Actuarial Notes

Martingale

Study guide: Exam INV-201

A martingale is a process X_t whose best forecast of any future value, given everything known today, is today's value: a fair game with no drift. In derivative pricing, the central result is that the price of every traded asset, measured in units of a chosen numeraire, is a martingale under the measure that goes with that numeraire. With the money-market account as numeraire, that measure is the risk-neutral measure \mathbb{Q}, and pricing a derivative reduces to taking an expectation.

E\left[X_t \mid \mathcal{F}_s\right] = X_s, \quad s \le t

dX_t = \sigma_t\,dW_t \;\Longrightarrow\; X_t \text{ is a martingale}

\frac{f_t}{g_t} = E^{\mathbb{Q}_g}\!\left[\frac{f_T}{g_T}\,\Big|\,\mathcal{F}_t\right]

  • Definition: \mathcal{F}_s is the information available at time s (the history of the process up to s). A martingale must also be adapted (its value is known at each time) and integrable (E|X_t| < \infty). The first block uses conditional expectation, and taking expectations of both sides gives E[X_t] = X_0, so a martingale's mean never changes.
  • Sub- and supermartingales: If E[X_t \mid \mathcal{F}_s] \ge X_s, the process is a submartingale and drifts up on average. If E[X_t \mid \mathcal{F}_s] \le X_s, it is a supermartingale and drifts down. A stock with expected return \alpha > r, discounted at r, is a submartingale under the real-world measure \mathbb{P}.
  • Driftless Itō processes: An Itō process is a martingale when its dt term is zero (second block), provided E\int_0^T \sigma_t^2\,dt < \infty. To test a process, apply Itō's lemma and check whether the drift vanishes, as Hull does.
  • Equivalent martingale measure: Hull's central result is the third block. For any traded security f and any traded numeraire g > 0 there is a measure \mathbb{Q}_g under which f/g is a martingale. With g_t = e^{rt} it says that e^{-rt}f_t is a \mathbb{Q}-martingale, so f_0 = e^{-rT}E^{\mathbb{Q}}[f_T].
  • Which prices are martingales: Only the prices of traded, self-financing assets are martingales after normalization. A dividend-paying stock's price S_t is not, but the value of a fund that reinvests the dividends, S_te^{\delta t}, is. An interest rate, a volatility or an index level is not a traded price, so it need not be a martingale under any pricing measure.
  • Why it matters: A martingale price process rules out arbitrage, because no trading strategy can earn a positive expected gain in normalized units from a zero start. Conversely, the fundamental theorem of asset pricing says that no arbitrage implies such a measure exists. Changing the measure from \mathbb{P} to \mathbb{Q} is done with Girsanov's theorem.

Showing that W Squared Minus t Is a Martingale Let W_t be a standard Brownian motion and M_t = W_t^2 - t. Show that M_t is a martingale, first from the definition and then with Itō's lemma.

Answer For s < t, write W_t = W_s + D, where D = W_t - W_s \sim N(0,\,t-s) is independent of \mathcal{F}_s. Then E[D] = 0 and E[D^2] = t - s.

\begin{align*}
E\left[W_t^2 \mid \mathcal{F}_s\right] &= E\left[W_s^2 + 2W_sD + D^2 \mid \mathcal{F}_s\right] \\
&= W_s^2 + 2W_s(0) + (t - s) \\
E\left[M_t \mid \mathcal{F}_s\right] &= W_s^2 + (t - s) - t \\
&= W_s^2 - s \\
&= M_s
\end{align*}

With Itō's lemma on G = W^2 - t, where G_t = -1, G_x = 2W and G_{xx} = 2, the drift is -1 + \tfrac12(2) = 0, so dM_t = 2W_t\,dW_t has no dt term. W_t^2 alone is a submartingale, since its mean t grows, and subtracting t removes that drift exactly.

