Quant interviews are notoriously rigorous. Candidates are expected to demonstrate not just surface-level knowledge of financial models, but also a deep understanding of the underlying mathematics: stochastic calculus, partial differential equations (PDEs), measure theory, and numerical methods.

Among these topics, Forward and Backward Equations - also known as Kolmogorov Forward and Backward Equations - are absolutely foundational. They appear in pricing derivatives, modeling stochastic processes, building transition probabilities for Markov processes, and connecting SDEs to PDEs.

This article aims to provide a complete, detailed, and intuitive guide to Forward and Backward Equations, why they matter in quant finance, and exactly what interviewers expect you to know.

Table of Contents

1. Introduction

2. Core Mathematical Foundations

   • Stochastic Processes

   • Markov Property

   • Stochastic Differential Equations (SDEs)

   • Infinitesimal Generators

3. The Kolmogorov Forward (Fokker–Planck) Equation

   • Derivation

   • Interpretation

   • Examples

   • Real-World Applications

4. The Kolmogorov Backward Equation

   • Derivation

   • Conceptual Meaning

   • Relationship to Option Pricing

   • Examples

5. Forward vs. Backward: Key Differences

6. Connection to the Feynman–Kac Formula

7. Forward and Backward PDEs in Derivative Pricing

   • Black–Scholes Equation as a Backward Equation

   • Risk-Neutral Density as Forward Equation

   • Local Volatility Models

   • Stochastic Volatility Models

8. Using Forward and Backward Equations in Risk, Trading, and Modeling

9. Typical Quant Interview Questions and Solutions

10. Additional Advanced Concepts

11. Conclusion

1. Introduction

Forward and Backward Equations are two of the most important mathematical tools in modern quantitative finance.

You can think of them as two sides of the same coin:

• The Forward Equation describes how probability distributions evolve over time.

• The Backward Equation describes how expected values evolve backward from a terminal payoff.

Together, they form a bridge between:

• Stochastic processes (probabilities)

• Partial differential equations (deterministic mathematics)

• Derivative pricing models (finance)

• Transition densities used in Monte Carlo simulation

• Risk management applications like CVA, VaR, and scenario simulations

Understanding these equations - and how to apply them - is essential for quant interviews across trading, quant research, risk, and derivatives modeling.

2. Core Mathematical Foundations

Before diving into the equations, let’s build the foundational concepts.

2.1 Stochastic Processes

In quant finance, most assets (equities, FX, rates, commodities) are modeled as stochastic processes. A stochastic process is a collection of random variables indexed by time:

\(X_t, t≥0\)

In continuous-time models, \(X_t\) typically follows an SDE.

2.2 The Markov Property

A process is Markovian if:

\(P(Xt+h∣Ft)=P(Xt+h∣Xt)\)

This means the future only depends on the present - not the past. Most models used in quant finance (GBM, OU process, CIR process, Heston volatility) have the Markov property. The Markov property leads directly to the Kolmogorov equations.

2.3 Stochastic Differential Equations (SDEs)

The general Itô diffusion used in quant finance is:

\(dXt=μ(Xt,t) dt+σ(Xt,t) dWt​\)

where:

• μ = drift

• σ = volatility

• Wt = Brownian motion

A huge number of real models fit in this form.

2.4 Infinitesimal Generator

The infinitesimal generator \(\mathcal{L}\) of the process is a differential operator:

\(\mathcal{L}f(x,t)= \mu(x,t)\frac{\partial f}{\partial x} + \frac{1}{2}\sigma^2(x,t)\frac{\partial^2 f}{\partial x^2}\)

Think of the generator as the “engine” behind both Forward and Backward Equations.

3. The Kolmogorov Forward Equation (Fokker–Planck Equation)

✔ What it describes:

The evolution of the probability density function (pdf) over time.

