## Any of Jane Street’s main career paths can be pursued in New York, London, or Hong Kong.

Traders work in teams to seek out and take advantage of pricing inefficiencies, develop models, manage risk, investigate new products, and push into new business areas. Experienced traders mentor newer colleagues, whose responsibilities increase with their capability.

Ideal candidates will be:

• Excellent quantitatively, with a strong understanding of probability and statistics
• Effective communicators in close-knit team settings
• Motivated, competitive, and eager to learn and teach
• Excited to engage in impromptu and exploratory debate on trading strategy and risk

Previous experience or coursework in finance, business, or economics is not required. We’re more interested in how you think and learn than what you know.

#### Technology

The technology group is actively hiring in multiple different areas. For all of these, no previous experience or coursework in finance, business, or economics is required. We’re more interested in how you think and learn than what you already know.

##### Software Development

The software development team works closely with the rest of the firm, building tools, exploring trading ideas, and designing and maintaining the firm’s software systems.

For dev roles, you should have:

• Top-notch programming skills, with an interest in (but not necessarily experience with) functional programming languages.
• Deep experience with—and love for—technology. There’s no specific checklist; we use software to attack a variety of problems, so we’re interested in everything from machine learning to systems administration to programming language design.
• Strong interpersonal skills. Software development at Jane Street is highly collaborative, and we are looking for people who can work effectively in small, close-knit teams.
• A commitment to the practical. We produce production software on a continuous basis, affecting day-to-day operations in every area of the firm.

Summer 2018 Software Development internship applications are currently closed, though we are still accepting applications for the software development summer internship for Australasia-based students. For all other locations, please check back to apply for Summer 2019.

##### Information Technology and Networking

This group builds and maintains the firm’s technological nervous system. Group members keep our networks fast and robust, manage counterparty connectivity, and provide the glue to hold together the firm’s far-flung activities.

For information technology and networking roles, you should have:

• A love for automation. We don’t like solving the same problem manually over and over, and we generally try to program our way out of it.
• A careful approach. The systems we manage trade billions of dollars a day, so the stakes are high. We need people who can make quick decisions when necessary, but at the same time maintain a sense of humility about the limits of their own knowledge.
• Strong interpersonal skills. Jane Street is a highly collaborative place, and we are looking for people who can work effectively with people across the firm.
View open Software Development positions
View Open IT & Networking Positions

#### Quantitative Research

Researchers at Jane Street are responsible for building models, strategies, and systems that price and trade a variety of financial instruments. As a mix of the trading and software developer roles, this work involves many things: analyzing large datasets, building and testing models, creating new trading strategies, and writing the code that implements them.

Researchers at Jane Street care about:

• Deploying the right tool (mathematical, statistical, computational) for the job. We see ourselves as generalists.
• Writing quality production-level code. Researchers spend most of their day writing code.
• Thinking and communicating precisely and openly. We believe great solutions come from the interaction between diverse groups of people across the firm.
• Having a real impact on the firm by always striving to keep our work relevant.

We’re looking for:

• An ability to apply logical and mathematical thinking to all kinds of problems. Asking great questions is more important than knowing all the answers.
• A desire to write great code, mostly in OCaml. You don’t need functional programming experience, but you should want to learn.
• Good taste in quantitative research. The problems we work on rarely have clean definitive answers. You should be comfortable pushing in new and unknown directions while maintaining clarity of purpose.
• Smart and curious people–previous experience or coursework in finance, business, or economics is not required. We’re more interested in how you think and learn than what you know.

View Open Quantitative Research Positions

#### Institutional Services

Jane Street offers our unique liquidity and trade execution services to the largest institutional investors and asset managers in the world.  Roles in Institutional Services are generally client-facing to some degree.

Specific skill sets for each Institutional Services position will vary, but all include strong capital markets knowledge, effective communication skills, and a desire to build and nurture client relationships.

Positions include:

• Institutional Marketing
View Open Institutional Services Positions

#### Infrastructure

Our infrastructure areas are key to the firm’s success. Roles in these areas offer endless opportunities to contribute directly and tangibly to our business. Everyone at Jane Street sits together on one trading floor, collaborating closely with each other to ensure that the firm’s critical functions operate efficiently while at the same time being constantly improved.

The specific skill set for each infrastructure position varies, but all include strong organizational, interpersonal and communication skills, detail orientation, and the desire to work in a team-based environment.

