Sources and References

Supplementary texts — W.E. Boyce & R.C. DiPrima, Elementary Differential Equations and Boundary Value Problems (11th ed., Wiley, 2017); M. Tenenbaum & H. Pollard, Ordinary Differential Equations (Dover, 1985); P. Wilmott, S. Howison & J. Dewynne, The Mathematics of Financial Derivatives (Cambridge, 1995); S.E. Shreve, Stochastic Calculus for Finance II: Continuous-Time Models (Springer, 2004)

Online resources — MIT OCW 18.03 Differential Equations; Paul’s Online Math Notes (tutorial.math.lamar.edu); Khan Academy Finance series

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Chapter 1: First-Order Ordinary Differential Equations

The central object of this course is the differential equation — an equation that relates an unknown function to its own rates of change. Rather than asking “what is the value of some quantity?”, a differential equation asks “how does that quantity evolve?” This shift in perspective is enormously powerful: most natural and economic phenomena are described far more easily in terms of rates than in terms of static values. A bank account earns interest proportional to its balance. A market price moves toward equilibrium in response to excess demand. A population grows at a rate depending on how large it already is. In each case, the governing law is a differential equation, and the solution is the quantity itself as a function of time.

The simplest differential equations can be solved by direct integration. If \(\frac{dy}{dx} = f(x)\), then \(y = \int f(x)\,dx + C\). The constant \(C\) represents the freedom we have in specifying where the solution starts — it is pinned down only when we impose an initial condition \(y(x_0) = y_0\). The combination of a differential equation plus an initial condition is called an initial-value problem (IVP).

Before computing solutions, we should ask whether a solution even exists, and whether it is unique. The following foundational theorem answers both questions for first-order equations.

The theorem tells us that pathological behaviour — solutions that blow up, split, or fail to exist — can only arise when the hypotheses fail, for instance when \(f\) is discontinuous or \(\partial f / \partial y\) is unbounded. In practice, the theorem covers nearly every equation we encounter in business and economics applications, and we can proceed to find solutions with confidence.

1.1 Separable Equations

Many important first-order ODEs have the form \(y' = f(x)\,g(y)\), where the right-hand side factors into a function of \(x\) alone times a function of \(y\) alone. Such equations are called separable because we can separate the variables onto opposite sides of the equation.

The separation step is formally justified by writing \(dy = g(y)\,f(x)\,dx\) and dividing through by \(g(y)\) (wherever \(g(y) \neq 0\)). Note that any value \(y = y^*\) with \(g(y^*) = 0\) is a constant (equilibrium) solution, and it should be checked separately.

1.2 First-Order Linear ODEs

Not all first-order equations are separable. The next most tractable class is first-order linear equations, which arise naturally in finance whenever the rate of change of a quantity depends linearly on both the quantity itself and an external forcing term.

The key technique for solving first-order linear ODEs is the integrating factor. The idea is to multiply both sides by a cleverly chosen function \(\mu(x)\) so that the left-hand side becomes a perfect derivative.

1.3 Logistic Growth and Newton’s Law of Cooling

Not every important first-order ODE is linear. Two nonlinear equations that appear throughout economics, ecology, and applied science are the logistic equation and Newton’s law of cooling.

\[ \frac{dP}{P(1 - P/K)} = r\,dt. \]

Using partial fractions: \(\frac{1}{P(1-P/K)} = \frac{1}{P} + \frac{1/K}{1 - P/K}\). Integrating both sides and applying the initial condition \(P(0) = P_0\):

\[ P(t) = \frac{K}{1 + \left(\dfrac{K - P_0}{P_0}\right)e^{-rt}}. \]

1.4 Substitution Methods

Some equations are neither separable nor linear in their original form, but a change of variables transforms them into one of these tractable classes.

A Bernoulli equation has the form \(y' + P(x)y = Q(x)y^n\) for some \(n \neq 0, 1\). The substitution \(v = y^{1-n}\) linearises it: differentiating, \(v' = (1-n)y^{-n}y'\), and substituting into the Bernoulli equation yields a linear ODE for \(v\).

\[ v + xv' = f(v) \implies \int \frac{dv}{f(v) - v} = \int \frac{dx}{x}. \]

Dimensional homogeneity is a useful modelling principle: every term in a physical or economic equation must have the same units. This constrains the form of equations and provides a check on algebra. For example, in an equation relating dollars, time, and rates, each additive term must have units of dollars (or dollars per unit time, consistently). Violations of dimensional homogeneity indicate errors in model formulation.

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Chapter 2: Applications to Finance and Economics

The differential equations developed in Chapter 1 are not merely mathematical abstractions — they are the natural language for modelling how economic quantities evolve over time. This chapter develops several core applications, building intuition for how the choice of equation structure encodes economic assumptions.

2.1 Supply-Demand Dynamics

In introductory economics, supply and demand determine a static equilibrium price. But in reality, prices adjust over time in response to excess demand. A simple dynamic model captures this:

\[ \frac{dp}{dt} \approx \alpha\left[D'(P^*) - S'(P^*)\right]p. \]

In a normal market, demand slopes downward (\(D' < 0\)) and supply slopes upward (\(S' > 0\)), so \(D'(P^*) - S'(P^*) < 0\). Setting \(\beta = -\alpha[D'(P^*) - S'(P^*)] > 0\), the linearised equation is \(dp/dt = -\beta p\), with solution \(p(t) = p(0)e^{-\beta t} \to 0\). The equilibrium is stable: prices converge exponentially to \(P^*\).

