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Quantitative finance often begins with randomness but ends with structure. That structure takes the form of partial differential equations (PDEs). These equations describe how the probability of future prices evolves. From stochastic processes, we arrive at PDEs. From PDEs, we find solutions that support option pricing, hedging, and risk management.

Taylor Series and the Idea of Local Approximation

Start with a function f(x). To understand its behavior near a point x_0, we can write:

This local approximation allows us to analyze nonlinear behavior by focusing on the first few derivatives. In finance, we apply this idea to value functions, payoff profiles, and density functions. The second-order term becomes essential when modeling variance, which reflects uncertainty.

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A Trinomial Random Walk

Before jumping into calculus, return to a discrete model. A simple way to represent price evolution is a trinomial tree. At each time step, the price moves up, down, or stays the same:

• Up: S→S⋅u

• Down: S→S⋅d

• No change: S→S

We assign probabilities p_u, p_d, and p_m. The sum equals 1. This random walk converges to a diffusion process as the step size shrinks. The convergence provides the intuition behind stochastic differential equations.

The key is to track the evolution of probability distributions. That leads us to transition density functions.

Transition Density Functions

Let X(t) be a stochastic process. The transition density p(x,t∣x0,t0) gives the probability density of X(t)=x given that X(t0)=x0. For a Markov process, this density contains all the information needed to understand future states.

The transition density evolves over time. Its dynamics can be expressed using a partial differential equation. That equation comes from the underlying stochastic process.

Our First Stochastic Differential Equation

Consider a basic stochastic differential equation (SDE):

Here, μ is the drift, σ is the diffusion, and W_t is a Wiener process. This equation describes how the state X_t changes over time under uncertainty.

Solving this SDE gives paths. But we often want something stronger: the evolution of the probability density across all possible paths. That brings us to the Fokker-Planck equation.

The Fokker-Planck Equation

The Fokker-Planck equation (also called the forward Kolmogorov equation) describes how the probability density p(x,t) evolves:

This PDE gives the transition density of the process. It shows how the drift and diffusion terms influence the spread of probability over time. You can use it to compute the likelihood of reaching a certain state or to evaluate expected values under uncertainty.

Solving PDEs with Similarity Reduction

Some PDEs admit closed-form solutions. Many do not. In those cases, we try to reduce the PDE to a simpler form. Similarity reduction transforms a PDE into an ordinary differential equation (ODE) by changing variables.

For example, suppose a PDE involves u(x,t). We define new variables:

Then we rewrite the PDE in terms of ξ and v. This often collapses the time and space derivatives into a single variable. The technique works best when the PDE has scaling symmetry. Many transition density PDEs — especially those derived from geometric Brownian motion — have this property.

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Why This Matters in Finance

Option pricing requires solving PDEs. Risk modeling requires transition densities. Term structure models of interest rates rest on solutions to SDEs. Every modern financial model touches this chain: stochastic process → transition density → PDE → solution.

The most famous application is the Black-Scholes equation. This backward PDE prices a derivative V(S,t) with terminal condition at maturity. The solution involves a transformation that reduces the PDE to the heat equation, then applies a similarity reduction to get a closed-form expression.

That same path — random walk, density, PDE, reduction — applies to many problems in finance. Transition densities describe where a price might go. PDEs describe how that distribution evolves. Solving the PDE gives actionable insights: prices, hedges, or risk measures.

Every stochastic model has an underlying density. That density evolves according to a PDE. Understanding and solving that PDE turns uncertainty into structure. In finance, we do not eliminate randomness. We work with it — through transition densities, differential equations, and boundary conditions.

With each step, the model becomes more precise. From trinomial trees to continuous processes. From random paths to density functions. From density to price.

import numpy as np
import matplotlib.pyplot as plt

# Set up consistent matplotlib style
plt.rcParams.update({
    "font.family": "serif",
    "axes.spines.top": False,
    "axes.spines.right": False
})

# Taylor expansion (2nd order)
def taylor_expand(f, df, d2f, x0, x):
    return f(x0) + df(x0) * (x - x0) + 0.5 * d2f(x0) * (x - x0)**2

# Trinomial tree simulation (one step)
def trinomial_step(S, u, d, m, pu, pd, pm):
    outcome = np.random.choice([u, m, d], p=[pu, pm, pd])
    return S * outcome

# Fokker-Planck for Ornstein-Uhlenbeck
def steady_state_ou(mu, theta, sigma, x_range):
    variance = sigma**2 / (2 * theta)
    return (1 / np.sqrt(2 * np.pi * variance)) * np.exp(-(x_range - mu)**2 / (2 * variance))

# Discretized Fokker-Planck evolution
def fokker_planck_evolution(mu, sigma, x, t, dx, dt, steps):
    p = np.exp(-x**2)  # Initial guess
    p /= np.sum(p) * dx
    for _ in range(steps):
        dpdx = np.gradient(p, dx)
        d2pdx2 = np.gradient(dpdx, dx)
        drift_term = -mu * dpdx
        diffusion_term = 0.5 * sigma**2 * d2pdx2
        p += dt * (drift_term + diffusion_term)
        p = np.maximum(p, 0)
        p /= np.sum(p) * dx  # Renormalize
    return p

# Define function and derivatives for Taylor expansion
f = np.sin
df = np.cos
d2f = lambda x: -np.sin(x)

x0 = 0
x_vals = np.linspace(-2, 2, 200)
approx = taylor_expand(f, df, d2f, x0, x_vals)

plt.figure(figsize=(10, 4))
plt.plot(x_vals, f(x_vals), label="f(x) = sin(x)")
plt.plot(x_vals, approx, label="2nd-order Taylor Expansion", linestyle='--')
plt.title("Taylor Series Expansion")
plt.xlabel("x")
plt.ylabel("f(x)")
plt.grid(False)
plt.savefig("taylor_series.png")
plt.show()

# Simulate one path of trinomial tree
S = 100
u, d, m = 1.1, 0.9, 1.0
pu, pd, pm = 0.3, 0.3, 0.4
steps = 100
prices = [S]
for _ in range(steps):
    S = trinomial_step(S, u, d, m, pu, pd, pm)
    prices.append(S)

plt.figure(figsize=(10, 4))
plt.plot(prices, label="Trinomial Random Walk")
plt.xlabel("Step")
plt.ylabel("Price")
plt.title("Simulated Trinomial Price Walk")
plt.grid(False)
plt.savefig("trinomial_walk.png")
plt.show()

# Fokker-Planck approximation for Gaussian diffusion
x = np.linspace(-5, 5, 500)
dx = x[1] - x[0]
dt = 0.001
p = fokker_planck_evolution(mu=0.0, sigma=1.0, x=x, t=0, dx=dx, dt=dt, steps=100)

plt.figure(figsize=(10, 4))
plt.plot(x, p, label="Approx. Transition Density")
plt.plot(x, steady_state_ou(0, 1, 1, x), label="Steady State (Ornstein-Uhlenbeck)", linestyle='--')
plt.title("Fokker-Planck Evolution and Steady State")
plt.xlabel("x")
plt.ylabel("Probability Density")
plt.grid(False)
plt.savefig("fokker_planck.png")
plt.show()