If the density of \(X_t\)​ is \(p(x,t)\), then the Forward Equation is:

\(\frac{\partial p(x,t)}{\partial t} = -\frac{\partial}{\partial x}(\mu(x,t)p(x,t)) +\frac{1}{2}\frac{\partial^2}{\partial x^2}(\sigma^2(x,t)p(x,t))\)

This is also known as:

• Fokker–Planck Equation

• Kolmogorov Forward Equation

3.1 Intuition

The forward equation tells us:

“Given today’s distribution of the asset, how will the distribution look a moment later?”

It is forward in time, starting from an initial distribution.

3.2 When is it used?

It is strongly used in:

• Forecasting probability distributions

• Transition densities

• Calibration of local volatility models (Dupire)

• Implied volatility surface modeling

• Risk-neutral density estimation

• Filtering problems (Kalman, particle filters)

3.3 Example: Forward Equation for Geometric Brownian Motion (GBM)

GBM:

\(dS_t = \mu S_t dt + \sigma S_t dW_t.\)

Forward equation becomes:

\(\frac{\partial p}{\partial t} = -\frac{\partial}{\partial S}(\mu S p) +\frac{1}{2}\frac{\partial^2}{\partial S^2}(\sigma^2 S^2 p)\)

The solution to this PDE is lognormal density:

\(S_t \sim \text{Lognormal}(\ldots)\)

3.4 Real-World Applications

1. Dupire Local Volatility Model (Critically Important)

Dupire derived a forward PDE for option prices:

\(\frac{\partial C}{\partial T} = \frac{1}{2} \sigma_\text{loc}^2(K,T) K^2 \frac{\partial^2 C}{\partial K^2}\)

The entire local volatility surface comes from the forward equation.

This is a must-know interview question.

2. Risk Management - Distribution Forecasting

For VaR, you need the forward evolution of returns.

The Forward Equation helps transition probability densities over horizons.

3. Particle Filtering and Hidden Markov Models

Filtering problems use Fokker–Planck PDEs to propagate state distributions.

4. The Kolmogorov Backward Equation

✔ What it describes:

How expected values of functions evolve backward from maturity to the present.

Given a function:

\(u(x,t) = \mathbb{E}[g(X_T) \mid X_t = x]\)

the backward equation is:

\(\frac{\partial u}{\partial t} + \mathcal{L} u = 0\)

Where:

\(\mathcal{L}u = \mu(x,t) u_x + \frac{1}{2} \sigma^2(x,t) u_{xx}\)

Terminal condition:

\(u(x,T)=g(x)\)

4.1 Intuition

“Start from the terminal payoff and go backward in time to find today’s price.”

This is the heart of derivative pricing.

4.2 Example: Pricing a European Call

Payoff:

\(g(S_T) = \max(S_T - K, 0)\)

Then:

\(u(S,t) = \mathbb{E}^{\mathbb{Q}}\left[e^{-r(T-t)}g(S_T)\mid S_t=S\right]\)

Backward equation becomes:

\(\frac{\partial u}{\partial t} + rS\frac{\partial u}{\partial S} + \frac{1}{2}\sigma^2 S^2\frac{\partial^2 u}{\partial S^2} -ru = 0\)

This is exactly the Black-Scholes PDE.

4.3 Why is it called “backward”?

Because it starts from the terminal condition at t=T and moves backward to t=0.

4.4 Real-World Applications

• Pricing options and derivatives

• Solving PDEs for exotic payoffs

• CVA/DVA computation

• American option pricing (through extension)

• Interest rate models (Hull–White, CIR)

• Heston model pricing

Nearly every PDE-based pricer in finance uses the backward equation.

5. Forward vs. Backward: Key Differences

Most interviewers want you to explain these differences quickly and clearly.

6. Connection to the Feynman–Kac Formula

This formula connects:

• Backward PDEs

• Stochastic expectations

It states:

\(u(x,t) = \mathbb{E}\left[ e^{-\int_t^T r(s)\,ds} g(X_T) \mid X_t = x\right]\)

solves the PDE:

\(\frac{\partial u}{\partial t} + \mathcal{L}u - r u = 0\)

Feynman–Kac is extremely common in quant interviews, especially link between SDE and PDE.