2.2 The Domar Model of Public Debt Sustainability

A classic application of first-order linear ODEs in macroeconomics is the Domar (1944) model of public debt dynamics. If the government runs a primary deficit that grows at rate \(g\) and the existing debt accumulates interest at rate \(r\), the evolution of the debt stock \(D(t)\) is:

\[ \frac{dD}{dt} = rD + G_0 e^{gt}, \]

where \(G_0 e^{gt}\) is the (growing) primary deficit. This is a first-order linear ODE with \(P(t) = -r\) and \(Q(t) = G_0 e^{gt}\). The integrating factor is \(\mu(t) = e^{-rt}\).

Multiplying through: \(\frac{d}{dt}[e^{-rt}D] = G_0 e^{(g-r)t}\).

\[ e^{-rt}D = \frac{G_0}{g - r}e^{(g-r)t} + C. \]\[ D(t) = e^{rt}\left[D_0 - \frac{G_0}{g-r}\right] + \frac{G_0}{g-r}e^{gt}. \]

Case 2: \(g = r\). Then \(\frac{d}{dt}[e^{-rt}D] = G_0\), giving \(D(t) = e^{rt}(D_0 + G_0 t)\) — debt grows faster than exponential.

The key quantity is the debt-to-GDP ratio \(d = D/Y\). If GDP grows at rate \(g\) (so \(Y = Y_0 e^{gt}\)), then \(d\) stabilises when the growth rate of \(D\) equals \(g\). Analysis of the solution shows:

• If \(r < g\): the term \(e^{rt}\) grows slower than \(e^{gt}\), and the deficit term dominates. The debt ratio stabilises at a finite level.
• If \(r > g\): the compounding interest on debt outpaces GDP growth, and the debt-to-GDP ratio grows without bound — debt is unsustainable.

This result — that \(r < g\) is the sustainability condition — has been central to debates about fiscal policy since Olivier Blanchard’s 2019 American Economic Association presidential lecture, in which he argued that for most advanced economies in recent decades, \(r < g\) has held, making some level of deficit spending sustainable.

2.3 The Evans Price Adjustment Model

\[ P'' + \alpha P' + \beta P = \gamma, \]

this is a second-order constant-coefficient ODE, analysed in Chapter 3. The roots of the characteristic equation determine whether prices oscillate to equilibrium (complex roots with negative real part) or converge monotonically (real negative roots). The Evans model thus predicts price cycles in commodity markets as a natural consequence of inventory dynamics, not market irrationality.

2.4 The Solow Growth Model

One of the most influential models in macroeconomics is the Solow growth model, which describes how capital accumulates in an economy over time.

\[ \frac{dk}{dt} = sf(k) - (n + \delta)k. \]

The steady state \(k^*\) satisfies \(sf(k^*) = (n+\delta)k^*\) — investment exactly offsets the dilution of capital from population growth and depreciation. Linearising near \(k^*\) shows the steady state is stable when \(sf'(k^*) < n + \delta\), which is guaranteed by the concavity of \(f\). This ODE is a Bernoulli equation when \(f(k) = k^\alpha\), and can be solved explicitly by the substitution \(v = k^{1-\alpha}\).

2.5 The IS-LM Model as a System of ODEs

The IS-LM model of macroeconomic equilibrium — the workhorse of undergraduate macro — can be cast as a dynamic system. Let \(Y(t)\) be national income and \(r(t)\) the interest rate. In a simple dynamic version:

\[ \frac{dY}{dt} = \alpha\left[I(r) + G - S(Y)\right], \qquad \frac{dr}{dt} = \beta\left[L(Y, r) - M\right], \]

where \(I(r)\) is investment (decreasing in \(r\)), \(G\) government spending, \(S(Y)\) saving (increasing in \(Y\)), \(L(Y,r)\) money demand (increasing in \(Y\), decreasing in \(r\)), and \(M\) the money supply. Near the IS-LM equilibrium, linearising gives a 2D linear system whose eigenvalues determine whether adjustment is monotone or oscillatory. In typical parameterisations both eigenvalues are negative (stable node or spiral), so the economy converges to its IS-LM equilibrium.

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Chapter 3: Higher-Order Linear ODEs

First-order equations describe systems with a single degree of freedom — one piece of initial data. Many physical and economic systems have richer dynamics: oscillations, resonance, and complex transients that require higher-order equations to capture.

A central result is that the solution set of the homogeneous equation forms a vector space of dimension \(n\): there exist \(n\) linearly independent solutions \(y_1, \ldots, y_n\), and every solution is a linear combination \(y = c_1 y_1 + \cdots + c_n y_n\). The general solution of the inhomogeneous equation is then \(y = y_h + y_p\), where \(y_h\) is the general homogeneous solution and \(y_p\) is any particular solution.

3.1 The Wronskian

The question of whether \(n\) solutions are linearly independent is answered by the Wronskian determinant.

3.2 The Damped Oscillator: Three Cases

\[ y'' + 2\gamma y' + \omega_0^2 y = 0, \]\[ r = -\gamma \pm \sqrt{\gamma^2 - \omega_0^2}. \]

The behaviour splits into three qualitatively different cases:

Economic interpretation (Cobweb model): In the cobweb model of agricultural markets, suppliers make production decisions based on last period’s price. The second-order ODE analogy captures the oscillatory price dynamics: overdamped markets have sluggish convergence (e.g., housing markets with slow supply response); underdamped markets exhibit price cycles (e.g., hog cycles in agricultural economics); critically damped markets would represent the ideal — fastest convergence without overshoot. The damping parameter \(\gamma\) is related to the relative slopes of supply and demand.

3.3 Resonance and Investment Cycles

3.5 Constant-Coefficient Homogeneous Equations

When the coefficients \(a_i\) are constants, we can solve the homogeneous equation completely by substituting the trial solution \(y = e^{rx}\).

