Chapter 5 — Explicitly Solvable Models

Guiding question

Which SDEs can be solved in closed form? What techniques reduce a nonlinear SDE to a linear one, and what do the solutions reveal about mean-reversion, fat tails, and the geometry of financial dynamics?

Most SDEs require numerical simulation. But a small, important family can be solved exactly using Itô's formula and algebraic tricks. These exact solutions are the calibration targets for more complex models and the benchmarks for numerical schemes.

Three canonical models appear throughout quantitative finance:

Model	SDE	Used for
GBM	$dS = \mu S\,dt + \sigma S\,dW$	Equity prices, Black–Scholes
Ornstein–Uhlenbeck	$dX = \kappa(\theta-X)\,dt + \sigma\,dW$	Interest rates, spreads, log-vol
CIR	$dX = \kappa(\theta-X)\,dt + \sigma\sqrt{X}\,dW$	Interest rates, variance

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5.1 Geometric Brownian Motion

We derived in Chapter 3 that

$$dS_t = \mu S_t\,dt + \sigma S_t\,dW_t,\qquad S_0 > 0,$$

has the explicit solution

$$S_t = S_0\exp\!\left[\left(\mu - \tfrac12\sigma^2\right)t + \sigma W_t\right].$$

Let us now study this solution carefully.

Log-normal distribution. Since $W_t\sim N(0,t)$, we have

$$\log S_t \sim N\!\left(\log S_0 + (\mu-\tfrac12\sigma^2)t,\ \sigma^2 t\right).$$

The price $S_t$ is log-normally distributed: $S_t = e^{\text{Normal}}$.

Mean and variance.

$$\mathbb E[S_t] = S_0 e^{\mu t}, \qquad \text{Var}(S_t) = S_0^2 e^{2\mu t}(e^{\sigma^2 t}-1).$$

Median.

$$\text{Median}(S_t) = S_0 e^{(\mu-\sigma^2/2)t},$$

which is less than the mean for all $\sigma > 0$. This is Jensen's inequality: $\mathbb E[e^X] \ge e^{\mathbb E[X]}$ for convex functions.

Volatility drag: why higher vol hurts median growth

Imagine two assets, both with $\mu = 10\%$. Asset A has $\sigma = 10\%$, Asset B has $\sigma = 40\%$.

After $T=10$ years:

• $\mathbb E[S_T/S_0] = e^{0.1\times 10} \approx 2.72$ (same for both)
• Median A: $e^{(0.10-0.005)\times 10} = e^{0.95}\approx 2.59$
• Median B: $e^{(0.10-0.08)\times 10} = e^{0.20}\approx 1.22$

Asset B's expected value is the same, but more than half the scenarios produce less than 22% gain over 10 years. High volatility depresses the typical outcome even when the mean is unchanged. This is the practical meaning of the $-\sigma^2/2$ Itô correction.

Geometric Brownian Motion: S_t = S₀ exp((μ − σ²/2)t + σW_t)

μ = 0.08σ = 0.25Log scale

GBM log-returns are normal: log(S_t/S_0) ~ N((μ−σ²/2)t, σ²t). Notice that higher σ widens the band but also drags the median below the mean — Jensen's inequality in action.

Return autocorrelation. Log-returns $r_k = \log(S_{t_{k+1}}/S_{t_k})$ are i.i.d. $N((\mu-\sigma^2/2)\Delta t, \sigma^2\Delta t)$. No autocorrelation, by the independence of Brownian increments.

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5.2 The Ornstein–Uhlenbeck process

OU process: explicit solution

The SDE

$$dX_t = \kappa(\theta - X_t)\,dt + \sigma\,dW_t, \qquad X_0 = x_0,$$

has the unique strong solution

$$X_t = \theta + (x_0-\theta)e^{-\kappa t} + \sigma\int_0^t e^{-\kappa(t-s)}\,dW_s.$$

Derivation via integrating factor

Define $Y_t = e^{\kappa t}(X_t - \theta)$. By Itô's product rule:

$$dY_t = \kappa e^{\kappa t}(X_t-\theta)\,dt + e^{\kappa t}dX_t.$$

Substituting $dX_t = \kappa(\theta-X_t)\,dt + \sigma\,dW_t$:

$$dY_t = e^{\kappa t}[\kappa(X_t-\theta)\,dt + \kappa(\theta-X_t)\,dt + \sigma\,dW_t] = \sigma e^{\kappa t}\,dW_t.$$

Integrating: $Y_t = Y_0 + \sigma\int_0^t e^{\kappa s}\,dW_s$.

