Martingales and No-Arbitrage¶

In the unifying framework of this section, no-arbitrage is the requirement that demands control — it is the economic reason we must upgrade local martingales to true martingales.

This section develops the deep connection between no-arbitrage and martingale pricing. The central insight of modern asset pricing theory is that absence of arbitrage is equivalent to the existence of a probability measure under which discounted prices are martingales.

Prerequisites

This section assumes familiarity with:

Local Martingales — the distinction between local and true martingales
Stochastic Exponential — the tool for constructing measure changes

Roadmap

This section provides conceptual foundations. The technical machinery (Girsanov's theorem) and applications (risk-neutral pricing) follow in subsequent sections.

The Setting: Continuous-Time Financial Markets¶

Consider a financial market with:

A money market account (risk-free asset) with dynamics:

\[ dB_t = r_t B_t\,dt, \quad B_0 = 1 \]

For constant \(r\): \(B_t = e^{rt}\).

A risky asset with price process \(S_t\) following:

\[ dS_t = \mu_t S_t\,dt + \sigma_t S_t\,dW_t \]

under the physical (real-world) probability measure \(\mathbb{P}\).

The discounted price process is:

\[ \tilde{S}_t = \frac{S_t}{B_t} = e^{-\int_0^t r_s\,ds} S_t \]

Using Itô's formula:

\[ d\tilde{S}_t = (\mu_t - r_t)\tilde{S}_t\,dt + \sigma_t \tilde{S}_t\,dW_t \]

The drift term \((\mu_t - r_t)\) is called the excess return or risk premium.

What is Arbitrage?¶

Informal Definition¶

An arbitrage opportunity is "free money"—a trading strategy that:

Costs nothing to enter
Cannot lose money
Has a positive probability of making money

Trading Strategies and Portfolios¶

A trading strategy is a pair of adapted processes \((\phi_t, \psi_t)\) representing holdings in the risky asset and money market account.

The portfolio value at time \(t\) is:

\[ V_t = \phi_t S_t + \psi_t B_t \]

Self-Financing Strategy

A strategy \((\phi, \psi)\) is self-financing if the portfolio value changes only due to asset price movements, not from injecting or withdrawing cash:

\[ dV_t = \phi_t\,dS_t + \psi_t\,dB_t \]

Equivalently, in discounted terms:

\[ d\tilde{V}_t = \phi_t\,d\tilde{S}_t \]

where \(\tilde{V}_t = V_t/B_t\) is the discounted portfolio value.

The self-financing condition means all gains come from trading, not from external funding.

The Gains Process¶

For a self-financing strategy, the gains process is:

\[ G_t(\phi) = \int_0^t \phi_s\,d\tilde{S}_s \]

This is the cumulative profit (in discounted terms) from holding \(\phi_s\) shares.

Admissible Strategy

A self-financing strategy \(\phi\) is admissible if the gains process is bounded below:

\[ G_t(\phi) \geq -M \quad \text{for some } M > 0 \text{ and all } t \in [0,T] \]

This rules out "doubling strategies" that require unlimited borrowing.

Formal Definition of Arbitrage¶

Arbitrage Opportunity

An arbitrage opportunity is an admissible self-financing strategy \(\phi\) with:

\(V_0 = 0\) (zero initial cost)
\(V_T \geq 0\) almost surely (no possibility of loss)
\(\mathbb{P}(V_T > 0) > 0\) (positive probability of gain)

NFLVR: The Technically Correct Condition¶

In continuous time, the naive arbitrage definition above is insufficient. The correct condition is NFLVR (No Free Lunch with Vanishing Risk):

NFLVR

NFLVR holds if there is no sequence of admissible strategies \((\phi^n)\) such that:

\(V_0^n = 0\) for all \(n\)
\(V_T^n \geq -1/n\) (losses vanish)
\(V_T^n \to f\) in probability, where \(f \geq 0\) and \(\mathbb{P}(f > 0) > 0\)

NFLVR rules out not just exact arbitrages but also sequences of "approximate arbitrages" that converge to free money. This is necessary because in continuous time, exact arbitrages may not exist while approximate ones do.

For most practical purposes, we work with the simpler definition, but NFLVR is what appears in the rigorous FTAP statement.

