PDE Pricing

Hey students! 👋 Ready to dive into one of the most fascinating intersections of mathematics and finance? Today we're exploring how partial differential equations (PDEs) help us price financial derivatives like options. You'll discover how mathematical tools originally developed for physics problems became the backbone of modern financial markets, learn about the famous Black-Scholes equation that revolutionized trading, and understand the beautiful connection between PDEs and probability through the Feynman-Kac theorem. By the end of this lesson, you'll see how elegant mathematics transforms complex financial problems into solvable equations! 📈

The Foundation: Why PDEs in Finance?

Imagine you're trying to figure out the fair price of a stock option - a contract that gives you the right to buy Apple stock at $150 in three months, even if it's trading at $200 then! 🍎 This seems like a gambling problem, but mathematicians discovered something amazing: option prices follow predictable patterns that can be described using partial differential equations.

The breakthrough came in 1973 when Fischer Black, Myron Scholes, and Robert Merton showed that under certain assumptions, the price of an option must satisfy a specific PDE. This wasn't just theoretical - it gave traders a concrete formula to calculate fair prices, revolutionizing financial markets overnight!

Here's the intuition: an option's value depends on multiple changing variables - the current stock price, time remaining until expiration, interest rates, and volatility. When all these factors change simultaneously, the option price changes in a way that follows mathematical laws, just like how heat spreads through a metal rod or how waves propagate through water.

The key insight is risk-neutral pricing. In a world where everyone is indifferent to risk, the expected return on any investment equals the risk-free rate. This assumption, combined with the ability to continuously trade and hedge, leads directly to PDEs that govern option prices.

The Black-Scholes PDE: The Crown Jewel

The most famous PDE in mathematical finance is the Black-Scholes equation. For a European option with value $V(S,t)$ depending on stock price $S$ and time $t$, it looks like this:

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

Don't let the symbols intimidate you, students! Let's break this down:

• $V(S,t)$ is the option value
• $\sigma$ is the stock's volatility (how much it jumps around)
• $r$ is the risk-free interest rate
• The partial derivatives represent how the option value changes

This equation tells us that the rate of change in option value over time, plus the effect of stock price movements (including the "gamma" effect from the second derivative), plus the drift from interest rates, must equal the interest we could earn on the option's value.

Real-world example: When Tesla's stock was highly volatile in 2021 (high $\sigma$), Tesla options were expensive because the $\frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2}$ term was large. The equation perfectly captured why volatile stocks have pricier options! ⚡

The beauty is in the boundary conditions. For a call option with strike price $K$ expiring at time $T$:

• At expiration: $V(S,T) = \max(S-K, 0)$
• If stock price goes to zero: $V(0,t) = 0$
• If stock price goes to infinity: $V(S,t) \approx S - Ke^{-r(T-t)}$

The Feynman-Kac Connection: PDEs Meet Probability

Here's where things get really cool, students! 🎯 The Feynman-Kac theorem creates a bridge between the deterministic world of PDEs and the probabilistic world of random stock movements. It's like having a universal translator between two mathematical languages!

The theorem states that if we have a PDE like:

$$\frac{\partial u}{\partial t} + \frac{1}{2}\sigma^2 \frac{\partial^2 u}{\partial x^2} + \mu \frac{\partial u}{\partial x} - ru = 0$$

Then the solution can be written as an expectation:

$$u(x,t) = E\left[e^{-r(T-t)} g(X_T) | X_t = x\right]$$

where $X_t$ follows a specific stochastic process and $g$ is the boundary condition.

In finance terms, this means: The option price equals the expected discounted payoff under risk-neutral probability! This is profound because it connects the PDE approach with Monte Carlo simulation methods. You can either solve the differential equation or run millions of random stock price simulations - both give the same answer!

Consider pricing a barrier option that becomes worthless if the stock ever drops below $80. Using Feynman-Kac, we can either solve a complex PDE with moving boundaries or simulate thousands of stock paths and count how many hit the barrier. Financial engineers use both approaches depending on the problem complexity.

Boundary Conditions: The Rules of the Game

Boundary conditions in financial PDEs aren't just mathematical technicalities - they represent real economic constraints! 💼 Think of them as the "rules of the game" that define what happens in extreme situations.

For American options (exercisable anytime), we get free boundary problems. The option holder can exercise whenever it's optimal, creating a moving boundary where $V(S,t) = \max(S-K, 0)$ for calls. This optimal exercise boundary satisfies additional conditions - it's smooth (no arbitrage opportunities) and satisfies the "smooth pasting" condition where $\frac{\partial V}{\partial S} = 1$ at the boundary.

Exotic options create fascinating boundary conditions. Asian options (payoff depends on average stock price) lead to PDEs in multiple dimensions. Lookback options (payoff depends on maximum stock price reached) create PDEs where one variable only increases. Barrier options have boundaries where the option value jumps to zero.

Real trading example: During the 2008 financial crisis, many structured products had complex barrier features. Banks used sophisticated PDE solvers to price these instruments, but when correlations broke down and boundaries were hit simultaneously across multiple assets, the models failed spectacularly. This shows both the power and limitations of PDE pricing! 📉

Advanced Applications: Beyond Black-Scholes

Modern financial markets demand more sophisticated PDEs, students! The basic Black-Scholes model assumes constant volatility and interest rates, but reality is messier. 🌪️

Stochastic volatility models like Heston lead to PDEs in two dimensions:

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

Here, volatility $v$ itself follows a random process, creating a cross-derivative term that captures correlation between stock moves and volatility changes.

Jump-diffusion models add discontinuous price movements, leading to partial integro-differential equations (PIDEs):

$$\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV + \lambda \int_{-\infty}^{\infty} [V(Se^y,t) - V(S,t)] f(y) dy = 0$$

The integral term captures sudden jumps in stock price with intensity $\lambda$ and jump size distribution $f(y)$.

Credit derivatives use intensity-based models where default risk creates additional PDE terms. Interest rate derivatives often require solving PDEs on curved surfaces representing the yield curve evolution.

Conclusion

We've journeyed through the elegant world of PDE pricing, students! You've seen how partial differential equations transform the seemingly random world of financial markets into mathematically tractable problems. The Black-Scholes PDE revolutionized finance by providing the first rigorous option pricing framework, while the Feynman-Kac theorem beautifully connects deterministic PDEs with probabilistic expectations. Boundary conditions encode the economic realities of different financial contracts, and modern extensions handle the complexities of real markets. These mathematical tools don't just price derivatives - they reveal the deep mathematical structures underlying financial risk and return! 🎓

Study Notes

• Black-Scholes PDE: $\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV = 0$

• Risk-neutral pricing: Option prices equal expected discounted payoffs under risk-neutral probability

• Feynman-Kac theorem: Connects PDEs with stochastic processes - PDE solutions equal expectations of stochastic integrals

• Boundary conditions for European call: $V(S,T) = \max(S-K,0)$, $V(0,t) = 0$, $V(\infty,t) = S - Ke^{-r(T-t)}$

• American options: Create free boundary problems with optimal exercise boundaries

• Volatility smile: Real market prices deviate from Black-Scholes, requiring stochastic volatility models

• Jump-diffusion models: Add integral terms to PDEs, creating partial integro-differential equations (PIDEs)

• Numerical methods: Finite difference, finite element, and Monte Carlo simulation all solve the same PDE

• Greeks: First and second derivatives of option price ($\Delta$, $\Gamma$, $\Theta$, $\Vega$, $\Rho$) come directly from PDE structure

• Smooth pasting condition: At optimal exercise boundary, option value and its derivative must be continuous (no arbitrage)