Integral Calculus & Differential Equations in Finance

A deposit schedule tells you how quickly money enters an account, but how much arrives over five years? A changing discount curve creates the reverse problem: what is a future cash flow worth today? Integrals turn specified rates into accumulated amounts, while differential equations describe how balances evolve under those rates.

These tools support many continuous-time finance models. For a time-varying instantaneous continuously compounded rate, integrating the rate and exponentiating gives the gross accumulation factor. Differential equations describe models in which asset prices drift and diffuse over time. The Black-Scholes framework, developed later in the book, includes a partial differential equation for an option's model value.

This chapter develops integral calculus and differential equations through financial applications. You will compute accumulated cash flows and continuously compounded growth factors, then model continuous growth, decay, and mean reversion.

The definite integral of a rate gives its net accumulated change over an interval. If you know the flow rate through a pipe, integrating it gives the volume that passed through. Similarly, if $f(t)$ is a financial rate of change in dollars per year, $\int_a^b f(t)\,dt$ gives the change in dollars. Integrating a balance itself would instead have units of dollar-years, not dollars.

Geometrically, the definite integral equals the signed area under the curve $f(t)$ between $t = a$ and $t = b$. Areas above the horizontal axis contribute positively, while areas below contribute negatively. This interpretation has direct financial meaning. If $f(t)$ represents a continuous cash flow rate, the integral gives the total cash accumulated over the period. Positive net flows increase the tracked account's cash balance, while negative flows decrease it. Transfers between your own accounts or loan repayments need not change total net worth by the same amount.

The challenge, of course, is computing these areas precisely when the curve is not a simple geometric shape. This is where the formal definition through Riemann sums becomes essential, and gives a rigorous foundation for calculating accumulated quantities even when the rate of change follows complex patterns.

The intuition behind Riemann sums is straightforward: we approximate a complex shape using familiar ones. Just as ancient mathematicians approximated the area of a circle by inscribing polygons with ever more sides, we approximate the area under a curve by filling it with thin rectangles. Each rectangle has width $\Delta t$ and height determined by the function value at some point within that slice. The product $f(t_i^*) \Delta t$ gives the area of each rectangle, and summing all rectangles approximates the total area. As we use more rectangles (larger $n$), each rectangle becomes narrower ($\Delta t$ shrinks), and the sums converge for a Riemann-integrable function, though successive errors need not decrease monotonically.

For a continuous $f$ on $[a,b]$, the Fundamental Theorem of Calculus connects differentiation and integration: differentiating the accumulation function $\int_a^t f(s)\,ds$ recovers $f(t)$. Integrating a derivative instead recovers the original function's change; an indefinite antiderivative is determined only up to an additive constant. If $F'(t)=f(t)$, then:

This theorem turns the problem of computing integrals into finding antiderivatives, which is often more tractable than evaluating limits of sums directly. Rather than computing an infinite limit of rectangle sums, we can simply find a function whose derivative equals our integrand, then evaluate the difference at the endpoints. This is why antiderivative tables and integration techniques are so useful: they provide shortcuts to what would otherwise require tedious limiting arguments.

Consider a savings account with a time-varying deposit rate $d(t)$ measured in dollars per year. This rate might change because your income grows over time, or because you commit to increasing your contributions annually. Over a small time interval $\Delta t$, you deposit approximately $d(t) \cdot \Delta t$ dollars, the rate times the duration. To find the total amount deposited from time $0$ to time $T$, we must sum all these infinitesimal contributions, which is precisely what the integral computes:

If $d(t)=1000+100t$ dollars per year, with $t$ in years, the deposit rate increases continuously by $100 per year for each elapsed year. Over five years:

You would deposit a total of $6,250 over the five-year period. Notice how the antiderivative $1000t + 50t^2$ captures both the base contribution ($1000t$) and the cumulative effect of the increasing contributions ($50t^2$). The Fundamental Theorem lets us compute this by simple evaluation rather than summing five years of daily deposits.

The numerical result confirms our analytical calculation. The scipy.integrate.quad function uses adaptive QUADPACK routines and returns an error estimate. Requested tolerances and the subdivision limit constrain the algorithm. Integrand behavior and problem conditioning also matter, as does floating-point arithmetic.

Continuous compounding is an important application of integral calculus in finance. Interest accrual, compounding, and scheduled crediting depend on the contract. Continuous compounding supplies a mathematical convention for the models below.

