Chapter 13

Hard

Martingale Pricing, Risk-Neutral Valuation, and Completeness

00 · Symbol Glossary

$\tilde V_t = e^{-rt}V_t$discounted price process

The value process $V_t$ rescaled by the risk-free bond $e^{-rt}$ . Chapters 10–12's central claim is a statement about $\tilde V_t$ , not $V_t$ directly.

$\mathbb{Q}$risk-neutral (equivalent martingale) measure

The measure from Chapter 10's Girsanov shift under which every discounted traded asset price is a martingale. "Risk-neutral" describes a mathematical property of $\mathbb{Q}$ , not an assumption that any investor is indifferent to risk.

$\{H_t\}$trading strategy (holdings)

An adapted process specifying how many units of each traded asset are held at time $t$ . Self-financing means the portfolio's value changes only from asset price moves — no cash added or withdrawn after $t=0$ .

$n,\ d$assets versus sources of randomness

$n$ : number of independently traded risky assets. $d$ : number of independent Brownian motions driving them. Completeness (Section 04) turns on the relationship between these two counts.

01 · The Discounted Risk-Neutral Expectation

Chapters 11–12 solved one PDE for one payoff. The martingale pricing formula states the general principle underneath both:

Definition — Risk-Neutral Pricing Formula

For any traded claim paying $V_T$ at $T$ ,

V_t = e^{-r(T-t)}\,\mathbb{E}^{\mathbb{Q}}\left[V_T\mid\mathcal{F}_t\right]

equivalently, the discounted price process $\tilde V_t=e^{-rt}V_t$ is a $\mathbb{Q}$ -martingale.

This is Feynman–Kac (Chapter 11) with the PDE erased and only the expectation kept — and it is more general than a PDE, since it makes no reference to $V$ solving any particular differential equation, only that it is priced consistently with no-arbitrage trading. The Black-Scholes formula (Chapter 12) is the special case where $V_T=\max(S_T-K,0)$ and $S_T$ is log-normal under $\mathbb{Q}$ ; change the payoff or the dynamics and the same formula still applies, only the expectation changes.

Example — the martingale property, concretely

If $\tilde V_t$ is a $\mathbb{Q}$ -martingale, then $\mathbb{E}^{\mathbb{Q}}[\tilde V_T\mid\mathcal{F}_t]=\tilde V_t$ for every $t<T$ — today's discounted price is the best $\mathbb{Q}$ -forecast of every future discounted price, with no drift left over once discounting is applied. That "no drift left over" is exactly what Chapter 10 built by hand: choose $\theta$ so that $\mu$ becomes $r$ , and $e^{-rt}S_t$ becomes driftless.

02 · Why the Risk-Neutral Measure Exists

The existence of some $\mathbb{Q}$ making discounted prices martingales is not automatic — it is equivalent to the market having no arbitrage.

First Fundamental Theorem of Asset Pricing (informal)

A market model admits no arbitrage if and only if there exists an equivalent measure $\mathbb{Q}$ under which every discounted traded-asset price is a martingale. One direction is the easy one: if $\mathbb{Q}$ exists, any self-financing strategy's discounted value is also a $\mathbb{Q}$ -martingale (linearity of the construction), so it cannot start at $0$ and end up strictly positive with positive probability while never going negative — which is precisely what an arbitrage would require. The converse (no arbitrage $\Rightarrow$ $\mathbb{Q}$ exists) is the hard direction and is where Girsanov (Chapter 10) does the actual construction, by exhibiting the $\theta$ that produces $\mathbb{Q}$ explicitly for diffusion models.

The theorem is why Chapters 10–12 could always find a measure that worked: for a market driven by GBM with a single Brownian motion and a single risky asset plus a bond, no-arbitrage is essentially automatic for reasonable parameters, so $\mathbb{Q}$ exists and Girsanov constructs it by hand.

A second theorem, the Second Fundamental Theorem, answers a different question: not whether $\mathbb{Q}$ exists, but whether it is unique. That question turns out to be exactly Section 04's completeness condition — existence comes from no-arbitrage alone; uniqueness needs enough traded assets to span every source of randomness.

03 · Self-Financing Strategies and the No-Arbitrage Price

Definition — Self-Financing Replication

A trading strategy $\{H_t\}$ (units held in each traded asset) is self-financing if its value $\Pi_t=H_t\cdot(\text{asset prices at }t)$ changes only through price moves: $d\Pi_t = H_t\cdot dS_t$ (no injected or withdrawn cash). It replicates a claim if $\Pi_T=V_T$ for every path.

If a self-financing strategy replicates $V_T$ , then $\Pi_t$ and $V_t$ must be equal at every $t<T$ too — not just at $T$ . If $\Pi_t>V_t$ at some earlier time, sell the claim, buy the replicating strategy, pocket the difference risklessly, and unwind at $T$ where the two are equal by construction; if $\Pi_t<V_t$ , do the reverse. Either mismatch is a static arbitrage, so no-arbitrage forces $V_t=\Pi_t$ for all $t$ . This is the same replication logic Chapter 11 used to derive the PDE (a locally riskless hedge earning $r$ ), now stated as a global principle: the price of any replicable claim is forced, not chosen.

04 · Completeness: Spanning Payoffs

A market is complete if every payoff at $T$ can be replicated by some self-financing strategy in the traded assets. Whether that's possible is a counting question, and the counting is exactly the linear-algebra intuition of spanning a vector space with a basis.

Think of the space of possible terminal payoffs (one number per possible path, or per state of the world) as a vector space. Each traded asset, dynamically rebalanced, contributes one "direction" you can move value in, and each independent Brownian motion is one independent source of randomness that must be matched by a direction to be hedged — exactly as $n$ vectors in $\mathbb{R}^n$ span the whole space only if they are linearly independent and there are enough of them (Chapter 09 of Linear Algebra).