Positions include:

• Business Development: documenting and solving problems related to the implementation of complex trading strategies within regulatory rules and under operational and technological constraints
• Operations: P&L reconciliation, clearing & settlements (some other financial institutions describe this as back office)
• Finance: accounting, regulatory capital finance, accounts payable/receivable
• Tax
• Human Resources
• Recruiting
• Legal & Compliance
View Open Infrastructure Positions
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### Solution

$$\def\V{\mathrm{Var}}$$

Overview: the self-symmetry of the Sierpinski triangle leads to a linear equation that the covariance matrix $$\Sigma$$ must satisfy; specifically, $$\Sigma = \tfrac{1}{4}\Sigma + \tfrac{1}{24}I$$, which gives $$\fbox{\(\Sigma = \tfrac{1}{18}I$$}\).

Without rigorously defining the uniform distribution over the Sierpinski triangle, we can at least recognize that there are certain properties it should have. The first relevant property is that it's invariant under rotation by 120°; it follows that $$\Sigma = kI$$ for some $$k$$. (Consider: The ellipse $$\mathbf{x}^*\Sigma^{-1} \mathbf{x} = 1$$ will be invariant under this rotation, and the only ellipses invariant under this rotation are circles.) We have $$k = \Sigma_{11} = \V(X_1)$$, so we have reduced the problem to finding $$\V(X_1)$$.

The next relevant property of the distribution is that it's equal to a mixture of 3 equally weighted copies of itself, with each copy scaled by 1/2 and translated by $$(1/4,\text{something}),(-1/4,\text{something})$$, or $$(0,\text{something})$$. (The somethings are respectively $$-\sqrt{3}/12,-\sqrt{3}/12,\sqrt{3}/6$$, but we don't need them.) In particular, if $$T$$ is uniform over $$\{-1/4,0,1/4\}$$ (and independent of $$X_1$$), then $$X_1$$ has the same distribution as $$\tfrac{1}{2}X_1 + T$$. We can directly compute $$\V(T) = 1/24$$, and we then have \begin{align} k = \V(X_1) &= \V(\tfrac{1}{2}X_1 + T)\\ &= \tfrac{1}{4}\V(X_1) + \V(T)\\ &= \tfrac{1}{4}k + 1/24, \end{align} and solving for $$k$$ now yields $$k = 1/18$$, so $$\Sigma = \tfrac{1}{18}I$$.

A similar approach is to write $$X_1 = \sum_{n=0}^\infty \frac{1}{2^n}T_n$$ where the $$T_n$$ are independent copies of $$T$$. Then $$\V(X_1) = \sum_{n=0}^\infty \V(\frac{1}{2^n}T_n) = \frac{1}{24} \sum_{n=0}^\infty \frac{1}{4^n} = 1/18$$.

CLOSE

### Solution

Let $$C⊆ℝ^3$$ be a closed bounded convex set. Let $$B = ∂C$$ be its boundary. Let $$D = B + B$$ = $$\{ p + q | p, q ∈ B \}$$. Prove that $$D$$ is convex.

Actually, let us start by considering the 2-dimensional case. Let $$E = \{ (p + q)/2 | p, q ∈ B \}$$; this is just a scaled down $$D$$, so it is enough to show that $$E$$ is convex. It suffices to show $$E = C. E ⊆ C$$ follows by the convexity of $$C$$. For $$C ⊆ E$$, take any point $$p ∈ C$$. If $$p ∈ B$$, then $$p = (p + p)/2 ∈ E$$. If $$p ∉ B$$, imagine drawing a vertical line through $$p$$, and then continuously rotating that line counterclockwise (about $$p$$) by 180◦. It will intersect $$B$$ in two points, one to either side of $$p$$. By the Intermediate Value Theorem, there will be some angle at which these two points are equidistant from $$p$$; then $$p$$ is the average of those other two points, so $$p ∈ E$$. We now have $$E = C$$, so $$E$$ is convex.

In 3 (or more dimensions) you can apply essentially the same argument. When you reach the $$p ∉ B$$ case, just pick any plane containing $$p$$, and use the above argument within that plane.

(In the case $$C = ∅$$, the above still applies, though some aspects are vacuous.)

If you landed here from the 3Blue1Brown video and are curious for more on Fourier, our talk, Echoes of Fourier is available above and on YouTube. If you’re interested in more puzzles, click here.

If you’re interested in working at Jane Street, click here to apply.

4 puzzles