Actuarial Notes

Stochastic Differential Equation

Study guide: Exam INV-201

A stochastic differential equation (SDE) specifies a process X_t by its instantaneous drift \mu and diffusion (volatility) \sigma in terms of a Brownian motion W_t. It is shorthand for an integral equation whose last term is an Itō integral, and its solution is a whole distribution of paths. SDEs are how stochastic models of stock prices, short rates and volatilities are written, simulated and used for derivative pricing.

dX_t = \mu(X_t,t)\,dt + \sigma(X_t,t)\,dW_t

X_t = X_0 + \int_0^t \mu(X_s,s)\,ds + \int_0^t \sigma(X_s,s)\,dW_s

S_T = S_0\exp\!\left[\left(\mu - \tfrac12\sigma^2\right)T + \sigma W_T\right]

r_T = b + (r_0 - b)e^{-aT} + \sigma\int_0^T e^{-a(T-s)}\,dW_s

E[r_T] = b + (r_0 - b)e^{-aT}

\operatorname{Var}^{\text{Vas}}(r_T) = \frac{\sigma^2}{2a}\left(1 - e^{-2aT}\right)

\begin{aligned} \operatorname{Var}^{\text{CIR}}(r_T) = {} & \frac{r_0\sigma^2}{a}\left(e^{-aT} - e^{-2aT}\right) \\ & + \frac{b\sigma^2}{2a}\left(1 - e^{-aT}\right)^2 \end{aligned}

  • Symbols: W_t is a standard Brownian motion, \mu is the drift (expected change per unit time) and \sigma is the diffusion coefficient. A solution is a process adapted to the history of W that satisfies the integral equation. Coefficients that are Lipschitz in x and grow at most linearly guarantee a unique solution.
  • Geometric Brownian motion (GBM): The SDE dS_t = \mu S_t\,dt + \sigma S_t\,dW_t has proportional drift and volatility. Itō's lemma on \ln S solves it in closed form (third block), so S_T is lognormal with E[S_T] = S_0e^{\mu T} and \operatorname{Var}(S_T) = S_0^2e^{2\mu T}\left(e^{\sigma^2T} - 1\right). It is the stock model of the Black-Scholes-Merton framework.
  • Vasicek (Ornstein-Uhlenbeck): Hull's chapter on equilibrium short-rate models writes it dr_t = a(b - r_t)\,dt + \sigma\,dW_t, where b is the long-run level and a is the speed of mean reversion. Multiplying by the integrating factor e^{at} and integrating solves it (fourth block), so r_T is normal with the mean and variance in the fifth and sixth blocks. Because r_T is normal, it can go negative.
  • Cox-Ingersoll-Ross (CIR): The model dr_t = a(b - r_t)\,dt + \sigma\sqrt{r_t}\,dW_t has the same drift as Vasicek, so r_T has the same mean. The volatility shrinks as the rate falls, so the rate stays non-negative, and it stays strictly positive when 2ab \ge \sigma^2. Its distribution is a scaled non-central chi-square rather than a normal, and its variance (last block) depends on r_0 and b.
  • Measure: The same SDE can be written under the real-world measure \mathbb{P} or the risk-neutral measure \mathbb{Q}. Girsanov's theorem changes the drift by the market price of risk and leaves \sigma alone, so a stock's drift \mu becomes r under \mathbb{Q}.
  • When there is no closed form: Most SDEs, such as those for stochastic volatility or for a short rate with a level-dependent volatility, cannot be solved explicitly. They are simulated on a time grid with the Euler-Maruyama scheme, X_{t+h} \approx X_t + \mu h + \sigma\sqrt{h}\,Z, or from an exact transition density where one is known.

Solving the GBM Equation for a Fund Value The equity fund behind a variable annuity follows dS_t = 0.10\,S_t\,dt + 0.30\,S_t\,dW_t with S_0 = 50. Solve the SDE and give the mean, standard deviation and median of S_1.

