M.3.5 Martingales

Basic Prerequisites: 04 · Conditional Expectation.

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1. Intuition & Practical Objective

A martingale is a fair game: given everything you know at time $s$, the expected value of the future state is the present value, $\mathbb{E}[M_t\mid {\mathcal{F}}_s]=M_s$. It is the mathematical backbone of no-arbitrage pricing because “prices under the risk-neutral measure are martingales” is modern pricing (Glasserman eq. 1.39). Crucially, “fair” means fair in conditional expectation - a martingale can be enormously volatile (Brownian motion, a discounted price) and still be fair.

The practical objective: understand the three linked faces - (1) the martingale/supermartingale/submartingale hierarchy and the discounted-stock martingale under the risk-neutral measure; (2) the exponential martingale $e^{\sigma W-\frac{1}{2}{\sigma }^{2}t}$; (3) the change of measure (Radon–Nikodym) that moves you from the physical measure $\mathbb{P}$ to the pricing measure $\tilde{\mathbb{P}}$ without changing which events are impossible. This is the discrete skeleton of Girsanov’s theorem in continuous time (Stochastic Calculus).

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2. Mathematical Ground Truth & Derivations

2.1 Martingales, super-, sub- (Shreve I §2.4)

With filtration ${\mathcal{F}}_0\subseteq \cdots \subseteq {\mathcal{F}}_n$ and adapted integrable $\{M_k\}$:

$$\text{martingale: }\mathbb{E}[M_{k+1}\mid {\mathcal{F}}_k]=M_k;\text{super-}:\le ;\text{sub-}:\ge .$$

Binomial consequences (Shreve I §2.4): if $\mathbb{E}[S_{k+1}\mid {\mathcal{F}}_k]=(pu+qd)S_k$, then $(pu+qd)=1$ ⇒ martingale, $>1$ ⇒ submartingale, $<1$ ⇒ supermartingale.

2.2 The discounted stock is the risk-neutral martingale (Shreve I §3.3; Glasserman §1.2)

Under the risk-neutral measure $\tilde{\mathbb{P}}$, the discounted stock and every discounted self-financing wealth process are martingales (Shreve I §3.3, verified):

$$\tilde{\mathbb{E}}\left[\frac{S_{k+1}}{(1+r{)}^{k+1}}\right|{\mathcal{F}}_k]=\frac{S_k}{(1+r{)}^k},$$

equivalently $\tilde{\mathbb{E}}[S_{k+1}\mid {\mathcal{F}}_k]=(1+r)S_k$. This is the cornerstone pricing equation in continuous time, $V(0)=\tilde{\mathbb{E}}[e^{-rT}V(T)]$ (Glasserman eq. 1.39). Pricing under $\mathbb{P}$ would depend on the drift $\mu$; under $\tilde{\mathbb{P}}$ it depends only on $r$ and $\sigma$.

2.3 Exponential martingale (Shreve II Thm 3.6.1; Shreve I Thm 9.41)

$$Z(t)=e^{\sigma W(t)-\frac{1}{2}{\sigma }^{2}t}\Rightarrow \mathbb{E}[Z(t)\mid \mathcal{F}(s)]=Z(s).$$

Proof: factor $Z(s)$, condition on the independent increment $W(t)-W(s)$, use its MGF $\mathbb{E}{e}^{\sigma \Delta W}=e^{\frac{1}{2}{\sigma }^2(t-s)}$ to cancel the correction. This process is the seed of the change of measure - and of Girsanov (Stochastic Calculus 05 · Girsanov).