Discrete compounding at a constant nominal annual rate $r$, $n$ times per year for $t$ years, gives the following value at compounding dates, where $nt$ is an integer:

Each completed period credits interest at rate $r/n$. The comparison below fixes this nominal quote as frequency changes; between credit dates, its curves use fractional-power interpolation, not continuous crediting. As the compounding frequency $n$ increases, moving from annual to quarterly to daily to hourly, this expression approaches a limit. Taking $n \to \infty$ yields continuous compounding:

where:

• $FV$: future value of the investment
• $PV$: present value (initial investment)
• $e \approx 2.71828$: Euler's number, the base of natural logarithms
• $r$: continuously compounded annual rate in $e^{rt}$; the preceding discrete formula uses the same number as a nominal annual quote, not an effective annual return
• $t$: time in years

The exponential appears because the balance changes in proportion to its current value at a constant rate. A positive rate produces growth, zero rate leaves the balance constant, and a negative rate produces decay. The continuous formula is a limit of multiplicative discrete factors, not literal infinitely frequent contractual payments.

Every solution of $y'=ky$ with constant $k$ has the form $y(t)=A e^{kt}$. An initial value fixes $A$; in particular, $e^t$ solves $y'=y$ with $y(0)=1$.

Let's derive the formula for a positive initial balance $V_0$ and constant continuously compounded rate $r$. The balance $V(t)$ changes at rate $r$ times its current value, growing for $r>0$ and shrinking for $r<0$:

where:

• $V(t)$: account value at time $t$
• $\frac{dV}{dt}$: instantaneous rate of change of account value
• $r$: continuous interest rate
• $r \cdot V(t)$: interest earned per unit time, proportional to current balance

Solving this differential equation gives an exponential family; the initial balance selects one path. On the positive-balance branch, separation of variables moves the $V$ and $t$ terms to opposite sides:

Integrating both sides:

Here $C$ is the integration constant. Because we assumed a positive balance, exponentiating $\ln|V|$ gives $V$, rather than $-V$:

Setting $t = 0$ and letting $V(0) = V_0$ be the initial value:

Therefore, $e^C = V_0$, and the solution is:

The final formula also solves the ODE directly for zero or negative initial values; $V_0=0$ gives the separate equilibrium $V(t)=0$. With a time-varying deterministic coefficient, $y'=k(t)y$ instead gives $y(t)=y(0)\exp(\int_0^t k(s)\,ds)$. Random shocks require a stochastic model rather than this ODE alone.

While the difference between daily and continuous compounding is small (about $0.19 on a $1,000 investment over 20 years), continuous compounding provides mathematical elegance that simplifies many calculations in quantitative finance.

The following examples use several integration techniques. They are useful when a model lacks a convenient antiderivative. We begin with substitutions and products before turning to numerical integration.

Integration by substitution reverses the chain rule of differentiation, allowing us to simplify integrals by changing variables. The core insight is that a complex integrand might become simple under the right transformation. If $g(t)$ is a differentiable function and $f$ is continuous, then:

where $u = g(t)$.

The power of substitution lies in recognizing when an integrand contains a function and its derivative: the signature of the chain rule applied in reverse. When we spot this pattern, we can simplify the integral by treating the inner function as a new variable.

Consider pricing a bond with continuous coupon payments where the coupon rate decays exponentially. This structure might arise when modeling a company whose cash flows are expected to decline over time. If the coupon rate at time $t$ is $C(t) = C_0 e^{-\alpha t}$ and the discount factor is $D(t) = e^{-rt}$, we need to compute the present value of this decaying stream of payments:

where:

• $C_0$: initial coupon cash-flow rate at $t=0$, in currency per year
• $\alpha$: decay rate of the coupon (how fast it decreases)
• $r$: risk-free discount rate
• $T$: maturity time

The present value of coupons from $0$ to $T$ is:

Here we used $e^a \cdot e^b=e^{a+b}$ to combine the exponentials. When $r+\alpha\ne0$, substitution with $u=-(r+\alpha)t$ gives the first branch below. At $r+\alpha=0$, the integrand is the constant $C_0$:

For the usual nonnegative discount and decay rates, $r+\alpha$ combines the two forces that reduce future coupon value. For $C_0>0$, the finite perpetuity limit $C_0/(r+\alpha)$ exists only when $r+\alpha>0$. At equality the value grows linearly as $C_0T$; when $r+\alpha<0$, it grows exponentially rather than converging. A zero stream has value zero for every rate.

