Free lesson · Differential equations

Partial differential equations and the bridge to current research

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Start with the idea

A partial differential equation relates partial derivatives across several independent variables.
A heat equation links change over time to curvature over space.
An initial condition and boundary conditions complete the problem; the PDE alone is not enough.

Symbols, units & horizon

u(x,t): temperature field
u_t: time partial
u_xx: second spatial partial
D: diffusion coefficient in distance squared per time
V(S,t): option value
S: spot price
V_S,V_SS: delta and gamma partials
σ: return volatility per square-root time
r: continuous risk-free rate
t: calendar time
g(S): specified terminal payoff at time T
V_t: time partial holding spot fixed
π: circle constant, approximately 3.14159
sin: sine function, with its argument in radians
x: spatial coordinate, dimensionless in the worked example

When and why to use this

Use PDEs to understand how local sensitivity terms form an entire evolution or pricing equation. Read the existing stochastic-calculus module next for the Itô and replication steps.

For heat diffusion on a rod, u(x,t) represents temperature and a positive coefficient D controls the diffusion rate.
Where the temperature profile curves upward, the local point gains heat relative to its neighbours.
Initial temperature describes the starting profile; endpoint conditions specify how the rod interacts with its environment.
Option-pricing PDEs reuse time and curvature derivatives under a different interpretation.
Under the no-dividend Black–Scholes assumptions, a delta hedge cancels the random spot term, while financing links the remaining deterministic change to the risk-free rate.
A payoff at expiry and suitable spatial boundary conditions complete that pricing problem.

u_{t} = D u_{x x}; V_{t} + \frac{1}{2} σ^{2} S^{2} V_{S S} + r S V_{S} - r V = 0

Calculus: derivation and arithmetic

Partial differential equations and the bridge to current research

For u=exp(−Dπ²t)sin(πx) on a dimensionless interval x∈[0,1], differentiate: u_t=−Dπ²u and u_xx=−π²u. Thus u_t=D u_xx, with zero endpoint temperatures and initial profile sin(πx).
For the pricing PDE, Itô gives dV=V_t dt+V_S dS+.5σ²S²V_SS dt. A local self-financing hedge holding one option and −V_S units of stock removes the dS term.
The hedged value V−S V_S earns r under the idealised financing assumptions. Equate V_t+.5σ²S²V_SS to r(V−S V_S) and rearrange. These replication assumptions are additional to the calculus.

Work it by hand

At x=.5,t=0,D=.1, temperature u=1, u_t=−.1π²≈−.98696 and D u_xx is the same. For a pricing check, V=S has V_S=1, V_SS=0,V_t=0, so the PDE residual is rS−rS=0.

An analogy to remember

Neighbouring points on a hot rod exchange heat. Curvature measures whether a point sits above or below its neighbours. The same differential operator can appear in a different model without the underlying economics becoming heat flow.

How this becomes a building block

The heat equation is a complete physical model only after its conditions are specified. The pricing PDE combines partial derivatives, stochastic dynamics and no-arbitrage assumptions. Modern solvers may approximate these objects with neural networks, but the derivative definitions and boundary conditions remain essential.

Research checkpoint · reviewed 11 September 2026

The definitions in this learning path are established mathematics. Current research helps show where these tools are being used and where implementation can fail. The two papers below are arXiv preprints; this review inspected their latest abstracts and version metadata, not their full experiments or code.

Differentiating a model with discontinuous payoffs. Glasserman and Karmarkar’s Differential ML with a Difference (version 2, 22 April 2026) reports that biased pathwise sensitivities for digital and barrier options can worsen differential-learning errors. The authors investigate likelihood-ratio labels and hybrid delta/gamma estimates. Scope: simulation-based derivatives pricing and risk; market sample dates are not stated in the abstract. This is evidence about estimator design in the reported experiments, not realised strategy returns.

Further reading: Glasserman & Karmarkar · arXiv:2512.05301v2 ↗

Solving systems with shared randomness. Antunes, Saporito and Jaimungal’s Deep Learning and Elicitability for McKean–Vlasov FBSDEs With Common Noise (version 2, 11 June 2026) combines iterative solving and neural approximations. Reported applications include an analytically solvable interbank systemic-risk model and an economic-growth model. These are model-based numerical studies; the abstract supplies no historical trading sample or after-cost investment results. Full experimental robustness has not been evaluated here.

Further reading: Antunes, Saporito & Jaimungal · arXiv:2512.14967v2 ↗

How to use this reading. Our teaching inference is to establish derivatives, conditional expectations and exact benchmark solutions before choosing a complex solver. Check smoothness and estimator bias, then approximation error and model assumptions. A newer method is a candidate to evaluate for a defined problem, rather than evidence of a universally best implementation.

Research sources, review dates and limitations

Extend the research question

Separate model instability from numerical instability. Freeze the horizon and inputs while refining the time step; document which output should converge.

Continue with the connected research module →

Python implementation

Self-contained teaching example. Python 3.10+; dependencies and input conventions are shown in the code and notation. Run in your own Python environment.

from math import exp,sin,pi

def heat_mode(x,time,diffusion):
    u=exp(-diffusion*pi*pi*time)*sin(pi*x)
    return u,-diffusion*pi*pi*u,-pi*pi*u

def pricing_pde_residual(value,time_partial,spot_partial,spot_second,spot,vol,rate):
    return time_partial+.5*vol*vol*spot*spot*spot_second+rate*spot*spot_partial-rate*value

print(heat_mode(.5,0,.1),pricing_pde_residual(100,0,1,0,100,.2,.05))

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