Step-by-step — Matching assets to randomness

Single asset, single Brownian motion (

n=1

d=1

, as in Black-Scholes): one traded risky asset, continuously rebalanced, exactly spans the one source of randomness. This is why the hedge in Chapter 11 used only

S_t

and a bond, and it was enough.

Fewer assets than sources of randomness (

n<d

): some direction of randomness has no traded asset moving with it. Payoffs depending on that direction cannot be replicated — an unhedgeable risk, and the market is incomplete.

More assets than sources of randomness, but not independent (

n>d

with a rank-deficient volatility matrix): extra assets are redundant combinations of the others — like adding a vector already in the span of a basis. They add no new hedging power.

Exactly matched and independent (

n=d

, volatility matrix invertible): assets act as an invertible "basis" mapping trading positions to payoff exposures, and every payoff is spanned — completeness, with a unique replicating strategy and hence a unique no-arbitrage price.

Practice — Count the sources

A market has a stock, a bond, and a plain-vanilla option all trading on the same single Brownian motion ( $d=1$ ). Is this market complete? Does adding the option as a third traded instrument change completeness relative to just the stock and bond? Explain using the $n$ versus $d$ counting above, and note what changes if a second, independent Brownian motion (say, driving stochastic volatility) enters the picture.

05 · Incomplete Markets

❌ Assuming every derivative has one 'true' no-arbitrage price

Treating the risk-neutral pricing formula as always pinning down a single number, regardless of what drives the underlying.

Why it breaks: completeness (Section 04) is a genuine restriction, not a default. Models with jump risk, stochastic volatility, or any source of randomness not matched one-for-one by a traded, independent asset leave directions unspanned. When the market is incomplete, multiple equivalent martingale measures $\mathbb{Q}$ satisfy Section 01's formula, and different choices give different, individually arbitrage-free prices for the same claim.

Consequence: the no-arbitrage argument only produces a range of prices — bounded by the cheapest and most expensive super-replicating strategies — not a single value. Picking one specific price within that range requires an additional input beyond no-arbitrage, such as a utility function, a calibration to other traded option prices, or a market convention for that particular source of unhedged risk.

06 · Exercises

EXERCISE 13.1

Write $\tilde V_t=e^{-rt}V_t$ and use the tower property of conditional expectation with the definition in Section 01.

$\mathbb{E}^{\mathbb{Q}}[\tilde V_T\mid\mathcal{F}_t]=\mathbb{E}^{\mathbb{Q}}[e^{-rT}V_T\mid\mathcal{F}_t]=e^{-rt}\left(e^{-r(T-t)}\mathbb{E}^{\mathbb{Q}}[V_T\mid\mathcal{F}_t]\right)=e^{-rt}V_t=\tilde V_t$ , using Section 01's formula $V_t=e^{-r(T-t)}\mathbb{E}^{\mathbb{Q}}[V_T\mid\mathcal{F}_t]$ in the middle step. This is exactly the martingale property.

Show directly that the risk-neutral pricing formula of Section 01 is equivalent to $\tilde V_t=e^{-rt}V_t$ being a $\mathbb{Q}$ -martingale.

EXERCISE 13.2

Compare a claim on a single GBM asset ( $n=d=1$ ) against a claim whose payoff depends on whether a separate, untraded coin-flip event occurs.

The GBM-only claim is spanned by the stock and bond ( $n=d=1$ , Section 04 step 1), so it is replicable and has a unique price. A payoff depending on an extra untraded random event introduces a second source of randomness ( $d=2$ ) with only one risky traded asset ( $n=1$ ), so $n<d$ : that extra randomness is unhedgeable, the market (restricted to just stock and bond) is incomplete with respect to that claim, and no unique no-arbitrage price exists without adding another traded instrument correlated with the coin-flip.

Explain why a claim written purely on GBM $S_t$ has a unique price, while a claim also depending on an independent, untraded random event does not, using the $n$ versus $d$ counting of Section 04.

EXERCISE 13.3

Use the static replication argument of Section 03: compare two hypothetical prices for the same replicable claim.

Suppose $V_t^A>V_t^B$ for two claims with identical replicable payoff $V_T$ at $T$ . Sell the $A$ -priced claim, buy the replicating strategy for $B$ (cost $V_t^B$ ), pocket $V_t^A-V_t^B>0$ today. At $T$ , the replicating strategy delivers exactly $V_T$ , which exactly offsets the sold claim's payoff. Riskless profit with zero net future obligation — an arbitrage — so no-arbitrage forces $V_t^A=V_t^B$ .

Using the self-financing replication argument of Section 03, show that two claims with the same replicable terminal payoff $V_T$ must have the same price at every $t<T$ .

07 · Chapter Summary

Concept	Meaning
Martingale pricing	$V_t=e^{-r(T-t)}\mathbb{E}^{\mathbb{Q}}[V_T\mid\mathcal{F}_t]$ ; $\tilde V_t$ is a $\mathbb{Q}$ -martingale
Existence of $\mathbb{Q}$	equivalent to no-arbitrage (First Fundamental Theorem)
Self-financing replication	forces $V_t=\Pi_t$ for all $t$ , or arbitrage
Completeness	every payoff spanned by traded assets; needs $n=d$ , full rank
Incomplete markets	multiple valid $\mathbb{Q}$ 's; price is a range, not a number
Unifies	Chapters 10–12 as one principle: reweight, discount, replicate

Next: this closes the core Stochastic Calculus sequence — the natural continuation is Time Series Chapter 06, Volatility Clustering and GARCH, which drops the constant- $\sigma$ assumption this chapter still relies on and models volatility empirically instead of assuming it.