Mathematical Foundations for Finance: Handbook

This reference guide follows one continuous line of reasoning:

$$\text{information} \longrightarrow \text{trading} \longrightarrow \text{no arbitrage} \longrightarrow \text{martingale measures} \longrightarrow \text{replication}$$$$\text{replication} \longrightarrow \text{stochastic calculus} \longrightarrow \text{Black-Scholes}.$$

The primary source is M. Schweizer and E. W. Farkas, Lecture Notes: Mathematical Foundations for Finance, version December 22, 2022, abbreviated below as LN. The weekly exercises, solutions, and exam simulation are from ETH Zürich, Fall 2023, abbreviated as ES 1-11 and SE. Definitions, theorem statements, and proof structures attributed to these sources are paraphrased. Paragraphs labelled Interpretation, Study note, Worked example, or Pitfall are supplementary explanations added for this guide.

AI-use note. AI tools assisted with transcription, restructuring, cross-referencing, notation checks, and LaTeX validation. The cited lecture notes, exercise sheets, and solutions remain the authoritative sources.

How to use this reference

Use the chapter narrative to understand an argument, the shaded reference cards to check hypotheses and statements, and the procedure blocks to solve standard problem types. Every card cites its LN chapter, original lecture label, and PDF page; its ref-... anchor stays unique even when the lecture reuses a number such as “Theorem 2.1.” A reference card is the canonical formal statement. Narrative formulas explain or derive it, while the formula sheet is only a memory aid.

For results whose hypotheses are easy to overlook, the card separates Assumptions, Conclusion, Prerequisites, and Common misuse. Check every assumption before using the conclusion.

Fast entrances: Notation · Find by task · Assumption checklist · Formula sheet · Theorem index · Alphabetical index.

Notation and conventions

In every market chapter, an undecorated price or value such as $S$, $V$, or $H$ is discounted by the tradable reference asset; a tilde marks an undiscounted quantity. In the probability and stochastic-calculus chapters, no discounting is implied. A subscript $k$ denotes a discrete trading date and a subscript $t$ continuous time. The symbol $T$ is an integer terminal date in discrete time and a positive real horizon in continuous time. Martingale statements always depend on both a measure and a filtration.

Symbol	Meaning	First relevant entry
$(\Omega,\mathcal F,\mathbb P)$	sample space, events, and physical probability	Conditional expectation
$\mathbb F=(\mathcal F_k)$ or $(\mathcal F_t)$	information available over time	Martingale
$\mathbb P,\mathbb Q$ and $\mathbb E_{\mathbb P},\mathbb E_{\mathbb Q}$	probability measures and their expectations	EMM and ELMM
$\mathbb Q\ll\mathbb P$	absolute continuity: every $\mathbb P$-null event is $\mathbb Q$-null	Density processes
$\mathbb Q\approx\mathbb P$	equivalence: the two measures have the same null events	EMM and ELMM
$\mathbb Q\overset{\mathrm{loc}}{\approx}\mathbb P$	equivalence on every finite-time $\sigma$-field $\mathcal F_t$	Continuous Girsanov theorem
$\widetilde S^0,\widetilde S$	undiscounted reference asset and risky prices	Trading strategy
$S^0\equiv1,\ S=\widetilde S/\widetilde S^0$	discounted asset prices	EMM and ELMM
$\varphi=(\varphi^0,\vartheta)$	bank-account and risky-asset holdings	Trading strategy
$\vartheta_k$	risky holding selected at $k-1$ and held over $(k-1,k]$	Predictable integral
$V(\varphi),G(\vartheta),C(\varphi)$	wealth, cumulative trading gains, and cumulative external cost	Gains and cost
$\Delta X_k$	one-step increment $X_k-X_{k-1}$	Predictable integral
$H$	terminal contingent claim; $K$ and $J$ denote stochastic-integral integrands	European payoff
$\mathcal P^e(S),\mathcal P^{e,\mathrm{loc}}(S)$	sets of EMMs and ELMMs	EMM and ELMM
$D_T:=\frac{d\mathbb Q}{d\mathbb P}\text{ on }\mathcal F_T,\ Z_t=\mathbb E_{\mathbb P}[D_T\mid\mathcal F_t]$	terminal density and density process	Density processes
$W$	Brownian motion	Brownian motion
$[X]$ and $[X,Y]$	quadratic variation and covariation of semimartingales	Optional quadratic variation
$\langle M\rangle$	predictable bracket of a locally square-integrable martingale; $[M]=\langle M\rangle$ when $M$ is continuous	Optional quadratic variation
$L^2(M),L^2_{\mathrm{loc}}(M)$	square-integrable and locally square-integrable integrands	Local square integrability
$\mathcal E(X)$	stochastic exponential	Stochastic exponential
$\mathbb Q^*,q^*$	binomial or Black-Scholes martingale measure and binomial up probability, as context indicates	Binomial EMM
$r$	one-period simple bank return in CRR; continuously compounded rate in Black-Scholes	CRR pricing

Find by task

If you need to…	Check this entry first	Then use
test a discrete market for arbitrage	First FTAP	NA and EMM procedure
find multinomial or binomial risk-neutral probabilities	Binomial EMM	NA and EMM procedure
move an expectation from $\mathbb Q$ to $\mathbb P$	Density-process Bayes formula	Continuous-time Bayes formula
decide whether a claim can be replicated	Attainability criterion	Attainable payoff
price and hedge an attainable claim	Valuation theorem	CRR pricing and hedging procedure
test market completeness	Second FTAP	Itô representation
apply optional stopping	Stopping theorem	Brownian hitting-time transform
construct or estimate an Itô integral	Elementary integrands	Itô isometry
differentiate a stochastic function	Continuous Itô formula	Itô procedure
change a Brownian drift	Continuous Girsanov theorem	Girsanov procedure
price a European call in Black-Scholes	Call formula	Black-Scholes procedure

Assumption checklist

This table is a gatekeeper, not a substitute for the linked statement.

Result	Minimum checks before use
Conditional expectation	The variable is integrable; the candidate is $\mathcal G$-measurable and integrable.
First FTAP	Discounted frictionless market, finite discrete time, predictable self-financing strategies, and the stated admissibility convention.
Attainable valuation	Finite discrete time, NA, trivial $\mathcal F_0$, and an attainable claim with an admissible replicating strategy.
Second FTAP	Finite discrete time, NA, trivial $\mathcal F_0$, and $\mathcal F_T=\mathcal F$.
Optional stopping	Right-continuous true martingale; $\sigma\leq\tau$; either bounded $\tau$ or uniform integrability.
Itô isometry	Square-integrable martingale and an admissible predictable integrand.
Itô formula	Continuous semimartingale and a $C^2$ function; use the jump formula when jumps are present.
Stochastic exponential as a density	Strict positivity and a true martingale with terminal expectation one; Novikov is only a sufficient condition in the continuous case.
Continuous Girsanov	Locally equivalent measures and a valid continuous density process; track the drift sign convention.
Itô representation	Integrable claim measurable in the augmented Brownian filtration; traded assets must span the Brownian directions before inferring completeness.
Black-Scholes valuation	Frictionless Brownian-filtration model, $\sigma>0$, integrable claim, consistent discounting, and the same Brownian risk driving the traded stock.

1. Probability, information, and conditional expectation

Companion reading. For a Chinese, example-driven introduction to probability spaces, sigma-algebras, and filtrations, see 从掷硬币理解概率空间、σ-代数与域流.

1.1 The filtered probability space

In finite discrete time, trading dates are

$$k=0,1,\ldots,T, \qquad T\in\mathbb N.$$

Uncertainty is described by a probability space $(\Omega,\mathcal F,\mathbb P)$. Information is described by a filtration

$$\mathbb F=(\mathcal F_k)_{k=0}^T, \qquad \mathcal F_k\subseteq\mathcal F_\ell \quad\text{for }k\leq\ell.$$

The objects have different roles:

Object	Mathematical role	Financial interpretation
$\Omega$	set of possible outcomes	all market scenarios or paths
$\mathcal F$	$\sigma$-algebra of events	all events admitted by the model
$\mathbb P$	probability measure	physical assessment of events
$\mathcal F_k$	information available at time $k$	events distinguishable by time $k$

A process $X=(X_k)_{k=0}^T$ is adapted if $X_k$ is $\mathcal F_k$-measurable. It is predictable if $X_k$ is $\mathcal F_{k-1}$-measurable for $k\geq1$. Asset prices are normally adapted: the current price is observable now. A position held over $(k-1,k]$ must be predictable: it must be selected before the price move is observed. (LN, p. 5.)

Interpretation. A $\sigma$-algebra is a collection of yes-or-no questions. If $A\in\mathcal F_k$, then at time $k$ an observer can determine whether the realised path lies in $A$. The inclusion $\mathcal F_k\subseteq\mathcal F_{k+1}$ says that recorded information is not forgotten.

1.2 Measurability and generated information

A $\sigma$-algebra $\mathcal F$ on a nonempty set $\Omega$ contains $\Omega$ and is closed under complements and countable unions. A mapping $X:\Omega\to\mathbb R$ is a random variable when

$$\{X\in B\}=X^{-1}(B)\in\mathcal F \qquad\text{for every }B\in\mathcal B(\mathbb R).$$

The generated $\sigma$-algebra $\sigma(X)$ is the smallest $\sigma$-algebra that makes $X$ measurable. It represents exactly the information revealed by observing $X$. For a family $\mathcal A$ of events,

$$\sigma(\mathcal A) = \bigcap_{\substack{\mathcal G\text{ a }\sigma\text{-algebra}\\ \mathcal A\subseteq\mathcal G}} \mathcal G.$$

This works because arbitrary intersections of $\sigma$-algebras are again $\sigma$-algebras. Unions do not have the same property. (LN, pp. 155-157; ES 1, Exercises 1.1-1.2.)

A statement holds $\mathbb P$-almost surely, abbreviated $\mathbb P$-a.s., when its failure set has probability zero. Random variables in $L^p(\mathbb P)$ are therefore equivalence classes under a.s. equality, not pointwise equality.

1.3 Conditional expectation

For $X\in L^1(\mathbb P)$, the conditional expectation $\mathbb E[X\mid\mathcal G]$ is the integrable $\mathcal G$-measurable summary of $X$ that preserves expectations on every event already observable in $\mathcal G$. The precise definition and existence statement are in the two cards below. (LN, pp. 158-160.) The rules used throughout the course are

$$\begin{aligned} \mathbb E[aX+bY\mid\mathcal G] &=a\mathbb E[X\mid\mathcal G]+b\mathbb E[Y\mid\mathcal G],\\ X\text{ is }\mathcal G\text{-measurable} &\Longrightarrow \mathbb E[X\mid\mathcal G]=X,\\ \mathbb E\!\left[\mathbb E[X\mid\mathcal G]\right] &=\mathbb E[X],\\ X\text{ independent of }\mathcal G &\Longrightarrow \mathbb E[X\mid\mathcal G]=\mathbb E[X],\\ \mathcal H\subseteq\mathcal G &\Longrightarrow \mathbb E[X\mid\mathcal H] =\mathbb E\!\left[\mathbb E[X\mid\mathcal G]\mid\mathcal H\right]. \end{aligned}$$

If $A_1,\ldots,A_n$ form a finite partition with $\mathbb P[A_i]\gt 0$ and $\mathcal G=\sigma(A_1,\ldots,A_n)$, then

$$\boxed{ \mathbb E[X\mid\mathcal G] = \sum_{i=1}^n \frac{\mathbb E[X\mathbf 1_{A_i}]}{\mathbb P[A_i]} \mathbf 1_{A_i}. }$$

The expression is constant on each atom $A_i$. This is the finite-tree version of conditional expectation: at a node, average over successor paths that are still indistinguishable. (ES 1, Exercise 1.3 and solution, pp. 3-4.)

Conditional expectation

Scope: Probability space $(\Omega,\mathcal F,\mathbb P)$ with a sub-$\sigma$-field $\mathcal G$. Assumptions: $U\in L^1(\mathbb P)$.

Definition: A conditional expectation of $U$ given $\mathcal G$ is a $\mathcal G$-measurable $Y\in L^1(\mathbb P)$ satisfying

$$\mathbb E_{\mathbb P}[U\mathbf 1_A]=\mathbb E_{\mathbb P}[Y\mathbf 1_A]\qquad\text{for every }A\in\mathcal G.$$

It is denoted $\mathbb E_{\mathbb P}[U\mid\mathcal G]$ and is unique up to $\mathbb P$-a.s. equality.

Use: Verify a proposed conditional expectation using measurability and test events. Source: LN Chapter 8, Definition and Theorem 2.1, p. 158. Nearby: Conditional expectation: existence and uniqueness

Conditional expectation: existence and uniqueness

Setting: Probability space with a sub-$\sigma$-field $\mathcal G$. Statement and assumptions: If $U\in L^1(\mathbb P)$, then $\mathbb E[U\mid\mathcal G]$ exists, belongs to $L^1(\mathbb P)$, and is unique up to almost-sure equality.