The Connection Between Drift and Arbitrage¶

Under the physical measure \(\mathbb{P}\), the discounted price has drift:

\[ d\tilde{S}_t = \underbrace{(\mu - r)\tilde{S}_t\,dt}_{\text{systematic drift}} + \sigma \tilde{S}_t\,dW_t \]

Why Drift Matters¶

Drift in the discounted price process violates the martingale property:

\[ \mathbb{E}^{\mathbb{P}}[\tilde{S}_T \mid \mathcal{F}_t] \neq \tilde{S}_t \]

For \(\mu > r\), the discounted price drifts upward on average. This creates expected profit from holding the stock (financed by borrowing), which is the essence of a risk premium.

From Expected Profit to Arbitrage¶

Expected profit is not the same as arbitrage (which requires riskless profit). However, drift enables arbitrage through limiting arguments:

Heuristic argument (full proof via Delbaen-Schachermayer FTAP): Consider a strategy that takes a small position when the drift is favorable and scales it up. In continuous time, one can construct sequences of such strategies where:

The expected gain remains positive
The variance of the gain shrinks to zero
The probability of loss vanishes

This is precisely what NFLVR rules out. The rigorous statement is given by the Fundamental Theorem of Asset Pricing (Delbaen and Schachermayer, 1994), which establishes that NFLVR holds if and only if there exists an equivalent local martingale measure.

The Key Insight¶

\[ \boxed{ \text{Drift in discounted prices} \iff \text{Violation of martingale property} \iff \text{Potential for arbitrage} } \]

To eliminate arbitrage, we need discounted prices to have no drift—i.e., to be martingales under some equivalent measure.

The Martingale Approach to No-Arbitrage¶

Core Observation¶

If the discounted price \(\tilde{S}_t\) is a martingale under some measure \(\mathbb{Q}\), then for any admissible strategy \(\phi\):

\[ G_t(\phi) = \int_0^t \phi_s\,d\tilde{S}_s \]

is a local martingale under \(\mathbb{Q}\) (as an Itô integral with adapted integrand).

The Supermartingale Argument¶

If \(G_t(\phi)\) is bounded below by \(-M\) (admissibility), we can show it's a supermartingale:

Step 1: Define \(H_t = G_t(\phi) + M \geq 0\).

Step 2: Since \(G_t\) is a local martingale, so is \(H_t = G_t + M\) (adding a constant preserves the local martingale property).

Step 3: A non-negative local martingale is a supermartingale (see Local Martingales).

Step 4: Therefore \(H_t\) is a supermartingale, which means:

\[ \mathbb{E}^{\mathbb{Q}}[H_T] \leq H_0 = M \]

Step 5: Thus \(\mathbb{E}^{\mathbb{Q}}[G_T(\phi)] = \mathbb{E}^{\mathbb{Q}}[H_T] - M \leq 0\).

Why This Rules Out Arbitrage¶

For an arbitrage, we need: - \(G_T(\phi) \geq 0\) almost surely - \(\mathbb{Q}(G_T(\phi) > 0) > 0\)

But if \(G_T \geq 0\) and \(\mathbb{E}^{\mathbb{Q}}[G_T] \leq 0\), then we must have \(G_T = 0\) almost surely.

Since \(\mathbb{Q}\) is equivalent to \(\mathbb{P}\) (they have the same null sets), \(G_T = 0\) \(\mathbb{P}\)-almost surely as well.

Conclusion: No arbitrage is possible!

Martingale Implies No Arbitrage

If there exists a probability measure \(\mathbb{Q} \sim \mathbb{P}\) such that discounted asset prices are \(\mathbb{Q}\)-local martingales, then no arbitrage opportunities exist.

Why "Equivalent" Measure?¶

The requirement \(\mathbb{Q} \sim \mathbb{P}\) (equivalence) is crucial:

Equivalent Measures

Two measures \(\mathbb{P}\) and \(\mathbb{Q}\) are equivalent (\(\mathbb{Q} \sim \mathbb{P}\)) if they have the same null sets:

\[ \mathbb{P}(A) = 0 \iff \mathbb{Q}(A) = 0 \]

Why equivalence matters:

Same "possible" outcomes: If \(\mathbb{Q}\) assigned zero probability to some \(\mathbb{P}\)-possible event, we could have \(G_T > 0\) on that event under \(\mathbb{P}\) but \(\mathbb{Q}(G_T > 0) = 0\), breaking the argument.
Economic interpretation: An equivalent measure preserves which scenarios can occur; it only reweights their probabilities.
Arbitrage detection: The supermartingale argument relies on \(G_T = 0\) \(\mathbb{Q}\)-a.s. implying \(G_T = 0\) \(\mathbb{P}\)-a.s.