Integration by parts reverses the product rule, enabling us to integrate products of functions by transferring the derivative from one factor to another. This technique is especially valuable when one function becomes simpler upon differentiation while the other has a known antiderivative. For continuously differentiable functions $u(t)$ and $v(t)$:

where:

• $u$: a function chosen to simplify when differentiated
• $v$: an antiderivative of the factor represented by $dv$, so $dv=v'(t)\,dt$
• $dv$: the differential of $v$
• $du$: the differential of $u$

The strategy is to choose $u$ and $dv$ so that $\int v \, du$ is simpler than the original integral. A useful mnemonic for choosing $u$ is LIATE (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential), which suggests the order of preference for what to differentiate.

This technique is important when integrating products, especially in option pricing where payoffs multiply probability densities. When computing expected values, we often encounter integrals of the form $\int x \cdot f(x) \, dx$, where $f(x)$ is some probability density.

For an option-pricing application, let $F$ and $p$ be the CDF and density of a nonnegative terminal price $S_T$. Provided the expectation is finite and $(s-K)[1-F(s)]\to0$ as $s\to\infty$, integration by parts turns a call's expected payoff into a tail-probability integral. Choose

Then the boundary term is explicit:

Consider computing the expected payoff of a call option under a uniform distribution (a simplified model for illustration). If the stock price $S_T$ at expiration is uniformly distributed on $[80, 120]$ with strike $K = 100$, the expected payoff is:

Direct integration gives the same value as integrating the uniform survival function $(120-s)/40$ from 100 to 120:

The Gaussian integral is fundamental to quantitative finance because normal distributions appear throughout risk modeling and option pricing. Some mean-variance analyses assume normal arithmetic returns; the mean-variance criterion itself uses means and covariances rather than requiring normality. Under Black-Scholes, continuously compounded (log) returns are normal and terminal prices are lognormal; other option models need not use normal returns. In each normal-model setting, we encounter integrals involving the exponential of a quadratic. The basic Gaussian integral states:

For this particular unnormalized Gaussian, the area is $\sqrt{\pi}$; changing its height or width changes that area. One derivation squares the integral and integrates $e^{-(x^2+y^2)}$ over the plane. Polar coordinates give $2\pi\int_0^\infty \rho e^{-\rho^2}\,d\rho=\pi$, so the positive original integral is $\sqrt{\pi}$.

More generally, for the standard normal density:

This integral equals 1 because it represents the total probability under the standard normal density. A probability density, when one exists, integrates to 1. Discrete probability masses are normalized by summation; more generally, a probability measure assigns total mass 1 to its sample space. The factor $1/\sqrt{2\pi}$ is precisely the normalization constant that ensures this property.

The cumulative distribution function of the standard normal:

where:

• $\Phi(z)$: cumulative distribution function (CDF) of the standard normal distribution
• $z$: upper limit of integration, representing the number of standard deviations
• $t$: dummy variable of integration
• $\frac{1}{\sqrt{2\pi}} e^{-t^2/2}$: the standard normal probability density function

has no expression in elementary functions. It can be written with the special function $\operatorname{erf}$ as $\Phi(z)=\tfrac12[1+\operatorname{erf}(z/\sqrt2)]$ and is evaluated numerically in practice. It appears directly in the Black-Scholes formula for option pricing, where call and put prices are expressed in terms of $\Phi(d_1)$ and $\Phi(d_2)$.

The value $\Phi(-2) \approx 0.023$ means that a standard normal variable falls below $-2$ with probability about 2.3%. Under a model in which a return is normal and has been standardized by subtracting its mean and dividing by its standard deviation, this is the model probability of a return more than two standard deviations below its mean.

An important application of integration in finance is computing present values when interest rates vary over time. In the real world, interest rates are not constant. Short-term rates typically differ from long-term rates, creating a term structure that may be quoted as par yields, zero or spot rates, forward rates, or discount factors. The integration formula below applies directly to the instantaneous forward curve. A continuously compounded zero rate maps directly to a discount factor, while par yields must first be transformed into zero rates or forward rates.