Use: Use existence and almost-sure uniqueness of conditional expectation. Source: LN Chapter 8, Theorem 2.1, p. 158. Nearby: Conditional expectation · Conditioning with an independent variable

Conditioning with an independent variable

Setting: Probability space with a sub-$\sigma$-field $\mathcal G$. Statement and assumptions: If $U$ is $\mathcal G$-measurable, $V$ is independent of $\mathcal G$, and $F:\mathbb R^2\to[0,\infty]$ is measurable, then

$$\mathbb E[F(U,V)\mid\mathcal G]=f(U),\qquad f(u)=\mathbb E[F(u,V)].$$

Use: Condition on known data while averaging an independent variable. Source: LN Chapter 8, Lemma 2.2, p. 160. Nearby: Conditional expectation: existence and uniqueness · Conditional Fatou and dominated convergence

Conditional Fatou and dominated convergence

Setting: Probability space with a sub-$\sigma$-field $\mathcal G$. Statement and assumptions: If $U_n\geq X$ for an integrable $X$, then

$$\mathbb E[\liminf_nU_n\mid\mathcal G]\leq\liminf_nE[U_n\mid\mathcal G].$$

If $U_n\to U$ almost surely and $|U_n|\leq X$ for an integrable $X$, then

$$\mathbb E[U\mid\mathcal G]=\lim_{n\to\infty}\mathbb E[U_n\mid\mathcal G]\qquad \mathbb P\text{-a.s.}$$

Use: Pass limits through conditional expectations under the stated hypotheses. Source: LN Chapter 8, Theorem 2.3, pp. 160-161. Nearby: Conditioning with an independent variable

1.4 Current state versus accumulated history

Consider $\Omega=\{UU,UD,DU,DD\}$ with uniform probability and a price that starts at $8$ and is multiplied by $2$ after $U$ and by $1/2$ after $D$:

Path	$X_0$	$X_1$	$X_2$
$UU$	8	16	32
$UD$	8	16	8
$DU$	8	4	8
$DD$	8	4	2

The natural filtration $\mathcal F_k=\sigma(X_0,\ldots,X_k)$ retains the path history. At time 2 it distinguishes $UD$ from $DU$ because their time-1 values differ. The collection $\mathcal G_k=\sigma(X_k)$ only records the current snapshot. Since $X_2=8$ on both $UD$ and $DU$, $\mathcal G_2$ forgets information present in $\mathcal G_1$, so $(\mathcal G_k)$ is not a filtration. (ES 1, Exercise 1.4 and solution, pp. 4-5.)

Pitfall. The fact that $X_k$ is measurable with respect to $\sigma(X_k)$ for every $k$ does not imply that $(\sigma(X_k))_k$ is a filtration.

2. Financial markets and self-financing trading

2.1 Discounting and model assumptions

Let $\widetilde S^0$ be a strictly positive adapted reference asset and let

$$\widetilde S=(\widetilde S^1,\ldots,\widetilde S^d)$$

be the risky asset prices. Using $\widetilde S^0$ as numeraire gives discounted prices

$$S^0_k=1, \qquad S_k=\frac{\widetilde S_k}{\widetilde S^0_k}.$$

Discounting changes units. In finite discrete time it does not change whether a strategy is self-financing. The reference asset must itself be tradable. (LN, pp. 9-10; ES 2, Exercise 2.1.)

The baseline market is frictionless: no bid-ask spread or transaction costs, unrestricted borrowing and short selling, divisible assets, and no market impact. These are model assumptions, not empirical conclusions.

2.2 Trading strategies, value, cost, and gain

A strategy is $\varphi=(\varphi^0,\vartheta)$, where $\varphi^0$ is the bank-account holding and $\vartheta=(\vartheta^1,\ldots,\vartheta^d)$ is a predictable risky-asset holding with $\vartheta_0=0$. Its discounted value is

$$V_k(\varphi) = \varphi^0_k+\vartheta_k^{\mathsf T}S_k.$$

Rebalancing from $\varphi_k$ to $\varphi_{k+1}$ at time-$k$ prices costs

$$\Delta C_{k+1}(\varphi) = \varphi^0_{k+1}-\varphi^0_k +(\vartheta_{k+1}-\vartheta_k)^{\mathsf T}S_k.$$

Add and subtract $\vartheta_{k+1}^{\mathsf T}S_{k+1}$:

$$\Delta C_{k+1}(\varphi) = \Delta V_{k+1}(\varphi) -\vartheta_{k+1}^{\mathsf T}\Delta S_{k+1}.$$

This identity identifies the gains process

$$G_k(\vartheta) := \sum_{j=1}^k\vartheta_j^{\mathsf T}\Delta S_j$$

and therefore

$$C_k(\varphi)=V_k(\varphi)-G_k(\vartheta).$$

All of this is bookkeeping, not an optional mathematical convention. (LN, pp. 11-14.)

Trading strategy and value process

Setting: Finite discrete-time filtered market. Statement and assumptions: A trading strategy is an $\mathbb R^{d+1}$-valued process $\varphi=(\varphi^0,\vartheta)$ such that $\varphi^0$ is adapted, $\vartheta$ is predictable, and $\vartheta_0=0$. Its discounted value is

$$V_k(\varphi)=\varphi^0_k+\vartheta_k^{\mathsf T}S_k,\qquad k=0,\ldots,T.$$

Use: Identify which holdings are allowed before computing portfolio value. Source: LN Chapter 1, Definition, p. 11. Nearby: Gains and cost processes

Gains and cost processes

Setting: Finite discrete-time filtered market. Statement and assumptions: For a trading strategy $\varphi=(\varphi^0,\vartheta)$,

$$G_k(\vartheta)=\sum_{j=1}^k\vartheta_j^{\mathsf T}\Delta S_j,\qquad C_k(\varphi)=V_k(\varphi)-G_k(\vartheta).$$

Thus $G$ records gains caused by price changes, while $C$ records cumulative external funding.

Use: Separate gains from price changes from deposits and withdrawals. Source: LN Chapter 1, Definition, pp. 13-14. Nearby: Trading strategy and value process

2.3 The self-financing constraint

A strategy is self-financing when $C(\varphi)$ is constant. Equivalently,

$$\boxed{ V_k(\varphi)=V_0(\varphi)+G_k(\vartheta) =V_0(\varphi)+\sum_{j=1}^k\vartheta_j^{\mathsf T}\Delta S_j. }$$

Once $V_0$ and the predictable risky holding $\vartheta$ are given, the bank-account holding is forced:

$$\varphi^0_k = V_0+G_k(\vartheta)-\vartheta_k^{\mathsf T}S_k = V_0+G_{k-1}(\vartheta)-\vartheta_k^{\mathsf T}S_{k-1}.$$

The second expression is $\mathcal F_{k-1}$-measurable, so the bank-account component is predictable as well. (LN, pp. 14-16.)

Interpretation. Self-financing means that portfolio changes are funded entirely by selling some holdings and buying others. A deposit or withdrawal is a change in the cost process, not a trading gain.

Cross-reference: First Fundamental Theorem of Asset Pricing.

Self-financing strategy

Setting: Finite discrete-time filtered market. Statement and assumptions: A strategy is self-financing when $C(\varphi)$ is constant. Equivalently,

$$V_k(\varphi)=V_0(\varphi)+G_k(\vartheta),\qquad k=0,\ldots,T.$$

Use: Check whether rebalancing requires external cash. Source: LN Chapter 1, Definition, pp. 14-15. Nearby: Parametrisation of self-financing strategies

Parametrisation of self-financing strategies

Setting: Finite discrete-time filtered market. Statement and assumptions: A self-financing strategy is uniquely determined by its initial wealth $V_0$ and predictable risky position $\vartheta$. Conversely, every pair consisting of an $\mathcal F_0$-measurable $V_0$ and a predictable $\mathbb R^d$-valued $\vartheta$ with $\vartheta_0=0$ determines a unique self-financing strategy, represented by the pair $(V_0,\vartheta)$. Its bank-account component is predictable from time $1$ onward.

Use: Construct the bank-account holding from initial wealth and risky holdings. Source: LN Chapter 1, Proposition 2.3, p. 15. Nearby: Self-financing strategy

2.4 Stopping times and admissibility

A random time $\tau$ is a stopping time when

$$\{\tau\leq k\}\in\mathcal F_k \qquad\text{for every }k.$$

Then $\vartheta_k=\mathbf 1_{\{k\leq\tau\}}$ is predictable and implements “hold until $\tau$”. The resulting wealth is the stopped process $S_{k\wedge\tau}$. A last passage time usually fails this test because deciding that a passage was the last one requires future information. (LN, pp. 17-18; ES 2, Exercises 2.2-2.3.)

A strategy is $a$-admissible if

$$V_k(\varphi)\geq-a \quad\mathbb P\text{-a.s. for every }k,$$

and admissible if such an $a\geq0$ exists. Admissibility excludes doubling-type strategies whose apparent high probability of profit is financed by unbounded downside. (LN, pp. 19-21.)

Admissibility

Setting: Finite discrete-time filtered market. Statement and assumptions: For $a\geq0$, a strategy is $a$-admissible if $V_k(\varphi)\geq-a$ almost surely for every $k$. It is admissible if it is $a$-admissible for some finite $a$.

Use: Rule out strategies whose wealth has uncontrolled downside. Source: LN Chapter 1, Definition, p. 20.

3. Martingales and discrete stochastic integrals

A martingale is a process whose conditional value does not change when more information is revealed; the exact integrability and conditioning requirements are in the card below. Reversing the equality gives the corresponding supermartingale and submartingale inequalities. The property depends on both the filtration and the probability measure. (LN, p. 22.)

If $X$ is a martingale and $\vartheta$ is bounded and predictable, then

$$(\vartheta\mathbin{\bullet} X)_k := \sum_{j=1}^k\vartheta_j^{\mathsf T}\Delta X_j$$

is a martingale. The one-step proof is the core calculation:

$$\mathbb E_{\mathbb Q} [\vartheta_{k+1}^{\mathsf T}\Delta X_{k+1}\mid\mathcal F_k] = \vartheta_{k+1}^{\mathsf T} \mathbb E_{\mathbb Q}[\Delta X_{k+1}\mid\mathcal F_k] =0.$$

Predictability permits the holding to be taken outside the conditional expectation. A local martingale integrator gives a local martingale integral; in finite discrete time, a stochastic integral that is uniformly bounded below is a true martingale. (LN, pp. 23-26.)

In the binomial model,

$$\frac{\widetilde S^1_k}{\widetilde S^1_{k-1}} =Y_k = \begin{cases} 1+u,&\text{with probability }p,\\ 1+d,&\text{with probability }1-p, \end{cases} \qquad \widetilde S^0_k=(1+r)^k.$$

The discounted stock is a $\mathbb P$-martingale exactly when

$$r=pu+(1-p)d.$$

This condition is about the physical probability $p$. No-arbitrage pricing will replace $p$ with a different probability chosen to make the discounted stock a martingale.

3.1 Discrete martingale reference statements

Martingale, supermartingale, and submartingale

Setting: Finite discrete-time filtered market. Statement and assumptions: An adapted process $X$ is a $\mathbb Q$-martingale if each $X_k$ is integrable and

$$\mathbb E_{\mathbb Q}[X_\ell\mid\mathcal F_k]=X_k,\qquad k\leq\ell.$$

Replacing equality by $\leq$ gives a supermartingale and by $\geq$ a submartingale.

Use: Check fair-game conditional expectations under a chosen measure. Source: LN Chapter 1, Definition, p. 22. Nearby: Discrete-time local martingale

Discrete-time local martingale

Setting: Finite discrete-time filtered market. Statement and assumptions: An adapted process $X$ with $X_0=0$ is a local martingale if there are stopping times $\tau_n$ increasing stationarily to $T$ such that every stopped process $X^{\tau_n}$ is a martingale. The sequence $(\tau_n)$ is a localising sequence.

Use: Localise a process before treating it as a martingale. Source: LN Chapter 1, Definition, p. 23. Nearby: Martingale, supermartingale, and submartingale · Predictable stochastic integrals

Predictable stochastic integrals

Setting: Finite discrete-time filtered market. Statement and assumptions: If $X$ is an $\mathbb R^d$-valued local martingale null at zero and $\vartheta$ is predictable, then

$$\left(\vartheta\mathbin{\bullet}X\right)_k=\sum_{j=1}^k\vartheta_j^{\mathsf T}\Delta X_j$$

is a real-valued local martingale null at zero. If $X$ is a martingale and $\vartheta$ is bounded, the integral is a true martingale.

Use: Show predictable gains inherit a local martingale property. Source: LN Chapter 1, Theorem 3.1, pp. 23-24. Nearby: Discrete-time local martingale · Stopping a martingale

Stopping a martingale

Setting: Finite discrete-time filtered market. Statement and assumptions: If $X$ is a martingale and $\tau$ is a stopping time, then $X^\tau$ is a martingale. In particular, $\mathbb E_{\mathbb Q}[X_{k\wedge\tau}]=\mathbb E_{\mathbb Q}[X_0]$ for every $k$.

