Affine Diffusion Models

General Affine Diffusions

A jump-diffusion process is a Markov process solving the stochastic differential equationd

\[Y_{t}=\mu\left(Y_{t},\theta_{0}\right)dt +\sigma\left(Y_{t},\theta_{0}\right)dW_{t}.\]

A discount-rate function \(R:D\to\mathbb{R}\) is an affine function of the state

\[R\left(Y\right)=\rho_{0}+\rho_{1}\cdot Y,\]

for \(\rho=\left(\rho_{0},\rho_{1}\right)\in\mathbb{R} \times\mathbb{R}^{N}\).

The affine dependence of the drift and diffusion coefficients of \(Y\) are determined by coefficients \(\left(K,H\right)\) defined by:

\[\mu\left(Y\right)=K_{0}+K_{1}Y,\]

for \(K=\left(K_{0},K_{1}\right)\in\mathbb{R}^{N}\times\mathbb{R}^{N\times N}\),

and

\[\left[\sigma\left(Y\right)\sigma\left(Y\right)^{\prime}\right]_{ij} =\left[H_{0}\right]_{ij}+\left[H_{1}\right]_{ij}\cdot Y,\]

for \(H=\left(H_{0},H_{1}\right)\in\mathbb{R}^{N\times N} \times\mathbb{R}^{N\times N\times N}\).

Here

\[\left[H_{1}\right]_{ij}\cdot Y=\sum_{k=1}^{N}\left[H_{1}\right]_{ijk}Y_{k}.\]

A characteristic \(\chi=\left(K,H,\rho\right)\) captures both the distribution of \(Y\) as well as the effects of any discounting.

Geometric Brownian Motion (GBM)

Suppose that \(S_{t}\) evolves according to

\[\frac{dS_{t}}{S_{t}}=\mu dt+\sigma dW_{t}.\]

In logs:

\[d\log S_{t}=\left(\mu-\frac{1}{2}\sigma^{2}\right)dt+\sigma dW_{t}.\]

After integration on the interval \(\left[t,t+h\right]\):

\[r_{t,h}=\log\frac{S_{t+h}}{S_{t}} =\left(\mu-\frac{1}{2}\sigma^{2}\right)h +\sigma\sqrt{h}\varepsilon_{t+h},\]

where \(\varepsilon_{t}\sim N\left(0,1\right)\).

Vasicek

Suppose that \(r_{t}\) evolves according to

\[dr_{t}=\kappa\left(\mu-r_{t}\right)dt+\eta dW_{t}.\]

Cox-Ingersoll-Ross (CIR)

Suppose that \(r_{t}\) evolves according to

\[dr_{t}=\kappa\left(\mu-r_{t}\right)dt+\eta\sqrt{r_{t}}dW_{t}.\]

Feller condition for positivity of the process is \(\kappa\mu>\frac{1}{2}\eta^{2}\).

Heston

The model is

\[\begin{split}dp_{t}&=\left(r+\left(\lambda_r-\frac{1}{2}\sigma_{t}^{2}\right)\right)dt +\sigma_{t}dW_{t}^{r},\\ d\sigma_{t}^{2}&=\kappa\left(\mu-\sigma_{t}^{2}\right)dt +\eta\sigma_{t}dW_{t}^{\sigma},\end{split}\]

with \(p_{t}=\log S_{t}\), and \(Corr\left[dW_{s}^{r},dW_{s}^{\sigma}\right]=\rho\), or in other words

\[W_{t}^{\sigma}=\rho W_{t}^{r}+\sqrt{1-\rho^{2}}W_{t}^{v}.\]

Feller condition for positivity of the volatility process is \(\kappa\mu>\frac{1}{2}\eta^{2}\).

Central Tendency (CT)

The model is

\[\begin{split}dp_{t}&=\left(r+\left(\lambda-\frac{1}{2}\right) \sigma_{t}^{2}\right)dt+\sigma_{t}dW_{t}^{r},\\ d\sigma_{t}^{2}&=\kappa_{\sigma}\left(v_{t}^{2}-\sigma_{t}^{2}\right)dt +\eta_{\sigma}\sigma_{t}dW_{t}^{\sigma},\\ dv_{t}^{2}&=\kappa_{v}\left(\mu-v_{t}^{2}\right)dt+\eta_{v}v_{t}dW_{t}^{v},\end{split}\]

with \(p_{t}=\log S_{t}\), and \(Corr\left[dW_{s}^{r},dW_{s}^{\sigma}\right]=\rho\), or in other words \(W_{t}^{\sigma}=\rho W_{t}^{r}+\sqrt{1-\rho^{2}}W_{t}^{v}\). Also let \(R\left(Y_{t}\right)=r\).