Converting back: $X_t - \theta = e^{-\kappa t}Y_t = (x_0-\theta)e^{-\kappa t} + \sigma\int_0^t e^{-\kappa(t-s)}\,dW_s$.

Distribution. Since the stochastic integral is a Gaussian with mean zero and variance $\sigma^2\int_0^t e^{-2\kappa(t-s)}\,ds = \frac{\sigma^2}{2\kappa}(1-e^{-2\kappa t})$:

$$X_t \sim N\!\left(\theta + (x_0-\theta)e^{-\kappa t},\ \frac{\sigma^2}{2\kappa}(1-e^{-2\kappa t})\right).$$

Stationary distribution. As $t\to\infty$, $X_t$ converges in distribution to

$$X_\infty \sim N\!\left(\theta,\ \frac{\sigma^2}{2\kappa}\right).$$

This is the unique stationary distribution of the OU process.

Mean reversion as exponential decay

The deterministic part of the solution is $(x_0-\theta)e^{-\kappa t}$: the displacement from the long-run mean $\theta$ decays exponentially at rate $\kappa$. The half-life of deviations is $\log 2/\kappa$. The stochastic part is the accumulated noise weighted by the exponential kernel $e^{-\kappa(t-s)}$ — older noise matters less. The OU process is a memory-fading random process: its past is discounted, not forgotten.

Ornstein–Uhlenbeck: dX = κ(θ − X)dt + σ dW

κ = 2.0θ = 1.0σ = 0.50

All paths are pulled toward θ at rate κ. Increasing κ tightens the band and speeds mean-reversion. The stationary distribution is N(θ, σ²/2κ). In finance, κ models interest-rate mean reversion speed.

Autocovariance. For $s\le t$:

$$\text{Cov}(X_s,X_t) = \frac{\sigma^2}{2\kappa}e^{-\kappa(t-s)}(1-e^{-2\kappa s}).$$

In the stationary regime ($s\to\infty$, $t-s = \tau$ fixed):

$$\text{Cov}(X_{s+\tau},X_s) \approx \frac{\sigma^2}{2\kappa}e^{-\kappa\tau}.$$

The autocovariance decays exponentially — a characteristic signature of mean reversion.

OU in finance: log-volatility, spreads, convenience yields

The OU process models quantities that fluctuate around a long-run level:

• Interest rates (Vasicek model): short rate $r_t$ reverts to $\theta$
• Log-volatility in rough/stochastic vol models: $\log\sigma_t$ follows OU
• Commodity convenience yield: the net benefit of holding physical commodity
• Credit spreads: OU on spreads captures mean reversion after credit events

The key feature is that extreme values are self-correcting: high values have negative expected drift, low values have positive expected drift.

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5.3 The CIR process

The Cox–Ingersoll–Ross (CIR) process is:

$$dX_t = \kappa(\theta - X_t)\,dt + \sigma\sqrt{X_t}\,dW_t, \qquad X_0 > 0.$$

Unlike OU, the diffusion coefficient $\sigma\sqrt{X_t}$ vanishes at $X_t = 0$, which prevents the process from going negative (under the Feller condition).

Feller condition for positivity

If $2\kappa\theta \ge \sigma^2$ (the Feller condition), then $X_t > 0$ for all $t$, almost surely.

If $2\kappa\theta < \sigma^2$, then $X_t$ touches zero and reflects, but the SDE remains well-posed.

Why the CIR diffusion coefficient matters. In the OU process, noise is constant: $\sigma\,dW$. In CIR, noise scales as $\sigma\sqrt{X_t}\,dW$: when $X_t$ is large, noise is proportionally larger; when $X_t\to 0$, noise vanishes, and the drift $\kappa\theta > 0$ pushes the process back up.