Use: Stop a discrete martingale without changing its martingale property. Source: LN Chapter 1, Corollary 3.2, p. 25. Nearby: Predictable stochastic integrals · A lower-bounded integral is a martingale

A lower-bounded integral is a martingale

Setting: Finite discrete-time filtered market. Statement and assumptions: Let $X$ be an $\mathbb R^d$-valued local $\mathbb Q$-martingale null at zero and let $\vartheta$ be predictable. If $\vartheta\mathbin{\bullet}X$ is uniformly bounded below, then it is a true $\mathbb Q$-martingale.

Use: Upgrade a lower-bounded discrete stochastic integral to a true martingale. Source: LN Chapter 1, Theorem 3.3, p. 25. Nearby: Stopping a martingale

4. Arbitrage and the first fundamental theorem

4.1 Arbitrage

For discounted prices, an arbitrage is an admissible self-financing strategy $\varphi$ with $V_0(\varphi)=0$ such that

$$V_T(\varphi)\geq0\quad\mathbb P\text{-a.s.}, \qquad \mathbb P[V_T(\varphi)\gt 0]\gt 0.$$

In finite discrete time, this is equivalent to

$$\mathcal G' \cap L^0_+(\mathcal F_T) =\{0\},$$

where

$$\mathcal G' = \{G_T(\vartheta):\vartheta\text{ predictable}\}$$

is the linear space of terminal gains obtainable from zero initial capital. (LN, pp. 35-38.)

Cross-reference: First Fundamental Theorem of Asset Pricing.

Arbitrage opportunity and NA

Setting: Discounted market in finite discrete time. Statement and assumptions: An arbitrage opportunity is an admissible self-financing strategy $\varphi$ with $V_0(\varphi)=0$ and

$$V_T(\varphi)\geq0\quad \mathbb P\text{-a.s.},\qquad \mathbb P[V_T(\varphi)\gt0]\gt0.$$

The market is arbitrage-free, or satisfies NA, if no such strategy exists.

Use: Check the exact zero-cost terminal payoff criterion for arbitrage. Source: LN Chapter 2, Definition, p. 35. Nearby: Equivalent forms of NA in finite discrete time

Equivalent forms of NA in finite discrete time

Setting: Discounted market in finite discrete time. Statement and assumptions: The following are equivalent: the market satisfies NA; no self-financing zero-cost strategy, admissible or otherwise, has a nonnegative and nonzero terminal value; every zero-cost self-financing strategy with nonnegative terminal value has terminal value zero; and

$$\mathcal G'\cap L^0_+(\mathcal F_T)=\{0\},\qquad \mathcal G'=\{G_T(\vartheta):\vartheta\text{ predictable}\}.$$

Use: Replace the strategy definition of NA by a terminal-gains criterion. Source: LN Chapter 2, Proposition 1.1, p. 37. Nearby: Arbitrage opportunity and NA

4.2 Equivalent martingale measures

A probability measure $\mathbb Q$ is equivalent to $\mathbb P$ on $\mathcal F_T$, written $\mathbb Q\approx\mathbb P$, when they have the same null sets. It is an equivalent martingale measure (EMM) for $S$ if the discounted price process $S$ is a $\mathbb Q$-martingale.

If an EMM exists, arbitrage is impossible. Indeed, for an admissible zero-cost strategy,

$$V(\varphi)=\vartheta\mathbin{\bullet} S$$

is a $\mathbb Q$-martingale in the finite discrete-time setting, so

$$\mathbb E_{\mathbb Q}[V_T]=V_0=0.$$

If $V_T\geq0$ $\mathbb P$-a.s., equivalence gives $V_T\geq0$ $\mathbb Q$-a.s. A nonnegative random variable with zero expectation must vanish a.s., so a strictly positive profit with positive probability is impossible. (LN, p. 39.)

The converse is the first fundamental theorem of asset pricing in finite discrete time:

$$\boxed{ \text{No arbitrage} \quad\Longleftrightarrow\quad \mathcal P^e(S)\neq\varnothing. }$$

Here $\mathcal P^e(S)$ is the set of EMMs. The proof for a finite state space interprets no arbitrage as the separation of the gain subspace $\mathcal G'$ from the positive orthant. The positive normal vector of a separating hyperplane becomes, after normalisation, the density of an EMM. (LN, pp. 43-48.)

Interpretation. No arbitrage is an economic restriction. The martingale-measure condition is a probabilistic restriction. The theorem says that, in this setting, they are two descriptions of the same geometry.

Cross-reference: Second Fundamental Theorem of Asset Pricing.

A martingale measure excludes arbitrage

Scope: Discounted financial market in finite discrete time. Assumptions: $\mathbb Q\approx\mathbb P$ on $\mathcal F_T$; $S$ is a local $\mathbb Q$-martingale; arbitrage is defined using admissible self-financing strategies.

Conclusion: $S$ satisfies NA.

Use: Prove that an EMM or ELMM prevents arbitrage. Prerequisites: Admissibility · EMM and ELMM Common misuse: Equivalence and the admissibility lower bound are essential. With only $\mathbb Q\ll\mathbb P$, a profit event of positive $\mathbb P$-probability may be $\mathbb Q$-null. The martingale property must hold for discounted prices. Source: LN Chapter 2, Lemma 1.2, p. 39. Nearby: First Fundamental Theorem of Asset Pricing

EMM and ELMM

Scope: Discounted market on a filtered probability space. Definition: An equivalent martingale measure for $S$ is a probability measure $\mathbb Q\approx\mathbb P$ on $\mathcal F_T$ under which $S$ is a $\mathbb Q$-martingale. Replacing martingale by local martingale gives an equivalent local martingale measure. Their sets are denoted $\mathcal P^e(S)$ and $\mathcal P^{e,\mathrm{loc}}(S)$.

Use: Distinguish an EMM from an ELMM. Common misuse: “Equivalent” refers to the terminal $\sigma$-field and the martingale property refers to the stated filtration. The price process must already be discounted. Source: LN Chapter 2, Definition, p. 43. Nearby: A martingale measure excludes arbitrage · First Fundamental Theorem of Asset Pricing

First Fundamental Theorem of Asset Pricing

Scope: Frictionless discounted market $S^0\equiv1,S$ in finite discrete time, with predictable self-financing strategies and the stated admissibility convention. Assumptions: The market and strategy class are those defined in Chapters 1-2 of LN.

Conclusion:

$$\mathrm{NA}\quad\Longleftrightarrow\quad\mathcal P^e(S)\neq\varnothing\quad\Longleftrightarrow\quad\mathcal P^{e,\mathrm{loc}}(S)\neq\varnothing.$$

Use: Test no-arbitrage by existence of an equivalent martingale measure. Prerequisites: Arbitrage and NA · EMM and ELMM Common misuse: The measure must be equivalent to $\mathbb P$ on $\mathcal F_T$, and the martingale condition concerns discounted prices. In continuous time, plain NA is not the standard condition equivalent to existence of an ELMM. Source: LN Chapter 2, Theorem 2.1, p. 43. Nearby: Second Fundamental Theorem of Asset Pricing

4.3 Multinomial and binomial conditions

Suppose the one-period risky return takes values $y_1\lt \cdots\lt y_m$, while the bank account earns $r$. An EMM requires strictly positive weights $q_1,\ldots,q_m$ such that

$$\sum_{j=1}^m q_j=1, \qquad \sum_{j=1}^m q_jy_j=r.$$

Such weights exist exactly when

$$y_1\lt r\lt y_m.$$

In the binomial case, the solution is unique:

$$q^*=\mathbb Q^*[Y_k=1+u] = \frac{r-d}{u-d}, \qquad 1-q^*=\frac{u-r}{u-d},$$

which requires $d\lt r\lt u$. The physical probability $p$ disappears. (LN, pp. 40-42; ES 3, Exercise 3.2.)

Cross-reference: The binomial martingale measure · First Fundamental Theorem of Asset Pricing.

Martingale measures in the multinomial model

Setting: Canonical multinomial or binomial model with its natural filtration and full-support physical transition probabilities. Statement and assumptions: For return levels $y_1\lt\cdots\lt y_m$ and risk-free rate $r$, an equivalent martingale measure exists exactly when

$$y_1\lt r\lt y_m.$$

Use: Check whether the risk-free return lies inside the multinomial return range. Source: LN Chapter 2, Corollary 1.4, p. 41. Nearby: The binomial martingale measure

The binomial martingale measure

Setting: Canonical multinomial or binomial model with its natural filtration and full-support physical transition probabilities. Statement and assumptions: For $u\gt d$, an equivalent martingale measure exists exactly when $u\gt r\gt d$. It is unique and makes the returns i.i.d. with

$$\mathbb Q[Y_k=1+u]=q^*=\frac{r-d}{u-d},\qquad \mathbb Q[Y_k=1+d]=1-q^*.$$

Use: Compute the unique binomial risk-neutral up probability. Source: LN Chapter 2, Corollary 1.5, p. 42. Nearby: Martingale measures in the multinomial model · Multinomial no-arbitrage condition

Multinomial no-arbitrage condition

Setting: Canonical multinomial or binomial model with its natural filtration and full-support physical transition probabilities. Statement and assumptions: The multinomial model is arbitrage-free exactly when $y_1\lt r\lt y_m$.

Use: Recognise NA in a multinomial model. Source: LN Chapter 2, Corollary 2.2, p. 48. Nearby: The binomial martingale measure · Binomial no-arbitrage condition

Binomial no-arbitrage condition

Setting: Canonical multinomial or binomial model with its natural filtration and full-support physical transition probabilities. Statement and assumptions: The binomial model is arbitrage-free exactly when $u\gt r\gt d$; in that case its EMM is the unique measure from Corollary 1.5.

Use: Recognise NA in a binomial model. Source: LN Chapter 2, Corollary 2.3, p. 48. Nearby: Multinomial no-arbitrage condition

Procedure: test no-arbitrage and find an EMM

Use when. The market is finite and discrete and the task is to test NA or construct risk-neutral probabilities. Read this with the first FTAP.

Steps.

1. Discount every traded price by the bank account.
2. At each non-terminal node introduce strictly positive conditional probabilities $q_1,\ldots,q_m$ with $\sum_jq_j=1$.
3. Impose the conditional martingale equations for every risky asset: $S_k=\sum_jq_jS_{k+1}^{(j)}$. In a one-period return model, this is $\sum_jq_jy_j=r$.
4. Solve the linear system and check strict positivity. A solution defines an EMM and proves NA; if none exists, the first FTAP implies arbitrage.
5. Check uniqueness separately. Under the second FTAP assumptions, one ELMM means completeness and several mean incompleteness.

Worked example. In the one-period trinomial model of ES 4, Exercise 4.2, take returns $d=-0.5$, $m=0$, $u=0.25$, and $r=0$. The equations are

$$q_1+q_2+q_3=1,\qquad -0.5q_1+0.25q_3=0.$$

Thus

$$q_1=\alpha,\qquad q_2=1-3\alpha,\qquad q_3=2\alpha,\qquad 0\lt\alpha\lt\frac13.$$

For example, $(q_1,q_2,q_3)=(1/4,1/4,1/2)$ is an EMM, so the market is arbitrage-free. The family is not a singleton, so the model is incomplete.

Common failure points. Allowing $q_j=0$ gives only absolute continuity; using physical probabilities instead of solving the martingale equations; checking only an unconditional equation in a multi-period tree; interpreting several EMMs as arbitrage rather than incompleteness.

5. Equivalent measures and density processes

If $\mathbb Q\approx\mathbb P$ on $\mathcal F$, the Radon-Nikodym density

$$D_T=\left.\frac{d\mathbb Q}{d\mathbb P}\right|_{\mathcal F_T}\gt 0$$

satisfies

$$\mathbb E_{\mathbb Q}[Y] = \mathbb E_{\mathbb P}[DY]$$

for nonnegative or integrable $Y$. Along a filtration, define

$$Z_k = \mathbb E_{\mathbb P}[D_T\mid\mathcal F_k].$$

Then $Z$ is a strictly positive $\mathbb P$-martingale and is the density process of $\mathbb Q$ with respect to $\mathbb P$. Bayes’ formula is

$$\boxed{ \mathbb E_{\mathbb Q}[U_k\mid\mathcal F_j] = \frac{1}{Z_j} \mathbb E_{\mathbb P}[Z_kU_k\mid\mathcal F_j], \qquad j\leq k. }$$

An adapted process $N$ is a $\mathbb Q$-martingale if and only if $ZN$ is a $\mathbb P$-martingale. (LN, pp. 49-50.)

