widetilde{\mathcal F}_t\), so we may freeze \(\widetilde S_t=s\). With \(\widetilde W_T-\widetilde W_t\sim N(0,T-t)\):

Computing the integral gives the Black–Scholes formula:

Only the time to maturity \(\tau=T-t\) enters, not calendar time.

Step 2. Verify directly that this \(c\) satisfies the PDE and \(c(T,s)=(s-K)^+\) (a homework exercise; the identity checked in Section 7 below is the key step).

6. The Greeks

Think of \(c=c(t,s;T,K,r,\sigma)\). Sensitivities to these inputs are the Greeks. Write \(\tau=T-t\) and \(d_\pm=d_\pm(\tau,s)\).

Delta. A call is worth more when the stock is higher. The replicating strategy is \(\Delta_t=c_s(t,S_t)\): to first order in \(S\) (pretend \(dt=0\)), \(dX_t=c_s\,dS_t=dc(t,S_t)\), so the hedge matches the option’s moves — we are delta neutral.

Gamma. \(c_{ss}>0\): the call is convex in the stock price. To second order in \(S\),

Intuition for “long convexity”. Own the call and short \(c_s\) shares. A delta hedge is a tangent line to the convex price curve; whichever way the stock jumps, the curve ends up above the tangent, so the hedged position gains \(\frac12c_{ss}(dS)^2\ge0\) from moves of either sign. Gamma is largest near the strike and becomes more concentrated as maturity approaches.

Theta. \(c_t<0\): with everything else fixed, the call loses value as time passes. That is the price paid for long gamma: with all second-order changes included,

and the PDE says these are equal — a perfect hedge. Theta exactly pays for the gamma gains plus financing.

Vega. \(c_\sigma>0\): higher volatility means bigger moves; up-moves help the call, down-moves hurt it only down to zero. Positive vega = long volatility.

7. Check: the hedge is perfect

With \(c=SN(d_+)-Ke^{-r\tau}N(d_-)\), \(c_s=N(d_+)\), \(c_{ss}=\frac{N'(d_+)}{\sigma S\sqrt\tau}\) and \(c_t\) from the table, we need \((c-c_sS)\,r=c_t+\frac12c_{ss}\sigma^2S^2\):

The theta term and the gamma term cancel exactly.

8. Implied volatility

Under Black–Scholes, conditional mean and variance of returns do not change over time:

In reality volatility is not constant. Two ways to estimate it:

• historical volatility from past returns (backward-looking);
• implied volatility from option prices (forward-looking).

For a European call with maturity \(T\) and strike \(K\), let \(C_{BS}(\sigma,K,T)\) be the Black–Scholes price and \(C_m(K,T)\) the observed market price. The implied volatility \(\sigma_{imp}=\sigma_{imp}(K,T)\) solves

Because vega \(>0\), \(C_{BS}\) is increasing in \(\sigma\), so the solution is unique. Options are typically quoted in implied volatility, not price. If the world were exactly Black–Scholes, \(\sigma_{imp}(K,T)=\sigma\) for every \(K\) and \(T\). Real data instead show \(\sigma_{imp}\) varying with strike and maturity — the volatility smile / skew — which is the market telling us the model is only an approximation.

Series contents

1. 01 · Probability Spaces, Filtrations, Conditional Expectation and Martingales
2. 02 · Random Walks, Quadratic Variation and Brownian Motion
3. 03 · Stochastic Integration
4. 04 · Itô’s Formula, Itô Processes and Geometric Brownian Motion
5. 05 · Black–Scholes: Replication, the PDE and the Greeks (this post)
6. 06 · Risk-Neutral Pricing, Change of Measure and Girsanov
7. 07 · SDEs, Feynman–Kac and Stopping Times

← Previous: Itô’s Formula, Itô Processes and Geometric Brownian Motion　·　Next: Risk-Neutral Pricing, Change of Measure and Girsanov →

This series collects my notes from QST MF 790 Stochastic Calculus at Boston University’s Questrom School of Business, in 7 parts. The content follows the order of the course; wording, intuition and a few corrections are my own. Throughout, \(N(\mu,\sigma^2)\) uses the variance as its second argument. Corrections are welcome. Chinese version of this post: 中文版.

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.

1. Why the ordinary chain rule fails

For a smooth path \(g\), \(f(g_t)-f(g_0)=\int_0^tf'(g_u)\,g'_u\,du\). The naive stochastic analogue would be \(f(W_t)=f(W_0)+\int_0^tf'(W_u)\,dW_u\). Test it with \(f(x)=\frac12x^2\), which would predict \(\frac12W_t^2=\int_0^tW_u\,dW_u\). Compute directly with a partition of \([0,t]\):

(Check the algebra: \(W_{t_{j-1}}W_{t_j}-W_{t_{j-1}}^2+\frac12(W_{t_j}-W_{t_{j-1}})^2=\frac12W_{t_j}^2-\frac12W_{t_{j-1}}^2\).)

The extra \(\frac12t\) is the quadratic variation. So the chain rule needs a second-order correction.

2. Itô’s formula

For \(W\) a Brownian motion and \(f\) smooth:

Shorthand: \(df(W_t)=f'(W_t)\,dW_t+\frac12f''(W_t)\,dt\). Note that “\(df(W_t)\)” by itself has no meaning — it is shorthand for the integral equation.

Heuristic derivation. Taylor to second order, \(f(x+y)-f(x)=f'(x)y+\frac12f''(x)y^2\), with \(x=W_t\), \(y=dW_t\) and \((dW_t)^2=d[W,W]_t\):

This is not rigorous — a two-term Taylor expansion is not exact, and plugging \(dW_t\) into it is formal — but it gives the right answer, and it is how everyone uses the formula.

Two examples.

• \(f(x)=e^x\): \(e^{W_t}=1+\int_0^te^{W_u}\,dW_u+\frac12\int_0^te^{W_u}\,du\).
• \(f(x)=\sin x\): \(\sin W_t=\int_0^t\cos W_u\,dW_u-\frac12\int_0^t\sin W_u\,du\). The \(du\) term changes sign, so \(\sin W_t\) is neither a sub-, super-, nor martingale.

Rule of thumb: the \(dW\) part is a martingale; the \(dt\) part is a drift. A function of \(W\) is a martingale exactly when its \(dt\) term vanishes.

3. Time-dependent Itô

If \(f(t,x)\) has continuous \(f_t,f_x,f_{xx}\):

Heuristic. Two-variable Taylor,

with \(s=dt\), \(y=dW_t\), and the multiplication table \(dt^2=0\), \(dt\,dW_t=0\), \((dW_t)^2=dt\). Why do the two mixed terms die? As \(\Vert \Pi\Vert \to0\):

The exponential martingale (important)

\(f(t,x)=e^{\alpha x-\frac12\alpha^2t}\) has \(f_t=-\frac12\alpha^2f\), \(f_x=\alpha f\), \(f_{xx}=\alpha^2f\), so the \(dt\) terms cancel:

is a martingale. The \(-\frac12\alpha^2t\) is exactly the “Itô correction” needed to kill the drift that \(e^{\alpha W_t}\) would otherwise have.

