Stochastic calculus interview questions, and the toolkit that answers them

Reference article

Square a small move of size 0.01 and it becomes 0.0001, so ordinary calculus throws it away. A Brownian path makes so many small moves that their squares add up to the time elapsed, and they cannot be thrown away. Almost every stochastic calculus interview question tests whether you keep that one extra term. Keep it, and the questions turn into short calculations.

Stochastic calculus is calculus for quantities that move at random, such as a share price. In quantitative finance, it is the language of option pricing, so interviews for researcher, derivatives trader and pricing developer roles often include it. This page sorts the questions into five kinds, gives the six tools that answer them, and works four examples in full. Everything on this page is written in our own words; some set-ups are classic problems.

The five kinds of question

We sort the stochastic calculus questions we see in interview preparation books and in our own question bank into five kinds. This is our grouping, not a standard one.

1. Properties of Brownian motion. What is the distribution of $W_t$? Are two increments independent? What is the covariance of the path at two times?
2. Apply Ito's lemma. Write the dynamics of $f(W_t)$ or $f(t, S_t)$ for a given $f$.
3. Is it a martingale? Decide whether a process is a fair game, and fix it if it is not.
4. Moments of a stochastic integral. Find the mean and the variance of something like $\int_0^T W_t \, dW_t$.
5. Pricing. Derive the Black-Scholes equation, or change from the real-world measure to the risk-neutral measure.

Kinds 1 to 4 are the toolkit. Kind 5 puts the toolkit to work.

Tool 1: Brownian motion, in four properties

A standard Brownian motion $W$ starts at zero and has three more properties: its increments over time windows that do not overlap are independent; an increment over a window of length $t - s$ has the normal distribution with mean 0 and variance $t - s$; and the path is continuous. The full treatment, with the properties that surprise people, is on our free Brownian motion page.

Two facts follow and come up again and again:

• $W_t$ has mean $0$ and variance $t$, so its typical size is $\sqrt{t}$, not $t$.
• For $s \le t$, write $W_t = W_s + (W_t - W_s)$. The second part is independent of the first. This one split answers most questions about the path at two times.

Tool 2: $(dW)^2 = dt$

Cut $[0, T]$ into $n$ equal steps. Each squared increment has mean $T/n$. There are $n$ of them, so their sum has mean $T$. Its variance shrinks to zero as the steps shrink. So the sum of squared increments is not random in the limit: it is exactly $T$. This sum is called the quadratic variation.

In shorthand, $(dW)^2 = dt$. The other products vanish: $dt \cdot dW = 0$ and $(dt)^2 = 0$, because they shrink faster than $dt$. This multiplication table is the whole difference between ordinary calculus and stochastic calculus.

Tool 3: Ito's lemma

Let $X$ follow $dX_t = \mu_t \, dt + \sigma_t \, dW_t$, and let $f(t, x)$ be smooth. Expand $f$ to second order and apply the table above:

The term with $\tfrac12 \sigma_t^2$ is the one ordinary calculus does not have. It comes from $(dX)^2 = \sigma_t^2 \, dt$. The full statement and its proof sketch are on our Ito's lemma page.

For two processes, the product rule gains the same kind of extra term: $d(XY) = X \, dY + Y \, dX + dX \, dY$.

Tool 4: the zero-drift test for a martingale

A martingale is a process whose best forecast of its future value is its value now. For an Ito process the test is short: write its $dt$ term. If the $dt$ term is not zero, the process is not a martingale. If it is zero, the process is a local martingale, and it is a true martingale under a mild size condition, for example $\mathbb{E} \int_0^T \sigma_t^2 \, dt < \infty$.

Interviewers often want both halves: the drift calculation, and a sentence on the condition.

Tool 5: the Ito isometry

For a process $H$ that does not look into the future,

The stochastic integral also has mean zero under the same size condition. Together these two facts turn a variance question about an integral into an ordinary integral.

Tool 6: Girsanov, the change of drift

Girsanov's theorem says that you can change the probability weights so that a Brownian motion with a drift becomes one without a drift. Under the new weights, $\widetilde W_t = W_t + \theta t$ is a standard Brownian motion, for a constant $\theta$, when the new measure has density $\exp(-\theta W_T - \tfrac12 \theta^2 T)$ with respect to the old one on $[0, T]$. The volatility does not change. Only the drift does. Our Girsanov theorem page gives the general statement.

Four worked examples

These four are written by us to show the tools in use. Try each one before you read the answer.

Example 1. Is $W_t^2 - t$ a martingale?

Apply Ito's lemma to $f(x) = x^2$ with $X = W$: $f' = 2x$, $f'' = 2$, so $d(W_t^2) = 2 W_t \, dW_t + dt$. Then $d(W_t^2 - t) = 2 W_t \, dW_t$. The $dt$ term is zero. The size condition holds because $\mathbb{E}\int_0^T 4 W_t^2 \, dt = 2T^2$ is finite. So yes, it is a martingale. Note what this says about $W_t^2$ alone: its drift is $1$, so it is not one.