Writing

$$\xi_k:=\frac{Z_k}{Z_{k-1}}$$

gives one-step conditional densities with

$$\mathbb E_{\mathbb P}[\xi_k\mid\mathcal F_{k-1}]=1.$$

To make $S$ a $\mathbb Q$-martingale, one additionally imposes

$$\mathbb E_{\mathbb P}[\xi_k\Delta S_k\mid\mathcal F_{k-1}]=0.$$

This is the discrete-time prototype of Girsanov’s theorem: change probability by a density chosen to remove the discounted drift. (LN, pp. 51-54; SE, Question 3.)

5.1 Density process and Bayes reference statement

Density processes and Bayes’ formula

Scope: Finite filtered probability space with $\mathbb Q\approx\mathbb P$. Assumptions: Let $D_T=\left.d\mathbb Q/d\mathbb P\right|_{\mathcal F_T}$ and $Z_k=\mathbb E_{\mathbb P}[D_T\mid\mathcal F_k]$. For the formula below, $U_k$ is $\mathcal F_k$-measurable and is either nonnegative or belongs to $L^1(\mathbb Q)$.

Conclusion: $Z_k$ is the density of $\mathbb Q$ relative to $\mathbb P$ on $\mathcal F_k$, and for $j\leq k$,

$$\mathbb E_{\mathbb Q}[U_k\mid\mathcal F_j]=\frac{1}{Z_j}\mathbb E_{\mathbb P}[Z_kU_k\mid\mathcal F_j].$$

An adapted process $N$ is a $\mathbb Q$-martingale if and only if $ZN$ is a $\mathbb P$-martingale.

Use: Move expectations and martingales between equivalent measures. Common misuse: If the density is originally defined on a larger $\mathcal F$, then $Z_T=D$ only when $\mathcal F_T=\mathcal F$; otherwise $Z_T=\mathbb E_{\mathbb P}[D\mid\mathcal F_T]$. Source: LN Chapter 2, Lemma 3.1, p. 50.

6. Valuation, replication, and complete markets

6.1 Attainable claims

A discounted contingent claim is an $\mathcal F_T$-measurable payoff $H\geq0$. It is attainable if there is an admissible self-financing strategy $\varphi$, represented by $(V_0,\vartheta)$, with

$$V_T(\varphi)=H.$$

If the underlying market is arbitrage-free and $\mathcal F_0$ is trivial, every attainable claim has the unique arbitrage-free price process

$$\boxed{ V^H_k = \mathbb E_{\mathbb Q}[H\mid\mathcal F_k] = V_k(V_0,\vartheta) }$$

for every EMM $\mathbb Q$ and every replicating strategy. (LN, pp. 57-62.)

The reasoning is direct. The claim and its replicating portfolio have the same terminal payoff. If their earlier prices differed, buy the cheaper object and sell the more expensive one. Their price difference would be an arbitrage. Under an EMM, the admissible replicating value is a martingale, giving the conditional-expectation formula.

For an unattainable claim, $\mathbb E_{\mathbb Q}[H]$ generally varies across EMMs. Selecting one EMM then adds a modelling or preference choice; no-arbitrage alone no longer determines a unique price. (LN, pp. 62-64.)

Cross-reference: First Fundamental Theorem of Asset Pricing · Second Fundamental Theorem of Asset Pricing.

European payoff

Scope: Finite discrete-time market with terminal $\sigma$-field $\mathcal F_T$. Definition: A general European option, payoff, or contingent claim is a nonnegative $\mathcal F_T$-measurable random variable $H\in L^0_+(\mathcal F_T)$.

Use: Identify the maturity payoff being valued. Source: LN Chapter 3, Definition, p. 57. Nearby: Attainable payoff

Attainable payoff

Scope: Discounted finite discrete-time market. Definition: A payoff $H$ is attainable if an admissible self-financing strategy $\varphi$ satisfies $V_T(\varphi)=H$ almost surely. Such a strategy replicates $H$.

Use: Decide whether a claim has an admissible replicating strategy. Source: LN Chapter 3, Definition, p. 60. Nearby: European payoff · Arbitrage-free valuation of attainable payoffs

Arbitrage-free valuation of attainable payoffs

Scope: Discounted financial market in finite discrete time. Assumptions: $S$ satisfies NA, $\mathcal F_0$ is trivial, and $H$ is attainable by an admissible self-financing strategy $\varphi$.

Conclusion: $H$ has a unique arbitrage-free price process, and for every ELMM $\mathbb Q$ and every replicating strategy,

$$V_k^H=\mathbb E_{\mathbb Q}[H\mid\mathcal F_k]=V_k(\varphi),\qquad k=0,\ldots,T.$$

The result is independent of the chosen ELMM and replicating strategy.

Use: Price and hedge a claim that is already attainable. Prerequisites: Attainable payoff · First FTAP Common misuse: For an unattainable claim in an incomplete market, $\mathbb E_{\mathbb Q}[H\mid\mathcal F_k]$ generally depends on the chosen ELMM. Source: LN Chapter 3, Theorem 1.1, p. 61. Nearby: Characterisation of attainable payoffs

Characterisation of attainable payoffs

Scope: Discounted financial market in finite discrete time. Assumptions: $S$ satisfies NA, $\mathcal F_0$ is trivial, and $H\in L^0_+(\mathcal F_T)$.

Conclusion: The following are equivalent: $H$ is attainable; $\sup_{\mathbb Q\in\mathcal P^{e,\mathrm{loc}}(S)}\mathbb E_{\mathbb Q}[H]$ is finite and attained; and $\mathbb Q\mapsto\mathbb E_{\mathbb Q}[H]$ is finite and constant over all ELMMs.

Use: Test attainability by comparing expectations across all ELMMs. Common misuse: Equality must be finite and hold over the entire ELMM set. Comparing two selected measures is conclusive only after the whole set of ELMMs has been characterised. Source: LN Chapter 3, Theorem 1.2, p. 62. Nearby: Arbitrage-free valuation of attainable payoffs

6.2 Completeness and the second fundamental theorem

A market is complete if every nonnegative $\mathcal F_T$-measurable payoff is attainable. Under no arbitrage, trivial $\mathcal F_0$, and $\mathcal F_T=\mathcal F$,

$$\boxed{ \text{Completeness} \quad\Longleftrightarrow\quad \#\mathcal P^e(S)=1. }$$

Existence of an EMM characterises no arbitrage; under the stated assumptions, uniqueness of the ELMM characterises completeness, and the unique ELMM is then also the unique EMM. (LN, pp. 65-66.)

The intuition is linear algebra. Terminal payoffs form a vector space. Trading spans a subspace. A unique state-price vector means the traded assets determine the value of every state-contingent payoff; multiple state-price vectors reveal directions that trading does not span.

Cross-reference: First Fundamental Theorem of Asset Pricing · Arbitrage-free valuation of attainable payoffs.

Complete market

Scope: Discounted finite discrete-time market. Definition: The market is complete if every $H\in L^0_+(\mathcal F_T)$ is attainable; otherwise it is incomplete.

Use: Decide whether every admitted payoff is hedgeable. Common misuse: Completeness is relative to the filtration and traded assets. It neither follows merely by counting assets nor implies NA by definition. Source: LN Chapter 3, Definition, p. 65. Nearby: Second Fundamental Theorem of Asset Pricing

Valuation and hedging in complete markets

Scope: Discounted financial market in finite discrete time. Assumptions: $\mathcal F_0$ is trivial and $S$ is arbitrage-free and complete.

Conclusion: Every payoff $H\in L^0_+(\mathcal F_T)$ has a unique arbitrage-free price process. For every ELMM $\mathbb Q$ and every replicating strategy $\varphi$,

$$V_k^H=\mathbb E_{\mathbb Q}[H\mid\mathcal F_k]=V_k(\varphi),\qquad k=0,\ldots,T.$$

Use: Price any admitted payoff in an arbitrage-free complete market. Common misuse: The price is unique because every admitted payoff is attainable. In an incomplete market, selecting one ELMM does not by itself produce a unique no-arbitrage price. Source: LN Chapter 3, Theorem 2.1, p. 65. Nearby: Complete market · Second FTAP

Second Fundamental Theorem of Asset Pricing

Scope: Discounted frictionless market in finite discrete time. Assumptions: $S$ satisfies NA, $\mathcal F_0$ is trivial, and $\mathcal F_T=\mathcal F$.

Conclusion: The market is complete if and only if $\mathcal P^{e,\mathrm{loc}}(S)$ is a singleton. In that case $\mathcal P^e(S)=\mathcal P^{e,\mathrm{loc}}(S)$ and both contain the same unique measure.

Use: Test completeness by uniqueness of the ELMM. Prerequisites: First FTAP · Complete market · EMM and ELMM Common misuse: Existence characterises NA; uniqueness characterises completeness only under the stated assumptions. Multiple ELMMs mean incompleteness, not arbitrage. The condition $\mathcal F_T=\mathcal F$ matters because completeness concerns terminal-measurable claims. Source: LN Chapter 3, Theorem 2.2, pp. 65-66. Nearby: Valuation and hedging in complete markets

6.3 Backward pricing and hedging in the CRR model

In the CRR model with $d\lt r\lt u$, the unique martingale probability is $q^*=(r-d)/(u-d)$. For undiscounted successor values $\widetilde v_k^u$ and $\widetilde v_k^d$,

$$\boxed{ \widetilde v_{k-1} = \frac{1}{1+r} \left(q^*\widetilde v_k^u+(1-q^*)\widetilde v_k^d\right). }$$

The hedge ratio at the node is

$$\boxed{ \vartheta_k = \frac{\widetilde v_k^u-\widetilde v_k^d}{\widetilde S_k^u-\widetilde S_k^d}. }$$

For a Markov payoff $\widetilde H=\widetilde h(\widetilde S_T)$, write $s=\widetilde S_{k-1}$ at the current node. The recursion becomes

$$\widetilde v(k-1,s) = \frac{q^*\widetilde v(k,s(1+u)) +(1-q^*)\widetilde v(k,s(1+d))}{1+r},$$

with terminal condition $\widetilde v(T,s)=\widetilde h(s)$, and

$$\vartheta_k = \frac{\widetilde v(k,s(1+u))-\widetilde v(k,s(1+d))}{(u-d)s}.$$

Pricing and hedging are therefore computed together by backward induction. (LN, pp. 68-74; ES 4, Exercise 4.3.)

Undiscounted binomial valuation

Setting: Arbitrage-free finite discrete-time market unless stated otherwise. Statement and assumptions: In the binomial model with $u\gt r\gt d$, every undiscounted payoff $\widetilde H$ has arbitrage-free price

$$\widetilde V_k^{\widetilde H}=\widetilde S^0_k\mathbb E_{\mathbb Q^*}\!\left[\frac{\widetilde H}{\widetilde S^0_T}\middle|\mathcal F_k\right]=\frac{\widetilde S^0_k}{\widetilde S^0_T}\mathbb E_{\mathbb Q^*}[\widetilde H\mid\mathcal F_k].$$

Use: Convert risk-neutral prices into undiscounted binomial prices. Source: LN Chapter 3, Corollary 3.1, p. 68.

Procedure: price and hedge a claim in the CRR model

Use when. A claim is defined on a binomial tree with $d\lt r\lt u$. Read this with undiscounted binomial valuation.

Steps.

1. Write the terminal claim values.
2. Compute $q^*=(r-d)/(u-d)$.
3. Work backwards at every node using $\widetilde v=(q^*\widetilde v^u+(1-q^*)\widetilde v^d)/(1+r)$.
4. Compute the stock holding $\vartheta=(\widetilde v^u-\widetilde v^d)/(\widetilde S^u-\widetilde S^d)$.
5. Recover the bank-account holding from the portfolio value and verify both successor-state values.

Worked example. Let $\widetilde S_0=100$, $u=0.20$, $d=-0.10$, $r=0.05$, and $\widetilde H=(\widetilde S_1-100)^+$. Then $q^*=0.5$, the successor payoffs are $20$ and $0$, and

$$\widetilde V_0=\frac{0.5(20)+0.5(0)}{1.05}=9.5238,\qquad \vartheta_1=\frac{20-0}{120-90}=\frac23.$$

The initial bank position is $\varphi^0_0=9.5238-(2/3)100=-57.1429$. At maturity it is worth $-60$, so the portfolio pays $(2/3)120-60=20$ in the up state and $(2/3)90-60=0$ in the down state.

Common failure points. Mixing discounted and undiscounted values; using the physical up probability; omitting $1/(1+r)$; dividing by $u-d$ instead of the stock-price difference; using one hedge ratio at every node of a multi-period tree.

6.4 Optional exercise topic: the Snell envelope

For an adapted integrable payoff process $Y=(Y_k)$, define

$$U_T=Y_T, \qquad U_k=\max\left(Y_k,\mathbb E[U_{k+1}\mid\mathcal F_k]\right).$$

$U$ is the smallest supermartingale dominating $Y$. The first time

$$\tau^*=\inf\{k:U_k=Y_k\}$$

is optimal, and $U^{\tau^*}$ is a martingale. This is the discrete-time foundation of American-option valuation. (ES 5, Exercise 5.2 and solution.)