3.6 Method of Undetermined Coefficients

When the inhomogeneous term \(g(x)\) is a polynomial, exponential, or sinusoidal function (or a product of these), we can find a particular solution by guessing its form and determining the coefficients.

The guessing rules are:

• If \(g(x) = P_m(x)e^{ax}\) (polynomial of degree \(m\) times exponential), guess \(y_p = x^s Q_m(x) e^{ax}\) where \(s\) is the multiplicity of \(a\) as a root of the characteristic equation (0 if not a root).
• If \(g(x) = e^{ax}(P_m(x)\cos bx + R_k(x)\sin bx)\), guess \(y_p = x^s e^{ax}(Q_n(x)\cos bx + S_n(x)\sin bx)\) where \(n = \max(m,k)\) and \(s\) is the multiplicity of \(a+bi\) as a characteristic root.

The factor of \(x^s\) corrects for resonance — when the driving frequency matches a natural frequency of the homogeneous equation.

3.7 Variation of Parameters

The method of undetermined coefficients works only for special forms of \(g(x)\). A more general technique, variation of parameters, works whenever the homogeneous solutions are known and \(g(x)\) is continuous.

3.8 Reduction of Order

If one solution \(y_1\) of a homogeneous second-order ODE is known, a second linearly independent solution can always be found. The substitution \(y = y_1(x) v(x)\) reduces the ODE to a first-order equation for \(v'\). Setting \(w = v'\), we obtain a first-order linear ODE for \(w\), which can be solved by the integrating factor method. The second solution is \(y_2 = y_1 \int v'(x)\,dx\).

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Chapter 4: Systems of Linear ODEs

Many economic models involve not one but several interacting quantities — prices in multiple markets, stocks of multiple goods, or components of a macroeconomic model — that evolve simultaneously. The natural framework is a system of first-order linear ODEs.

Any \(n\)-th order linear ODE can be converted to a first-order system of dimension \(n\) by introducing new variables for the successive derivatives. This means the theory of systems is fully general.

4.1 Converting a Higher-Order ODE to a System

Any \(n\)-th order linear ODE can be written as a first-order system by introducing state variables for the successive derivatives. This reduction is important both theoretically (it unifies the theory) and computationally (numerical solvers work with first-order systems).

This reduction clarifies why the three cases of the damped oscillator (Section 3.2) correspond to node, spiral, and centre/node in the phase portrait of the companion system: overdamped gives a stable node; critically damped gives a degenerate node; underdamped gives a stable spiral.

4.2 The Lotka-Volterra Trade Model

The eigenvalue method applies beyond ecological systems to economic models of international trade and comparative advantage.

4.3 Homogeneous Systems: The Eigenvalue Method

\[ \lambda e^{\lambda t}\mathbf{v} = A e^{\lambda t}\mathbf{v} \implies A\mathbf{v} = \lambda\mathbf{v}. \]

So \(\lambda\) must be an eigenvalue of \(A\) and \(\mathbf{v}\) a corresponding eigenvector.

For \(2 \times 2\) systems, the phase portrait gives qualitative insight: if both eigenvalues are real negative, trajectories converge to the origin (stable node); if real positive, they diverge (unstable node); if of opposite sign, the origin is a saddle (unstable); if complex with negative real part, a stable spiral; if purely imaginary, a centre (neutrally stable).

4.4 Inhomogeneous Systems: Variation of Parameters

\[ \mathbf{X}_p(t) = \Phi(t)\int \Phi^{-1}(t)\mathbf{F}(t)\,dt. \]

The general solution is \(\mathbf{X} = \Phi(t)\mathbf{c} + \mathbf{X}_p(t)\), where \(\mathbf{c}\) is a constant vector determined by initial conditions.

4.5 The Samuelson Multiplier-Accelerator Model

The Samuelson (1939) multiplier-accelerator model of business cycles generates oscillatory output through the interaction of consumption multiplier effects and investment accelerator effects. In continuous time, if \(Y(t)\) is aggregate income, \(c \in (0,1)\) the marginal propensity to consume, and \(a > 0\) the accelerator coefficient:

\[ Y'' - (c - 1 + a)Y' + (1 - c)Y = G_0, \]

where \(G_0\) is autonomous government expenditure. This is a second-order constant-coefficient ODE with forcing. The characteristic roots are:

\[ r = \frac{(c - 1 + a) \pm \sqrt{(c-1+a)^2 - 4(1-c)}}{2}. \]

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Chapter 5: Introduction to Partial Differential Equations

So far, all functions have depended on a single independent variable. Many phenomena in finance and physics — heat diffusion, wave propagation, option pricing — involve functions of two or more variables, and the equations governing them involve partial derivatives. These are partial differential equations (PDEs).

The new feature of PDEs, compared to ODEs, is that initial/boundary data must be specified along curves or surfaces, and the structure of the solution depends crucially on the geometry of these data sets relative to the equation’s characteristics.

5.1 First-Order Linear PDEs and the Method of Characteristics

The simplest first-order linear PDE in two variables is \(a u_x + b u_t = c\), where \(a, b, c\) may be functions of \(x\), \(t\), and \(u\). The method of characteristics converts this PDE into a system of ODEs.

5.2 Classification of Second-Order Linear PDEs

\[ A u_{xx} + B u_{xt} + C u_{tt} + D u_x + E u_t + Fu = G. \]

This classification is not merely taxonomic — it determines which boundary conditions are appropriate, what solution methods apply, and what physical behaviours are possible. The Black-Scholes equation, as we will see, is parabolic, which is why its solution methods mirror those of the heat equation.