Why "Local" Martingale?¶

In continuous time, discounted prices are typically local martingales, not necessarily true martingales.

The Issue¶

Even under the risk-neutral measure, the gains process:

\[ G_t(\phi) = \int_0^t \phi_s\,d\tilde{S}_s \]

is only guaranteed to be a local martingale (as an Itô integral with adapted integrand).

When Does Local Suffice?¶

For admissible strategies (gains bounded below), local martingales that are bounded below are supermartingales. This is enough to rule out arbitrage, as shown above.

When Local Fails: Bubbles¶

If the discounted price is a strict local martingale (local but not true martingale), then:

\[ \mathbb{E}^{\mathbb{Q}}[\tilde{S}_T] < \tilde{S}_0 \]

This corresponds to a financial bubble—the current price exceeds the expected discounted payoff. See Local Martingales for detailed examples and implications.

The Fundamental Theorem of Asset Pricing¶

The converse is also true: if there's no arbitrage, then a martingale measure must exist.

Fundamental Theorem of Asset Pricing (FTAP)

The following are equivalent:

No arbitrage (NFLVR holds)
Existence of ELMM: There exists a probability measure \(\mathbb{Q} \sim \mathbb{P}\) such that discounted asset prices are \(\mathbb{Q}\)-local martingales

The measure \(\mathbb{Q}\) is called an Equivalent Local Martingale Measure (ELMM).

Direction 1 (ELMM ⟹ No Arbitrage): Proved above using the supermartingale argument.

Direction 2 (No Arbitrage ⟹ ELMM): This is the hard direction, requiring functional analysis (Hahn–Banach theorem, separating hyperplanes). See FTAP for the full proof.

How Measure Change Removes Drift¶

What Changes Under Q¶

A change from \(\mathbb{P}\) to \(\mathbb{Q}\) does not alter:

The paths of asset prices (same sample space \(\Omega\))
The available trading strategies
The economic payoffs

It only alters:

The probabilities assigned to different scenarios
The drift of stochastic processes

The Mechanism¶

Under \(\mathbb{P}\):

\[ d\tilde{S}_t = (\mu - r)\tilde{S}_t\,dt + \sigma \tilde{S}_t\,dW_t^{\mathbb{P}} \]

Under \(\mathbb{Q}\) (via Girsanov):

\[ d\tilde{S}_t = \sigma \tilde{S}_t\,dW_t^{\mathbb{Q}} \]

where \(W_t^{\mathbb{Q}} = W_t^{\mathbb{P}} + \int_0^t \theta_s\,ds\) is a \(\mathbb{Q}\)-Brownian motion, with \(\theta_t = (\mu_t - r_t)/\sigma_t\).

The drift has been absorbed into the change of Brownian motion!

The Market Price of Risk¶

The quantity:

\[ \theta_t = \frac{\mu_t - r_t}{\sigma_t} \]

is called the market price of risk or Sharpe ratio. It measures the excess return per unit of volatility.

Girsanov's theorem says: shifting Brownian motion by \(\int \theta\,dt\) removes the drift from discounted prices.

Example: Black–Scholes Model¶

Under the Physical Measure P¶

\[ dS_t = \mu S_t\,dt + \sigma S_t\,dW_t^{\mathbb{P}} \]

Discounted:

\[ d\tilde{S}_t = (\mu - r)\tilde{S}_t\,dt + \sigma \tilde{S}_t\,dW_t^{\mathbb{P}} \]

This has drift \(\mu - r \neq 0\) (typically), so the martingale property fails under \(\mathbb{P}\).

Under the Risk-Neutral Measure Q¶

The market price of risk is constant: \(\theta = (\mu - r)/\sigma\).

Define the Radon–Nikodym derivative:

\[ \frac{d\mathbb{Q}}{d\mathbb{P}}\bigg|_{\mathcal{F}_T} = \exp\left(-\theta W_T^{\mathbb{P}} - \frac{\theta^2 T}{2}\right) \]

This is the stochastic exponential \(\mathcal{E}(-\theta W^{\mathbb{P}})_T\).