7. Brownian motion

7.1 Definition and scaling

A Brownian motion $W=(W_t)_{t\geq0}$ relative to $(\mathbb P,\mathbb F)$ is adapted, starts at zero, has continuous paths, and satisfies

$$W_t-W_s\sim\mathcal N(0,t-s), \qquad W_t-W_s\text{ independent of }\mathcal F_s$$

for $0\leq s\leq t$. (LN, pp. 75-76.) Important transformations include sign reversal, time shifts, time reversal on a finite interval, and the scaling relation

$$\left(cW_{t/c^2}\right)_{t\geq0} \overset{d}{=} (W_t)_{t\geq0}.$$

Brownian paths are continuous but a.s. nowhere differentiable. They grow more slowly than linearly, but oscillate at every scale. (LN, pp. 77-81.)

Brownian motion

Setting: Brownian motion and continuous-time martingales. Statement and assumptions: A process $W$ adapted to $(\mathcal F_t)$ is a Brownian motion if $W_0=0$, $W_t-W_s$ is independent of $\mathcal F_s$ and distributed as $N(0,t-s)$ for $s\leq t$, and its sample paths are continuous. In $\mathbb R^m$, the increment law is $N(0,(t-s)I_m)$.

Use: Check whether a continuous process qualifies as Brownian motion. Source: LN Chapter 4, Definition, pp. 75-76. Nearby: Brownian transformations

Brownian transformations

Setting: Brownian motion and continuous-time martingales. Statement and assumptions: If $W$ is Brownian motion, so are $-W$; $(W_{t+T}-W_T)_{t\geq0}$; $(cW_{t/c^2})_{t\geq0}$ for $c\neq0$; $(W_{T-t}-W_T)_{0\leq t\leq T}$ on $[0,T]$; and the time-inverted process defined by $W^5_0=0$ and $W^5_t=tW_{1/t}$ for $t\gt0$.

Use: Rescale, restart, reverse, or invert Brownian motion. Source: LN Chapter 4, Proposition 1.1, pp. 76-77. Nearby: Brownian motion · Brownian asymptotics and the LIL

Brownian asymptotics and the LIL

Setting: Brownian motion and continuous-time martingales. Statement and assumptions: Almost surely, $W_t/t\to0$ as $t\to\infty$. Moreover,

$$\limsup_{t\to\infty}\frac{W_t}{\sqrt{2t\log\log t}}=1,\qquad \liminf_{t\to\infty}\frac{W_t}{\sqrt{2t\log\log t}}=-1,$$

and, for every fixed $t\geq0$, the analogous local limits for $(W_{t+h}-W_t)/\sqrt{2h\log\log(1/h)}$ are $1$ and $-1$ as $h\downarrow0$.

Use: Estimate long-time or local Brownian oscillations. Source: LN Chapter 4, Proposition 1.2, pp. 77-78. Nearby: Brownian transformations · Path regularity

Path regularity

Setting: Brownian motion and continuous-time martingales. Statement and assumptions: Almost every Brownian sample path is continuous everywhere and differentiable nowhere.

Use: Recall why a Brownian path cannot be differentiated. Source: LN Chapter 4, Proposition 1.3, p. 78. Nearby: Brownian asymptotics and the LIL

7.2 Quadratic variation

For a refining sequence of deterministic partitions with mesh tending to zero,

$$\sum_i \left(W_{t_{i+1}\wedge t}-W_{t_i\wedge t}\right)^2 \longrightarrow t \quad\text{a.s.}$$

Thus

$$[W]_t=\langle W\rangle_t=t.$$

The mnemonic $(dW_t)^2=dt$ records this result, but it is not itself a proof. Brownian paths have infinite total variation; otherwise their quadratic variation would vanish. (LN, pp. 79-81.)

Cross-reference: Itô isometry for elementary integrands · Itô’s formula: continuous one-dimensional case.

Brownian quadratic variation

Setting: Brownian motion and continuous-time martingales. Statement and assumptions: Along any refining sequence of deterministic partitions whose mesh tends to zero, almost surely and simultaneously for every $t\geq0$,

$$\sum_{t_i\in\Pi_n}(W_{t_i\wedge t}-W_{t_{i-1}\wedge t})^2\longrightarrow t.$$

Thus $\langle W\rangle_t=t$.

Use: Replace sums of squared Brownian increments by elapsed time. Source: LN Chapter 4, Theorem 1.4, p. 80.

7.3 Brownian martingales and stopping

The following are martingales:

$$W_t, \qquad W_t^2-t, \qquad \exp\left(\alpha W_t-\frac12\alpha^2t\right).$$

The exponential martingale follows from the moment-generating function of a normal increment. (LN, pp. 82-85.)

If $M$ is a right-continuous martingale and $\sigma\leq\tau$ are stopping times, then under boundedness of $\tau$ or uniform integrability of $M$,

$$\mathbb E[M_\tau\mid\mathcal F_\sigma]=M_\sigma.$$

The conditions matter. For

$$\tau=\inf\{t\geq0:W_t\lt -1\},$$

one has $W_\tau=-1$ a.s., so $\mathbb E[W_\tau]\neq\mathbb E[W_0]$. Here $\tau$ is unbounded and $W$ is not uniformly integrable. (LN, pp. 82-87; ES 7.)

Brownian motion also has the Markov and strong Markov properties: after a fixed time, or after an a.s. finite stopping time, the future increment process is a fresh Brownian motion independent of the accumulated past. (LN, pp. 88-89.)

Optional stopping

Scope: Continuous-time filtered probability space; Brownian motion is not required. Assumptions: $M=(M_t)_{t\geq0}$ is a right-continuous $(\mathbb P,\mathbb F)$-martingale, $\sigma\leq\tau$ are stopping times, and either $\tau\leq T_0$ a.s. for a deterministic $T_0\lt\infty$ or $M$ is uniformly integrable.

Conclusion: $M_\sigma,M_\tau\in L^1(\mathbb P)$ and

$$\mathbb E_{\mathbb P}[M_\tau\mid\mathcal F_\sigma]=M_\sigma\qquad\mathbb P\text{-a.s.}$$

Use: Stop a martingale while preserving conditional expectation. Prerequisites: Stopping times, $\mathcal F_\tau$, stopped processes, and uniform integrability. Common misuse: Almost-sure finiteness of $\tau$, or $\mathbb E[\tau]\lt\infty$, is not sufficient by itself. A local martingale is not enough, and passing from $M_{\tau\wedge t}$ to $M_\tau$ requires a valid limiting argument. Source: LN Chapter 4, Theorem 2.2, p. 83. Nearby: Continuous-time local martingale

Continuous-time local martingale null at zero

Scope: Continuous-time filtered probability space. Definition: An adapted process $X$ with $X_0=0$ is a local martingale if stopping times $\tau_n\uparrow\infty$ exist such that every $X^{\tau_n}$ is a martingale.

Use: Localise a continuous-time adapted process. Common misuse: A local martingale need not be integrable or a true martingale. “Local” refers to stopping in time, not locality in state or path space. Source: LN Chapter 4, Definition, p. 84. Nearby: Optional stopping · Basic Brownian martingales

Basic Brownian martingales

Setting: Brownian motion and continuous-time martingales. Statement and assumptions: For any $\alpha\in\mathbb R$, the processes

$$W_t,\qquad W_t^2-t,\qquad \exp\!\left(\alpha W_t-\frac12\alpha^2t\right)$$

are martingales.

Use: Recognise the standard Brownian martingale test processes. Source: LN Chapter 4, Proposition 2.3, pp. 84-85. Nearby: Continuous-time local martingale · Brownian hitting-time transforms

Brownian hitting-time transforms

Setting: Brownian motion and continuous-time martingales. Statement and assumptions: For $a,b,\lambda\gt0$, set $\tau_a=\inf\{t\geq0:W_t\gt a\}$ and $\sigma_{a,b}=\inf\{t\geq0:W_t\gt a+bt\}$. Then

$$\mathbb E[e^{-\lambda\tau_a}]=e^{-a\sqrt{2\lambda}},\qquad \mathbb E[e^{-\lambda\sigma_{a,b}}\mathbf 1_{\{\sigma_{a,b}\lt\infty\}}]=e^{-a(b+\sqrt{b^2+2\lambda})}.$$

Use: Compute Laplace transforms of Brownian hitting times. Source: LN Chapter 4, Proposition 2.4, p. 86. Nearby: Basic Brownian martingales

8. Stochastic integration and semimartingales

8.1 Why the Itô integral uses left endpoints

Continuous-time trading gains should be the limit of sums

$$\sum_i K_{t_i} \left(M_{t_{i+1}}-M_{t_i}\right).$$

The integrand is evaluated at the left endpoint because a trading position must be chosen before the price increment is known. Midpoint evaluation produces the Stratonovich integral; right-endpoint evaluation uses future information. (LN, pp. 91-92.)

Bounded elementary integrands

Setting: RCLL processes and stochastic integration in continuous time. Statement and assumptions: The class $b\mathcal E$ consists of bounded processes

$$K=\sum_{i=0}^{n-1}h_i\mathbf 1_{(t_i,t_{i+1}]}$$

with $h_i$ bounded and $\mathcal F_{t_i}$-measurable. For any process $X$,

$$\int_0^tK_s\,dX_s=\sum_{i=0}^{n-1}h_i(X_{t_{i+1}\wedge t}-X_{t_i\wedge t}).$$

Use: Construct an Itô integral from predictable step processes. Source: LN Chapter 5, Definition, p. 95.

8.2 Brackets and Itô isometry

For a local martingale $M$ null at zero, the optional quadratic variation $[M]$ is the unique adapted increasing RCLL process with

$$\Delta[M]=(\Delta M)^2$$

such that $M^2-[M]$ is a local martingale. For two local martingales,

$$[M,N] = \frac14\left([M+N]-[M-N]\right).$$

For a square-integrable martingale and an elementary predictable process

$$K=\sum_{i=0}^{n-1}h_i\mathbf 1_{(t_i,t_{i+1}]},$$

define

$$\int_0^tK_s\,dM_s = \sum_{i=0}^{n-1}h_i \left(M_{t_{i+1}\wedge t}-M_{t_i\wedge t}\right).$$

Then

$$\boxed{ \mathbb E\left[\left(\int_0^\infty K_s\,dM_s\right)^2\right] = \mathbb E\left[\int_0^\infty K_s^2\,d[M]_s\right]. }$$

This isometry extends the integral by completion from elementary processes to

$$L^2(M) = \left\{K\text{ predictable}: \mathbb E\left[\int_0^\infty K_s^2\,d[M]_s\right]\lt \infty \right\}.$$

Localisation then gives $L^2_{\mathrm{loc}}(M)$. For Brownian motion,

$$K\in L^2_{\mathrm{loc}}(W) \quad\Longleftrightarrow\quad \int_0^tK_s^2\,ds\lt \infty \quad\text{a.s. for every }t.$$

(LN, pp. 93-103.)

Cross-reference: Itô’s formula: continuous one-dimensional case.

Optional quadratic variation

Scope: Càdlàg local martingales in continuous time. Assumptions: $M$ is a local martingale with $M_0=0$.

Conclusion: There is a unique adapted increasing càdlàg process $[M]$ null at zero such that $\Delta[M]=(\Delta M)^2$ and $M^2-[M]$ is a local martingale. Along a suitable sequence of partitions, $[M]$ is the limit of sums of squared increments. If $\sup_{s\leq T}|M_s|\in L^2$, then $[M]_T\in L^1$ and $M^2-[M]$ is a martingale on $[0,T]$.

Use: Identify the quadratic variation of a local martingale. Common misuse: $[M]$ exists for every local martingale, whereas the predictable bracket $\langle M\rangle$ needs local square integrability. They coincide for continuous locally square-integrable martingales, not in general. The convergence statement uses a suitable partition sequence, not every refining sequence. Source: LN Chapter 5, Theorem 1.1, p. 93. Nearby: Itô isometry for elementary integrands

Itô isometry for elementary integrands

Setting: RCLL processes and stochastic integration in continuous time. Statement and assumptions: If $M$ is a square-integrable martingale and $K\in b\mathcal E$, then $K\mathbin{\bullet}M$ is a square-integrable martingale,

$$[K\mathbin{\bullet}M]=\int K^2\,d[M],\qquad \mathbb E\!\left[\left(\int_0^\infty K_s\,dM_s\right)^2\right]=\mathbb E\!\left[\int_0^\infty K_s^2\,d[M]_s\right].$$

Use: Compute the second moment of an elementary stochastic integral. Source: LN Chapter 5, Lemma 1.3, pp. 95-96. Nearby: Optional quadratic variation · Extension of the stochastic integral

Extension of the stochastic integral

Setting: RCLL processes and stochastic integration in continuous time. Statement and assumptions: If $M\in\mathcal M^2_0$, then $b\mathcal E$ is dense in $L^2(M)$. Consequently, the integral $K\mathbin{\bullet}M$ is defined for every $K\in L^2(M)$, belongs to $\mathcal M^2_0$, and satisfies the Itô isometry.

