Stochastic Processes
This page states the stochastic processes used throughout the project as mathematical objects: their driving equations, parameters, and the closed-form moments used for validation. Each process is defined once here. Simulation schemes and implementation notes are documented in the solver and market_model sections, not here.
Throughout, \(W_t\) is a standard Brownian motion and \(dW_t\) its increment. See [@shreve2004stochastic, @oksendal2013stochastic] for the stochastic calculus foundations.
Geometric Brownian motion
The log-normal price model with drift \(\mu \in \mathbb{R}\) and volatility \(\sigma > 0\):
\[dS_t = \mu S_t\,dt + \sigma S_t\,dW_t.\]
The strong solution is
\[S_t = S_0 \exp\!\Big(\big(\mu - \tfrac12 \sigma^2\big) t + \sigma W_t\Big),\]
with moments
\[\mathbb{E}[S_t] = S_0 e^{\mu t}, \qquad \operatorname{Var}(S_t) = S_0^2 e^{2\mu t}\big(e^{\sigma^2 t} - 1\big).\]
Ornstein-Uhlenbeck
The mean-reverting Gaussian diffusion with speed \(\theta > 0\), long-run mean \(\mu\), and volatility \(\sigma > 0\):
\[dX_t = \theta(\mu - X_t)\,dt + \sigma\,dW_t.\]
The solution is Gaussian with moments
\[\mathbb{E}[X_t] = X_0 e^{-\theta t} + \mu\big(1 - e^{-\theta t}\big), \qquad \operatorname{Var}(X_t) = \frac{\sigma^2}{2\theta}\big(1 - e^{-2\theta t}\big).\]
As \(t \to \infty\) the distribution converges to \(\mathcal{N}\big(\mu, \tfrac{\sigma^2}{2\theta}\big)\). See [@vasicek1977equilibrium] for the original term-structure application.
Cox-Ingersoll-Ross
The non-negative mean-reverting diffusion used for variance and intensity, with speed \(\kappa > 0\), long-run level \(\theta > 0\), and vol-of-vol \(\sigma > 0\):
\[dv_t = \kappa(\theta - v_t)\,dt + \sigma\sqrt{v_t}\,dW_t.\]
The process stays non-negative when the Feller condition \(2\kappa\theta \ge \sigma^2\) holds. Its moments are
\[\mathbb{E}[v_t] = v_0 e^{-\kappa t} + \theta\big(1 - e^{-\kappa t}\big),\]
\[ \begin{aligned} \operatorname{Var}(v_t) &= v_0\frac{\sigma^2}{\kappa}\big(e^{-\kappa t} - e^{-2\kappa t}\big)\\ &\quad + \frac{\theta\sigma^2}{2\kappa}\big(1 - e^{-\kappa t}\big)^2. \end{aligned} \]
Heston stochastic volatility
A two-factor model coupling a log-normal price \(S_t\) to a CIR variance \(v_t\), with price drift \(\mu\), mean-reversion speed \(\kappa\), long-run variance \(\theta\), vol-of-vol \(\sigma\), and spot-variance correlation \(\rho \in [-1,1]\):
\[ \begin{aligned} dS_t &= \mu S_t\,dt + \sqrt{v_t}\,S_t\,dW_t^S,\\ dv_t &= \kappa(\theta - v_t)\,dt + \sigma\sqrt{v_t}\,dW_t^v, \end{aligned} \qquad d\langle W^S, W^v \rangle_t = \rho\,dt. \]
The variance moments are the CIR moments above; the price has \(\mathbb{E}[S_t] = S_0 e^{\mu t}\). The correlation enters only through the quadratic covariation of the two Brownian drivers. See [@heston1993closed] for the original model and [@lord2010comparison] for simulation-scheme comparisons.
Jump diffusion (Merton)
Geometric Brownian motion with compound Poisson jumps. The price follows
\[ \begin{aligned} dS_t &= \mu S_t\,dt + \sigma S_t\,dW_t + dJ_t,\\ dJ_t &= (e^{Z} - 1)\,dN_t, \qquad Z \sim \mathcal{N}(\mu_J, \sigma_J^2), \qquad N_t \sim \operatorname{Poisson}(\lambda t), \end{aligned} \]
where \(N_t\) is a Poisson process with intensity \(\lambda\) and the jump sizes \(e^{Z} - 1\) are log-normal with \(Z\) normal of mean \(\mu_J\) and variance \(\sigma_J^2\), independent of the diffusion.
Bates
Heston stochastic volatility with Merton log-normal jumps on the price:
\[dS_t = \mu S_t\,dt + \sqrt{v_t}\,S_t\,dW_t^S + dJ_t,\]
with the same jump structure as jump diffusion above and the same variance dynamics as Heston. The model reduces to Heston when \(\lambda = 0\).
Hawkes process
A self-exciting point process with conditional intensity
\[\lambda^*(t) = \mu + \sum_{t_i < t} \phi(t - t_i) = \mu + \int_{-\infty}^{t-} \phi(t - s)\,dN_s,\]
where \(\mu > 0\) is the baseline intensity and the kernel \(\phi \ge 0\) encodes self-excitation. Two kernels are used:
- Exponential: \(\phi(t) = \alpha e^{-\beta t}\) with \(\alpha, \beta > 0\), equivalent to the SDE representation \(d\lambda(t) = \beta\big(\mu - \lambda(t)\big)\,dt + \alpha\,dN_t\).
- Power law: \(\phi(t) = \dfrac{\alpha}{(c + t)^p}\) with \(p > 1\).
Stationarity requires the branching ratio to be subcritical,
\[\int_0^\infty \phi(u)\,du < 1,\]
which for the exponential kernel is \(\alpha/\beta < 1\). The stationary mean intensity of the exponential kernel is
\[\lambda^* = \frac{\mu}{1 - \alpha/\beta}.\]
See [@hawkes2018hawkes] for the finance review and [@ogata1988statistical] for the thinning (Ogata) simulation algorithm.
Rough Ornstein-Uhlenbeck
Fractional Ornstein-Uhlenbeck with Hurst parameter \(H \in (0,1)\) and mean-reversion speed \(\theta\):
\[dX_t = -\theta X_t\,dt + \sigma\,dW_t^H,\]
where \(W_t^H\) is a fractional Brownian motion. The driving kernel \(t^{H - 1/2}\) is approximated by a finite sum of exponentials (Abi Jaber 2019), yielding a multi-factor Markovian representation whose accuracy is controlled by the number of factors.