Then \(W_t^{\mathbb{Q}} = W_t^{\mathbb{P}} + \theta t\) is a \(\mathbb{Q}\)-Brownian motion, and:

\[ dS_t = rS_t\,dt + \sigma S_t\,dW_t^{\mathbb{Q}} \]

Discounted:

\[ d\tilde{S}_t = \sigma \tilde{S}_t\,dW_t^{\mathbb{Q}} \]

No drift! The discounted price is a \(\mathbb{Q}\)-martingale.

Verification via Novikov¶

Since \(\theta\) is constant:

\[ \mathbb{E}^{\mathbb{P}}\left[\exp\left(\frac{1}{2}\int_0^T \theta^2\,ds\right)\right] = e^{\theta^2 T/2} < \infty \]

Novikov's condition is satisfied, so the stochastic exponential is a true martingale, and \(\mathbb{Q}\) is a valid probability measure equivalent to \(\mathbb{P}\).

See Novikov and Kazamaki Conditions for general criteria.

Economic Interpretation¶

Two Perspectives on Prices¶

Measure	Interpretation	Use
\(\mathbb{P}\) (Physical)	Real-world probabilities; reflects beliefs and risk preferences	Forecasting, risk management
\(\mathbb{Q}\) (Risk-Neutral)	Pricing probabilities; reflects no-arbitrage constraints	Derivative pricing

What Q Does NOT Mean¶

The risk-neutral measure \(\mathbb{Q}\) does not imply:

Investors are actually risk-neutral
Expected returns equal the risk-free rate in reality
\(\mathbb{Q}\)-probabilities are "correct" forecasts

What Q DOES Mean¶

The measure \(\mathbb{Q}\) encodes the prices that prevent arbitrage:

\(\mathbb{Q}\)-expectations give arbitrage-free prices
\(\mathbb{Q}\) adjusts probabilities so that fair prices emerge
\(\mathbb{Q}\) is determined by market prices of traded instruments, not by beliefs

Complete vs. Incomplete Markets¶

Complete Markets¶

A market is complete if every contingent claim can be replicated by a self-financing strategy.

Second Fundamental Theorem

An arbitrage-free market is complete if and only if the ELMM \(\mathbb{Q}\) is unique.

In complete markets:

Every derivative has a unique arbitrage-free price
Hedging is perfect (zero residual risk)
The Martingale Representation Theorem holds: every martingale is a stochastic integral w.r.t. traded assets

Example: Black–Scholes model (one stock driven by one Brownian motion)

Incomplete Markets¶

When markets are incomplete:

Multiple ELMMs exist
Derivative prices lie in an interval \([\underline{V}, \overline{V}]\)
Additional criteria needed to select a price (utility maximization, variance-optimal, minimal entropy)

Example: Stochastic volatility models (two sources of randomness—stock and volatility—but only stock is traded)

The connection to the Martingale Representation Theorem: incompleteness means some martingales cannot be represented as integrals w.r.t. traded assets, so perfect hedging is impossible.

See Martingale Representation Theorem for the mathematical details.

Summary: The No-Arbitrage ⟺ Martingale Connection¶

\[ \boxed{ \text{No Arbitrage (NFLVR)} \iff \exists\, \mathbb{Q} \sim \mathbb{P} : \tilde{S}_t \text{ is a } \mathbb{Q}\text{-local martingale} } \]

Concept	Mathematical Statement
Arbitrage-free	NFLVR holds
Risk-neutral measure	ELMM \(\mathbb{Q}\) exists
Drift removal	\(d\tilde{S} = \sigma \tilde{S}\,dW^{\mathbb{Q}}\) (no \(dt\) term)
Market price of risk	\(\theta = (\mu - r)/\sigma\)
Complete market	\(\mathbb{Q}\) is unique
Incomplete market	Multiple \(\mathbb{Q}\)'s exist

What Comes Next¶

Subsequent Sections

Risk-Neutral Valuation Principle: How to use \(\mathbb{Q}\) for pricing
Girsanov's Theorem: The technical machinery for measure change
Novikov & Kazamaki: When is the stochastic exponential a true martingale?
FTAP (Ch5): The complete proof

Exercises¶

Exercise 1. Let \(V_t = \phi_t S_t + \psi_t B_t\) be a portfolio value.