5.3 The Heat Equation

The heat equation \(u_t = \kappa u_{xx}\) (where \(\kappa > 0\) is the diffusivity) models the spread of heat in a rod or, more abstractly, the diffusion of any quantity — concentration of a chemical, probability distributions in stochastic processes.

\[ \frac{T'}{T\kappa} = \frac{X''}{X} = -\lambda \quad (\text{separation constant}). \]\[ u(x,t) = \sum_{n=1}^\infty b_n \sin\!\left(\frac{n\pi x}{L}\right) e^{-\kappa (n\pi/L)^2 t}, \]

where the coefficients \(b_n = \frac{2}{L}\int_0^L f(x)\sin(n\pi x/L)\,dx\) are determined by the initial condition \(u(x,0) = f(x)\). The exponential decay factor ensures the solution smooths out and decays to zero as \(t \to \infty\).

5.4 Steady-State Solutions of the Heat Equation

\[ u_{xx} = 0. \]

This result is the one-dimensional version of Laplace’s equation. The steady state represents the long-time behaviour of the heat equation: regardless of the initial condition, the solution converges to \(u_{ss}(x)\) as \(t \to \infty\) (assuming fixed, nonzero boundary conditions).

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Chapter 6: The Fourier Transform

Separation of variables works beautifully on a finite interval, but many problems — including the Black-Scholes equation — are posed on the entire real line. The right tool for these problems is the Fourier Transform, which converts the spatial PDE into an ODE in time, exactly as the Laplace transform converts ODEs into algebraic equations.

The Fourier Transform has several key properties that make it ideal for solving constant-coefficient linear PDEs:

• Linearity: \(\widehat{af + bg} = a\hat{f} + b\hat{g}\).
• Derivative rule: \(\widehat{f^{(n)}}(\omega) = (i\omega)^n \hat{f}(\omega)\). In particular, differentiation in \(x\) corresponds to multiplication by \(i\omega\) in frequency space.
• Convolution theorem: if \(h(x) = \int f(x-y)g(y)\,dy\), then \(\hat{h}(\omega) = \hat{f}(\omega)\hat{g}(\omega)\). Convolution in \(x\)-space is multiplication in \(\omega\)-space.

6.1 Solving the Heat Equation on the Real Line

\[ u_t = \kappa u_{xx}, \quad x \in \mathbb{R},\; t > 0, \quad u(x,0) = f(x). \]\[ \frac{\partial \hat{u}}{\partial t} = \kappa(i\omega)^2 \hat{u} = -\kappa\omega^2 \hat{u}. \]\[ \hat{u}(\omega, t) = \hat{f}(\omega)\,e^{-\kappa\omega^2 t}. \]\[ G(x,t) = \frac{1}{\sqrt{4\pi\kappa t}}\,e^{-x^2/(4\kappa t)}. \]\[ u(x,t) = \int_{-\infty}^{\infty} f(y)\,G(x-y,t)\,dy = \frac{1}{\sqrt{4\pi\kappa t}}\int_{-\infty}^{\infty} f(y)\,\exp\!\left(-\frac{(x-y)^2}{4\kappa t}\right)dy. \]

This formula is not just elegant — it is precisely the structure that underpins the solution to the Black-Scholes equation. The analogy between heat diffusion and the uncertainty in future asset prices is deep and deliberate.

6.2 The Black-Scholes Equation as a Heat Equation

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

with terminal condition \(V(S,T) = \text{Payoff}(S)\) at expiry \(T\). The variable coefficient \(S^2\) makes this look different from the heat equation, but a change of variables transforms it exactly.

\[ C(S,t) = S\,N(d_1) - K e^{-r(T-t)} N(d_2), \]\[ d_1 = \frac{\ln(S/K) + (r + \sigma^2/2)(T-t)}{\sigma\sqrt{T-t}}, \qquad d_2 = d_1 - \sigma\sqrt{T-t}. \]

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Chapter 7: The Black-Scholes Equation

The previous chapters have been building toward one of the most celebrated results in financial mathematics: the Black-Scholes model for option pricing. This chapter derives the Black-Scholes PDE from first principles, solves it using the heat equation connection, and interprets the result economically. The mathematical tools developed throughout the course — ODEs, linear algebra, separation of variables, the Fourier transform — all converge here.

7.1 Stochastic Models for Asset Prices

A fundamental challenge in finance is modelling the future price of an asset. Prices are not deterministic — they are subject to random fluctuations. The standard model is Geometric Brownian Motion (GBM), which posits that the log-return of an asset over a short time interval is normally distributed and independent of the past.

The Wiener process \(W_t\) has the properties: \(W_0 = 0\); increments \(W_{t+s} - W_t \sim \mathcal{N}(0, s)\) for \(s > 0\); increments over non-overlapping intervals are independent. The \(\sigma S\,dW_t\) term introduces randomness proportional to the current price — consistent with the empirical observation that percentage changes in price, not absolute changes, are approximately normally distributed.

The solution to the GBM SDE is \(S(t) = S(0)\exp\!\left((\mu - \tfrac{1}{2}\sigma^2)t + \sigma W_t\right)\), so \(\ln S(t)\) is normally distributed — hence the name “geometric” Brownian motion.

7.2 Itô’s Lemma

To price derivatives — contracts whose payoff depends on \(S\) — we need to know how smooth functions of \(S\) evolve over time. In ordinary calculus, if \(V = V(S)\) and \(S\) evolves deterministically, then \(dV = V'(S)\,dS\). For a stochastic process, however, the ordinary chain rule is modified by a second-order correction term, because \((dW_t)^2 = dt\) in the Itô calculus (not zero as in ordinary calculus).

The crucial extra term \(\frac{1}{2}\sigma^2 S^2 V_{SS}\,dt\) is the Itô correction. It vanishes in the deterministic limit \(\sigma \to 0\), recovering ordinary calculus.