Use: Extend the integral from step processes to $L^2(M)$. Source: LN Chapter 5, Proposition 1.4, p. 100. Nearby: Itô isometry for elementary integrands · Local square integrability

Local square integrability

Setting: RCLL processes and stochastic integration in continuous time. Statement and assumptions: A local martingale $M$ is locally square-integrable, written $M\in\mathcal M^2_{0,\mathrm{loc}}$, if some stopping times $\tau_n\uparrow\infty$ make every $M^{\tau_n}$ an element of $\mathcal M^2_0$. A predictable $K$ lies in $L^2_{\mathrm{loc}}(M)$ if some such sequence makes $K\mathbf 1_{(0,\tau_n]}\in L^2(M)$ for every $n$.

Use: Check whether a local martingale and integrand can be integrated locally. Source: LN Chapter 5, Definition, p. 101. Nearby: Extension of the stochastic integral

8.3 Calculation rules

For admissible integrands, stochastic integration is linear and associative, respects stopping, and satisfies

$$\left[\int K\,dM,\int J\,dN\right] = \int KJ\,d[M,N],$$

so in particular

$$\left[\int K\,dM\right] = \int K^2\,d[M].$$

The jumps obey

$$\Delta\left(\int K\,dM\right)_t = K_t\Delta M_t.$$

An important example is

$$\boxed{ \int_0^tW_s\,dW_s = \frac12W_t^2-\frac12t. }$$

The correction $-t/2$ is the contribution of Brownian quadratic variation. (LN, pp. 104-107; ES 10.)

8.4 Semimartingales

A semimartingale has a decomposition

$$X=X_0+M+A,$$

where $M$ is a local martingale and $A$ is adapted, RCLL, and of finite variation. It is special if $A$ can be chosen predictable; then the predictable decomposition is unique. The integral is defined by

$$\int K\,dX = \int K\,dM+\int K\,dA,$$

combining stochastic integration for $M$ and pathwise Lebesgue-Stieltjes integration for $A$. Semimartingales are stable under $C^2$ transformations and equivalent changes of measure. They form the natural class of frictionless price processes. (LN, pp. 108-112.)

Semimartingale

Setting: RCLL processes and stochastic integration in continuous time. Statement and assumptions: A process $X$ is a semimartingale if

$$X=X_0+M+A,$$

where $M$ is a local martingale null at zero and $A$ is an adapted finite-variation RCLL process null at zero. It is special if a decomposition exists with $A$ predictable.

Use: Decide whether stochastic integration with respect to $X$ is available. Source: LN Chapter 5, Definition, p. 108.

9. Itô calculus

9.1 Itô’s formula

If $X$ is a continuous semimartingale and $f\in C^2(\mathbb R)$, then

$$\boxed{ f(X_t) = f(X_0) +\int_0^t f'(X_s)\,dX_s +\frac12\int_0^t f''(X_s)\,d[X]_s. }$$

In differential notation,

$$df(X_t) = f'(X_t)\,dX_t +\frac12f''(X_t)\,d[X]_t.$$

The second-order term distinguishes stochastic calculus from the ordinary chain rule. A Taylor expansion over a partition explains it: first-order increments converge to the stochastic integral, squared increments converge to quadratic variation, and higher orders vanish. (LN, pp. 113-118.)

For a continuous $d$-dimensional semimartingale,

$$df(X_t) = \sum_i f_{x_i}(X_t)\,dX_t^i +\frac12\sum_{i,j}f_{x_ix_j}(X_t) \,d[X^i,X^j]_t.$$

The product rule becomes

$$d(XY)=Y\,dX+X\,dY+d[X,Y].$$

(LN, pp. 119-124.)

Cross-reference: Brownian quadratic variation · Itô isometry for elementary integrands.

Itô’s formula: continuous one-dimensional case

Scope: Continuous semimartingale calculus. Assumptions: $X$ is a continuous real semimartingale and $f\in C^2(\mathbb R)$.

Conclusion: $f(X)$ is a continuous semimartingale and

$$f(X_t)=f(X_0)+\int_0^tf'(X_s)\,dX_s+\frac12\int_0^tf''(X_s)\,d[X]_s.$$

Use: Apply the chain rule to a continuous one-dimensional semimartingale. Prerequisites: Semimartingale · Quadratic variation Common misuse: Ordinary calculus omits the quadratic-variation term. If $X$ has jumps, use the jump formula instead. Source: LN Chapter 6, Theorem 1.1, p. 116. Nearby: Itô’s formula: multidimensional and jump cases · Itô procedure

Itô’s formula: multidimensional and jump cases

Scope: Multidimensional continuous and one-dimensional càdlàg semimartingale calculus. Assumptions: For the continuous case, $X$ is an $\mathbb R^d$-valued continuous semimartingale and $f\in C^2(\mathbb R^d)$. For the jump case, $X$ is a real càdlàg semimartingale and $f\in C^2(\mathbb R)$.

Conclusion: In the continuous case,

$$f(X_t)=f(X_0)+\sum_i\int_0^tf_{x_i}(X_s)\,dX_s^i+\frac12\sum_{i,j}\int_0^tf_{x_ix_j}(X_s)\,d[X^i,X^j]_s.$$

For jumps, with $R_f(x,y)=f(x+y)-f(x)-f'(x)y-\frac12f''(x)y^2$,

$$f(X_t)=f(X_0)+\int_0^tf'(X_{s-})\,dX_s+\frac12\int_0^tf''(X_{s-})\,d[X]_s+\sum_{0\lt s\leq t}R_f(X_{s-},\Delta X_s).$$

Use: Apply multidimensional Itô calculus or account for jumps. Common misuse: The jump formula uses left limits and $[X]$. Do not apply the continuous formula unchanged when jumps are present. Source: LN Chapter 6, Theorem 1.2, p. 119. Nearby: Itô’s formula: continuous one-dimensional case · Itô procedure

Procedure: apply Itô’s formula

Use when. A process satisfies $dX_t=b_t\,dt+a_t\,dW_t$ and a function of $t$ and $X_t$ must be differentiated.

Steps.

1. Identify $b_t$, $a_t$, and $f(t,x)$.
2. Compute $f_t$, $f_x$, and $f_{xx}$.
3. Record $d[X]_t=a_t^2\,dt$.
4. Substitute into

$$df(t,X_t)=\left(f_t+b_tf_x+\frac12a_t^2f_{xx}\right)dt+a_tf_x\,dW_t.$$

5. Separate finite-variation and local-martingale terms, then integrate.

Worked example. For $X=W$ and $f(x)=x^2$,

$$d(W_t^2)=2W_t\,dW_t+dt,$$

so

$$W_t^2-t=2\int_0^tW_s\,dW_s.$$

This also identifies $W_t^2-t$ as a martingale.

Common failure points. Omitting $f_t$ for time-dependent functions; omitting $a_t^2$ in the second-order term; using $(dW)^2=dt$ as a proof; failing to separate drift and stochastic-integral terms.

9.2 Geometric Brownian motion

Consider

$$d\widetilde S_t = \mu\widetilde S_t\,dt +\sigma\widetilde S_t\,dW_t.$$

Applying Itô’s formula to the exponential gives

$$\boxed{ \widetilde S_t = \widetilde S_0 \exp\left( \sigma W_t+\left(\mu-\frac12\sigma^2\right)t \right). }$$

The $-\sigma^2/2$ term compensates for the quadratic-variation contribution of the exponential. (LN, pp. 120-121.)

9.3 Stochastic exponentials

For a continuous semimartingale $X$ with $X_0=0$,

$$\mathcal E(X)_t = \exp\left(X_t-\frac12[X]_t\right)$$

is the unique solution of

$$dZ_t=Z_t\,dX_t, \qquad Z_0=1.$$

Stochastic exponentials will serve as density processes for equivalent changes of measure. (LN, pp. 121-122; ES 11, Exercise 11.1.)

Stochastic exponential

Scope: Real càdlàg semimartingale $X$ with $X_0=0$. Definition: The stochastic exponential $\mathcal E(X)$ is the unique solution of

$$dZ_t=Z_{t-}\,dX_t,\qquad Z_0=1.$$

If $X$ is continuous, $\mathcal E(X)=\exp(X-\frac12[X])$. With jumps, $\mathcal E(X)_t=\mathcal E(X)_{t-}(1+\Delta X_t)$.

Use: Solve a linear stochastic differential equation and construct candidate density processes. Prerequisites: Semimartingale · stochastic integration · quadratic variation. Common misuse: The stochastic exponential of a local martingale is generally only a local martingale. To define an equivalent probability on $\mathcal F_T$, require strict positivity and a true martingale with $\mathbb E_{\mathbb P}[Z_T]=1$; Novikov is one sufficient condition in the continuous case. Source: LN Chapter 6, Definition, p. 122.

10. Girsanov’s theorem and martingale representation

10.1 Continuous-time Bayes formula

If $\mathbb Q\approx\mathbb P$ on $\mathcal F_T$ and

$$Z_t = \mathbb E_{\mathbb P} \left[ \left.\frac{d\mathbb Q}{d\mathbb P}\right|\mathcal F_t \right],$$

then

$$\mathbb E_{\mathbb Q}[U_t\mid\mathcal F_s] = \frac{1}{Z_s} \mathbb E_{\mathbb P}[Z_tU_t\mid\mathcal F_s].$$

Moreover, $Y$ is a local $\mathbb Q$-martingale if and only if $ZY$ is a local $\mathbb P$-martingale. (LN, pp. 126-127.)

Continuous-time Bayes formula

Scope: Fixed horizon $[0,T]$ with $\mathbb Q\approx\mathbb P$ on $\mathcal F_T$ and strictly positive càdlàg density process $Z$. Assumptions: $s\leq t\leq T$ and $U_t$ is $\mathcal F_t$-measurable and either nonnegative or in $L^1(\mathbb Q)$.

Conclusion:

$$\mathbb E_{\mathbb Q}[U_t\mid\mathcal F_s]=\frac1{Z_s}\mathbb E_{\mathbb P}[Z_tU_t\mid\mathcal F_s].$$

An adapted process $Y$ is a local or true $\mathbb Q$-martingale exactly when $ZY$ is, respectively, a local or true $\mathbb P$-martingale.

Use: Convert continuous-time conditional expectations between equivalent measures. Common misuse: Do not apply the formula to a signed nonintegrable variable whose conditional expectation is undefined. Source: LN Chapter 6, Lemma 2.1, pp. 126-127.

10.2 Girsanov’s drift transformation

If $\mathbb Q$ is locally equivalent to $\mathbb P$ with continuous density

$$Z=Z_0\mathcal E(L),$$

then for every local $\mathbb P$-martingale $M$,

$$\widetilde M = M-\langle L,M\rangle$$

is a local $\mathbb Q$-martingale. If

$$L_t=\int_0^t\nu_s\,dW_s,$$

then

$$\widetilde W_t = W_t-\int_0^t\nu_s\,ds$$

is a $\mathbb Q$-Brownian motion. (LN, pp. 127-130.)

To construct $\mathbb Q$, $Z=\mathcal E(L)$ must be a true martingale, not merely a local martingale. A standard sufficient condition is Novikov’s condition:

$$\mathbb E_{\mathbb P} \left[ \exp\left(\frac12\langle L\rangle_T\right) \right] \lt \infty.$$

Then $d\mathbb Q=Z_Td\mathbb P$ defines an equivalent probability measure. (LN, p. 130.)

Cross-reference: Itô martingale representation theorem.

Girsanov’s theorem: general case

Scope: Continuous-time filtered probability space; $\mathbb Q\overset{\mathrm{loc}}{\approx}\mathbb P$ with strictly positive càdlàg density process $Z$. Assumptions: $M$ is a local $\mathbb P$-martingale with $M_0=0$.

Conclusion:

$$\widetilde M=M-\int\frac1Z\,d[Z,M]$$

is a local $\mathbb Q$-martingale with $\widetilde M_0=0$. In particular, equivalent measures have the same semimartingales.

Use: Find the local-martingale correction after a change of measure. Prerequisites: Local equivalence, density processes, and optional covariation $[Z,M]$. Common misuse: In this optional-covariation version the denominator is $Z$, not $Z_-$. The integral is a pathwise Lebesgue-Stieltjes integral against the finite-variation process $[Z,M]$; do not mix it with predictable-bracket formulations. Source: LN Chapter 6, Theorem 2.2, pp. 127-128. Nearby: Girsanov’s theorem: continuous-density case

Girsanov’s theorem: continuous-density case

Scope: $\mathbb Q\overset{\mathrm{loc}}{\approx}\mathbb P$ has a continuous density process $Z=Z_0\mathcal E(L)$, where $L$ is its continuous local-$\mathbb P$-martingale stochastic logarithm. Assumptions: $M$ is a local $\mathbb P$-martingale with $M_0=0$.