4. Itô processes

An Itô process has both a regular and a stochastic integral:

with \(\Theta,\Delta\) adapted so that the integrals exist and \(X_0\) constant. \(A\) is never a martingale unless \(\Theta\equiv0\).

Examples: \(\Theta\equiv0,\ \Delta\equiv1,\ X_0=x\) gives \(X_t=x+W_t\), Brownian motion started at \(x\); \(\Delta\equiv0\) gives a smooth process \(X_t=X_0+\int_0^t\Theta_u\,du\).

Quadratic variation. Only the \(dW\) part contributes:

Integrating against an Itô process. For adapted \(\Gamma\),

provided \(\int_0^t\vert\Gamma_u\Theta_u\vert du<\infty\) and \(E\int_0^t\Gamma_u^2\Delta_u^2du<\infty\). If \(X\) is a share price, this is the P/L of trading \(\Gamma\) shares. The result \(Y_t=\int_0^t\Gamma_u\,dX_u\) is again an Itô process (a martingale only if \(\Theta\equiv0\)), with \([Y,Y]_t=\int_0^t\Gamma_u^2\Delta_u^2\,du\).

5. Itô’s formula for Itô processes

For nice \(f(t,x)\) (smooth, bounded derivatives):

Plugging in \(dX_t=\Theta_tdt+\Delta_tdW_t\) and \(d[X,X]_t=\Delta_t^2dt\):

So a function of an Itô process is again an Itô process — but the martingale property is not preserved in general (e.g. \(W\) is a martingale, \(W^2\) is a sub-martingale).

How to remember it: pretend the second-order Taylor expansion is exact, then apply the multiplication table, using \(dt\,dX_t=dt(\Theta_tdt+\Delta_tdW_t)=0\) and \((dX_t)^2=\Delta_t^2dt\).

Discounting (used all the time in Parts 05–06). If \(dY_t=\Theta_tdt+\Delta_tdW_t\), apply the formula to \(f(t,y)=e^{-rt}y\) (\(f_t=-rf\), \(f_y=e^{-rt}\), \(f_{yy}=0\)):

6. Geometric Brownian motion

Let \(\mu=\{\mu_t\}\), \(\sigma=\{\sigma_t\}\) be adapted with \(\int_0^t\vert\mu_s\vert ds<\infty\) and \(E\int_0^t\sigma_s^2ds<\infty\). A (generalized) geometric Brownian motion, written \(S\sim GBM(\mu,\sigma)\), is

It is the most common stock-price model. With constant \(\mu,\sigma\),

Dynamics. Let \(X_t=\int_0^t(\mu_u-\frac12\sigma_u^2)du+\int_0^t\sigma_udW_u\), so \(S_t=f(X_t)\) with \(f(x)=S_0e^x\) (\(f=f'=f''\)):

\(dS_t/S_t\) is the instantaneous return over \([t,t+dt]\). Since \(\mu_t,\sigma_t\) are \(\mathcal F_t\)-measurable and \(dW_t\perp\!\!\!\perp\mathcal F_t\),

so \(\mu\) and \(\sigma\) are the drift and volatility of returns. The \(-\frac12\sigma^2\) in the exponent is the Itô correction again: the log-price grows more slowly than the expected return.

7. Application: integrals of deterministic functions are Gaussian

Claim. If \(f\) is deterministic (non-random), then

The variance \(\int_0^tf_u^2du\) is exactly what the Itô isometry predicts; the new information is that the distribution is normal. We will use this for the Ornstein–Uhlenbeck process in Part 07.

Series contents

1. 01 · Probability Spaces, Filtrations, Conditional Expectation and Martingales
2. 02 · Random Walks, Quadratic Variation and Brownian Motion
3. 03 · Stochastic Integration
4. 04 · Itô’s Formula, Itô Processes and Geometric Brownian Motion (this post)
5. 05 · Black–Scholes: Replication, the PDE and the Greeks
6. 06 · Risk-Neutral Pricing, Change of Measure and Girsanov
7. 07 · SDEs, Feynman–Kac and Stopping Times

← Previous: Stochastic Integration　·　Next: Black–Scholes: Replication, the PDE and the Greeks →

This series collects my notes from QST MF 790 Stochastic Calculus at Boston University’s Questrom School of Business, in 7 parts. The content follows the order of the course; wording, intuition and a few corrections are my own. Throughout, \(N(\mu,\sigma^2)\) uses the variance as its second argument. Corrections are welcome. Chinese version of this post: 中文版.

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 \(\mathrm{Var}[I_t]=E\int_0^t\Delta_s^2ds\) · \([I,I]_t=\int_0^t\Delta_s^2ds\).

1. Discrete time: the integral as trading P/L

Let \(M=\{M_n\}\) be a martingale (think: a share price). For a trading strategy \(\Delta=\{\Delta_n\}\) define

\(\Delta_j\,dM_j\) is the P/L over \([j-1,j]\) and \(I_n\) the cumulative P/L over \([0,n]\).

Predictable strategies. We hold \(\Delta_j\) over \([j-1,j]\), and we cannot look into the future, so \(\Delta_j\) must be known at time \(j-1\): \(\Delta_j\) is \(\mathcal F_{j-1}\)-measurable. Such a \(\Delta=\{\Delta_n\}_{n\ge1}\) is called predictable (\(\Delta_0\) is not defined since \(\mathcal F_{-1}\) does not exist).

Properties of \(I=\sum\Delta\,dM\) (\(\Delta\) predictable, \(M\) a martingale)

1. Linear in \(\Delta\): \(I(\alpha\Delta+\beta\psi)=\alpha I(\Delta)+\beta I(\psi)\) — just split the sum.
2. Adapted: \(\Delta_1,\dots,\Delta_n\) are known by time \(n-1\) and \(M_0,\dots,M_n\) by time \(n\), so \(I_n\) is \(\mathcal F_n\)-measurable.

3. Martingale: \(I_{n+1}-I_n=\Delta_{n+1}(M_{n+1}-M_n)\) and \(\Delta_{n+1}\) is \(\mathcal F_n\)-measurable, so

   \[E[I_{n+1}-I_n\mid\mathcal F_n]=\Delta_{n+1}\,E[M_{n+1}-M_n\mid\mathcal F_n]=0.\]
4. Quadratic variation: \([I,I]_n=\sum_j\Delta_j^2(dM_j)^2=\sum_j\Delta_j^2\,d[M,M]_j\).

5. Variance: since \(I\) is a martingale with \(I_0=0\), Fact 1 of Part 02 gives

   \[\mathrm{Var}[I_n]=E\big[[I,I]_n\big]=E\Big[\sum_{j=1}^n\Delta_j^2\,d[M,M]_j\Big].\]

For the random walk, \(d[M,M]_j=1\), so \([I,I]_n=\sum_j\Delta_j^2\) and \(\mathrm{Var}[I_n]=E\sum_j\Delta_j^2\).

Intuition for “martingale”: you cannot beat a fair game by choosing your bet size, as long as you choose it before the coin is tossed.