7.3 Delta-Hedging and the Black-Scholes PDE

\[ \Pi = V - \Delta\cdot S. \]\[ d\Pi = dV - \Delta\,dS = \left(V_t + \mu S V_S + \tfrac{1}{2}\sigma^2 S^2 V_{SS}\right)dt + \sigma S V_S\,dW_t - \Delta\left(\mu S\,dt + \sigma S\,dW_t\right). \]\[ d\Pi = \left(V_t + \tfrac{1}{2}\sigma^2 S^2 V_{SS}\right)dt. \]\[ d\Pi = r\Pi\,dt = r(V - \Delta S)\,dt = r(V - S V_S)\,dt. \]

Setting the two expressions for \(d\Pi\) equal and rearranging:

Notice the structure: this is a second-order linear PDE, backward in time (the terminal condition is given at \(t = T\), and we solve backward to \(t < T\)). The coefficients involve \(S\), making it variable-coefficient — but a change of variables will transform it into the heat equation.

7.4 Reduction to the Heat Equation

We transform the Black-Scholes PDE into the standard heat equation through a sequence of changes of variables.

Step 1: Let \(S = e^x\), so \(x = \ln S \in \mathbb{R}\). The log-price is a more natural variable since it ranges over all of \(\mathbb{R}\).

Step 2: Reverse time. Let \(\tau = T - t \geq 0\). The terminal condition at \(t = T\) becomes an initial condition at \(\tau = 0\).

Step 3: Remove the exponential growth by writing \(V = e^{\alpha x + \beta \tau} u(x, \tau)\) for constants \(\alpha\) and \(\beta\) to be determined.

\[ \frac{\partial u}{\partial \tau} = \frac{\sigma^2}{2}\frac{\partial^2 u}{\partial x^2}. \]

This is exactly the heat equation with diffusivity \(\kappa = \sigma^2/2\)! The initial condition \(u(x, 0)\) is determined by the payoff function.

7.5 The Black-Scholes Formula

Applying the heat kernel solution (Chapter 6) to the transformed initial value problem and reversing the change of variables yields the celebrated Black-Scholes formula.

The formula has a beautiful economic interpretation. \(N(d_2)\) is the risk-neutral probability that the option expires in the money (\(S_T > K\)) — the probability, computed under a risk-neutral measure that replaces the drift \(\mu\) with the risk-free rate \(r\). The term \(Ke^{-r(T-t)}N(d_2)\) is the discounted expected strike payment. The term \(S\,N(d_1)\) accounts for the expected receipt of the stock, adjusted for the probability weighting. The option price is the present value of the expected net benefit from exercising.

7.6 The Greeks

The partial derivatives of the option price with respect to its parameters are called the Greeks, and they quantify the sensitivities used by traders to manage risk.

Note that the Black-Scholes PDE can be written in terms of the Greeks as \(\Theta + \frac{1}{2}\sigma^2 S^2 \Gamma + rS\Delta - rC = 0\). This form has the interpretation: the time decay \(\Theta\) is offset by the gain from Gamma (the curvature of the option’s value in \(S\)) and the return earned on the delta hedge.

7.7 Assumptions and Limitations

The Black-Scholes model rests on several idealising assumptions worth understanding critically:

• Constant volatility: In reality, implied volatility varies with strike and maturity (the “volatility smile” or “volatility surface”). Models such as stochastic volatility (Heston) or local volatility (Dupire) extend Black-Scholes to address this.
• Continuous trading: The delta-hedging argument requires instantaneous and costless rebalancing. In practice, transaction costs and discrete rebalancing introduce hedging error.
• Log-normal distribution: Empirical returns exhibit heavier tails than the normal distribution — so-called “fat tails” or excess kurtosis — meaning extreme events occur more often than the model predicts.
• No dividends, constant interest rate: These can be incorporated by straightforward modifications to the formula.
• Complete markets: The assumption that all risk can be perfectly hedged is an idealisation.

Despite these limitations, Black-Scholes remains the universal benchmark in options markets. Practitioners quote option prices in terms of “implied volatility” — the value of \(\sigma\) that makes the Black-Scholes formula reproduce the observed market price — as a common language, even when more sophisticated models are used for actual pricing.

7.8 Numerical Black-Scholes Example

7.9 The Arc from ODEs to Black-Scholes

It is worth pausing to appreciate the mathematical arc of this course. We began with the simplest differential equations — separable ODEs describing exponential growth in a bank account — and progressively built the tools needed to handle more complex phenomena. First-order linear ODEs gave us the integrating factor and debt dynamics. Higher-order linear ODEs introduced the characteristic equation and superposition. Systems of ODEs introduced eigenvalues and phase portraits. PDEs introduced the method of characteristics and the classification of equations. The Fourier Transform gave us the heat kernel. And finally, Itô’s lemma and the delta-hedging argument produced the Black-Scholes PDE — which, through a clever change of variables, reduces to the very heat equation we solved in Chapter 6.

The Black-Scholes formula is thus the culmination of a long chain of mathematical ideas, each building on the last. The diffusion of heat and the diffusion of uncertainty in financial markets are not merely analogous — they are governed by the same equation, and the same mathematical techniques solve both.

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Chapter 8: Phase Plane Analysis and Nonlinear Systems

The eigenvalue method of Chapter 4 gives exact solutions for linear systems. But most real systems — ecological interactions, epidemic spread, market dynamics — are genuinely nonlinear, and exact solutions are rarely available. The phase plane is the tool that compensates for this: it provides a complete qualitative picture of all trajectories without requiring closed-form expressions.