Conclusion:

$$\widetilde M=M-[L,M]=M-\langle L,M\rangle$$

is a local $\mathbb Q$-martingale. If $W$ is $\mathbb P$-Brownian motion, $\widetilde W$ is $\mathbb Q$-Brownian motion. In particular, when $L=\int\nu\,dW$,

$$\widetilde W_t=W_t-\int_0^t\nu_s\,ds.$$

Use: Remove or add Brownian drift under an equivalent measure. Prerequisites: Density process, stochastic exponential and logarithm, and quadratic covariation. Common misuse: The theorem transforms an already valid density process. Starting with a candidate $L$ does not define $\mathbb Q$ until $\mathcal E(L)$ is a strictly positive true martingale. Track the sign in $W^{\mathbb Q}=W-\int\nu\,ds$. Source: LN Chapter 6, Theorem 2.3, p. 129. Nearby: Girsanov’s theorem: general case · Girsanov procedure

Procedure: change a Brownian drift

Use when. A drift must be removed or replaced by changing to an equivalent probability measure.

Steps.

1. Choose the desired drift under the new measure.
2. Under the convention $Z=\mathcal E(\int\nu\,dW)$ and $W_t^{\mathbb Q}=W_t-\int_0^t\nu_sds$, solve for $\nu$. For $dX_t=b_tdt+\sigma_tdW_t$, the new drift is $b_t+\sigma_t\nu_t$.
3. Form $Z_t=\exp(\int_0^t\nu_s\,dW_s-\frac12\int_0^t\nu_s^2ds)$.
4. Prove that $Z$ is a true martingale, for example by Novikov’s condition, and define $d\mathbb Q/d\mathbb P=Z_T$.
5. Substitute $dW_t=dW_t^{\mathbb Q}+\nu_tdt$ and check the resulting drift.

Worked example. Let $X_t=W_t+at$ and choose $\nu=-a$. Then

$$Z_t=\exp\left(-aW_t-\frac12a^2t\right)$$

is a true martingale, and under $d\mathbb Q/d\mathbb P=Z_T$,

$$W_t^{\mathbb Q}=W_t+at=X_t$$

is a $\mathbb Q$-Brownian motion.

Common failure points. Reversing the sign of $\nu$; using a positive local martingale as a density without proving expectation one; confusing $d\mathbb Q/d\mathbb P$ with $d\mathbb P/d\mathbb Q$; forgetting that discounted traded prices need zero $\mathbb Q$-drift.

10.3 Itô representation and completeness

Let $\mathbb F^W$ be the augmented filtration generated by an $m$-dimensional Brownian motion. Every $H\in L^1(\mathcal F_\infty^W)$ has a unique representation

$$\boxed{ H = \mathbb E[H] +\int_0^\infty\psi_s\,dW_s, }$$

where the integral is a uniformly integrable martingale on the closed interval. Consequently, every local martingale in a Brownian filtration is a stochastic integral with respect to $W$ and is continuous. (LN, pp. 133-134.)

Interpretation. A Brownian filtration contains no martingale risk orthogonal to the driving Brownian motion. If traded assets expose all Brownian directions, every integrable claim measurable in that filtration can be replicated. This is the structural reason behind Black-Scholes completeness.

Cross-reference: Second Fundamental Theorem of Asset Pricing.

Itô martingale representation theorem

Scope: The $\mathbb P$-augmented natural filtration $\mathbb F^W$ of an $\mathbb R^m$-valued Brownian motion $W$. Assumptions: $H\in L^1(\mathcal F^W_\infty,\mathbb P)$.

Conclusion: There is a unique representation

$$H=\mathbb E_{\mathbb P}[H]+\int_0^\infty\psi_s\,dW_s,$$

where $\psi\in L^2_{\mathrm{loc}}(W)$ and the integral is a uniformly integrable martingale on the closed interval $[0,\infty]$. The integrand is unique up to $\mathbb P\otimes dt$-a.e. equality.

Use: Represent an integrable Brownian-filtration claim by a unique integrand. Prerequisites: Augmented Brownian filtration, $L^2_{\mathrm{loc}}(W)$, and uniformly integrable martingales. Common misuse: The result may fail in a larger filtration containing independent randomness. Financial completeness also requires traded assets to span every Brownian direction and the resulting strategy to be admissible. Source: LN Chapter 6, Theorem 3.1, pp. 133-134. Nearby: Martingales in a Brownian filtration · Second FTAP

Martingales in a Brownian filtration

Scope: The augmented natural filtration $\mathbb F^W$ of an $\mathbb R^m$-valued Brownian motion. Conclusion: Every real local martingale in $\mathbb F^W$ has the form

$$L=L_0+\int\nu\,dW$$

for some $\nu\in L^2_{\mathrm{loc}}(W)$. Consequently, every local martingale in this filtration is continuous.

Use: Represent any Brownian-filtration local martingale as an integral. Common misuse: If the filtration contains extra independent information, martingales driven by that information need not be representable using $W$. Source: LN Chapter 6, Corollary 3.2, p. 134. Nearby: Itô martingale representation theorem · Dudley’s representation theorem

Dudley’s representation theorem

Setting: Continuous-time semimartingales and equivalent measures. Statement and assumptions: If $W$ is Brownian motion relative to a filtration $\mathbb F$, every finite $\mathcal F_\infty$-measurable random variable $H$ can be written

$$H=\int_0^\infty\psi_s\,dW_s$$

for some $\psi\in L^2_{\mathrm{loc}}(W)$. Here the integral need not be a martingale on $[0,\infty]$, and the integrand need not be unique.

Use: Distinguish pathological terminal integral representation from a true martingale representation. Source: LN Chapter 6, Theorem 3.3, pp. 134-135. Nearby: Martingales in a Brownian filtration

11. The Black-Scholes model

11.1 Model and discounted dynamics

Fix $T\gt 0$. The model assumes a Brownian filtration, a bank account, and one stock:

$$\widetilde S^0_t=e^{rt},$$$$\widetilde S^1_t = \widetilde S^1_0 \exp\left( \sigma W_t+\left(\mu-\frac12\sigma^2\right)t \right), \qquad \sigma\gt 0.$$

Equivalently,

$$\frac{d\widetilde S^0_t}{\widetilde S^0_t}=r\,dt, \qquad \frac{d\widetilde S^1_t}{\widetilde S^1_t} = \mu\,dt+\sigma\,dW_t.$$

After discounting,

$$S^1_t = \frac{\widetilde S^1_t}{\widetilde S^0_t}, \qquad dS^1_t = S^1_t\left((\mu-r)dt+\sigma dW_t\right).$$

(LN, pp. 137-139.)

The assumptions are strong: constant coefficients, continuous paths, frictionless trading, and a filtration generated by the same Brownian motion that drives the stock. The model is a reference model, not a literal description of markets.

11.2 The risk-neutral measure

Define the market price of risk

$$\lambda=\frac{\mu-r}{\sigma}$$

and the density process

$$Z_t^* = \exp\left(-\lambda W_t-\frac12\lambda^2t\right).$$

Set $d\mathbb Q^*/d\mathbb P=Z_T^*$. By Girsanov’s theorem,

$$W_t^*=W_t+\lambda t$$

is a $\mathbb Q^*$-Brownian motion, and

$$dS^1_t = \sigma S^1_t\,dW_t^*.$$

Thus the discounted stock is a $\mathbb Q^*$-martingale. Itô representation in the Brownian filtration implies that this EMM is unique. The model is therefore complete. (LN, pp. 139-142.)

For a discounted claim $H\geq0$ with $H\in L^1(\mathbb Q^*)$,

$$V_t = \mathbb E_{\mathbb Q^*}[H\mid\mathcal F_t] = V_0+\int_0^t\psi_s\,dW_s^*.$$

Since $dS^1_t=\sigma S^1_t\,dW_t^*$, the stock holding

$$\vartheta_t = \frac{\psi_t}{\sigma S^1_t}$$

replicates $H$. This is the continuous-time analogue of solving the two successor-state equations at every binomial node.

11.3 Markovian claims and the pricing PDE

For a discounted payoff $H=h(S^1_T)$,

$$V_t=v(t,S^1_t),$$

where

$$v(t,x) = \mathbb E_{\mathbb Q^*} \left[ h\left( x\exp\left( \sigma\sqrt{T-t}\,Y -\frac12\sigma^2(T-t) \right) \right) \right], \qquad Y\sim\mathcal N(0,1).$$

Applying Itô’s formula to $v(t,S^1_t)$ gives

$$dV_t = v_x(t,S^1_t)\,dS^1_t +\left( v_t+\frac12\sigma^2(S^1_t)^2v_{xx} \right)dt.$$

$V$ is a $\mathbb Q^*$-martingale, so its finite-variation drift must vanish:

$$\boxed{ v_t(t,x) +\frac12\sigma^2x^2v_{xx}(t,x) =0, \qquad v(T,x)=h(x). }$$

The hedge is

$$\boxed{ \vartheta_t=v_x(t,S^1_t). }$$

In undiscounted variables, the PDE becomes

$$\boxed{ \widetilde v_t +rx\widetilde v_x +\frac12\sigma^2x^2\widetilde v_{xx} -r\widetilde v =0, \qquad \widetilde v(T,x)=\widetilde h(x). }$$

(LN, pp. 145-149.)

11.4 European call formula and delta

For

$$\widetilde H=(\widetilde S^1_T-K)^+,$$

the arbitrage-free value is

$$\boxed{ \widetilde V_t = \widetilde S^1_t\Phi(d_1) -Ke^{-r(T-t)}\Phi(d_2), }$$

where

$$d_1 = \frac{ \log(\widetilde S^1_t/K) +(r+\frac12\sigma^2)(T-t) }{ \sigma\sqrt{T-t} }, \qquad d_2=d_1-\sigma\sqrt{T-t}.$$

The replicating stock holding is

$$\boxed{ \Delta_t = \frac{\partial\widetilde v}{\partial x} (t,\widetilde S^1_t) = \Phi(d_1). }$$

(LN, pp. 150-151.)

The physical drift $\mu$ does not appear in the price. Replication removes exposure to that drift; the remaining price is determined by the traded financing rate, volatility, payoff, and time. This parallels the disappearance of the physical up probability $p$ from the CRR price.

Study note. The formula is the final calculation. The main result is the method: specify a tradable market, rule out arbitrage, identify a martingale measure, represent the claim as a stochastic integral, and read the integrand as a hedge.

Procedure: price and hedge a Black-Scholes claim

Use when. The payoff is measurable with respect to the Brownian filtration and depends on the terminal stock price.

Steps.

1. Write the bank account $\widetilde S^0_t=e^{rt}$, discount the stock, and identify $\lambda=(\mu-r)/\sigma$.
2. Construct the unique risk-neutral measure $\mathbb Q^*$, under which $d\widetilde S^1_t/\widetilde S^1_t=r\,dt+\sigma\,dW_t^*$.
3. Discount the payoff and price it by

$$\widetilde V_t=\widetilde S^0_t\mathbb E_{\mathbb Q^*}\!\left[\frac{\widetilde H}{\widetilde S^0_T}\middle|\mathcal F_t\right].$$

4. For $\widetilde V_t=\widetilde v(t,\widetilde S^1_t)$, compute the expectation or solve the pricing PDE.
5. Read the stock holding from $\Delta_t=\widetilde v_x(t,\widetilde S^1_t)$ and put the remaining value in the bank account.
6. Check the terminal condition and confirm that the physical drift $\mu$ has disappeared.

Worked example. For a one-year at-the-money call with $\widetilde S^1_0=K=100$, $r=5\%$, and $\sigma=20\%$,

$$d_1=0.35,\qquad d_2=0.15.$$

Using $\Phi(0.35)\approx0.6368$ and $\Phi(0.15)\approx0.5596$ gives

$$\widetilde V_0\approx100(0.6368)-100e^{-0.05}(0.5596)=10.45,$$

and the replicating stock holding is $\Delta_0\approx0.6368$.

Common failure points. Taking a risk-neutral expectation of an undiscounted payoff without the discount factor; leaving $\mu$ in the final price; confusing $T-t$ with $T$; computing a price without extracting delta; mixing discounted portfolio values and undiscounted stock holdings.

12. Formula sheet

These are lookup formulas, with their setting beside them. The linked statement supplies the full assumptions. Use the undiscounted formula only when a tilde is displayed.