(a) Apply the product rule to \(d(\phi_t S_t)\) and \(d(\psi_t B_t)\), then use the self-financing condition \(dV_t = \phi_t\,dS_t + \psi_t\,dB_t\) to show:

\[ S_t\,d\phi_t + B_t\,d\psi_t = 0 \]

(b) Interpret this condition economically: what does it say about rebalancing the portfolio?

Solution to Exercise 1

(a) By the product rule:

\[ d(\phi_t S_t) = \phi_t\,dS_t + S_t\,d\phi_t + d\phi_t\,dS_t \]

\[ d(\psi_t B_t) = \psi_t\,dB_t + B_t\,d\psi_t + d\psi_t\,dB_t \]

Since \(\phi_t\) and \(\psi_t\) are of bounded variation (trading strategies), the cross terms \(d\phi_t\,dS_t\) and \(d\psi_t\,dB_t\) vanish. Adding:

\[ dV_t = \phi_t\,dS_t + S_t\,d\phi_t + \psi_t\,dB_t + B_t\,d\psi_t \]

The self-financing condition states \(dV_t = \phi_t\,dS_t + \psi_t\,dB_t\). Subtracting:

\[ S_t\,d\phi_t + B_t\,d\psi_t = 0 \]

(b) This says that any rebalancing of the portfolio must be self-funding: if you buy more stock (increasing \(\phi_t\)), you must finance it by selling bonds (decreasing \(\psi_t\)), and vice versa. The total cost of rebalancing at current prices is zero — no external cash is injected or withdrawn.

Exercise 2. Show that for a self-financing strategy, the discounted portfolio value satisfies:

\[ \tilde{V}_t = \tilde{V}_0 + \int_0^t \phi_s\,d\tilde{S}_s \]

Hint: Use Itô's product rule on \(\tilde{V}_t = V_t/B_t = V_t \cdot B_t^{-1}\).

Solution to Exercise 2

Define \(\tilde{V}_t = V_t / B_t = V_t \cdot B_t^{-1}\). By Itô's product rule:

\[ d\tilde{V}_t = V_t\,d(B_t^{-1}) + B_t^{-1}\,dV_t + dV_t\,d(B_t^{-1}) \]

Since \(B_t\) is differentiable (no stochastic term), the cross-variation vanishes: \(dV_t\,d(B_t^{-1}) = 0\). Now \(d(B_t^{-1}) = -r_t B_t^{-1}\,dt\), and by self-financing \(dV_t = \phi_t\,dS_t + \psi_t\,dB_t\). Substituting:

\[ d\tilde{V}_t = -r_t V_t B_t^{-1}\,dt + B_t^{-1}(\phi_t\,dS_t + \psi_t\,dB_t) \]

\[ = -r_t \tilde{V}_t\,dt + \phi_t B_t^{-1}\,dS_t + \psi_t B_t^{-1} r_t B_t\,dt \]

\[ = -r_t \tilde{V}_t\,dt + \phi_t B_t^{-1}\,dS_t + r_t \psi_t\,dt \]

Since \(\tilde{V}_t = \phi_t \tilde{S}_t + \psi_t\), we have \(r_t\tilde{V}_t = r_t\phi_t\tilde{S}_t + r_t\psi_t\). Thus:

\[ d\tilde{V}_t = -r_t\phi_t\tilde{S}_t\,dt + \phi_t B_t^{-1}\,dS_t \]

Now \(d\tilde{S}_t = d(S_t B_t^{-1}) = B_t^{-1}\,dS_t - r_t \tilde{S}_t\,dt\), so \(B_t^{-1}\,dS_t = d\tilde{S}_t + r_t\tilde{S}_t\,dt\). Substituting:

\[ d\tilde{V}_t = -r_t\phi_t\tilde{S}_t\,dt + \phi_t(d\tilde{S}_t + r_t\tilde{S}_t\,dt) = \phi_t\,d\tilde{S}_t \]

Integrating from \(0\) to \(t\):

\[ \tilde{V}_t = \tilde{V}_0 + \int_0^t \phi_s\,d\tilde{S}_s \]

Exercise 3. Let \(G_t\) be a local martingale with \(G_t \geq -M\) for some \(M > 0\).

