Reference manual - version qle_version
Cross asset model
Reference:
Lichters, Stamm, Gallagher: Modern Derivatives Pricing and Credit Exposure Analysis, Palgrave Macmillan, 2015
The model is operated under the domestic LGM measure. There are two ways of calibrating the model:
The inter-parametrization correlation matrix specified here can not be calibrated currently, but is a fixed, external input.
The model does not own a reference date, the times given in the parametrizations are absolute and insensitive to shifts in the global evaluation date. The termstructures are required to be consistent with these times, i.e. should all have the same reference date and day counter. The model does not observe anything, so its update() method must be explicitly called to notify observers of changes in the constituting parametrizations, update these parametrizations and flushing the cache of the state process. The model ensures these updates during calibration though.
The cross asset model for \(n\) currencies is specified by the following SDE
\begin{eqnarray*} dz_0(t) &=& \alpha^z_0(t)\,dW^z_0(t) \\ dz_i(t) &=& \gamma_i(t)\,dt + \alpha^z_i(t)\,dW^z_i(t), \qquad i = 1,\dots, n-1 \\ dx_i(t) / x_i(t) &=& \mu^x_i(t)\, dt +\sigma_i^x(t)\,dW^x_i(t), \qquad i=1, \dots, n-1 \\ dW^a_i\,dW^b_j &=& \rho^{ab}_{ij}\,dt, \qquad a, b \in \{z, x\} \end{eqnarray*}
Factors \(z_i\) drive the LGM interest rate processes (index \(i=0\) denotes the domestic currency), and factors \(x_i\) the foreign exchange rate processes.
The no-arbitrage drift terms are
\begin{eqnarray*} \gamma_i &=& -H^z_i\,(\alpha^z_i)^2 + H^z_0\,\alpha^z_0\,\alpha^z_i\,\rho^{zz}_{0i} - \sigma_i^x\,\alpha^z_i\, \rho^{zx}_{ii}\\ \mu^x_i &=& r_0-r_i +H^z_0\,\alpha^z_0\,\sigma^x_i\,\rho^{zx}_{0i} \end{eqnarray*}
where we have dropped time-dependencies to lighten notation.
The short rate \(r_i(t)\) in currency \(i\) is connected with the instantaneous forward curve \(f_i(0,t)\) and model parameters \(H_i(t)\) and \(\alpha_i(t)\) via
\begin{eqnarray*} r_i(t) &=& f_i(0,t) + z_i(t)\,H'_i(t) + \zeta_i(t)\,H_i(t)\,H'_i(t), \quad \zeta_i(t) = \int_0^t \alpha_i^2(s)\,ds \\ \\ \end{eqnarray*}
Parameters \(H_i(t)\) and \(\alpha_i(t)\) (or alternatively \(\zeta_i(t)\)) are LGM model parameters which determine, together with the stochastic factor \(z_i(t)\), the evolution of numeraire and zero bond prices in the LGM model:
\begin{eqnarray*} N(t) &=& \frac{1}{P(0,t)}\exp\left\{H_t\, z_t + \frac{1}{2}H^2_t\,\zeta_t \right\} \\ P(t,T,z_t) &=& \frac{P(0,T)}{P(0,t)}\:\exp\left\{ -(H_T-H_t)\,z_t - \frac{1}{2} \left(H^2_T-H^2_t\right)\,\zeta_t\right\}. \end{eqnarray*}