8.1 Phase Portraits and Equilibrium Classification

For a two-dimensional autonomous system \(\mathbf{x}' = \mathbf{f}(\mathbf{x})\), a phase portrait is the family of solution curves in the \((x_1, x_2)\) plane, with arrows indicating the direction of increasing \(t\). The key objects are the nullclines — curves where one component of \(\mathbf{f}\) is zero — and the equilibria (or fixed points) where \(\mathbf{f}(\mathbf{x}^*) = \mathbf{0}\).

For linear systems \(\mathbf{x}' = A\mathbf{x}\), the origin is always an equilibrium. The character of all other trajectories is determined entirely by the eigenvalues of \(A\). Writing \(\tau = \mathrm{tr}(A)\) and \(\Delta = \det(A)\), the trace-determinant criterion gives a complete classification.

8.2 Competitive Species and Predator-Prey Models

The power of phase plane analysis lies in its application to genuinely nonlinear systems. A canonical example is the Lotka-Volterra competition model, which describes two species competing for the same resource.

Let \(x(t)\) and \(y(t)\) be the populations of two competing species. Each grows logistically in isolation, but each inhibits the other:

\[ \frac{dx}{dt} = r_1 x\!\left(1 - \frac{x}{K_1} - \frac{\alpha_{12} y}{K_1}\right), \qquad \frac{dy}{dt} = r_2 y\!\left(1 - \frac{y}{K_2} - \frac{\alpha_{21} x}{K_2}\right). \]

The parameter \(\alpha_{12}\) measures how much a unit of species 2 “costs” species 1 in terms of carrying-capacity units; \(\alpha_{21}\) is the reciprocal effect. Equilibria occur where both nullclines intersect. The \(x\)-nullclines are \(x = 0\) and \(x + \alpha_{12} y = K_1\); the \(y\)-nullclines are \(y = 0\) and \(\alpha_{21} x + y = K_2\).

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Chapter 9: Linearisation of Nonlinear Systems and the Hartman-Grobman Theorem

The trace-determinant analysis applies directly to linear systems. For nonlinear systems, the key insight is that near a hyperbolic equilibrium, the nonlinear system looks like its linearisation — and the phase portrait of the linearisation accurately represents that of the nonlinear system.

The Jacobian at an equilibrium is computed by differentiating \(\mathbf{f}\) and evaluating at the equilibrium point.

9.1 The SIR Epidemic Model

The SIR model (Kermack-McKendrick, 1927) is the canonical model of epidemic spread. A population of fixed size \(N\) is divided into: \(S\) (susceptibles), \(I\) (infectives), \(R\) (recovered/removed). The dynamics are:

\[ \frac{dS}{dt} = -\beta S I, \qquad \frac{dI}{dt} = \beta S I - \gamma I, \qquad \frac{dR}{dt} = \gamma I, \]

where \(\beta > 0\) is the transmission rate and \(\gamma > 0\) is the recovery rate. Since \(S + I + R = N\) is conserved, \(R\) is determined by \(S\) and \(I\), so we work in the two-dimensional \((S, I)\) plane.

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Chapter 10: The IS-LM Model as a Dynamical System

The IS-LM framework introduced in Chapter 2 deserves a more careful dynamical treatment. In the static version, national income \(Y\) and the interest rate \(r\) are simultaneously determined by goods market equilibrium (the IS curve) and money market equilibrium (the LM curve). In the dynamic version, these adjust at finite speeds, generating a system of ODEs.

10.1 Dynamic IS-LM

Let income and the interest rate satisfy:

\[ \frac{dY}{dt} = \alpha\bigl[I(r) + G - sY\bigr], \qquad \frac{dr}{dt} = \beta\bigl[kY - \bar{M}/P - hr\bigr], \]

where \(s\) is the marginal propensity to save, \(k > 0\) is the income sensitivity of money demand, \(h > 0\) is the interest sensitivity of money demand, \(\bar{M}/P\) is the real money supply, \(I(r) = I_0 - br\) is investment (decreasing in \(r\)), and \(\alpha, \beta > 0\) are adjustment speeds. Linearising near the IS-LM equilibrium \((Y^*, r^*)\):

\[ \frac{d}{dt}\begin{pmatrix}Y - Y^* \\ r - r^*\end{pmatrix} = \begin{pmatrix} -\alpha s & -\alpha b \\ \beta k & -\beta h \end{pmatrix}\begin{pmatrix}Y - Y^* \\ r - r^*\end{pmatrix}. \]

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Chapter 11: The Solow Model with Human Capital

The standard Solow model of Chapter 2 explains convergence in physical capital \(k\) but leaves an uncomfortably large residual — “total factor productivity” — unexplained. Mankiw, Romer, and Weil (1992) extended the model by adding human capital \(h\) (skills, education) as a separate accumulable factor. This MRW extension resolves the empirical puzzle and is solvable as a two-dimensional linear system after log-linearisation.

11.1 The MRW Equations

With a Cobb-Douglas production function \(Y = K^\alpha H^\phi L^{1-\alpha-\phi}\) (where \(\alpha + \phi < 1\)), fractions \(s_k\) and \(s_h\) of output invested in physical and human capital, and common depreciation and population growth \((n + \delta)\), the dynamics of \(k = K/L\) and \(h = H/L\) are:

\[ \frac{dk}{dt} = s_k k^\alpha h^\phi - (n+\delta)k, \qquad \frac{dh}{dt} = s_h k^\alpha h^\phi - (n+\delta)h. \]

The steady state \((k^*, h^*)\) satisfies both equations simultaneously. Taking logs and setting \(\hat{k} = \ln k - \ln k^*\) and \(\hat{h} = \ln h - \ln h^*\) (log-deviations from steady state), the linearised system is:

\[ \frac{d}{dt}\begin{pmatrix}\hat{k}\\\hat{h}\end{pmatrix} = (n+\delta)\begin{pmatrix}\alpha - 1 & \phi \\ \alpha & \phi - 1\end{pmatrix}\begin{pmatrix}\hat{k}\\\hat{h}\end{pmatrix}. \]

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Chapter 12: Laplace Transforms

While the Fourier Transform is the right tool for problems on the entire real line, the Laplace Transform is designed for initial-value problems on the half-line \(t \geq 0\). It converts an ODE with initial conditions into an algebraic equation, solves it in “transform space,” and inverts. The power of the Laplace Transform lies especially in handling discontinuous or impulsive forcing — a sudden policy change, a tax reform implemented at \(t = t_0\), or a sharp shock to investment.