(a) Define \(H_t = G_t + M\). Show that \(H_t\) is a non-negative local martingale.

(b) Use the fact that non-negative local martingales are supermartingales to conclude \(\mathbb{E}[H_T] \leq H_0\).

(c) Deduce that \(\mathbb{E}[G_T] \leq G_0\).

(d) If additionally \(G_T \geq 0\) a.s. and \(G_0 = 0\), show that \(G_T = 0\) a.s.

Solution to Exercise 3

(a) Since \(G_t \geq -M\), we have \(H_t = G_t + M \geq 0\). Since \(G_t\) is a local martingale with localizing sequence \(\{\tau_n\}\), the stopped process \(G_{t \wedge \tau_n}\) is a martingale. Then \(H_{t \wedge \tau_n} = G_{t \wedge \tau_n} + M\) is also a martingale (adding a constant preserves the martingale property). Hence \(H_t\) is a non-negative local martingale.

(b) Since \(H_t \geq 0\) is a non-negative local martingale, by Fatou's lemma:

\[ \mathbb{E}[H_T] = \mathbb{E}\left[\lim_{n \to \infty} H_{T \wedge \tau_n}\right] \leq \liminf_{n \to \infty} \mathbb{E}[H_{T \wedge \tau_n}] = \liminf_{n \to \infty} H_0 = H_0 \]

Therefore \(\mathbb{E}[H_T] \leq H_0 = G_0 + M\).

(c) Since \(\mathbb{E}[H_T] = \mathbb{E}[G_T + M] = \mathbb{E}[G_T] + M \leq G_0 + M\), we get:

\[ \mathbb{E}[G_T] \leq G_0 \]

(d) If \(G_T \geq 0\) a.s. and \(G_0 = 0\), then from part (c): \(\mathbb{E}[G_T] \leq 0\). But \(G_T \geq 0\) a.s. implies \(\mathbb{E}[G_T] \geq 0\). Together: \(\mathbb{E}[G_T] = 0\). Since \(G_T \geq 0\) a.s. and has zero expectation, we conclude \(G_T = 0\) almost surely.

Exercise 4. In the Black–Scholes model with \(\mu = 0.10\), \(r = 0.02\), \(\sigma = 0.20\):

(a) Compute the market price of risk \(\theta\).

(b) Write down the Radon–Nikodym derivative \(d\mathbb{Q}/d\mathbb{P}|_{\mathcal{F}_T}\) for \(T = 1\).

(c) Verify Novikov's condition is satisfied.

(d) Under \(\mathbb{Q}\), what is the drift of \(S_t\)?

Solution to Exercise 4

(a) The market price of risk is:

\[ \theta = \frac{\mu - r}{\sigma} = \frac{0.10 - 0.02}{0.20} = \frac{0.08}{0.20} = 0.4 \]

(b) The Radon–Nikodym derivative for \(T = 1\) is:

\[ \frac{d\mathbb{Q}}{d\mathbb{P}}\bigg|_{\mathcal{F}_1} = \exp\left(-\theta W_1^{\mathbb{P}} - \frac{\theta^2}{2}\right) = \exp\left(-0.4\, W_1^{\mathbb{P}} - 0.08\right) \]

(c) Novikov's condition requires:

\[ \mathbb{E}^{\mathbb{P}}\left[\exp\left(\frac{1}{2}\int_0^1 \theta^2\,ds\right)\right] = \exp\left(\frac{\theta^2}{2}\right) = \exp(0.08) \approx 1.0833 < \infty \]

Since \(\theta\) is constant, the integral \(\int_0^1 \theta^2\,ds = \theta^2 = 0.16\) is deterministic, and the exponential moment is trivially finite. Novikov's condition is satisfied.

(d) Under \(\mathbb{Q}\), the drift of \(S_t\) is \(r = 0.02\). Specifically:

\[ dS_t = rS_t\,dt + \sigma S_t\,dW_t^{\mathbb{Q}} = 0.02\,S_t\,dt + 0.20\,S_t\,dW_t^{\mathbb{Q}} \]

The physical drift \(\mu = 0.10\) has been replaced by the risk-free rate \(r = 0.02\).