2. Continuous time: why the obvious definition fails

We want \(I_t=\int_0^t\Delta_s\,dW_s\) (\(\Delta\): integrand, \(W\): integrator; same P/L reading). The natural guess is a Riemann sum,

If \(W\) were smooth, the mean value theorem would turn this into \(\int_0^t\Delta_s\dot W_s\,ds\). But \(W\) is nowhere differentiable (Part 02), so \(\dot W_s\) does not exist and path-by-path the limit does not exist in the ordinary sense. We need a new kind of limit.

3. Simple (buy-and-hold) integrands

First bring things back to almost discrete time. Fix \(0=t_0<t_1<\dots<t_n\) and \(\mathcal F_{t_j}\)-measurable r.v.’s \(\Delta_j\), and set

hold \(\Delta_{j-1}\) shares over \((t_{j-1},t_j]\). For such a simple integrand the integral is explicit:

Reading it piece by piece: for \(t_{j-1}\le t\le t_j\),

and for \(t\ge t_n\) nothing changes any more — we stopped trading.

Properties for simple \(\Delta\)

• Linear and adapted: as in discrete time.
• Continuous paths (new): each term \(t\mapsto\Delta_{j-1}(W_{t\wedge t_j}-W_{t\wedge t_{j-1}})\) is continuous because \(W\) is, and a finite sum of continuous processes is continuous.

• Martingale. For \(t_{j-1}<s<t_j<t\) (adjacent buckets),

  \[\begin{aligned} E[I_t-I_s\mid\mathcal F_s]&=E[\Delta_{j-1}(W_{t_j}-W_s)\mid\mathcal F_s]+E\big[E[\Delta_j(W_t-W_{t_j})\mid\mathcal F_{t_j}]\mid\mathcal F_s\big]\\ &=\Delta_{j-1}\cdot0+E[\Delta_j\cdot0\mid\mathcal F_s]=0. \end{aligned}\]

  It is crucial that the position is chosen at the beginning of each period.

• Itô isometry: \(\mathrm{Var}[I_t]=E\Big[\int_0^t\Delta_s^2\,ds\Big]\).
• Quadratic variation: \([I,I]_t=\int_0^t\Delta_s^2\,ds\).

Note the difference between the last two: \([I,I]_t\) is a random quantity path by path; the isometry is its expectation.

4. General integrands: approximate, then take an \(L^2\) limit

Fix \(T>0\). Let \(\mathcal L\) be the set of integrands \(\Delta\) that are

• adapted, and
• square-integrable: \(E\Big[\int_0^T\Delta_s^2\,ds\Big]<\infty\).

For \(\Delta\in\mathcal L\) and partitions \(\Pi^n\) of \([0,T]\) with mesh \(\to0\), freeze \(\Delta\) at the left endpoints:

\(\Delta^n\) is a simple “rebalance only at grid times” version of \(\Delta\), and \(I^n\) is perfectly well defined. For a fixed \(\omega\), \(\lim I^n_t(\omega)\) need not exist; instead we take the limit in mean square.

Step 1 — the approximation is good. A key result of the course is that for \(\Delta\in\mathcal L\),

written \(\Delta^n\to\Delta\) in \(L^2([0,T]\times\Omega)\). (Intuitively: as the rebalancing grid gets finer, the frozen strategy tracks the real one, on average over both time and scenarios.)

Step 2 — so the approximations are Cauchy. From \((a+b)^2\le2a^2+2b^2\),

Step 3 — the isometry transfers this to the integrals. By linearity, \(I^n_t-I^m_t=I(\Delta^n-\Delta^m)_t\), and \(\Delta^n-\Delta^m\) is simple, so

Step 4 — define the integral as the limit. The integrals \(I^n\) form a Cauchy sequence in mean square, so they converge (as processes) to a limit \(I\), and we define

Why the isometry is the hero. It says the map “strategy \(\mapsto\) P/L” preserves distance: two strategies that are close in \(E\int(\cdot)^2\) produce P/Ls that are close in variance. So “approximate the strategy” automatically gives “approximate the integral” — even though no single path converges in the classical sense.

5. Properties for general integrands

Everything carries over to \(\Delta\in\mathcal L\):

• linear: \(I(\alpha\Delta+\beta\psi)=\alpha I(\Delta)+\beta I(\psi)\);
• continuous sample paths, \(I_0=0\);
• adapted;
• martingale;
• Itô isometry: \(\mathrm{Var}[I_t]=E\Big[\int_0^t\Delta_s^2\,ds\Big]\);
• quadratic variation: \([I,I]_t=\int_0^t\Delta_s^2\,ds\) path by path.

In differential shorthand, \(dI_t=\Delta_t\,dW_t\) and \(d[I,I]_t=\Delta_t^2\,dt=(\Delta_t\,dW_t)^2\) — consistent with \((dW_t)^2=dt\).

Series contents

1. 01 · Probability Spaces, Filtrations, Conditional Expectation and Martingales
2. 02 · Random Walks, Quadratic Variation and Brownian Motion
3. 03 · Stochastic Integration (this post)
4. 04 · Itô’s Formula, Itô Processes and Geometric Brownian Motion
5. 05 · Black–Scholes: Replication, the PDE and the Greeks
6. 06 · Risk-Neutral Pricing, Change of Measure and Girsanov
7. 07 · SDEs, Feynman–Kac and Stopping Times

← Previous: Random Walks, Quadratic Variation and Brownian Motion　·　Next: Itô’s Formula, Itô Processes and Geometric Brownian Motion →

This series collects my notes from QST MF 790 Stochastic Calculus at Boston University’s Questrom School of Business, in 7 parts. The content follows the order of the course; wording, intuition and a few corrections are my own. Throughout, \(N(\mu,\sigma^2)\) uses the variance as its second argument. Corrections are welcome. Chinese version of this post: 中文版.

Overview

Brownian motion is easiest to understand as the limit of a coin-tossing game. This post starts from the random walk, introduces the one quantity that makes stochastic calculus different from ordinary calculus — quadratic variation — and then shows that Brownian motion inherits everything: it is a martingale, \(W_t^2-t\) is a martingale, and \([W,W]_t=t\).

The punchline to keep in mind: \((dW_t)^2=dt\). Ordinary smooth curves have zero quadratic variation; Brownian paths do not, and that single fact is where Itô’s extra term will come from.

1. The random walk

Take the infinite fair coin-toss space, \(\omega=(\omega_1,\omega_2,\dots)\), with independent tosses. Define

The \(Z_j\) are i.i.d. with \(E Z_j=0\), \(\mathrm{Var}(Z_j)=E Z_j^2=1\). \(M\) is the random walk and the \(Z_n=M_n-M_{n-1}\) are its increments.

Properties.