7. Forward and Backward PDEs in Derivative Pricing

Let’s explore a few major applications.

7.1 Black–Scholes Equation is a Backward Equation

\(\frac{\partial V}{\partial t} + rS \frac{\partial V}{\partial S} + \frac{1}{2}\sigma^2S^2 \frac{\partial^2 V}{\partial S^2} - rV = 0\)

All PDE-based derivative pricing uses backward equations.

7.2 Risk-Neutral Density Uses Forward Equation

The evolution of the risk-neutral density p(S,t) uses the Forward Equation:

\(\frac{\partial p}{\partial t} = -\frac{\partial (r S p)}{\partial S} +\frac{1}{2}\frac{\partial^2 (\sigma^2 S^2 p)}{\partial S^2}\)

This is crucial for:

• implied volatility extraction

• non-parametric density estimation

• model calibration

7.3 Local Volatility: Dupire Forward PDE

Dupire showed:

\(\sigma_\text{loc}^2(K,T) = \frac{\frac{\partial C}{\partial T}} {\frac{1}{2}K^2 \frac{\partial^2 C}{\partial K^2}}\)

Every quant trader must know this.

7.4 Heston Model

The joint distribution (St,vt) evolves via a two-dimensional forward PDE.

Simultaneously, pricing uses a two-dimensional backward PDE.

8. Forward and Backward Equations in Risk, Trading, and Modeling

Risk Management

• CVA uses backward PDE

• Expected positive exposure uses forward density

Trading

• Volatility traders use Dupire forward PDE

• Market makers model forward density to quote skew

Model Validation

• Check the forward PDE to validate calibrated models

• Check backward PDE to validate pricers

Monte Carlo

• Backward PDE gives expected value

• Forward PDE gives transition density

9. Typical Quant Interview Questions and Solutions

Here are the types of questions you will almost certainly encounter.

Question 1: What is the difference between the Forward and Backward Kolmogorov Equations?

Solution Summary:

• Forward = evolves probability density

• Backward = evolves expectations

• Forward starts with initial condition

• Backward starts with terminal condition

• Forward is Fokker–Planck equation

• Backward leads to Black–Scholes

Question 2: Derive the Black–Scholes PDE from the Backward Equation.

The derivation is essentially applying the generator under the risk-neutral measure.

Question 3: Write the Forward Equation for GBM.

Already covered above.

Question 4: What is the Feynman–Kac formula?

Explain the link between stochastic expectation and backward PDE.

Question 5: Explain the role of Forward Equation in local volatility calibration.

Answer:

• The Dupire formula is derived from the forward equation for option price densities.

• It gives \(\sigma_{loc}(K,T)\)

Question 6: In Heston model, why is the forward equation 2D?

Because the joint density of \((S_t, v_t)\) is 2-dimensional.

Question 7: What PDE does a European call satisfy?

The backward Black–Scholes PDE.

Question 8: If you know a forward density, how do you compute the option price?

Integrate the payoff against the density:

\(C = e^{-rT} \int_0^\infty (S - K)^+ p(S,T)\,dS.\)

10. Additional Advanced Concepts

For senior quant interviews, you may also discuss:

1. Adjoint Methods

Used for sensitivity calculations; backward equation formulation.

2. Kolmogorov Equations in Jump Diffusions

Forward equation becomes a partial integro-differential equation (PIDE).

3. PDE Solvers

Finite difference methods rely on backward equations.

4. Long-Term Behavior

Stationary distributions from forward equations.

11. Conclusion

Forward and Backward Equations sit at the heart of modern quantitative finance. Whether you’re pricing a vanilla option, calibrating a local volatility model, building a risk-neutral density surface, or developing a new stochastic volatility model, these equations are fundamental.

In a quant interview, demonstrating mastery of:

• derivations

• intuitive meaning

• real-world relevance

• and examples

will set you apart from the competition.

If you're preparing for quant roles involving derivative pricing, stochastic processes, or financial engineering, understanding these equations is non-negotiable.

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