A specified instantaneous forward curve determines the discount factors used in these cash-flow models. Slope alone does not determine a discount factor: the upward-sloping forward curve and flat curve below have identical ten-year discount factors, even though they differ at other maturities. The integral uses the full curve up to the chosen maturity.

For a time-zero term structure, let $f(0,t)$ denote the instantaneous forward rate for maturity $t$. We can no longer apply $e^{-rT}$ with one constant rate. Instead, the discount factor from time $0$ to time $T$ integrates the forward curve across maturities:

where:

• $D(0, T)$: discount factor from time $0$ to time $T$
• $f(0,s)$: time-zero instantaneous forward rate for maturity $s$
• $s$: dummy variable indexing maturity
• $\int_0^T f(0,s) \, ds$: accumulated continuously compounded forward rates through maturity $T$

The corresponding continuously compounded $T$-maturity spot, or zero, rate is the maturity average

Thus an instantaneous forward rate, a maturity-specific spot rate, and a future short rate are distinct objects. The same exponential formula can integrate a deterministic future short-rate path, but such a path is a model assumption rather than the observed time-zero spot curve.

Consider a time-zero instantaneous forward curve that starts at 3% and increases linearly to 5% at a 10-year maturity:

This is an illustrative upward-sloping term structure, not a causal account of why the curve slopes upward. The discount factor for a 10-year cash flow is:

Computing the integral:

Therefore:

A cash flow of $1,000 in ten years has a present value of approximately $670.32. Its maturity-specific zero rate is $R(0,10)=10^{-1}\int_0^{10}f(0,s)\,ds=4\%$, so the flat-4% factor agrees exactly at this maturity: $e^{-R(0,10)10}=e^{-0.4}$. This identity holds for any integrable forward curve; linearity makes the average easy to calculate. One maturity average does not determine discount factors at other maturities.

Consider a bond paying continuous coupon at rate $c$ per year on a principal of $P$, maturing at time $T$. Coupon bonds normally pay on discrete dates. A continuous-coupon stream is a simplifying model and can approximate a sufficiently frequent payment schedule when the coupon timing and discount curve are smooth. The present value combines the stream of coupons and the principal repayment:

where:

• $PV$: present value of the bond
• $c$: continuous coupon rate (per year)
• $P$: principal (face value)
• $T$: maturity time
• $D(0, t)$: discount factor from time $0$ to time $t$

The first term captures the present value of all coupon payments: at each instant $t$, the bond pays $c \cdot P$ per year, and this payment is discounted back to today using $D(0, t)$. The second term is the present value of the principal repayment at maturity.

With our upward-sloping instantaneous forward curve:

The bond is slightly above par, at about $1,005.54. Its cumulative average forward rate $R(0,t)=0.03+0.001t$ is below 4% for every $0<t<10$, equaling 4% at maturity. Thus each earlier coupon has a larger discount factor than under the flat-4% continuous-coupon bond, while principal has the same factor. The instantaneous forward itself is below 4% only during the first five years.

Differential equations describe relationships between quantities and their rates of change. Algebraic equations relate values directly, while differential equations relate a function to its derivatives and state the rules its path must satisfy. In finance, they model how portfolios, prices, and economic variables evolve over time, capturing the dynamic nature of markets and economies.

Differential equations encode rules of change that, when solved, reveal the trajectory of a system. If we know that interest rates revert toward a long-term mean, or that a portfolio grows proportionally to its value plus deposits, we can express these dynamics as differential equations and solve for the resulting paths.

A first-order linear ordinary differential equation has the form:

The term "first-order" means the equation involves only the first derivative of $y$, not higher derivatives. "Linear" means $y$ and its derivative appear only to the first power and are not multiplied together. On an interval where $P$ and $Q$ are continuous (or satisfy suitable integrability conditions), an integrating factor gives the general solution.

The general solution uses an integrating factor $\mu(t) = e^{\int P(t) dt}$:

where:

• $y(t)$: the unknown function we are solving for
• $P(t)$: coefficient function multiplying $y$
• $Q(t)$: forcing function (the non-homogeneous term)
• $\mu(t) = e^{\int P(t) dt}$: integrating factor that makes the left side a perfect derivative
• $C$: constant of integration, determined by initial conditions

The integrating factor works because multiplying by $\mu(t)$ converts the left side into $\frac{d}{dt}[\mu(t) y]$, allowing direct integration. This technique appears frequently in differential equations. By multiplying through by just the right function, we transform a seemingly intractable equation into one we can solve by straightforward integration.