2.4 Change of measure = Radon–Nikodym (Shreve II Thm 1.6.1; Shreve I Ch 9)

Let $Z\ge 0$ a.s. with $\mathbb{E}Z=1$; define $\tilde{\mathbb{P}}(A)={\int }_{A}Zd\mathbb{P}$. Then $\tilde{\mathbb{P}}$ is a probability measure, $\tilde{\mathbb{P}}\ll \mathbb{P}$, and if $Z>0$ a.s. the measures are equivalent (mutually absolutely continuous). Change of expectation:

$$\tilde{\mathbb{E}}X=\mathbb{E}[XZ],\mathbb{E}Y=\tilde{\mathbb{E}}[Y/Z].$$

On the finite market (Shreve I Ex 9.1) with physical $p=\frac{1}{3},q=\frac{2}{3}$ and risk-neutral $\tilde{p}=\tilde{q}=\frac{1}{2}$, the density is $Z(\omega )=\tilde{\mathbb{P}}(\omega )/\mathbb{P}(\omega )$ with $Z(HH)=\frac{9}{4},Z(HT)=Z(TH)=\frac{9}{8},Z(TT)=\frac{9}{16}$, and $Z_k=\mathbb{E}[Z\mid {\mathcal{F}}_k]$ is a $\mathbb{P}$-martingale. The state-price density is ${\zeta }_k=(1+r{)}^{-k}{Z}_k$ (Shreve I Ch 9).

2.5 The normal-recentering preview of Girsanov (Shreve II Ex 1.6.x)

If $X$ is standard normal under $\mathbb{P}$ and $Z=\exp \{-\theta X-\frac{1}{2}{\theta }^2\}$ ($\theta$ constant), then under $\tilde{\mathbb{P}}$ the shifted variable $Y=X+\theta$ is standard normal (completing the square: $e^{-\theta x-{\theta }^2/2}\phi (x)=\phi (x+\theta )$). This is exactly the discrete/finite version of the Girsanov drift shift - verified verbatim on Shreve II pp. 57–58.

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3. Computational Implementation - verify the martingale & the change of measure

Check the random-walk martingale property, the risk-neutral discounted-stock relation, the exponential martingale mean, and the normal-recentering change of measure. Stdlib only.

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4. Failure Modes & First-Principles Breakdowns

1. “Zero drift” ≠ “stays put.” A martingale can have enormous variance - BM, discounted prices. The property says only that the conditional mean of the future is the present, never that the path is flat. Treating “martingale” as “predictable at the level” misreads the definition.
2. Discounting under the wrong measure. Pricing needs expectation under $\tilde{\mathbb{P}}$ (drift $r$), not $\mathbb{P}$ (drift $\mu$). A $\mathbb{P}$-expectation of $e^{-rT}{V}_T$ depends on the unhedgeable drift - the exact mispricing that a change of measure fixes (Shreve I Ch 12; Glasserman §1.2).
3. Forgetting it is the discounted stock that is a martingale. Under $\tilde{\mathbb{P}}$, $S_k/(1+r{)}^k$ is the martingale, not $S_k$ itself - unless $r=0$. Dropping the discount breaks the pricing equation.
4. Equivalent measures share the “impossible.” $\tilde{\mathbb{P}}$ and $\mathbb{P}$ are equivalent ($Z>0$) so they agree on which events have probability zero; a model where $\mathbb{P}$ lets a price hit zero while $\tilde{\mathbb{P}}$ forces $S_t>0$ violates absolute continuity and the RN density blows up.
5. Optional sampling needs bounded stopping times. $\mathbb{E}[Y_{\tau }\mid {\mathcal{F}}_{\sigma }]=Y_{\sigma }$ for martingales holds for bounded $\sigma \le \tau$ (Shreve I §5.3); stopping at the wrong (unbounded) time and asserting constancy of expectation silently fails.

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5. References

• Shreve, Stochastic Calculus for Finance II
• Shreve, Stochastic Calculus for Finance I
• Glasserman, Monte Carlo Methods in Financial Engineering

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6. Connected Graph Bridges

• Back: 04 · Conditional Expectation · Index Hub
• Forward: 06 · Advanced Extensions (RN derivative, convergence theorems) · Stochastic Calculus (Girsanov, exponential martingale, FTA)
• Applications: Black–Scholes–Merton (risk-neutral valuation) · No-Arbitrage & Binomial Trees (the discrete discounted-stock martingale)