Answer Itō's lemma on \ln S gives d\ln S_t = \left(0.10 - \tfrac12(0.09)\right)dt + 0.30\,dW_t = 0.055\,dt + 0.30\,dW_t. Integrating from 0 to t and exponentiating solves the SDE.

\begin{align*}
S_t &= 50\exp\!\left(0.055\,t + 0.30\,W_t\right) \\
E[S_1] &= 50\,e^{0.10} \\
&= 55.26 \\
\operatorname{Var}(S_1) &= 50^2 e^{0.20}\left(e^{0.09} - 1\right) \\
&= 287.56 \\
\text{median}(S_1) &= 50\,e^{0.055} \\
&= 52.83
\end{align*}

The standard deviation is \sqrt{287.56} = 16.96. The median is below the mean because the lognormal distribution is skewed to the right, which matters for a guarantee such as a GMMB that pays in the lower tail.

Solving the Vasicek Equation and Finding the Expected Short Rate The short rate follows the Vasicek model dr_t = 0.2(0.05 - r_t)\,dt + 0.01\,dW_t with r_0 = 0.03. Solve the SDE, then find E[r_5], the standard deviation of r_5 and the probability that r_5 < 0.

Answer Apply the product rule to e^{at}r_t with a = 0.2. The r_t terms cancel, leaving d\left(e^{at}r_t\right) = ab\,e^{at}\,dt + \sigma e^{at}\,dW_t. Integrating from 0 to T and dividing by e^{aT} gives the fourth block above, whose stochastic integral has mean zero and, by the Itō isometry, variance \sigma^2\int_0^T e^{-2a(T-s)}\,ds.

\begin{align*}
E[r_5] &= 0.05 + (0.03 - 0.05)e^{-1} \\
&= 0.05 - 0.02(0.367879) \\
&= 0.042642 \\
\operatorname{Var}(r_5) &= \frac{0.01^2}{0.4}\left(1 - e^{-2}\right) \\
&= 0.000216166 \\
\Pr(r_5 < 0) &= N\!\left(\frac{-0.042642}{0.014703}\right) \\
&= N(-2.900) \\
&= 0.0019
\end{align*}

The expected rate has closed about 63% of the gap to b = 5\%, since 1 - e^{-1} = 0.632, and the standard deviation is 0.0147. A CIR model with the same drift and \sigma = 0.05 has the same mean and a variance of 0.000212, almost the same, but under CIR the probability of a negative rate is exactly zero.

How it appears on the exam

  • INV-201: The syllabus names stochastic differential equations first in its list of stochastic calculus techniques for pricing derivatives, with Hull's chapters on Itō's lemma, martingales and measures, and the equilibrium short-rate models (Vasicek and CIR) as readings.
  • Solving questions: A question gives a GBM or an Ornstein-Uhlenbeck SDE and asks for the explicit solution, or for E[X_T], \operatorname{Var}(X_T) or a probability. The work is Itō's lemma, an integrating factor and the Itō isometry.
  • Interpretation questions: A question may ask what each parameter means, such as the speed of mean reversion or the long-run level. It may also ask for the long-run distribution (as T \to \infty, Vasicek tends to N\left(b, \sigma^2/2a\right)) or why Vasicek allows negative rates and CIR does not.
  • Simulation and calibration: SDEs feed the later objectives on discretization (one Euler-Maruyama step) and on calibrating a short-rate model to a yield curve.

Exam traps

  • GBM's median drift: In S_T = S_0\exp\left[\left(\mu - \tfrac12\sigma^2\right)T + \sigma W_T\right], using \mu alone in the exponent gives the mean of \ln S_T wrong. The mean of S_T is still S_0e^{\mu T}, so do not subtract the \tfrac12\sigma^2 twice.
  • Vasicek's variance factor: The variance is \frac{\sigma^2}{2a}\left(1 - e^{-2aT}\right), with 2a in both places. Writing e^{-aT} inside the bracket, or \sigma^2 T as for a Brownian motion, overstates the spread of a mean-reverting rate.
  • Hull's a and b: In a(b - r), b is the level and a is the speed. Some texts write (\theta - ar) or \kappa(\theta - r), so the long-run level is \theta/a in the first form, not \theta.
  • CIR's variance is not Vasicek's: The CIR variance depends on r_0 and b because the volatility is \sigma\sqrt r. Reusing the Vasicek formula with the CIR \sigma of the example gives 0.0054, about 25 times too large, because the CIR \sigma multiplies \sqrt{r}, which is only about 0.2.