Chapter 5 Risk-Neutral Pricing and Martingales
5.1 Martingales: Fair Games
A stochastic process \(M(t)\) is a martingale if:
\[E[M(t) \mid \mathcal{F}_s] = M(s) \quad \text{for all } t \geq s\]
where \(\mathcal{F}_s\) represents all information up to time \(s\).
Intuitive meaning: The best prediction of tomorrow’s value is today’s value. No drift, just random fluctuations. It’s a “fair game” - you can’t make money on average.
Key property: If \(M(t)\) is a martingale, then \(E[M(t)] = E[M(0)]\) for all \(t\).
5.2 The Fundamental Theorem of Asset Pricing
This is the nuclear bomb of mathematical finance:
Theorem: A market is arbitrage-free if and only if there exists a probability measure \(\mathbb{Q}\) (the risk-neutral measure) under which all discounted asset prices are martingales.
Let me unpack this carefully.
5.2.1 What’s Arbitrage?
An arbitrage is a “free lunch” - a trading strategy that: 1. Costs nothing to set up 2. Never loses money 3. Sometimes makes money
Markets without arbitrage are called arbitrage-free. This is a minimal rationality assumption.
5.2.2 The Risk-Neutral Measure
In the real world (measure \(\mathbb{P}\)), a stock might follow:
\[dS = \mu S \, dt + \sigma S \, dB\]
where \(\mu\) is the real drift (could be 8%, 12%, whatever).
The discounted price is \(\tilde{S}(t) = e^{-rt}S(t)\) where \(r\) is the risk-free rate.
By Itô’s lemma: \[d\tilde{S} = \tilde{S}[(\mu - r)dt + \sigma \, dB]\]
This has drift \(\mu - r\). If \(\mu \neq r\), this is NOT a martingale under \(\mathbb{P}\).
The trick: Change to a different probability measure \(\mathbb{Q}\) where the drift vanishes!
5.2.3 Girsanov’s Theorem
Under \(\mathbb{Q}\), we define a new Brownian motion:
\[\tilde{B}(t) = B(t) + \frac{\mu - r}{\sigma}t\]
By Girsanov’s theorem (deep measure theory), \(\tilde{B}(t)\) is a Brownian motion under \(\mathbb{Q}\).
Then under \(\mathbb{Q}\): \[d\tilde{S} = \tilde{S}\sigma \, d\tilde{B}\]
No drift! The discounted stock price is a martingale under \(\mathbb{Q}\).
5.3 Risk-Neutral Pricing Formula
Since \(\tilde{S}(t) = e^{-rt}S(t)\) is a \(\mathbb{Q}\)-martingale:
\[e^{-rt}S(t) = E^{\mathbb{Q}}[e^{-rT}S(T) \mid \mathcal{F}_t]\]
For a derivative with payoff \(g(S(T))\) at time \(T\), no-arbitrage requires:
\[\boxed{V(t) = e^{-r(T-t)}E^{\mathbb{Q}}[g(S(T)) \mid S(t)]}\]
This is the universal pricing formula!
5.3.1 The Miracle
- The real drift \(\mu\) doesn’t appear!
- Risk preferences are already encoded in \(S(t)\)
- Only need: \(S(t)\), \(r\), \(\sigma\), and payoff \(g\)
Under \(\mathbb{Q}\), assets grow at rate \(r\) (not \(\mu\)):
\[dS = rS \, dt + \sigma S \, d\tilde{B}\]
This is why it’s called “risk-neutral” - it’s as if investors are indifferent to risk.
5.4 Why Does This Work?
The key insight: replication.
If you can replicate the derivative’s payoff using the underlying asset, then to avoid arbitrage, the derivative’s price must equal the replication cost.
The risk-neutral measure emerges from imposing no-arbitrage across all possible replication strategies.
5.5 Example: Forward Contracts
A forward contract obliges you to buy stock at price \(K\) at time \(T\). Payoff: \(S(T) - K\).
Price: \[V(t) = e^{-r(T-t)}E^{\mathbb{Q}}[S(T) - K]\] \[= e^{-r(T-t)}[E^{\mathbb{Q}}[S(T)] - K]\] \[= e^{-r(T-t)}[S(t)e^{r(T-t)} - K]\] \[= S(t) - Ke^{-r(T-t)}\]
The forward price (fair delivery price) is \(F = S(t)e^{r(T-t)}\).
5.6 Change of Numeraire
You can use any traded asset as the “numeraire” (unit of account).
If you use asset \(N(t)\) as numeraire, there exists a measure \(\mathbb{Q}^N\) under which all prices relative to \(N(t)\) are martingales:
\[\frac{S(t)}{N(t)} \text{ is a } \mathbb{Q}^N\text{-martingale}\]
Common choices: - Money market account: \(N(t) = e^{rt}\) → standard risk-neutral measure - Zero-coupon bond: Useful for interest rate derivatives - Stock itself: Useful for options on options
This technique is powerful for simplifying derivative pricing.
5.7 Market Price of Risk
The drift shift from \(\mathbb{P}\) to \(\mathbb{Q}\) is governed by the market price of risk \(\lambda\):
\[\lambda = \frac{\mu - r}{\sigma}\]
This measures the excess return per unit of volatility. Under \(\mathbb{Q}\):
\[d\tilde{B} = dB + \lambda \, dt\]
Different assets may have different market prices of risk. In a complete market, all risks are spanned by traded assets, so there’s a unique \(\mathbb{Q}\).
5.8 Incomplete Markets
If not all risks can be hedged (e.g., jumps, stochastic volatility without tradable volatility derivatives), the market is incomplete.
Then: - Multiple risk-neutral measures exist - No unique price for derivatives - Need to choose \(\mathbb{Q}\) based on additional criteria (e.g., minimize risk, match market prices)
This is a major research area in quantitative finance.
5.9 Summary
The martingale approach to pricing:
- Identify the source of randomness (Brownian motions)
- Find the risk-neutral measure (Girsanov theorem)
- Price as discounted expectation under \(\mathbb{Q}\)
This elegant framework unifies all of derivative pricing and connects probability theory to finance in a profound way.