Example: Portfolio Growth with Continuous Deposits

Suppose you have a portfolio earning continuous return $r$ and you make continuous deposits at rate $D$ per year. This models a retirement account where you contribute regularly while earning investment returns. The portfolio value $V(t)$ satisfies:

This equation says: the rate of change of your portfolio equals the investment return ($rV$, proportional to what you have) plus your deposits ($D$, a constant flow). This is a first-order linear ODE with $P(t) = -r$ and $Q(t) = D$. The integrating factor is $\mu(t) = e^{-rt}$.

Multiplying both sides by $\mu(t)$:

The left side is the derivative of $e^{-rt} V$:

This is the key insight: what looked like two separate terms is a single derivative. For $r \ne 0$, we can integrate directly:

With initial condition $V(0) = V_0$:

For $r \ne 0$, the solution is:

The first term represents growth of the initial investment, while the second represents accumulated deposits with their earned interest. This decomposition is intuitive. Your wealth at any time is the sum of what your original stake has grown to, plus what your contributions have accumulated.

For $r>0$, $D/r$ is the time-zero value of a perpetual continuous cash-flow rate $D$ discounted continuously at rate $r$. In the finite-horizon accumulation formula, the stable term is $D\,\operatorname{expm1}(rt)/r$. Its continuous limit at $r=0$ is $Dt$, so the zero-rate solution is $V(t)=V_0+Dt$ rather than a division by zero.

The power of compounding is evident: starting with $10,000 and depositing $6,000 per year, the portfolio grows to over $600,000 after 30 years, with investment gains far exceeding the total contributions.

Numerical methods can compute trajectories when a convenient known closed form is unavailable. Time-varying coefficients and nonlinearity do not automatically prevent analytical solutions. The examples below also use numerical evaluation when a benchmark formula is known. The SciPy function odeint wraps LSODA and uses requested local-error tolerances; success does not guarantee a chosen global error. SciPy recommends solve_ivp for new code.

Example: Mean-Reverting Interest Rates

The constant-parameter Vasicek model is the stochastic differential equation $dr_t=\kappa(\theta-r_t)dt+\sigma_r dW_t$. Assume $\kappa>0$ for mean reversion. The ODE below retains only its drift and also governs the conditional mean $m(t)=\mathbb{E}[r_t\mid r_0]$:

where:

• $m$: deterministic drift trajectory, equivalently the conditional mean rate at time $t$
• $\kappa>0$: speed of mean reversion (larger positive values mean faster reversion)
• $\theta$: long-term mean interest rate
• $\sigma_r$: short-rate volatility in the full SDE; $dW_t$ is a Brownian increment

Mean reversion is a modeling assumption, not a claim that central banks or arbitrage impose hard bounds. In the drift ODE, $m>\theta$ gives a negative derivative and $m<\theta$ gives a positive derivative; larger $\kappa$ means faster convergence of the conditional mean. For positive elapsed time and nonzero volatility, the constant-parameter Vasicek rate has a nondegenerate Gaussian law with support on the entire real line. At time zero or zero volatility, its conditional law is degenerate instead.

This conditional-mean ODE has the analytical solution $m(t)=\theta+(r_0-\theta)e^{-\kappa t}$, but we solve it numerically to demonstrate the technique:

For $y'=h(t)g(y)$, separation writes $dy/g(y)=h(t)\,dt$ on intervals where $g(y)\ne0$. Integrating that branch does not recover excluded equilibrium solutions, which must be checked in the original equation.

The deterministic component of the geometric Brownian motion model for stock prices is:

where:

• $S$: stock price at time $t$
• $\mu$: drift rate (expected return per unit time)
• $\frac{dS}{dt}$: rate of change of stock price

For the positive-price branch $S_0>0$, the proportionality to $S$ means percentage changes are constant: a $100 stock and a $200 stock both grow at the same percentage rate $\mu$. Percentage returns make changes comparable across scales, while dollar profit and loss still matter for position sizing, liabilities, and risk limits. A $10 gain on a $100 stock is a 10% return, whereas the same dollar gain on a $1,000 stock is a 1% return. Separating variables:

Integrating:

Under the geometric Brownian motion convention $dS_t=\mu S_tdt+\sigma S_tdW_t$, this deterministic solution equals the conditional mean $\mathbb{E}[S_t\mid S_0]=S_0e^{\mu t}$. It is not a path that individual prices oscillate or mean-revert around. Log-price variance is $\sigma^2t$, and variance relative to the squared mean is $e^{\sigma^2t}-1$, so these measures increase for nonzero $\sigma$. Absolute price variance need not increase for negative $\mu$. Deviations do not experience a restoring force.

