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/*
 Copyright (C) 2016 Quaternion Risk Management Ltd
 All rights reserved.
 
 This file is part of ORE, a free-software/open-source library
 for transparent pricing and risk analysis - http://opensourcerisk.org
 
 ORE is free software: you can redistribute it and/or modify it
 under the terms of the Modified BSD License.  You should have received a
 copy of the license along with this program.
 The license is also available online at <http://opensourcerisk.org>
 
 This program is distributed on the basis that it will form a useful
 contribution to risk analytics and model standardisation, but WITHOUT
 ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
 FITNESS FOR A PARTICULAR PURPOSE. See the license for more details.
*/
 
namespace QuantExt {
 
/*! CrossAssetAnalytics 
     
    This namesace provides a number of functions which compute analytical moments
    (expectations and covariances) of cross asset model factors.
    These are used in the exact propagation of cross asset model paths (i.e. without 
    time discretisation error).
 
    Reference:
    Lichters, Stamm, Gallagher: Modern Derivatives Pricing and Credit Exposure
    Analysis, Palgrave Macmillan, 2015
 
    See also the documentation in class CrossAssetModel.
 
    Section 16.1 in the reference above lists the analytical expectations and covariances 
    implemented in this namespace. 
    In the following we consider time intervals \f$(s,t)\f$. We aim at computing conditional 
    expectations of factors at time \f$t\f$ given their state at time \f$s\f$, likewise covariances of 
    factor moves \f$\Delta z\f$ and \f$\Delta x\f$ over time interval \f$(s,t)\f$. 
 
    Starting with the interest rate processes 
    \f{eqnarray*}{
    dz_i &=& \epsilon_i\,\gamma_i\,dt + \alpha^z_i\,dW^z_i, \qquad \epsilon_i = \left\{ \begin{array}{ll} 0 & i = 0  \\ 1 & i > 0 \end{array}\right.
    \f}
    we get the factor move by integration
    \f{eqnarray*}{
    \Delta z_i &=& -\int_s^t H^z_i\,(\alpha^z_i)^2\,du + \rho^{zz}_{0i} \int_s^t H^z_0\,\alpha^z_0\,\alpha^z_i\,du\\
    && - \epsilon_i  \rho^{zx}_{ii}\int_s^t \sigma_i^x\,\alpha^z_i\,du + \int_s^t \alpha^z_i\,dW^z_i. \\
    \f}
    Thus, conditional expectation and covariances are
    \f{eqnarray*}{
    \mathbb{E}[\Delta z_i] &=& -\int_s^t H^z_i\,(\alpha^z_i)^2\,du + \rho^{zz}_{0i} \int_s^t H^z_0\,\alpha^z_0\,\alpha^z_i\,du
    - \epsilon_i  \rho^{zx}_{ii}\int_s^t \sigma_i^x\,\alpha^z_i\,du \\
    \mathrm{Cov}[\Delta z_a, \Delta z_b] &=& \rho^{zz}_{ab} \int_s^t \alpha^z_a\,\alpha^z_b\,du
    \f}
 
    Proceeding similarly with the foreign exchange rate processes
    \f[
    dx_i / x_i = \mu^x_i \, dt +\sigma_i^x\,dW^x_i,
    \f]
    we get the following log-moves by integration
 
    \f{eqnarray*}{
    \Delta \ln x_i &=& \ln \left( \frac{P^n_0(0,s)}{P^n_0(0,t)} \frac{P^n_i(0,t)}{P^n_i(0,s)}\right) - \frac12 \int_s^t (\sigma^x_i)^2\,du + \rho^{zx}_{0i}\int_s^t H^z_0\, \alpha^z_0\, \sigma^x_i \,du\nonumber\\
    &&+\int_s^t \zeta^z_0\,H^z_0\, (H^z_0)^{\prime}\,du-\int_s^t \zeta^z_i\,H^z_i\, (H^z_i)^{\prime}\,du\nonumber\\
    &&+ \int_s^t \left(H^z_0(t)-H^z_0\right)\alpha_0^z\,dW^z_0+ \left(H^z_0(t)-H^z_0(s)\right) z_0(s) \nonumber\\
    &&- \int_s^t \left(H^z_i(t)-H^z_i\right)\alpha_i^z\,dW^z_i  -\left(H^z_i(t)-H^z_i(s)\right)z_i(s) \nonumber\\
    &&- \int_s^t \left(H^z_i(t)-H^z_i\right)\gamma_i\,du + \int_s^t\sigma^x_i dW^x_i \nonumber
    \f}
 