The Feller boundary at zero

The boundary $\{X=0\}$ is a reflecting boundary for the CIR process when the Feller condition fails ($2\kappa\theta < \sigma^2$), or an unattainable boundary when it holds. The scaling $\sqrt{X}$ of the diffusion near zero creates a repulsion from the boundary that is precisely calibrated by the Feller condition: when the drift $\kappa\theta$ is strong enough relative to $\sigma^2$, the boundary is never reached.

Distribution. The conditional distribution of $X_t$ given $X_0=x_0$ is a scaled non-central chi-squared:

$$\frac{4\kappa}{\sigma^2(1-e^{-\kappa t})}\,X_t \sim \chi^2\!\left(\frac{4\kappa\theta}{\sigma^2},\ \lambda\right),$$

where $\lambda = \frac{4\kappa x_0 e^{-\kappa t}}{\sigma^2(1-e^{-\kappa t})}$ is the non-centrality parameter.

Stationary distribution. As $t\to\infty$, $X_t$ converges to a Gamma distribution:

$$X_\infty \sim \text{Gamma}\!\left(\frac{2\kappa\theta}{\sigma^2},\ \frac{\sigma^2}{2\kappa}\right).$$

Mean $= \theta$, variance $= \frac{\sigma^2\theta}{2\kappa}$.

CIR in finance: interest rates and variance

Vasicek/CIR interest rates: The short rate $r_t$ in the CIR model stays positive (under the Feller condition), unlike Vasicek where negative rates are possible. CIR became a benchmark short-rate model for interest rate derivatives.

Heston stochastic volatility: The variance process $V_t$ in the Heston model satisfies $dV = \kappa(\theta-V)\,dt + \xi\sqrt{V}\,dW^v$. The Feller condition $2\kappa\theta > \xi^2$ ensures $V_t > 0$, keeping volatility well-defined.

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5.4 Linear SDEs and the integrating factor method

The technique that solved the OU process generalizes. A linear SDE has the form:

$$dX_t = [a(t) + b(t)X_t]\,dt + [c(t) + d(t)X_t]\,dW_t.$$

The solution uses the stochastic integrating factor $\Phi_t$ satisfying

$$d\Phi_t = -b(t)\Phi_t\,dt - d(t)\Phi_t\,dW_t.$$

By Itô's formula, this exponential process has the form

$$\Phi_t = \exp\!\left(-\int_0^t b_s\,ds - \int_0^t d_s\,dW_s - \tfrac12\int_0^t d_s^2\,ds\right),$$

and $d(\Phi_t X_t) = \Phi_t\,c(t)\,dW_t + \Phi_t\,a(t)\,dt$, which can then be integrated.

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5.5 Summary table of explicit solutions

Model	SDE	Solution
GBM	$dS = \mu S\,dt + \sigma S\,dW$	$S_t = S_0 e^{(\mu-\sigma^2/2)t+\sigma W_t}$
OU	$dX = \kappa(\theta-X)\,dt + \sigma\,dW$	$X_t = \theta + (x_0-\theta)e^{-\kappa t} + \sigma\int_0^t e^{-\kappa(t-s)}dW_s$
Brownian bridge	$dX = \frac{\theta-X_t}{T-t}\,dt + dW$	$X_t = x_0\frac{T-t}{T} + \theta\frac{t}{T} + \int_0^t\frac{T-t}{T-s}\,dW_s$
Affine diffusion	$dX = (a+bX)\,dt + (c+dX)\,dW$	Via integrating factor + Itô

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5.6 Problems

Problem 1: OU autocovariance

For the stationary OU process, compute $\text{Corr}(X_t,X_{t+\tau})$ and show it equals $e^{-\kappa\tau}$.

Problem 2: OU variance

Verify from the explicit solution that $\text{Var}(X_t) = \frac{\sigma^2}{2\kappa}(1-e^{-2\kappa t})$.

Problem 3: Log-normal moments

For GBM, compute $\mathbb P(S_T > K)$ in terms of the normal CDF $\Phi$. (This is the $N(d_2)$ term in Black–Scholes.)