1. \(M\) is a martingale. \(\mathcal F_n\) = first \(n\) tosses; \(Z_{n+1}\) only depends on toss \(n+1\), so \(Z_{n+1}\perp\!\!\!\perp\mathcal F_n\):

   \[E[M_{n+1}\mid\mathcal F_n]=E[M_n+Z_{n+1}\mid\mathcal F_n]=M_n+E[Z_{n+1}]=M_n.\]
2. Independent increments. For \(0\le m<n\le p<q\), \(M_n-M_m=\sum_{j=m+1}^nZ_j\) and \(M_q-M_p=\sum_{j=p+1}^qZ_j\) use disjoint tosses, so they are independent.

3. Increment mean and variance. \(E[M_n-M_m]=0\) and

   \[\mathrm{Var}(M_n-M_m)=\sum_{i,j=m+1}^nE[Z_iZ_j]=n-m,\qquad E[Z_iZ_j]=\begin{cases}1,&i=j\\0,&i\ne j.\end{cases}\]

2. The squared random walk

\(M_n^2\) is not a martingale — it drifts up:

So \(M^2\) is a sub-martingale, and subtracting the drift fixes it: \(Y_n:=M_n^2-n\) satisfies \(E[Y_{n+1}\mid\mathcal F_n]=M_n^2+1-(n+1)=Y_n\), a martingale.

3. Quadratic variation (discrete time)

For a process \(X=\{X_n\}\), the quadratic variation is the process

It adds up squared increments and measures how rough the path is. (The definition does not need \(X\) to be a martingale.)

• Random walk: every increment is \(\pm1\), so \([M,M]_n=n\) for every \(\omega\) — and this holds even for an unfair coin. This is special; in general \([X,X]_n\) depends on both \(n\) and \(\omega\).

• \(Y_n=M_n^2-n\): using \(M_j=M_{j-1}+Z_j\) and \(Z_j^2=1\),

  \[[Y,Y]_n=\sum_{j=1}^n\big(Z_j^2+2Z_jM_{j-1}-1\big)^2=\sum_{j=1}^n4M_{j-1}^2,\]

  which genuinely varies with \(\omega\).

Two facts about martingales and quadratic variation

Fact 1. If \(X\) is a martingale with \(X_0=0\), then \(\mathrm{Var}(X_n)=E\big[[X,X]_n\big]\).

Fact 2 (quadratic variation as compensator). If \(X\) is a martingale with \(E X_n^2<\infty\), then \(\{X_n^2-[X,X]_n\}\) is a martingale.

Since \([X,X]_n\) only goes up, \(X_n^2\) itself is a sub-martingale; we must compensate (subtract) \([X,X]\) to get a martingale. That is why \([X,X]\) is also called the compensator. \(M_n^2-n\) above is the special case \([M,M]_n=n\).

4. From random walk to Brownian motion

Random walk and Brownian motion share independent increments with mean \(0\) and variance equal to the time elapsed. They differ in that \(M\) lives in discrete time with integer jumps, while \(W\) lives in continuous time and takes any real value.

Scaled random walk. Speed up time by \(n\) and shrink jumps to \(1/\sqrt n\):

• Independent increments: inherited from \(M\).

• Right mean and variance: for \(ns,nt\) integers,

  \[E[W^{(n)}_t-W^{(n)}_s]=0,\qquad \mathrm{Var}[W^{(n)}_t-W^{(n)}_s]=\frac1n\,\mathrm{Var}[M_{nt}-M_{ns}]=\frac{nt-ns}{n}=t-s.\]

• Gaussian in the limit (CLT):

  \[W^{(n)}_t-W^{(n)}_s=\underbrace{\sqrt{\frac{\lfloor nt\rfloor-\lfloor ns\rfloor}{n}}}_{\to\sqrt{t-s}}\cdot\underbrace{\frac1{\sqrt{\lfloor nt\rfloor-\lfloor ns\rfloor}}\sum_{j=\lfloor ns\rfloor+1}^{\lfloor nt\rfloor}Z_j}_{\to N(0,1)\text{ by the CLT}}\ \Longrightarrow\ N(0,t-s).\]

So \(W^{(n)}\to W\): Brownian motion exists, and it is literally a coin-tossing game watched from very far away. (Be careful which “\(n\)” the CLT is applied to — it is the number of tosses in the window, \(\lfloor nt\rfloor-\lfloor ns\rfloor\).)

5. Brownian motion and its properties

Recall the definition from Part 01: \(W_0=0\); continuous paths; \(W_t-W_s\sim N(0,t-s)\); independent increments. \(W\) is a BM with respect to \(\mathbb F\) if additionally it is adapted and \(W_t-W_s\perp\!\!\!\perp\mathcal F_s\).

1. Martingale. Shown in Part 01. In continuous time, super/sub-martingales and the monotonicity of \(E X_t\) work exactly as in discrete time.

2. Covariance. For \(s<t\), add and subtract \(W_s\) to create an independent increment:

So \((W_s,W_t)\) is jointly normal with

3. Conditional law and the Markov property. Using \(E[e^{\alpha Y}]=e^{\alpha\mu+\frac12\alpha^2\sigma^2}\) for \(Y\sim N(\mu,\sigma^2)\):

Given everything up to time \(s\), the law of \(W_t\) depends only on \(W_s\) — the Markov property. The transition density is

read as “go from \(y\) to near \(x\) in time \(\tau\)”.

4. \(W_t^2-t\) is a martingale. With \(W_t^2=(W_t-W_s)^2+2(W_t-W_s)W_s+W_s^2\):

This is the continuous-time version of \(M_n^2-n\).

6. Quadratic variation in continuous time

Fix \(t\) and a partition \(\Pi=\{0=t_0<t_1<\dots<t_n=t\}\) with mesh \(\Vert \Pi\Vert =\max_j(t_j-t_{j-1})\). Define

Over a window, \([X,X]_{[s,t]}\) is defined the same way and quadratic variation accumulates: \([X,X]_t=[X,X]_s+[X,X]_{[s,t]}\).

Theorem. \([W,W]_t=t\) with probability one, and \([W,W]_{[s,t]}=t-s\).

Intuition. For the scaled walk, \([W^{(n)},W^{(n)}]_t=\frac1n\lfloor nt\rfloor\in[t-\frac1n,t]\), and \(W^{(n)}\to W\).

Fact 2 carries over: if \(X\) is a continuous martingale with \(E X_t^2<\infty\), then \(X_t^2-[X,X]_t\) is a martingale (proof needs Itô’s formula, Part 04).

7. Why Brownian paths must be rough

The first-order variation of a path is \(FV(X)_t=\lim_{\Vert \Pi\Vert \to0}\sum_j\vert X_{t_j}-X_{t_{j-1}}\vert\). For a smooth path, the mean value theorem gives

Finite first variation forces zero quadratic variation (for continuous paths):

Since \([W,W]_t=t>0\), Brownian paths cannot have finite first variation, and in fact they are nowhere differentiable with probability one.

This roughness is a feature, not a bug: a smooth price path would be locally predictable, and predictability means arbitrage.

The heuristic everyone uses. Differentiating \([W,W]_t=t\) gives \(d[W,W]_t=dt\), and people write

thinking of \([W,W]_{[t,t+dt]}\approx\sum(W_{t_j}-W_{t_{j-1}})^2\). Taking a square root to say “\(dW_t=\sqrt{dt}\)” is meaningless, but the multiplication rule \((dW_t)^2=dt\) is exactly right and drives all of Part 04.