The transform is linear: \(\mathcal{L}\{af + bg\} = a\mathcal{L}\{f\} + b\mathcal{L}\{g\}\). Its defining property for ODEs is the derivative rule: \(\mathcal{L}\{f'\}(s) = sF(s) - f(0)\), and more generally \(\mathcal{L}\{f^{(n)}\}(s) = s^n F(s) - s^{n-1}f(0) - \cdots - f^{(n-1)}(0)\). Initial conditions are automatically incorporated.

12.1 Table of Key Transforms

The following table collects the transforms needed for most applications:

\(f(t)\)	\(F(s) = \mathcal{L}\{f\}(s)\)	Condition
\(1\)	\(1/s\)	\(s > 0\)
\(e^{at}\)	\(1/(s-a)\)	\(s > a\)
\(t^n\)	\(n!/s^{n+1}\)	\(s > 0\)
\(\sin(\omega t)\)	\(\omega/(s^2+\omega^2)\)	\(s > 0\)
\(\cos(\omega t)\)	\(s/(s^2+\omega^2)\)	\(s > 0\)
\(e^{at}\sin(\omega t)\)	\(\omega/((s-a)^2+\omega^2)\)	\(s > a\)
\(e^{at}\cos(\omega t)\)	\((s-a)/((s-a)^2+\omega^2)\)	\(s > a\)
\(u_c(t) = \mathbf{1}_{t \geq c}\)	\(e^{-cs}/s\)	\(s > 0\)
\(u_c(t)f(t-c)\)	\(e^{-cs}F(s)\)	(Second shifting theorem)
\(\delta(t - c)\)	\(e^{-cs}\)	(Dirac delta)
\((f*g)(t) = \int_0^t f(\tau)g(t-\tau)d\tau\)	\(F(s)G(s)\)	(Convolution theorem)

Here \(u_c(t)\) is the Heaviside step function that switches on at \(t = c\): \(u_c(t) = 0\) for \(t < c\) and \(u_c(t) = 1\) for \(t \geq c\).

12.2 Solving an IVP with Discontinuous Forcing

12.3 Economic Application: Policy Shock

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Chapter 13: Power Series and the Method of Frobenius

Not every ODE with variable coefficients can be solved by the previous methods. When the coefficient functions are analytic (expressible as convergent power series), we can seek solutions as power series too. This is particularly relevant for the Euler equation and its generalisations, which arise in the Solow model and in spatial economics when problems are posed in polar or spherical coordinates.

13.1 The Euler Equation

\[ x^2 y'' + \alpha x y' + \beta y = 0, \quad x > 0. \]\[ r^2 + (\alpha - 1)r + \beta = 0. \]

The substitution \(x = e^t\) (i.e., \(t = \ln x\)) converts the Euler equation into a constant-coefficient ODE in \(t\), which can be solved by the characteristic equation method.

13.2 Power Series Solution Near an Ordinary Point

Near an ordinary point, the method of power series is simpler than Frobenius — no indicial equation is needed.

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Chapter 14: Travelling Waves and the Fisher-KPP Equation

The final topic brings together nonlinear ODEs, phase plane analysis, and PDE theory in an application with direct epidemiological and economic interpretations: travelling wave solutions to reaction-diffusion equations.

14.1 The Fisher-KPP Equation

Consider a population \(u(x,t)\) — the density of infected individuals in a spatial domain, or the density of adopters of a new technology — that both diffuses and grows logistically:

\[ \frac{\partial u}{\partial t} = D\frac{\partial^2 u}{\partial x^2} + ru\!\left(1 - u\right), \]

where we have scaled so that the carrying capacity is 1. This is the Fisher-KPP equation (Fisher 1937; Kolmogorov, Petrovsky, Piskunov 1937). The key feature is that it admits travelling wave solutions of the form \(u(x,t) = \varphi(x - ct)\) — a front that moves at constant speed \(c\) while maintaining its shape.

Substituting \(u = \varphi(\xi)\) into the PDE: \(-c\varphi' = D\varphi'' + r\varphi(1-\varphi)\), or

\[ D\varphi'' + c\varphi' + r\varphi(1-\varphi) = 0. \]

This is a second-order nonlinear ODE for \(\varphi(\xi)\). Setting \(p = \varphi'\), we convert to a first-order system in the phase plane \((\varphi, p)\):

\[ \frac{d\varphi}{d\xi} = p, \qquad \frac{dp}{d\xi} = -\frac{c}{D}p - \frac{r}{D}\varphi(1-\varphi). \]

The equilibria are \((\varphi, p) = (0, 0)\) and \((1, 0)\). A travelling wave corresponds to a trajectory in the \((\varphi, p)\) phase plane connecting \((1,0)\) to \((0,0)\) — a heteroclinic orbit.

