Join us · open applications

Work on problems that have equations.

We are always looking for mathematicians, physicists, computer scientists, engineers and quants. What we look for is curiosity, and the rigour to follow it through.

x y z = (1 + \frac{v}{2} cos \frac{u}{2}) cos u = (1 + \frac{v}{2} cos \frac{u}{2}) sin u = \frac{v}{2} sin \frac{u}{2}

u, v = mgrid[0:2*π:160j, -1:1:12j]
x = (1 + v/2 * cos(u/2)) * cos(u)
y = (1 + v/2 * cos(u/2)) * sin(u)
z = v/2 * sin(u/2)      # one side, one edge

Apply Read the research

the centre line
one edge, going round twice
u/2: the half-twist
research
engineering
two sides? Keep walking: there is only one
Listing and Möbius found it apart, in 1858; the name went to Möbius
cut it along the centre line: still one piece, twice as long, with two full twists
click the strip: an ant walks the centre line; one lap turns it upside down

§1 What we ask of a model

Model first. Then data. Then code.

Before any code: the variables, the assumptions and what would prove the model wrong. A model earns its place by surviving new data; most do not, and knowing why is part of the work. What survives becomes software. AION is the first example.

β^X^{T} X β^= β arg min i = 1 \sum n (y_{i} - x_{i}^{T} β)^{2} = X^{T} y

β, *_ = lstsq(X, y, rcond=None)
residuals = y - X @ β
print(residuals.std())
# the model, with its error

the model: one plane
a residual: what it can’t explain
(0.06, 0.40, −0.24), from 800 points
it prints 0.1787. We put in 0.18
Legendre, 1805. Gauss said he had it in 1795
why squares: the total error is a smooth bowl, and its lowest point solves XᵀXβ = Xᵀy
Gauss, 1809: with Gaussian noise, least squares is also the most likely fit
Ceres, lost in the Sun’s glare in 1801, was found where Gauss’s fit put it

§2 A problem for the margin

Why is this the heat equation?

Equation (1) prices an option V(S, t) on an asset with price S, volatility σ and risk-free rate r. A change of variables turns it into the equation of heat spreading along a rod. Which one, and why should it exist at all? Try it before you scroll.

\frac{\partial V}{\partial t} + \frac{1}{2} σ^{2} S^{2} \frac{\partial ^{2} V}{\partial S ^{2}} + r S \frac{\partial V}{\partial S} - r V = 0 (1)

v = σ*sqrt(τ)
d1 = (log(S/K) + (r + σ**2/2)*τ) / v
d2 = d1 - v
C = S*norm.cdf(d1) - K*exp(-r*τ)*norm.cdf(d2)
# (1), solved. Now: why is it the heat equation?

K: the kink at expiry
a diffusion coefficient, in disguise
x = S/K (crossed out): x = ln(S/K)
further from expiry: smoother
this margin is too narrow. The next screen isn’t
S = K = 100, a year from expiry: (1) prices the call at 14.23 (σ 30%, r 5%)
no μ in (1): the price does not depend on how fast the stock is expected to grow
Black and Scholes, 1973, with Merton the same year; the Nobel came in 1997, after Black’s death in 1995
1 · x = ln(S/K)
2 · τ = ½σ²(T − t)
3 · V = K e^(αx+βτ) u

§2 The answer

Heat, spreading along a rod.

Write x = ln(S/K) and measure time backwards from expiry, in units of variance: τ = ½σ²(T − t). Take out an exponential factor, V = K e^{αx + βτ}u. The kink of the payoff then spreads out exactly as heat would: what is left is u_τ = u_xx, Fourier’s equation of 1822.

u_{τ} = u_{xx} x = ln \frac{S}{K}, τ = \frac{1}{2} σ^{2} (T - t), V = K e^{α x + β τ} u (2)

k = 2*r / σ**2
α, β = -(k - 1)/2, -(k + 1)**2/4
V = lambda x, τ: K * exp(α*x + β*τ) * u(x, τ)
# u solves (2): the kink at K, smoothed

(1) ≡ (2), in other coordinates
τ = 0: a pulse of heat
later: lower, wider, smooth
time to expiry, rescaled
had it before scrolling? Say so when you apply
the heat kernel: u at time τ is u at 0 smoothed by a Gaussian of variance 2τ
Bachelier, 1900: prices as a random walk, five years before Einstein’s Brownian motion

§3 Who we look for

Curious, and rigorous.

Mathematics, physics, software engineering: the field matters less than the habit of stating a problem exactly, testing the answer and building it. Today most of our problems come from quantitative finance; they do not stop there.

r ¨_{i} = j \neq = i \sum G m_{j} \frac{r _{j} - r _{i}}{∣ r _{j} - r _{i} ∣ ^{3}}

def accel(r, G=1, m=1):
    a = zeros_like(r)
    for i, j in permutations(range(3), 2):
        d = r[j] - r[i]
        a[i] += G * m * d / norm(d)**3
    return a        # three bodies, one path

everyone pulls on everyone
three equal masses, one shared path
time runs up
its shadow: a figure eight
Moore found it in 1993; the proof came in 2000
some problems wait for the right people
a choreography: all three chase round the same eight, a third of a period apart
Chenciner and Montgomery proved it by minimising the action over paths like this

§4 Apply

Tell us about yourself.

An open application: what you have worked on, and what you would like to work on. Links to a CV, code or papers help.

Prefer email? Write to info@withequilibrium.com.

no answer to §2 required
links beat adjectives
there’s a riddle in the console. Optional, obviously