Series contents

1. 01 · Probability Spaces, Filtrations, Conditional Expectation and Martingales
2. 02 · Random Walks, Quadratic Variation and Brownian Motion (this post)
3. 03 · Stochastic Integration
4. 04 · Itô’s Formula, Itô Processes and Geometric Brownian Motion
5. 05 · Black–Scholes: Replication, the PDE and the Greeks
6. 06 · Risk-Neutral Pricing, Change of Measure and Girsanov
7. 07 · SDEs, Feynman–Kac and Stopping Times

← Previous: Probability Spaces, Filtrations, Conditional Expectation and Martingales　·　Next: Stochastic Integration →

This series collects my notes from QST MF 790 Stochastic Calculus at Boston University’s Questrom School of Business, in 7 parts. The content follows the order of the course; wording, intuition and a few corrections are my own. Throughout, \(N(\mu,\sigma^2)\) uses the variance as its second argument. Corrections are welcome. Chinese version of this post: 中文版.

Overview

These are my notes from QST MF 790 Stochastic Calculus at Boston University’s Questrom School of Business, cleaned up into 7 posts.

The whole course hangs on one sentence: “Let \(W\) be a Brownian motion adapted to a filtration \(\mathbb F\). Then \(W\) is a martingale.” This first post unpacks every word in it — probability space, random variable, filtration, adaptedness, conditional expectation and martingale — so that the rest of the series can just use them.

1. The probability space \((\Omega,\mathcal F,\mathbb P)\)

• \(\Omega\): the sample space; an element \(\omega\in\Omega\) is one outcome (“one scenario of the world”).
• \(\mathcal F\): the collection of events. An event is a set of outcomes, so each \(A\in\mathcal F\) is a subset of \(\Omega\).
• \(\mathbb P\): the probability function, \(\mathbb P:\mathcal F\to[0,1]\).

A point that is easy to gloss over: \(\mathbb P\) is a function on \(\mathcal F\), not on \(\Omega\). So \(\mathbb P[\omega]\) is not defined, and \(\mathbb P[\{\omega\}]\) is defined only if \(\{\omega\}\in\mathcal F\).

Example: finite space, equal likelihood. \(\Omega=\{1,2,\dots,N\}\) and \(\mathcal F=2^\Omega\) (the power set, all subsets of \(\Omega\)). Here every singleton is an event, and for \(A\in\mathcal F\)

2. Random variables, expectation, independence

A random variable (r.v.) is a function \(X:\Omega\to\mathbb R\): feed it an outcome, it outputs a number.

• Discrete: \(X\) only takes values in a countable set \(E_X=\{x_1,x_2,\dots\}\). Its pmf is \(p(x)=\mathbb P[\{\omega: X(\omega)=x\}]=\mathbb P[X=x]\), and \(p(x)=0\) for \(x\notin E_X\).
• Continuous with pdf \(f\): \(\mathbb P(X\in E)=\int_E f(x)\,dx\) for every (Borel) set \(E\subseteq\mathbb R\).

The expectation is the \(\mathbb P\)-weighted average of \(X\):

Mean \(=E(X)\), variance \(=E[(X-E X)^2]\), \(N\)-th central moment \(=E[\vert X-EX\vert ^N]\).

Independence. \(X\perp\!\!\!\perp Y\) iff for all (nice) functions \(f,g\)

Keep the superscript in \(E^{\mathbb P}\) in mind: in Part 06 we will change \(\mathbb P\), and every one of these averages will change with it.

3. Stochastic processes and sample paths

A stochastic process is a collection of random variables indexed by time:

• discrete time: \(X=\{X_0,X_1,X_2,\dots\}\);
• continuous time: \(X=\{X_t\}_{t\ge0}\).

It is cleanest to think of a process as a function of two arguments, \(X:[0,\infty)\times\Omega\to\mathbb R\):

• fix \(t\): \(\omega\mapsto X_t(\omega)\) is a random variable;
• fix \(\omega\): \(t\mapsto X_t(\omega)\) is a curve, called a sample path. Each scenario \(\omega\) draws its own curve, so a process is a “random curve”.

\(X\) has continuous sample paths if \(t\mapsto X_t(\omega)\) is continuous for every \(\omega\).

4. Brownian motion (first look)

A process \(W=\{W_t\}_{t\ge0}\) on \((\Omega,\mathcal F,\mathbb P)\) is a Brownian motion if

1. \(W_0=0\) with probability one;
2. the sample paths \(t\mapsto W_t\) are continuous with probability one;
3. for \(0\le s<t\): \(W_t-W_s\sim N(0,\,t-s)\);
4. for \(0\le s<t\le u<v\): \(W_v-W_u\) and \(W_t-W_s\) are independent.

(Throughout the series \(N(\mu,\sigma^2)\) means mean \(\mu\) and variance \(\sigma^2\). The increment over a window of length \(t-s\) has variance \(t-s\), i.e. standard deviation \(\sqrt{t-s}\).)

5. Filtrations: information that only grows

Discrete time. \(\mathbb F=\{\mathcal F_n\}_{n\ge0}\) is a filtration if each \(\mathcal F_n\subseteq\mathcal F\) is a \(\sigma\)-algebra and \(\mathcal F_m\subseteq\mathcal F_n\) for \(m\le n\).

Continuous time. \(\mathbb F=\{\mathcal F_t\}_{t\ge0}\) with \(\mathcal F_s\subseteq\mathcal F_t\subseteq\mathcal F\) for \(0\le s\le t\).

Read \(\mathcal F_t\) as “everything we know at time \(t\)”. The only requirement is that we never forget.

The filtration generated by a process. \(\mathcal F^X_n:=\sigma(X_0,\dots,X_n)\), or in continuous time \(\mathcal F^X_t:=\sigma(X_s:0\le s\le t)\) — the smallest \(\sigma\)-algebra containing what you learn by watching \(X\) up to now.

6. Adaptedness and measurability

\(X\) is adapted to \(\mathbb F\) if the information in \(\mathbb F\) is enough to know \(X\) at every time:

• \(X_t\) is \(\mathcal F_t\)-measurable (“mbl”), i.e.
• \(\{\omega: X_t(\omega)\in E\}\in\mathcal F_t\) for every nice \(E\subseteq\mathbb R\).

Intuition: at time \(t\), if you know \(\mathcal F_t\) you can read off the number \(X_t\). No peeking into the future is needed.

A filtration for a Brownian motion. \(\mathbb F\) (not necessarily \(\mathbb F^W\)) is a filtration for \(W\) if

• \(W\) is adapted to \(\mathbb F\), and
• \(W_t-W_s\) is independent of \(\mathcal F_s\) for all \(0\le s<t\).

The second condition says: everything we know today tells us nothing about tomorrow’s increment.

