Chapter 28: Martingales and Measures

The Market Price of Risk

When moving from the real world to the risk-neutral world, the expected growth rate of a traded asset changes from $\mu$ to $r$. The difference, adjusted for risk, is the market price of risk ($\lambda$):

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

For an asset that provides a return of $\mu$ with volatility $\sigma$, investors demand a risk premium of $\lambda$ per unit of risk. In a risk-neutral world, $\lambda =0$.

Extension to Dependence on Multiple Variables

When a derivative depends on $n$ state variables, each with its own market price of risk ${\lambda }_i$, the differential equation becomes:

$\frac{\partial f}{\partial t}+{\sum }_{i=1}^n\frac{\partial f}{\partial {\theta }_i}(m_i-{\lambda }_i{s}_i)+\frac{1}{2}{\sum }_{i=1}^n{\sum }_{j=1}^n\frac{{\partial }^{2}f}{\partial {\theta }_i\partial {\theta }_j}{\rho }_{ij}{s}_i{s}_j=rf$

where ${\theta }_i$ are state variables, $m_i$ are their real-world drifts, and $s_i$ are their volatilities. By setting ${\lambda }_i$ values appropriately, we can price in any risk-neutral world.

Several State Variables

Many derivatives depend on multiple variables (e.g., bond options depend on interest rates; quanto options depend on stock price and exchange rate). The multidimensional Itô's lemma generalizes the single-variable case:

$df=\left(\frac{\partial f}{\partial t}+{\sum }_i\frac{\partial f}{\partial x_i}{a}_i+\frac{1}{2}{\sum }_{i,j}\frac{{\partial }^{2}f}{\partial x_i\partial x_j}{b}_i{b}_j{\rho }_{ij}\right)dt+{\sum }_i\frac{\partial f}{\partial x_i}{b}_{i}d{z}_i$

Martingales

A martingale is a stochastic process with zero drift:

$E[X_{t+\Delta t}\mid {\mathcal{F}}_t]=X_t$

where ${\mathcal{F}}_t$ represents all information known at time $t$.

In a risk-neutral world, the discounted price of any traded security is a martingale:

$E[e^{-rt}{f}_t\mid {\mathcal{F}}_0]=f_0$

This fundamental property provides an alternative approach to pricing: find a measure under which discounted prices are martingales.

Equivalent Martingale Measure

The risk-neutral measure is an equivalent martingale measure — "equivalent" means it agrees with the real-world measure on events of zero probability. Multiple equivalent martingale measures can exist, especially when markets are incomplete (e.g., stochastic volatility).

Alternative Choices for the Numeraire

The numeraire is the unit in which values are expressed. A powerful result: if we can find a measure under which $f/g$ is a martingale (where $g$ is the numeraire price), then:

$\frac{f_0}{g_0}=E^g\left[\frac{f_T}{g_T}\right]$

where $E^g[\cdot ]$ denotes expectation under the measure associated with numeraire $g$.

Common Numeraire Choices

Forward Risk-Adjusted Measure

Under the $T$-forward measure (numeraire = $P(t,T)$), the forward price is a martingale:

$F_0=E^T[S_T]$

This is particularly useful for interest rate derivatives because discounting is moved outside the expectation.

Black's Model Revisited

Black's model can be interpreted using the forward risk-adjusted measure. The expected payoff under the forward measure is:

$c=P(0,T)\cdot E^T[\max (F_T-K,0)]$

Since $F_T$ is lognormal with volatility $\sigma$ under the forward measure, this yields Black's formula.

Option to Exchange One Asset for Another

Margrabe's formula (Chapter 26) can be elegantly derived using a change of numeraire. Under the measure where asset $U$ is the numeraire, we price the option to exchange $V$ for $U$. The ratio $V/U$ is lognormal under this measure.

Change of Numeraire

The Radon–Nikodym derivative connects different measures. For a change from measure $M$ to measure $N$, with numeraire $g$ to numeraire $h$:

$\frac{dN}{dM}=\frac{h_T/h_0}{g_T/g_0}$

This formalizes the machinery needed to switch between pricing measures and is a cornerstone of modern derivatives pricing theory.