Example 2. Solve geometric Brownian motion.

A share price in the Black-Scholes model follows $dS_t = \mu S_t \, dt + \sigma S_t \, dW_t$. Take $f(x) = \ln x$, so $f' = 1/x$ and $f'' = -1/x^2$. Ito's lemma gives

so $S_T = S_0 \exp\left( (\mu - \tfrac12 \sigma^2) T + \sigma W_T \right)$. The $-\tfrac12\sigma^2$ is the Ito term at work. A candidate who writes $S_T = S_0 e^{\mu T + \sigma W_T}$ has dropped it, and in our view that is the slip to guard against first.

One consequence worth saying: the median of $S_T$ is $S_0 e^{(\mu - \sigma^2/2) T}$, because $W_T$ has median zero. The median sits below $S_0 e^{\mu T}$ whenever $\sigma > 0$.

Example 3. The mean and variance of $\int_0^T W_t \, dW_t$.

The mean is zero, because the integral is a stochastic integral and the size condition holds. For the variance, use the isometry: $\mathbb{E}\left[ \left(\int_0^T W_t \, dW_t\right)^2 \right] = \int_0^T \mathbb{E}[W_t^2] \, dt = \int_0^T t \, dt = T^2/2$.

Check it a second way. Example 1 gives $\int_0^T W_t \, dW_t = \tfrac12 (W_T^2 - T)$. Since $W_T^2 / T$ is chi-squared with one degree of freedom, $\operatorname{Var}(W_T^2) = 2T^2$, so the variance is $\tfrac14 \cdot 2T^2 = T^2/2$. Two routes agree, which is what an interviewer hopes to see.

Example 4. What is $P(W_1 > 0 \text{ and } W_2 > 0)$?

Use the split from Tool 1. $W_1$ and $W_2$ are jointly normal, each with mean zero, with $\operatorname{Cov}(W_1, W_2) = \operatorname{Var}(W_1) = 1$, so the correlation is $\rho = 1/\sqrt{2}$. For two standard normals with correlation $\rho$, the chance that both are positive is $\tfrac14 + \arcsin(\rho) / (2\pi)$. Here $\arcsin(1/\sqrt2) = \pi/4$, so the answer is $\tfrac14 + \tfrac18 = \tfrac38$.

A sanity check: if the two values were independent the answer would be $\tfrac14$. They are positively linked, so the answer must be above $\tfrac14$, and it is.

From the toolkit to pricing

The Black-Scholes equation is the toolkit in one argument. Hold one option worth $V(t, S)$ and sell $\Delta$ shares. Ito's lemma gives the change in this portfolio as

Choose $\Delta = \partial V / \partial S$ and the random part vanishes. A portfolio with no risk must earn the risk-free rate $r$, or there is a free profit. Setting the change equal to $r (V - \Delta S) \, dt$ gives

The real-world drift $\mu$ does not appear. Girsanov explains why: with $\theta = (\mu - r)/\sigma$, the share price follows $dS_t = r S_t \, dt + \sigma S_t \, d\widetilde W_t$ under the risk-neutral measure, and prices are expectations under that measure. Our Black-Scholes equation and risk-neutral valuation pages take this further.

The slips that cost the most

In our view, these five slips cost the most marks.

• Dropping the Ito term. Example 2 is the classic case.
• Treating zero mean as enough. A process can have mean zero at every time and still not be a martingale. Use the drift test, not the mean.
• Mixing up $t$ and $\sqrt{t}$. The variance of $W_t$ is $t$, so its standard deviation is $\sqrt{t}$.
• Forgetting independence across windows. Split the path at the earlier time, as in Tool 1.
• Answering with a formula and no reason. Say which tool you are using, and why it applies.

How to practise

1. Rebuild the six tools on paper, without notes, until each takes under a minute.
2. Redo the four examples above, saying each step out loud.
3. Work the module Brownian motion and Ito calculus in Thuztra Learn, which is our product. Its free pages are Brownian motion and this one; the rest of the module is not free. Two textbook companions are listed in it: Oksendal, Stochastic Differential Equations and Williams, Probability with Martingales.
4. For the probability that sits underneath all of this, see the best books for quantitative finance interviews.

Membership is a paid subscription: see what Thuztra Learn is and what it costs. Nothing on this page promises an interview result or a job. Education only, not advice.

In short

Learn six tools: the properties of Brownian motion, the rule that dW squared is dt, Ito's lemma, the zero-drift test for a martingale, the Ito isometry, and the change of drift from Girsanov. Then practise saying each step out loud, because the interviewer marks the reasoning as well as the answer.

Terms used here

• Quantitative finance
• Probability
• Risk-neutral measure
• Filtration
• Volatility

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Topics

• interview preparation
• stochastic calculus
• brownian motion
• ito calculus
• martingales

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