7. Conditional expectation

Fix a r.v. \(X\) and a sub-\(\sigma\)-algebra \(\mathcal G\subseteq\mathcal F\). The conditional expectation \(E[X\mid\mathcal G]\) is a random variable with two properties:

1. Measurability: \(E[X\mid\mathcal G]\) is \(\mathcal G\)-measurable (knowing \(\mathcal G\) is enough to know it);
2. Partial averaging: for every \(A\in\mathcal G\),

Interpretation: \(E[X\mid\mathcal G]\) is our best guess of \(X\) given the information in \(\mathcal G\).

Intuition for partial averaging. Think of \(\mathcal G\) as a way of chopping \(\Omega\) into pieces you can tell apart. On each piece \(A\), the guess \(E[X\mid\mathcal G]\) must have the same average as \(X\) itself. So the conditional expectation is “\(X\), averaged within each piece you can distinguish” — it keeps exactly the detail that \(\mathcal G\) can see and smooths away the rest. Example: two coin tosses, \(X=\) number of heads, \(\mathcal G=\) “the first toss”. On \(\{H\ast\}\) the guess is \(1.5\), on \(\{T\ast\}\) it is \(0.5\); check \(E[\mathbf 1_{\{H\ast\}}X]=\tfrac14(2+1)=\tfrac34=\tfrac12\cdot1.5\).

Properties (use these constantly)

1. Notation. If \(\mathcal G=\sigma(Y)\), write \(E[X\mid Y]\).
2. Independence: if \(X\perp\!\!\!\perp\mathcal G\) then \(E[X\mid\mathcal G]=E[X]\).
3. Known: if \(X\) is \(\mathcal G\)-measurable then \(E[X\mid\mathcal G]=X\).
4. Linearity: \(E[\alpha X+\beta Y\mid\mathcal G]=\alpha E[X\mid\mathcal G]+\beta E[Y\mid\mathcal G]\).
5. Taking out what is known (TOWK): if \(X\) is \(\mathcal G\)-measurable, \(E[XY\mid\mathcal G]=X\,E[Y\mid\mathcal G]\).
6. Tower: if \(\mathcal H\subseteq\mathcal G\subseteq\mathcal F\), then \(E\big[E(X\mid\mathcal G)\mid\mathcal H\big]=E[X\mid\mathcal H]\).

Memory trick for Tower: the smaller information set wins, and in time-indexed form the later time sits inside.

On processes. With \(X\) adapted and \(0\le s<t<u\):

At time \(s\), \(X_s\) behaves like a constant; Tower is time-consistency of averaging.

8. Martingales

Definition (continuous time). An adapted \(X\) is a martingale if for all \(0\le s<t\)

Discrete time is the same with \(n,m\); there it is enough to check one step: \(E[X_{n+1}\mid\mathcal F_n]=X_n\) for all \(n\).

• Super-martingale (\(\searrow\)): \(E[X_t\mid\mathcal F_s]\le X_s\) — the future is smaller on average.
• Sub-martingale (\(\nearrow\)): \(E[X_t\mid\mathcal F_s]\ge X_s\).

Taking expectations (Tower), \(t\mapsto E[X_t]\) is non-increasing for a super-martingale, constant for a martingale, non-decreasing for a sub-martingale.

Intuition: a martingale is a fair game. Given everything you know, your best forecast of tomorrow’s value is today’s value.

Brownian motion is a martingale

Let \(\mathbb F\) be a filtration for \(W\) and \(s<t\):

The trick — write \(W_t=(W_t-W_s)+W_s\), “future increment plus what we know” — will be used again and again.

Series contents

1. 01 · Probability Spaces, Filtrations, Conditional Expectation and Martingales (this post)
2. 02 · Random Walks, Quadratic Variation and Brownian Motion
3. 03 · Stochastic Integration
4. 04 · Itô’s Formula, Itô Processes and Geometric Brownian Motion
5. 05 · Black–Scholes: Replication, the PDE and the Greeks
6. 06 · Risk-Neutral Pricing, Change of Measure and Girsanov
7. 07 · SDEs, Feynman–Kac and Stopping Times

Next: Random Walks, Quadratic Variation and Brownian Motion →

This series collects my notes from QST MF 790 Stochastic Calculus at Boston University’s Questrom School of Business, in 7 parts. The content follows the order of the course; wording, intuition and a few corrections are my own. Throughout, \(N(\mu,\sigma^2)\) uses the variance as its second argument. Corrections are welcome. Chinese version of this post: 中文版.

Overview

The last part of the series puts all of interval estimation and hypothesis testing into tables: the one-sample and two-sample normal cases, large-sample methods, and the u, t, χ² and F tests.

A trick for remembering these tables: intervals and tests use the same pivotal quantity. First get clear on “under which conditions, which statistic follows which distribution”, and you can derive both the interval and the rejection region yourself. At the end there is a summary of handy results derived in the exercises.

Notation: \(u_p\), \(t_p\), \(\chi^2_p\), \(F_p\) are all lower \(p\)-quantiles. The confidence level is \(1-\alpha\).

1. Confidence intervals and pivotal quantities (P343–P352, Guide P369)

Pivotal method: construct a function \(G\) (the pivot) that involves only the parameter of interest and whose distribution is known and free of parameters; solve \(P(a\le G\le b)=1-\alpha\) for the parameter to get the interval.

One normal population \(N(\mu,\sigma^2)\)

Two normal populations \(X\sim N(\mu_1,\sigma_1^2)\) (sample size \(m\)), \(Y\sim N(\mu_2,\sigma_2^2)\) (sample size \(n\))

Confidence intervals for \(\mu_1-\mu_2\)

Confidence interval for \(\sigma_1^2/\sigma_2^2\) (\(\mu_1,\mu_2\) unknown)

Large-sample confidence intervals (know the derivation)

For \(X\sim b(1,p)\), the CLT gives \(\dfrac{\bar x-p}{\sqrt{p(1-p)/n}}\ \dot\sim\ N(0,1)\). Replacing \(p\) in the denominator by \(\bar x\) gives

2. Hypothesis tests at a glance

Basic steps of a test (P357): see Part 1.
p-value: the probability, assuming \(H_0\) is true, of getting a result at least as extreme as the one observed. Reject \(H_0\) when the p-value is \(\le\alpha\).

Tests for the mean \(\mu\) of one normal population (P369)

(\(u_0\), \(t_0\) are the observed values of the statistics; \(T\sim t(n-1)\).)

Tests for the difference of two normal means (P373)

The null is always written \(H_0:\mu_1-\mu_2=0\) (or \(\le0\), \(\ge0\)); the direction of the rejection region follows the table above.

Tests for variances

Duality between intervals and tests: the two-sided test of \(\mu=\mu_0\) at level \(\alpha\) accepts if and only if \(\mu_0\) lies inside the \(1-\alpha\) confidence interval.

Power function (P360): e.g. for the u test of \(H_0:\mu\le\mu_0\) vs \(H_1:\mu>\mu_0\),

which is increasing in \(\mu\) and equals \(\alpha\) at \(\mu=\mu_0\).