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Appendix: Key Formulas and Solution Methods

First-Order ODEs: Solution Map

Equation type	Solution method	Result
\(y' = ry\)	Separation	\(y = y_0 e^{rt}\)
\(y' = ry(1-y/K)\)	Separation (logistic)	\(y = K/(1+Ae^{-rt})\), \(A = (K-y_0)/y_0\)
\(y' + P(x)y = Q(x)\)	Integrating factor \(\mu = e^{\int P}\)	\(y = \mu^{-1}(\int \mu Q\,dx + C)\)
\(y' + P(x)y = Q(x)y^n\)	Bernoulli: \(v = y^{1-n}\)	Linear ODE for \(v\)
Newton cooling \(T' = k(A-T)\)	Linear 1st order	\(T = A + (T_0-A)e^{-kt}\)

Second-Order ODE: Characteristic Roots

For \(ay'' + by' + cy = 0\) with discriminant \(\Delta = b^2 - 4ac\):

• \(\Delta > 0\) (overdamped / distinct real): \(y = c_1 e^{r_1 x} + c_2 e^{r_2 x}\)
• \(\Delta = 0\) (critically damped / repeated): \(y = (c_1 + c_2 x)e^{rx}\)
• \(\Delta < 0\) (underdamped / complex): \(y = e^{\alpha x}(A\cos\beta x + B\sin\beta x)\)

Resonance: multiply particular solution guess by \(x^s\) where \(s\) = multiplicity of the forcing frequency as a characteristic root.

Economic Models Summary

Model	ODE/System	Key result
Continuous compounding	\(A' = rA\)	\(A = A_0e^{rt}\)
Logistic adoption	\(P' = rP(1-P/K)\)	S-curve; inflection at \(P = K/2\)
Public debt (Domar)	\(D' = rD + G_0e^{gt}\)	Sustainable iff \(r < g\)
Solow growth	\(k' = sk^\alpha - (n+\delta)k\)	Stable \(k^*\) via Bernoulli substitution
Samuelson cycle	\(Y'' - (c-1+a)Y' + (1-c)Y = G_0\)	Oscillatory iff \((c-1+a)^2 < 4(1-c)\)
Price adjustment	\(P' = \alpha(D(P)-S(P))\)	Stable iff \(D'(P^*) < S'(P^*)\)
SIR epidemic	\(S' = -\beta SI,\; I' = \beta SI - \gamma I\)	Epidemic iff \(\mathcal{R}_0 = \beta N/\gamma > 1\)
IS-LM dynamic	\(Y' = \alpha[I(r)+G-sY],\; r' = \beta[kY-hr-\bar{M}/P]\)	Always stable; node/spiral by \(\tau^2 - 4\Delta\)
MRW (Solow + human capital)	\(dk/dt = s_k k^\alpha h^\phi - (n+\delta)k\)	Convergence rate \(\mu = (n+\delta)(1-\alpha-\phi)\)
Fisher-KPP (spatial spread)	\(u_t = Du_{xx} + ru(1-u)\)	Wave speed \(c^* = 2\sqrt{Dr}\)

Black-Scholes Quick Reference

PDE: \(V_t + \frac{1}{2}\sigma^2 S^2 V_{SS} + rSV_S - rV = 0\)

\[ C = SN(d_1) - Ke^{-r\tau}N(d_2), \quad d_1 = \frac{\ln(S/K)+(r+\sigma^2/2)\tau}{\sigma\sqrt{\tau}}, \quad d_2 = d_1 - \sigma\sqrt{\tau}. \]

Put-call parity: \(C - P = S - Ke^{-r\tau}\).

Delta: \(\Delta = N(d_1)\). Vega: \(\mathcal{V} = S\sqrt{\tau}N'(d_1)\).

Laplace Transform Quick Reference

Key pairs: \(\mathcal{L}\{e^{at}\} = 1/(s-a)\); \(\mathcal{L}\{\sin\omega t\} = \omega/(s^2+\omega^2)\); \(\mathcal{L}\{\cos\omega t\} = s/(s^2+\omega^2)\); \(\mathcal{L}\{t^n\} = n!/s^{n+1}\); \(\mathcal{L}\{u_c(t)f(t-c)\} = e^{-cs}F(s)\).

Derivative rule: \(\mathcal{L}\{y''\} = s^2Y - sy(0) - y'(0)\). Converts an IVP to an algebraic equation for \(Y(s)\).

Solving procedure: (1) Transform the ODE; (2) solve for \(Y(s)\) algebraically; (3) apply partial fractions; (4) invert using the table and shifting theorems.

Trace-Determinant Quick Reference

For \(\mathbf{x}' = A\mathbf{x}\) with \(\tau = \mathrm{tr}(A)\), \(\Delta = \det(A)\):

• \(\Delta < 0\): saddle (always unstable)
• \(\Delta > 0\), \(\tau < 0\): stable (\(\tau^2 > 4\Delta\): node; \(\tau^2 < 4\Delta\): spiral)
• \(\Delta > 0\), \(\tau > 0\): unstable (\(\tau^2 > 4\Delta\): node; \(\tau^2 < 4\Delta\): spiral)
• \(\Delta > 0\), \(\tau = 0\): centre

Euler Equation Quick Reference

\(x^2y'' + \alpha x y' + \beta y = 0\): substitute \(y = x^r\), indicial equation \(r^2 + (\alpha-1)r + \beta = 0\).

• Distinct real roots \(r_1 \neq r_2\): \(y = c_1 x^{r_1} + c_2 x^{r_2}\).
• Repeated root \(r\): \(y = x^r(c_1 + c_2\ln x)\).
• Complex roots \(r = \mu \pm \nu i\): \(y = x^\mu(c_1\cos(\nu\ln x) + c_2\sin(\nu\ln x))\).

Heat equation connection: \(x = \ln S\), \(\tau = T-t\), exponential substitution \(\Rightarrow\) \(u_\tau = \frac{\sigma^2}{2}u_{xx}\), solved by the heat kernel.