Exercise 5. A discrete-time market has two periods and a stock that can go up by factor \(u = 1.3\) or down by factor \(d = 0.8\) each period. The risk-free rate per period is \(R = 0.05\).

(a) Compute the risk-neutral probability \(q = (1 + R - d)/(u - d)\).

(b) The stock starts at \(S_0 = 100\). Compute the discounted stock price \(\tilde{S}_t = S_t / (1+R)^t\) at all nodes of the tree.

(c) Verify that \(\tilde{S}_t\) is a martingale under \(\mathbb{Q}\): check \(\mathbb{E}^{\mathbb{Q}}[\tilde{S}_1 \mid S_0] = \tilde{S}_0\) and \(\mathbb{E}^{\mathbb{Q}}[\tilde{S}_2 \mid S_1] = \tilde{S}_1\) at each node.

Solution to Exercise 5

(a) The risk-neutral probability is

\[ q = \frac{1 + R - d}{u - d} = \frac{1.05 - 0.8}{1.3 - 0.8} = \frac{0.25}{0.50} = 0.5 \]

(b) The stock prices at each node are:

\(t = 0\): \(S_0 = 100\), so \(\tilde{S}_0 = 100\).
\(t = 1\): \(S_1^u = 130\), \(\tilde{S}_1^u = 130 / 1.05 \approx 123.81\). \(S_1^d = 80\), \(\tilde{S}_1^d = 80 / 1.05 \approx 76.19\).
\(t = 2\): \(S_2^{uu} = 169\), \(\tilde{S}_2^{uu} = 169 / 1.05^2 \approx 153.29\). \(S_2^{ud} = 104\), \(\tilde{S}_2^{ud} = 104 / 1.1025 \approx 94.33\). \(S_2^{dd} = 64\), \(\tilde{S}_2^{dd} = 64 / 1.1025 \approx 58.05\).

(c) At \(t = 0\):

\[ \mathbb{E}^{\mathbb{Q}}[\tilde{S}_1] = q \cdot \tilde{S}_1^u + (1-q) \cdot \tilde{S}_1^d = 0.5 \times \frac{130}{1.05} + 0.5 \times \frac{80}{1.05} = \frac{0.5 \times 210}{1.05} = \frac{105}{1.05} = 100 = \tilde{S}_0 \;\checkmark \]

At \(t = 1\), node \(u\):

\[ \mathbb{E}^{\mathbb{Q}}[\tilde{S}_2 \mid S_1 = 130] = 0.5 \times \frac{169}{1.1025} + 0.5 \times \frac{104}{1.1025} = \frac{0.5 \times 273}{1.1025} = \frac{136.5}{1.1025} = \frac{130}{1.05} = \tilde{S}_1^u \;\checkmark \]

At \(t = 1\), node \(d\):

\[ \mathbb{E}^{\mathbb{Q}}[\tilde{S}_2 \mid S_1 = 80] = 0.5 \times \frac{104}{1.1025} + 0.5 \times \frac{64}{1.1025} = \frac{0.5 \times 168}{1.1025} = \frac{84}{1.1025} = \frac{80}{1.05} = \tilde{S}_1^d \;\checkmark \]

The discounted stock price is a \(\mathbb{Q}\)-martingale at every node.

Exercise 6. Consider a market with one traded stock driven by two independent Brownian motions:

\[ dS_t = \mu S_t\,dt + \sigma_1 S_t\,dW_t^1 + \sigma_2 S_t\,dW_t^2 \]

with risk-free rate \(r\).

(a) Write the risk premium equation \(\mu - r = \sigma_1\theta_1 + \sigma_2\theta_2\) and explain why it defines a line in \((\theta_1, \theta_2)\) space.

(b) Is this market complete or incomplete? Justify by counting assets and sources of risk.

(c) Pick two specific points on the line from (a) and explain why they define two different equivalent martingale measures, each producing a different price for a claim \(\Phi(W_T^2)\) that depends only on the second Brownian motion.

Solution to Exercise 6

(a) Under \(\mathbb{Q}^{\boldsymbol{\theta}}\), the discounted stock must have zero drift. The Girsanov shift \(W_t^{i,\mathbb{Q}} = W_t^{i,\mathbb{P}} + \theta_i t\) removes the drift when

\[ \mu - r = \sigma_1\theta_1 + \sigma_2\theta_2 \]

This is one linear equation in two unknowns \((\theta_1, \theta_2)\), so the solution set is a line in \(\mathbb{R}^2\) (an affine subspace of dimension 1).