Many vehicles and pieces of equipment lose economic value with age and use, although some physical assets can appreciate. A simple depreciation model assumes value declines in proportion to its current level:

where:

• $V$: asset value at time $t$
• $\delta$: constant continuous decay rate, in inverse years; its finite one-year loss fraction is $1-e^{-\delta}$
• $-\delta V$: negative because value decreases over time

Assets sharing a proportional decay rate lose the same percentage over a fixed interval, while a larger starting value gives a larger dollar loss. A discrete 15% annual loss removes $4,500 from $30,000 or $2,250 from $15,000. In the continuous model that annual loss corresponds to $\delta=-\ln(0.85)\approx0.16252$ per year, not $\delta=0.15$. The latter gives a one-year loss of about 13.93%. The continuous ODE has solution:

where $V_0$ is the initial asset value at $t = 0$.

For $V_0>0$ and $\delta>0$, the asset value decays exponentially, with half-life $t_{1/2} = \ln(2)/\delta$. The half-life formula follows directly from solving $V_0 e^{-\delta t_{1/2}} = V_0/2$, which gives $e^{-\delta t_{1/2}} = 1/2$ and thus $t_{1/2} = \ln(2)/\delta$. Zero value is a separate equilibrium; zero decay has no finite halving time. The following rates are illustrative continuous parameters, not estimated asset-class depreciation rates.

A fixed carrying capacity can be a useful assumption when modeling adoption in a finite market. The logistic equation reduces proportional growth as adoption approaches that capacity; it does not identify an empirical competition or resource mechanism by itself:

where:

• $P$: the quantity being modeled (population, market share, number or fraction of adopters)
• $t$: time
• $r$: intrinsic growth rate (growth rate when $P$ is small)
• $K$: carrying capacity (maximum sustainable value)
• $(1 - P/K)$: saturation factor that approaches zero as $P$ approaches $K$

For $r>0$ and $0<P<K$, the proportional rate $P'/P=r(1-P/K)$ falls as $P$ rises. Absolute growth $P'=rP(1-P/K)$ instead increases up to $K/2$ and decreases afterward. At $P=K$, growth is zero.

Assume $K>0$, $r>0$, and $0<P_0<K$ for an increasing path. A future accelerating-then-decelerating S-shaped segment additionally requires $P_0<K/2$. If $P_0\ge K/2$, the future starts at or beyond the inflection. $P_0=0$ is an equilibrium, while $P_0>K$ decreases toward $K$ from above.

Under $0<P_0<K$, the logistic equation has the analytical solution:

where:

• $P(t)$: quantity at time $t$
• $P_0$: initial quantity at $t = 0$
• $K$: carrying capacity
• $r$: intrinsic growth rate
• $(K - P_0)/P_0$: ratio determining how far the initial value is from capacity

The full logistic solution is sigmoid on a linear vertical scale. The example starts at $P_0=0.01K$, so its future includes both sides of the inflection. Its comparison figure uses a logarithmic vertical axis to retain the unmodified exponential curve; that geometry is not the ordinary-scale S shape. As $t\to\infty$, the exponential term tends to zero and $P(t)$ approaches $K$.

An S-shaped curve can model technology adoption, market penetration, or the spread of a financial product when a fixed carrying capacity is a reasonable assumption. In that model, early adoption is close to exponential, the pool of potential new adopters shrinks, and adoption approaches the specified capacity. The curve is a modeling choice, not a historical claim that every product follows a logistic path.

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:

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:

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:

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:

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.

Many complex, path-dependent, or high-dimensional pricing and risk problems require numerical integration or simulation because a convenient closed form is unavailable. The appropriate method depends on the payoff, state dimension, model dynamics, and accuracy target.

Numerical integration approximates definite integrals using weighted sums of function values.

where:

• $a, b$: integration bounds
• $f(x)$: the function being integrated
• $n$: number of evaluation points
• $x_i$: nodes (evaluation points) within $[a, b]$
• $w_i$: weights determining the contribution of each function value