3. Results derived in the exercises

Series contents

1. 01 · Core Definitions
2. 02 · Common Distributions
3. 03 · Random Vectors, Covariance and Conditional Expectation
4. 04 · Characteristic Functions, LLN and CLT
5. 05 · Sampling Distributions, Order Statistics and Sufficiency
6. 06 · Point Estimation
7. 07 · Interval Estimation, Hypothesis Testing and Exercise Results (this post)

That’s the end of the series — thanks for reading all the way here.　·　← Previous: Point Estimation

This series collects my review notes on Probability Theory and Mathematical Statistics by Mao Shisong, Cheng Yiming and Pu Xiaolong, in 7 parts. P××× refers to page numbers in the book; “Guide” refers to its companion study guide and solutions manual. Corrections are welcome. A Chinese version is on my Chinese homepage.

Overview

The properties in the estimation chapter are especially easy to mix up: is unbiasedness invariant? Is the method-of-moments estimator unique? What about the MLE?

This part first lines up all the “unique / invariant” results in one table, then collects the theorems you need to be able to prove — consistency, UMVUE, the sufficiency principle and the Cramér–Rao inequality — and ends with a conjugate-prior example of Bayesian estimation.

1. Two methods of point estimation

Method of moments (substitution principle): replace population moments with sample moments, and a function of population moments with the same function of sample moments.
Example: \(E(X)=\mu\Rightarrow\hat\mu=\bar x\); \(\mathrm{Var}(X)=\sigma^2\Rightarrow\hat\sigma^2=s_n^2\).

Maximum likelihood: \(L(\theta)=\prod_{i=1}^np(x_i;\theta)\); usually solve the log-likelihood equation \(\frac{\partial\ln L}{\partial\theta}=0\).

• Normal population (P316): \(\hat\mu=\bar x\), \(\hat\sigma^2=s_n^2=\frac1n\sum(x_i-\bar x)^2\)
• When the support depends on the parameter you cannot differentiate; look directly at how \(L\) behaves. Example: for \(U(0,\theta)\), \(L(\theta)=\theta^{-n}I_{\{\theta\ge x_{(n)}\}}\) is decreasing on \(\theta\ge x_{(n)}\), so \(\hat\theta=x_{(n)}\).

2. Properties side by side (what is “unique”, what is “invariant”)

3. Consistency

Theorem 6.2.1 (P310, know the proof): if \(\lim_{n\to\infty}E(\hat\theta_n)=\theta\) and \(\lim_{n\to\infty}\mathrm{Var}(\hat\theta_n)=0\), then \(\hat\theta_n\) is a consistent estimator of \(\theta\).

Theorem 6.2.2 (P311, know the proof): if \(\hat\theta_{n1},\dots,\hat\theta_{nk}\) are consistent for \(\theta_1,\dots,\theta_k\), and \(\eta=g(\theta_1,\dots,\theta_k)\) is continuous, then \(\hat\eta_n=g(\hat\theta_{n1},\dots,\hat\theta_{nk})\) is consistent for \(\eta\).

Asymptotic normality of the MLE (Guide P328): under suitable regularity conditions,

where \(n\) is the sample size and \(I(\theta)\) the Fisher information. So the MLE is both consistent and asymptotically normal, and its asymptotic variance attains the C-R lower bound (asymptotically efficient).

4. Mean squared error and efficiency

Mean squared error (P323)

For an unbiased estimator, MSE is just the variance. A biased estimator can actually have smaller MSE. Example: for a normal population, the estimator \(\frac1{n+1}\sum(x_i-\bar x)^2\) of \(\sigma^2\) has smaller MSE than \(s^2\).

5. UMVUE

Theorem 6.4.1 (P325, know the proof): let \(\hat\theta\) be unbiased for \(\theta\) with \(\mathrm{Var}(\hat\theta)<\infty\). Then \(\hat\theta\) is the UMVUE of \(\theta\) if and only if, for every statistic \(\varphi\) with \(E(\varphi)=0\) and \(\mathrm{Var}(\varphi)<\infty\) (an “unbiased estimator of zero”),

In one sentence: the UMVUE is uncorrelated with every unbiased estimator of zero.

Theorem 6.4.2, sufficiency principle (P327, know the proof): let \(T\) be a sufficient statistic, \(\hat\theta\) an unbiased estimator of \(\theta\), and \(\tilde\theta=E(\hat\theta\mid T)\). Then \(\tilde\theta\) is also unbiased for \(\theta\), and \(\mathrm{Var}(\tilde\theta)\le\mathrm{Var}(\hat\theta)\).

Intuition: a good estimator should depend only on the sufficient statistic, and conditioning on the sufficient statistic can only make an estimator better (or at least no worse). So to find the UMVUE, you only need to search among functions of the sufficient statistic (Rao–Blackwell).

6. Fisher information and the C-R inequality

Fisher information (P329)

For the second equality see Exercise 6.4.5; the second-derivative form is usually easier to compute. The larger \(I(\theta)\), the more information one observation carries about \(\theta\).

☆ Theorem 6.4.3, Cramér–Rao inequality (P329, know the proof): under regularity conditions, if \(T\) is an unbiased estimator of \(g(\theta)\), then

In particular, every unbiased estimator of \(\theta\) has variance \(\ge\dfrac1{nI(\theta)}\). An unbiased estimator attaining the bound is called efficient.

Fisher information of common distributions (Guide P341)

Example: for a Poisson population, \(\mathrm{Var}(\bar x)=\frac\lambda n=\frac{1}{nI(\lambda)}\), so \(\bar x\) is an efficient estimator of \(\lambda\).

7. Bayesian estimation

Posterior distribution

where \(\pi(\theta)\) is the prior and \(m\) is the marginal distribution of the sample. The posterior mean \(E(\theta\mid\boldsymbol x)\) is usually taken as the Bayes estimator of \(\theta\).

Conjugate prior example: \(x_i\sim b(1,p)\) with prior \(p\sim Be(a,b)\). The posterior is \(Be\left(a+\sum x_i,\ b+n-\sum x_i\right)\), and the Bayes estimator is

Intuition: the prior is like “having already seen \(a+b\) trials, \(a\) of them successes”; you pool those with the real data and compute the success rate.

Series contents

1. 01 · Core Definitions
2. 02 · Common Distributions
3. 03 · Random Vectors, Covariance and Conditional Expectation
4. 04 · Characteristic Functions, LLN and CLT
5. 05 · Sampling Distributions, Order Statistics and Sufficiency
6. 06 · Point Estimation (this post)
7. 07 · Interval Estimation, Hypothesis Testing and Exercise Results

← Previous: Sampling Distributions, Order Statistics and Sufficiency　·　Next: Interval Estimation, Hypothesis Testing and Exercise Results →

This series collects my review notes on Probability Theory and Mathematical Statistics by Mao Shisong, Cheng Yiming and Pu Xiaolong, in 7 parts. P××× refers to page numbers in the book; “Guide” refers to its companion study guide and solutions manual. Corrections are welcome. A Chinese version is on my Chinese homepage.

Overview

From here on we are in mathematical statistics. The sample mean \(\bar x\), the sample variance \(s^2\), and the three sampling distributions \(\chi^2\), t and F are the “raw materials” for all the interval estimation and hypothesis testing that follows.