(b) The market is incomplete: there is one traded risky asset but two independent sources of randomness (\(d = 2 > n = 1\)). The volatility matrix \(\Sigma = (\sigma_1, \sigma_2)\) is \(1 \times 2\) with rank 1, so the system \(\mu - r = \Sigma\boldsymbol{\theta}\) is underdetermined. By the Second Fundamental Theorem, the equivalent martingale measure is not unique, confirming incompleteness.

(c) Choose \(\theta_1 = 0\), giving \(\theta_2 = (\mu - r)/\sigma_2\). Alternatively, choose \(\theta_2 = 0\), giving \(\theta_1 = (\mu - r)/\sigma_1\). Both satisfy the risk premium equation, so both define valid ELMMs.

For a claim \(\Phi(W_T^2)\): under the first measure (\(\theta_1 = 0\), \(\theta_2 = (\mu-r)/\sigma_2\)), the risk-neutral \(W^{2,\mathbb{Q}}\) has a nontrivial Girsanov shift, so \(W_T^{2,\mathbb{P}} = W_T^{2,\mathbb{Q}} - \theta_2 T\) and the price depends on \(\theta_2\). Under the second measure (\(\theta_1 = (\mu-r)/\sigma_1\), \(\theta_2 = 0\)), the second Brownian motion is unchanged: \(W^{2,\mathbb{Q}} = W^{2,\mathbb{P}}\), and the claim is priced using the original distribution of \(W_T^2\).

These two prices differ because \(\Phi(W_T^2)\) has exposure to the non-traded factor \(W^2\), and each ELMM reweights this factor differently. This is the pricing ambiguity inherent in incomplete markets.

Exercise 7. Explain why the admissibility condition \(G_t(\phi) \geq -M\) is necessary to rule out arbitrage. Consider the following "doubling strategy" in discrete time: at each step \(k\), bet \(2^k\) on a fair coin flip. If you win, stop. If you lose, double the bet.

(a) Show that this strategy has zero initial cost, always eventually wins, and produces a guaranteed profit of 1 unit.

(b) Explain why the gains process \(G_t\) of this strategy is unbounded below.

(c) Relate this to the admissibility requirement in continuous time: why does the supermartingale argument from the text fail without a lower bound on \(G_t\)?

Solution to Exercise 7

(a) At step \(k\), the bet is \(2^k\) on a fair coin. If the first win occurs at step \(n\), the cumulative losses from steps \(0, \ldots, n-1\) are \(1 + 2 + \cdots + 2^{n-1} = 2^n - 1\). The win at step \(n\) pays \(2^n\). Net profit: \(2^n - (2^n - 1) = 1\). Since the coin is fair and independent, the probability of never winning is \(\lim_{n \to \infty} (1/2)^n = 0\). The strategy eventually wins with probability 1, costing nothing to enter and guaranteeing a profit of 1. This is an apparent arbitrage.

(b) Before winning, the cumulative loss after \(n\) steps is \(-(2^n - 1)\). This is unbounded: \(G_n \leq -(2^n - 1) \to -\infty\) as \(n \to \infty\). There is no finite \(M\) such that \(G_t \geq -M\) for all \(t\). The strategy requires the ability to borrow arbitrary amounts.

(c) The supermartingale argument in the text proceeds as follows: if \(G_t \geq -M\), then \(H_t = G_t + M \geq 0\) is a non-negative local martingale, hence a supermartingale, and \(\mathbb{E}[G_T] \leq 0\). This rules out \(G_T \geq 0\) with \(\mathbb{P}(G_T > 0) > 0\).

Without admissibility, we cannot form the non-negative process \(H_t\). The gains process \(G_t\) is a local martingale (under the fair-coin measure), but since it is not bounded below, it is not a supermartingale: \(\mathbb{E}[G_T]\) may not be \(\leq 0\), and in fact the strategy achieves \(G_T = 1\) a.s. The supermartingale inequality fails precisely because the unbounded losses allow probability mass to leak through the negative tail, circumventing the no-arbitrage conclusion. Admissibility is therefore not a technical nicety but an economic necessity: it restricts attention to strategies with bounded credit exposure.