Is the Discounted Stock Price a Martingale Under P or Q? A non-dividend stock follows dS_t = \alpha S_t\,dt + \sigma S_t\,dW^{\mathbb{P}}_t with \alpha = 0.09, \sigma = 0.20 and S_0 = 100, and the risk-free rate is r = 0.04. Find the dynamics of Y_t = e^{-rt}S_t, and say under which measure it is a martingale. Compute E[Y_1] under each measure.

Answer By the product rule, d\left(e^{-rt}S_t\right) = -re^{-rt}S_t\,dt + e^{-rt}\,dS_t. Under \mathbb{P} the drift does not vanish.

\begin{align*}
dY_t &= (\alpha - r)\,Y_t\,dt + \sigma\,Y_t\,dW^{\mathbb{P}}_t \\
E^{\mathbb{P}}[Y_1] &= 100\,e^{(0.09 - 0.04)(1)} \\
&= 105.13
\end{align*}

Under \mathbb{P}, Y_t is a submartingale that grows at the risk premium of 5%. Under \mathbb{Q} the stock drifts at r, so dY_t = \sigma Y_t\,dW^{\mathbb{Q}}_t and E^{\mathbb{Q}}[Y_1] = 100 = Y_0. The discounted price is a martingale under \mathbb{Q} only, which is why prices are expectations under \mathbb{Q} and not under \mathbb{P}.

Testing Three Candidate Processes Which of X_t = W_t^3, Y_t = W_t^3 - 3tW_t and Z_t = \exp\left(\sigma W_t - \tfrac12\sigma^2 t\right) are martingales?

Answer Apply Itō's lemma to each and read off the drift.

\begin{align*}
dX_t &= 3W_t\,dt + 3W_t^2\,dW_t \\
dY_t &= \left(-3W_t + 3W_t\right)dt + \left(3W_t^2 - 3t\right)dW_t \\
&= \left(3W_t^2 - 3t\right)dW_t \\
dZ_t &= \left(-\tfrac12\sigma^2 + \tfrac12\sigma^2\right)Z_t\,dt + \sigma Z_t\,dW_t \\
&= \sigma Z_t\,dW_t
\end{align*}

X_t has drift 3W_t, so it is not a martingale, while Y_t and Z_t are driftless and are martingales. Z_t is the exponential martingale that Girsanov's theorem uses as a change-of-measure density.

How it appears on the exam

  • INV-201: The syllabus lists martingales among the stochastic calculus techniques for pricing derivatives, and its next objective asks candidates to apply the risk-neutral measure, the forward measure and normalization. The readings are Hull's chapter on martingales and measures and the Chin, Nel and Olafsson problems.
  • Proof questions: A question asks candidates to show that a given process, such as W_t^2 - t, e^{\sigma W_t - \sigma^2t/2} or a discounted price, is or is not a martingale, by conditional expectation or by Itō's lemma.
  • Choosing the measure: Questions ask which normalized prices are martingales under which measure, such as a forward price under the forward measure, and then use that to price a derivative as an expectation.
  • Written answers: Candidates may need to explain why the real-world drift drops out of a price, or why a non-traded variable such as the short rate is not a martingale under \mathbb{Q}.

Exam traps

  • Constant mean is not enough: E[X_t] = X_0 for all t does not make X_t a martingale. The definition conditions on \mathcal{F}_s, and W_t^3 has mean zero but is not a martingale.
  • Forgetting the dividend: For a stock with dividend yield \delta, e^{-rt}S_t has \mathbb{Q}-drift -\delta. The martingale is e^{-(r-\delta)t}S_t, which is the reinvested fund value discounted at r.
  • The wrong measure: "The discounted price is a martingale" is true under \mathbb{Q}, not under \mathbb{P}. A question that gives \alpha is often testing whether you discount the real-world expectation at r, which is wrong.
  • Non-traded variables: The short rate r_t and the volatility \sigma_t are not prices, so a mean-reverting short rate is not a contradiction. Only f/g for a traded f and numeraire g must be a martingale.