Problem 4: Feller condition interpretation

For CIR with parameters $\kappa=2$, $\theta=0.04$, $\sigma=0.1$, check whether the Feller condition holds. What happens to the variance of the stationary distribution as $\sigma$ increases toward the Feller boundary $\sigma^2 = 2\kappa\theta$?

Problem 5: Brownian bridge

A Brownian bridge $B_t$ satisfies $B_0 = 0$ and $B_T = 0$. Show that $B_t = W_t - \frac{t}{T}W_T$ has this property. Find the SDE satisfied by $B_t$, and verify it matches the Brownian bridge entry in the table above (with $\theta = 0$).

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5.7 Selected solutions

Solution 1

In the stationary OU process, $X_s$ and $X_{s+\tau}$ are jointly Gaussian. Using the explicit solution:

$$\text{Cov}(X_s, X_{s+\tau}) = e^{-\kappa\tau}\text{Var}(X_s) = \frac{\sigma^2}{2\kappa}e^{-\kappa\tau}.$$

Since $\text{Var}(X_s) = \sigma^2/(2\kappa)$ in stationarity, $\text{Corr}(X_s,X_{s+\tau}) = e^{-\kappa\tau}$.

Solution 3

$S_T = S_0 e^{(\mu-\sigma^2/2)T + \sigma W_T}$. Then $S_T > K$ iff $\sigma W_T > \log(K/S_0) - (\mu-\sigma^2/2)T$, iff $W_T/\sqrt{T} > d_2$ where

$$d_2 = \frac{\log(K/S_0) - (\mu-\sigma^2/2)T}{\sigma\sqrt{T}}.$$

Wait, we want $S_T > K$:

$$\mathbb P(S_T > K) = \mathbb P\!\left(\frac{W_T}{\sqrt{T}} > \frac{\log(K/S_0)-(\mu-\sigma^2/2)T}{\sigma\sqrt{T}}\right) = \Phi(-d_2) = \Phi(d_2'),$$

where $d_2' = \frac{\log(S_0/K)+(\mu-\sigma^2/2)T}{\sigma\sqrt{T}}$. Under the risk-neutral measure ($\mu = r$), this is the $N(d_2)$ in Black–Scholes.

Solution 5

$B_t = W_t - \frac{t}{T}W_T$. Clearly $B_0 = 0$. To get the SDE, use Itô's formula for $B_t = W_t - \frac{t}{T}W_T$. Note $W_T = W_T$ is fixed at maturity $T$, not $W_t$. Write instead:

$$B_t = \int_0^t\left(1 - \frac{t}{T}\right)dW_s - \int_0^t \frac{s}{T}\,\frac{dW_s}{1} + \cdots$$

Actually, the SDE for the Brownian bridge is derived by noting $B_T = 0$ must hold, so the drift must absorb the Brownian path. Differentiating $B_t = W_t - \frac{t}{T}W_T$:

$$dB_t = dW_t - \frac{1}{T}W_T\,dt.$$

Now $W_T = B_T + \frac{T}{T}W_T = W_T$, and $\frac{W_T - B_t}{T-t}$ is the conditional expected value of what the remaining path must cover. The correct SDE form is:

$$dB_t = -\frac{B_t}{T-t}\,dt + dW_t,$$

with the drift pulling $B_t$ toward zero at rate $\frac{1}{T-t}$, which becomes infinite as $t\to T$, forcing the bridge to hit zero.

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5.8 What this chapter gives you

The three canonical models encode three fundamentally different behaviors:

• GBM: multiplicative noise, log-normal distribution, no mean reversion, used for asset prices.
• OU: additive noise, Gaussian distribution, exponential mean reversion, used for rates and spreads.
• CIR: multiplicative noise scaled by $\sqrt{X}$, gamma stationary distribution, positivity preservation, used for variances and rates.

Each is solved by the integrating factor method, which generalizes to all linear SDEs. The next chapter introduces the change-of-measure technique — Girsanov's theorem — which is the analytical foundation of risk-neutral pricing.