The single most important theorem is: for a normal sample, \(\bar x\) and \(s^2\) are independent (Theorem 5.4.1). Almost every pivotal quantity later is built on it, so make sure you can prove it.

1. Sample mean and sample variance

Sample variance (P267)

A computational identity (worth memorizing):

Often useful in proofs (Exercise 5.3.10): \(\left(\sum_{i=1}^nx_i\right)^2=\sum_{i=1}^nx_i^2+2\sum_{i<j}x_ix_j\).

Recursive update (Exercise 5.3.4): with \(\bar x_n=\frac1n\sum_{i=1}^nx_i\) and \(s_n^2=\frac1{n-1}\sum_{i=1}^n(x_i-\bar x_n)^2\),

The sample mean minimizes squared deviation (P264, Theorem 5.3.2): \(\sum(x_i-\bar x)^2\le\sum(x_i-c)^2\) for any constant \(c\).

Theorem 5.3.3 (P265): distribution of \(\bar x\)

• Normal population \(N(\mu,\sigma^2)\): \(\bar x\sim N(\mu,\sigma^2/n)\) (exactly)
• Arbitrary population with finite variance \(\sigma^2\): for large \(n\), \(\bar x\ \dot\sim\ N(\mu,\sigma^2/n)\) (by the CLT)

Theorem 5.3.4 (P268, know the proof): if the population has a finite second moment, then

2. Order statistics

Distribution of a single order statistic (P273, memorize): for a population with density \(p(x)\) and distribution function \(F(x)\),

In particular,

Note: the density formula only applies to continuous populations; it fails for discrete ones. In the discrete case, work from the distribution function: \(F_{(n)}(x)=[F(x)]^n\), \(F_{(1)}(x)=1-[1-F(x)]^n\).

Joint distribution of several order statistics (P274; see Exercises 5.3.22, 5.3.23 for applications of Theorem 5.3.6): the joint density of \((x_{(i)},x_{(j)})\), \(i<j\), is

The joint density of all order statistics is \(n!\prod_{i=1}^np(y_i)\), \(y_1<\dots<y_n\).
Example (Exercise 5.3.34): for \(X\sim U(0,\theta)\), the joint density of \((x_{(1)},\dots,x_{(n)})\) is \(\frac{n!}{\theta^n}\), \(0<u_1\le\dots\le u_n\le\theta\).

Order statistics of the uniform

• Example 5.3.8 (P274): for a \(U(0,1)\) population, \(x_{(k)}\sim Be(k,\ n-k+1)\)
• Example 5.3.9 (P275): for a \(U(0,1)\) population, the range \(x_{(n)}-x_{(1)}\sim Be(n-1,\ 2)\)

☆ Exercise 5.3.28 (turning any continuous population into a uniform): if the population distribution function \(F\) is continuous, let \(\eta_i=F(x_{(i)})\). Then

1. \(\eta_1\le\dots\le\eta_n\) are exactly the order statistics of a \(U(0,1)\) sample (since \(F(X)\sim U(0,1)\) and \(F\) is monotone);
2. \(E(\eta_i)=\frac{i}{n+1}\), \(\mathrm{Var}(\eta_i)=\frac{i(n+1-i)}{(n+1)^2(n+2)}\) (from the mean and variance of \(Be(i,n-i+1)\));
3. the covariance matrix of \(\eta_i,\eta_j\) (\(i<j\)) is

Asymptotic distribution of the sample \(p\)-quantile (Theorem 5.3.7, P276): if the population density \(p(x)\) is continuous at \(x_p\) with \(p(x_p)>0\), then

In particular, the sample median \(m_{0.5}\ \dot\sim\ N\left(x_{0.5},\ \frac{1}{4n\,p^2(x_{0.5})}\right)\).

The median is robust (P277): a few outliers barely affect the median, but can move the mean a lot.

3. The three sampling distributions

• Quantiles of the F distribution (P287): \(F_\alpha(n,m)=\dfrac{1}{F_{1-\alpha}(m,n)}\) (since \(F\sim F(m,n)\) implies \(\frac1F\sim F(n,m)\))
• \(t(1)\) is the standard Cauchy; \(t(n)\to N(0,1)\) as \(n\to\infty\); if \(T\sim t(n)\), then \(T^2\sim F(1,n)\)
• \(\chi^2(n)\) has mean \(n\) and variance \(2n\); for the means and variances of \(F(m,n)\) and \(t(n)\), see Part 2

☆ Theorem 5.4.1 (P284, know the proof): let \(x_1,\dots,x_n\) be a sample from \(N(\mu,\sigma^2)\). Then

1. \(\bar x\) and \(s^2\) are independent;
2. \(\bar x\sim N(\mu,\sigma^2/n)\);
3. \(\dfrac{(n-1)s^2}{\sigma^2}\sim\chi^2(n-1)\).

Corollaries (the building blocks of pivotal quantities)

• \[\dfrac{\sqrt n(\bar x-\mu)}{s}\sim t(n-1)\]
• Two independent normal samples: \(\dfrac{s_x^2/\sigma_1^2}{s_y^2/\sigma_2^2}\sim F(m-1,n-1)\)
• If \(\sigma_1=\sigma_2\): \(\dfrac{(\bar x-\bar y)-(\mu_1-\mu_2)}{s_w\sqrt{\frac1m+\frac1n}}\sim t(m+n-2)\), where \(s_w^2=\dfrac{(m-1)s_x^2+(n-1)s_y^2}{m+n-2}\)

4. Sufficient statistics

Theorem 5.5.1, factorization theorem (P297): \(T(\boldsymbol x)\) is sufficient for \(\theta\) if and only if there exist functions \(g(t;\theta)\) and \(h(\boldsymbol x)\) such that

where \(h\) does not involve \(\theta\), and \(g\) depends on the sample only through \(T\).

Sufficient statistics of common distributions (Guide P300)

• A one-to-one transformation of a sufficient statistic is still sufficient (Guide P300); e.g. \(\sum x_i\) and \(\bar x\) are equivalent.
• The pattern: for an exponential family, the sufficient statistics are the sums of the sample functions that multiply the parameters in the exponent; when the support depends on the parameter (e.g. the uniform), the sufficient statistics are the extremes.

Series contents

1. 01 · Core Definitions
2. 02 · Common Distributions
3. 03 · Random Vectors, Covariance and Conditional Expectation
4. 04 · Characteristic Functions, LLN and CLT
5. 05 · Sampling Distributions, Order Statistics and Sufficiency (this post)
6. 06 · Point Estimation
7. 07 · Interval Estimation, Hypothesis Testing and Exercise Results

← Previous: Characteristic Functions, LLN and CLT　·　Next: Point Estimation →

This series collects my review notes on Probability Theory and Mathematical Statistics by Mao Shisong, Cheng Yiming and Pu Xiaolong, in 7 parts. P××× refers to page numbers in the book; “Guide” refers to its companion study guide and solutions manual. Corrections are welcome. A Chinese version is on my Chinese homepage.