Integral Calculus and Differential Equations › Connecting to Continuous-Time Finance

The mathematical tools developed in this chapter form the foundation for continuous-time financial models. This section previews how these concepts connect to more advanced topics.

Under the stated Black-Scholes-Merton assumptions, a sufficiently smooth European option value satisfies this PDE before expiration. The model assumes a tradable non-dividend-paying underlying, constant risk-free rate and volatility, frictionless replication, and no arbitrage. For an American option, this equality describes its continuation region, not its exercise region:

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

where:

• $V$: option value (function of $S$ and $t$)
• $t$: time
• $S$: underlying asset price
• $\sigma$: volatility of the underlying asset
• $r$: risk-free interest rate
• $\frac{\partial V}{\partial t}$: rate of change of option value with respect to time (theta)
• $\frac{\partial V}{\partial S}$: rate of change of option value with respect to asset price (delta)
• $\frac{\partial^2 V}{\partial S^2}$: second derivative with respect to asset price (gamma)

This PDE combines ideas from this chapter.

The partial derivatives describe how option value $V$ changes with time $t$ and stock price $S$. Unlike ordinary differential equations where we have one independent variable, here $V$ depends on both $t$ and $S$, requiring partial derivatives to capture how $V$ responds to changes in each variable separately.

The equation structure resembles a diffusion equation with drift. The term involving the second derivative ($\frac12\sigma^2 S^2 \partial^2 V / \partial S^2$) captures how uncertainty diffuses through the option value. The first derivative term ($rS \partial V / \partial S$) captures drift.

Solving the PDE requires a terminal payoff condition, any applicable spatial boundary conditions, and an analytical transformation or a numerical PDE method. The expiration payoff supplies the terminal condition; integration alone is not a universal procedure for working backward to today's price.

The derivation and solution of this equation will be covered in detail in later chapters, and recognizing its structure as a differential equation prepares you for what follows.

Financial models include randomness. A stochastic differential equation (SDE) generalizes the ODEs in this chapter by adding a random component:

$$dS = \mu S \, dt + \sigma S \, dW$$

where:

• $dS$: infinitesimal change in asset price
• $S$: current asset price
• $\mu$: drift rate (expected return)
• $\sigma$: diffusion volatility; over an interval of length $\Delta t$, the conditional standard deviation of the log return is $\sigma\sqrt{\Delta t}$ in this model
• $dt$: infinitesimal time increment
• $dW$: increment of a Wiener process (Brownian motion)
• $\mu S \, dt$: deterministic drift component
• $\sigma S \, dW$: random diffusion component

The first term ($\mu S \, dt$) is the deterministic drift we analyzed earlier in this chapter, representing the expected growth of the stock price over time. The second term ($\sigma S \, dW$) adds random fluctuations driven by a Wiener process $W$, which is a mathematical model of Brownian motion. These random shocks cause the stock price to fluctuate unpredictably while maintaining an overall drift. Solving SDEs requires stochastic calculus, which extends the integration techniques from this chapter to handle these random components.

In an arbitrage-free complete model admitting a risk-neutral measure $\mathbb{Q}$, an integrable European payoff $f(S_T)$ with constant deterministic short rate $r$ has time-zero value:

$$V_0 = e^{-rT} \mathbb{E}^{\mathbb{Q}}[f(S_T)]$$

If the terminal stock price has a density $p_{\mathbb Q}$ on positive prices, the expectation becomes the following density integral. Discrete terminal models instead use weighted sums; in general, integrate against the terminal probability law:

$$V_0 = e^{-rT} \int_{0}^{\infty} f(s) \cdot p_{\mathbb{Q}}(s) \, ds$$

where:

• $V$: present value of the derivative
• $r$: risk-free interest rate
• $T$: time to expiration
• $e^{-rT}$: discount factor
• $f(s)$: payoff function (e.g., $\max(s - K, 0)$ for a call option)
• $s$: possible terminal stock price
• $p_{\mathbb{Q}}(s)$: risk-neutral density of the positive terminal stock price under the specified model
• $\int_{0}^{\infty} f(s) \cdot p_{\mathbb{Q}}(s) \, ds$: risk-neutral expected terminal payoff, assumed finite

Under the Black-Scholes assumptions for a non-dividend-paying stock and a European call, the payoff $\max(S_T-K,0)$ and its risk-neutral lognormal density make this terminal-state integral analytically tractable. Path-dependent claims instead require expectations of path functionals, and incomplete models may not select a unique risk-neutral price without additional assumptions.