12.1 Discrete-time market and valuation

Formula	Setting and link
$V_k=V_0+\sum_{j=1}^k\vartheta_j^{\mathsf T}\Delta S_j$	Discounted self-financing strategy; self-financing.
$\mathcal G'\cap L^0_+(\mathcal F_T)=\{0\}$	Finite discrete-time NA; equivalent NA forms.
$\mathrm{NA}\Longleftrightarrow\mathcal P^e(S)\neq\varnothing$	Discounted finite discrete-time market; first FTAP.
$q^*=(r-d)/(u-d)$	Binomial model with $d\lt r\lt u$; binomial EMM.
$\mathbb E_{\mathbb Q}[U_k\mid\mathcal F_j]=Z_j^{-1}\mathbb E_{\mathbb P}[Z_kU_k\mid\mathcal F_j]$	Equivalent measures, $j\leq k$; density-process Bayes formula.
$V_k^H=\mathbb E_{\mathbb Q}[H\mid\mathcal F_k]$	Attainable claim, NA, trivial $\mathcal F_0$; valuation theorem.
$\text{complete}\Longleftrightarrow\#\mathcal P^{e,\mathrm{loc}}(S)=1$	NA, trivial $\mathcal F_0$, $\mathcal F_T=\mathcal F$; second FTAP.
$\widetilde V_{k-1}=(q^*\widetilde V_k^u+(1-q^*)\widetilde V_k^d)/(1+r)$	CRR backward price; CRR pricing.
$\vartheta_k=(V_k^u-V_k^d)/(S_k^u-S_k^d)$	CRR node hedge; CRR hedging.

12.2 Brownian motion and stochastic calculus

Formula	Setting and link
$[W]_t=\langle W\rangle_t=t$	Brownian quadratic variation; Brownian bracket.
$\mathbb E[(\int K\,dM)^2]=\mathbb E[\int K^2\,d[M]]$	Square-integrable martingale and elementary $H$; Itô isometry.
$df(X_t)=f'(X_t)dX_t+\tfrac12f''(X_t)d[X]_t$	Continuous real semimartingale, $f\in C^2$; Itô formula.
$\mathcal E(X)=\exp(X-\tfrac12[X])$	Continuous semimartingale $X$ null at zero; stochastic exponential.
$\widetilde W_t=W_t-\int_0^t\nu_sds$	Continuous density with $L=\int\nu dW$; Girsanov.
$H=\mathbb E[H]+\int_0^\infty\psi_s\,dW_s$	Integrable claim in augmented Brownian filtration; Itô representation.
$\widetilde V_t=\widetilde S_t^1\Phi(d_1)-Ke^{-r(T-t)}\Phi(d_2)$	Black-Scholes European call; call formula and delta.

13. Theorem index

Lecture numbering restarts within chapters. The LN chapter and stable link distinguish repeated labels.

LN Chapter 1

Lecture label	Result	PDF page
Proposition 2.3	Parametrisation of self-financing strategies	LN p. 15
Theorem 3.1	Predictable stochastic integrals	LN pp. 23-24
Corollary 3.2	Stopping a martingale	LN p. 25
Theorem 3.3	A lower-bounded integral is a martingale	LN p. 25

LN Chapter 2

Lecture label	Result	PDF page
Proposition 1.1	Equivalent forms of NA in finite discrete time	LN p. 37
Lemma 1.2	A martingale measure excludes arbitrage	LN p. 39
Corollary 1.4	Martingale measures in the multinomial model	LN p. 41
Corollary 1.5	The binomial martingale measure	LN p. 42
Theorem 2.1	First Fundamental Theorem of Asset Pricing	LN p. 43
Corollary 2.2	Multinomial no-arbitrage condition	LN p. 48
Corollary 2.3	Binomial no-arbitrage condition	LN p. 48
Lemma 3.1	Density processes and Bayes’ formula	LN p. 50

LN Chapter 3

Lecture label	Result	PDF page
Theorem 1.1	Arbitrage-free valuation of attainable payoffs	LN p. 61
Theorem 1.2	Characterisation of attainable payoffs	LN p. 62
Theorem 2.1	Valuation and hedging in complete markets	LN p. 65
Theorem 2.2	Second Fundamental Theorem of Asset Pricing	LN pp. 65-66
Corollary 3.1	Undiscounted binomial valuation	LN p. 68

LN Chapter 4

Lecture label	Result	PDF page
Proposition 1.1	Brownian transformations	LN pp. 76-77
Proposition 1.2	Brownian asymptotics and the LIL	LN pp. 77-78
Proposition 1.3	Path regularity	LN p. 78
Theorem 1.4	Brownian quadratic variation	LN p. 80
Theorem 2.2	Optional stopping	LN p. 83
Proposition 2.3	Basic Brownian martingales	LN pp. 84-85
Proposition 2.4	Brownian hitting-time transforms	LN p. 86

LN Chapter 5

Lecture label	Result	PDF page
Theorem 1.1	Optional quadratic variation	LN p. 93
Lemma 1.3	Itô isometry for elementary integrands	LN pp. 95-96
Proposition 1.4	Extension of the stochastic integral	LN p. 100

LN Chapter 6

Lecture label	Result	PDF page
Theorem 1.1	Itô’s formula: continuous one-dimensional case	LN p. 116
Theorem 1.2	Itô’s formula: multidimensional and jump cases	LN p. 119
Lemma 2.1	Continuous-time Bayes formula	LN pp. 126-127
Theorem 2.2	Girsanov’s theorem: general case	LN p. 127
Theorem 2.3	Girsanov’s theorem: continuous-density case	LN p. 129
Theorem 3.1	Itô martingale representation theorem	LN pp. 133-134
Corollary 3.2	Martingales in a Brownian filtration	LN p. 134
Theorem 3.3	Dudley’s representation theorem	LN pp. 134-135

LN Chapter 8

Lecture label	Result	PDF page
Theorem 2.1	Conditional expectation: existence and uniqueness	LN p. 158
Lemma 2.2	Conditioning with an independent variable	LN p. 160
Theorem 2.3	Conditional Fatou and dominated convergence	LN pp. 160-161

14. Alphabetical index

Names link to the sole full statement; definitions and results appear together.

Letter	Entries
A	A lower-bounded integral is a martingale · A martingale measure excludes arbitrage · Admissibility · Arbitrage opportunity and NA · Arbitrage-free valuation of attainable payoffs · Attainable payoff
B	Basic Brownian martingales · Binomial no-arbitrage condition · Bounded elementary integrands · Brownian asymptotics and the LIL · Brownian hitting-time transforms · Brownian motion · Brownian quadratic variation · Brownian transformations
C	Characterisation of attainable payoffs · Complete market · Conditional expectation · Conditional expectation: existence and uniqueness · Conditional Fatou and dominated convergence · Conditioning with an independent variable · Continuous-time Bayes formula · Continuous-time local martingale
D	Density processes and Bayes’ formula · Discrete-time local martingale · Dudley’s representation theorem
E	EMM and ELMM · Equivalent forms of NA in finite discrete time · European payoff · Extension of the stochastic integral
F	First Fundamental Theorem of Asset Pricing
G	Gains and cost processes · Girsanov’s theorem: continuous-density case · Girsanov’s theorem: general case
I	Itô isometry for elementary integrands · Itô martingale representation theorem · Itô’s formula: continuous one-dimensional case · Itô’s formula: multidimensional and jump cases
L	Local square integrability
M	Martingale measures in the multinomial model · Martingale, supermartingale, and submartingale · Martingales in a Brownian filtration · Multinomial no-arbitrage condition
O	Optional quadratic variation · Optional stopping
P	Parametrisation of self-financing strategies · Path regularity · Predictable stochastic integrals
S	Second Fundamental Theorem of Asset Pricing · Self-financing strategy · Semimartingale · Stochastic exponential · Stopping a martingale
T	The binomial martingale measure · Trading strategy and value process
U	Undiscounted binomial valuation
V	Valuation and hedging in complete markets

15. Source and exercise map

Topic	Primary pages	Exercises or assessment
Probability, filtration, conditional expectation	LN pp. 5-8, 155-161	ES 1
Trading, discounting, self-financing, stopping	LN pp. 9-21	ES 2
Martingales and discrete stochastic integration	LN pp. 22-34	ES 3
Arbitrage, FTAP, equivalent measures	LN pp. 35-56	ES 3-5, SE Questions 1 and 3
Attainability, completeness, CRR valuation	LN pp. 57-74	ES 4-5, SE Question 2
Brownian motion	LN pp. 75-90	ES 6-7
Stochastic integration and semimartingales	LN pp. 91-112	ES 8-10
Itô formula, stochastic exponentials	LN pp. 113-125	ES 10-11, SE Question 4
Girsanov and representation	LN pp. 126-136	ES 11
Black-Scholes model and formula	LN pp. 137-154	SE Question 5

16. Appendices

16.1 Common mistakes and scope limits

1. Confusing $\mathbb P$ and $\mathbb Q$. $\mathbb P$ models physical frequencies; an EMM is chosen to make discounted traded prices martingales.
2. Calling every conditional expectation a price. In an incomplete market, $\mathbb E_{\mathbb Q}[H]$ depends on the chosen EMM unless $H$ is attainable.
3. Ignoring predictability. A strategy that uses the current price move to choose the holding for that same move is anticipative.
4. Treating self-financing as a definition that may be altered. It follows from portfolio bookkeeping.
5. Forgetting admissibility. Local martingale representation alone can generate pathological zero-cost terminal payoffs through wealth processes with uncontrolled downside.
6. Applying optional stopping without its hypotheses. Unbounded stopping times and non-uniformly-integrable martingales can invalidate $\mathbb E[M_\tau]=\mathbb E[M_0]$.
7. Using $(dW)^2=dt$ as a proof. It is a mnemonic for quadratic variation.
8. Assuming a local martingale is a martingale. A true martingale needs integrability and preservation of conditional expectations.
9. Assuming a stochastic exponential has expectation one. A positive local martingale is only a supermartingale in general; Novikov-type conditions ensure it is a true martingale.
10. Equating uniqueness of an EMM with realism. Uniqueness is a spanning property of the model. It can arise because the model omits relevant sources of risk.
11. Forgetting the filtration in completeness claims. Adding independent randomness enlarges the claim space and can destroy completeness.
12. Reading Black-Scholes as a market prediction. It is a replication result under idealised assumptions.

16.2 Review questions

1. Why does a risky holding for period $(k-1,k]$ have to be $\mathcal F_{k-1}$-measurable?
2. Derive $C_k=V_k-G_k$ from the rebalancing cost identity.
3. Explain why self-financing makes $(V_0,\vartheta)$ sufficient to determine the full strategy.
4. Prove that a bounded predictable integral of a discrete-time martingale is a martingale.
5. Prove that an equivalent martingale measure rules out arbitrage.
6. In a one-period multinomial model, interpret $\sum_jq_jy_j=r$ geometrically.
7. Why is the martingale probability unique in a binomial model but generally not in a trinomial model with one stock?
8. Derive Bayes’ formula for conditional expectations under an equivalent measure.
9. Explain why attainable claims have the same expectation under every EMM.
10. Derive the CRR backward price and hedge recursions.
11. State the assumptions needed for the optional stopping theorem used in these notes.
12. Why can Brownian paths have finite quadratic variation but infinite total variation?
13. Derive $\int_0^tW_s\,dW_s=(W_t^2-t)/2$ from Itô’s formula.
14. Verify the geometric Brownian motion solution by differentiating it with Itô’s formula.
15. What extra condition is needed before a stochastic exponential can define a probability measure?
16. How does Girsanov’s theorem remove a drift from Brownian motion?
17. Why does the Brownian martingale representation theorem imply market completeness only when the filtration contains no additional risk?
18. Derive the discounted Black-Scholes PDE from the martingale property.
19. Convert the discounted PDE to the undiscounted PDE.
20. Explain why $\mu$ disappears from the Black-Scholes call value while $\sigma$ remains.

16.3 Proof sketches

First FTAP: why no arbitrage produces a martingale measure

At a finite event tree, zero-cost terminal gains form a linear subspace. No arbitrage means this subspace does not meet the positive cone except at zero. A separating linear functional can therefore be chosen strictly positive on every possible state. Normalising its state weights gives a probability measure equivalent to the physical measure. Orthogonality to every trading gain makes each discounted asset a martingale. The full lecture theorem also treats general probability spaces.

Attainable valuation and the second FTAP

A claim and its admissible replicating portfolio have the same terminal payoff; if their earlier values differed, buying the cheaper and selling the dearer would create arbitrage. Under any ELMM the replicating wealth is a martingale, so its current wealth is the conditional expectation of terminal payoff. If every event indicator can be replicated, all ELMMs assign the same probability to every terminal event and therefore coincide. The converse uses the attainability criterion and requires the stated integrability conditions.

Itô isometry and Itô formula

For a predictable step integrand, each increment is chosen before its martingale increment occurs. Cross terms in the second moment vanish after conditioning on the past; the remaining diagonal terms are integrated against quadratic variation. The isometry then extends the integral by completion. For Itô’s formula, sum second-order Taylor expansions over a fine partition: the first-order sums converge to the stochastic integral, squared increments converge to the bracket, and the remainders vanish.

Girsanov drift correction

Bayes’ formula converts the Q-martingale question into a P-martingale question after multiplication by the density process. The product rule reveals the extra covariance term between the density and the original local martingale. Subtracting that term leaves a local Q-martingale. With a continuous exponential density driven by Brownian motion, the correction is the integral of the drift coefficient over time.