    Integration by parts yields
 
    \f{eqnarray*}{
    && \int_s^t \zeta^z_0\,H^z_0\, (H^z_0)^{\prime}\,du-\int_s^t \zeta^z_i\,H^z_i\, (H^z_i)^{\prime}\,du\\
    && = \frac12 \left((H^z_0(t))^2 \zeta^z_0(t) -  (H^z_0(s))^2 \zeta^z_0(s)- \int_s^t (H^z_0)^2 (\alpha^z_0)^2\,du\right)\nonumber\\
    &&\qquad {}- \frac12 \left((H^z_i(t))^2 \zeta^z_i(t) -  (H^z_i(s))^2 \zeta^z_i(s)-\int_s^t (H^z_i)^2 (\alpha^z_i)^2\,du \right)
    \f}
 
    so that the expectation is
    \f{eqnarray}{
    \mathbb{E}[\Delta \ln x_i] &=& \ln \left( \frac{P^n_0(0,s)}{P^n_0(0,t)} \frac{P^n_i(0,t)}{P^n_i(0,s)}\right) - \frac12 \int_s^t (\sigma^x_i)^2\,du + \rho^{zx}_{0i} \int_s^t H^z_0\, \alpha^z_0\, \sigma^x_i\,du\nonumber\\
    &&+\frac12 \left((H^z_0(t))^2 \zeta^z_0(t) -  (H^z_0(s))^2 \zeta^z_0(s)- \int_s^t (H^z_0)^2 (\alpha^z_0)^2\,du\right)\nonumber\\
    &&-\frac12 \left((H^z_i(t))^2 \zeta^z_i(t) -  (H^z_i(s))^2 \zeta^z_i(s)-\int_s^t (H^z_i)^2 (\alpha^z_i)^2\,du \right)\nonumber\\
    &&+ \left(H^z_0(t)-H^z_0(s)\right) z_0(s) -\left(H^z_i(t)-H^z_i(s)\right)z_i(s)\nonumber\\
    &&  - \int_s^t \left(H^z_i(t)-H^z_i\right)\gamma_i \,du, \label{eq:meanX} 
    \f}
 
    and IR-FX and FX-FX covariances are
      
    \f{eqnarray}{
    \mathrm{Cov}[\Delta \ln x_a, \Delta \ln x_b] &=&  \int_s^t \left(H^z_0(t)-H^z_0\right)^2 (\alpha_0^z)^2\,du \nonumber\\
    &&- \rho^{zz}_{0b}\int_s^t \left(H^z_0(t)-H^z_0\right)\alpha_0^z \left(H^z_b(t)-H^z_b\right)\alpha_b^z\,du \nonumber\\
    &&+ \rho^{zx}_{0b}\int_s^t \left(H^z_0(t)-H^z_0\right)\alpha_0^z \sigma^x_b\,du \nonumber\\
    && -\rho^{zz}_{0a} \int_s^t \left(H^z_a(t)-H^z_a\right) \alpha_a^z\left(H^z_0(t)-H^z_0\right) \alpha_0^z\,du \nonumber\\
    &&+ \rho^{zz}_{ab}\int_s^t \left(H^z_a(t)-H^z_a\right)\alpha_a^z \left(H^z_b(t)-H^z_b\right)\alpha_b^z\,du \nonumber\\
    &&- \rho^{zx}_{ab}\int_s^t \left(H^z_a(t)-H^z_a\right)\alpha_a^z \sigma^x_b,du\nonumber\\
    &&+ \rho^{zx}_{0a}\int_s^t \left(H^z_0(t)-H^z_0\right)\alpha_0^z\,\sigma^x_a\,du \nonumber\\
    &&- \rho^{zx}_{ba}\int_s^t \left(H^z_b(t)-H^z_b\right)\alpha_b^z\,\sigma^x_a\, du \nonumber\\
    &&+ \rho^{xx}_{ab}\int_s^t\sigma^x_a\,\sigma^x_b \,du \label{eq:covXX}\\
    &&\nonumber\\
    \mathrm{Cov} [\Delta z_a, \Delta \ln x_b]) &=& \rho^{zz}_{0a}\int_s^t \left(H^z_0(t)-H^z_0\right)  \alpha^z_0\,\alpha^z_a\,du \nonumber\\
    &&- \rho^{zz}_{ab}\int_s^t \alpha^z_a \left(H^z_b(t)-H^z_b\right) \alpha^z_b \,du \nonumber\\
    &&+\rho^{zx}_{ab}\int_s^t \alpha^z_a \, \sigma^x_b \,du. \label{eq:covZX}
    \f}
 
    Based on these expectations of factor moves and log-moves, respectively, we can work out the 
    conditonal expectations of the factor levels at time \f$t\f$. These expectations have state-dependent
    parts (levels at time \f$s\f$) and state-independent parts which we separate in the implementation, 
    see functions ending with "_1" and "_2", respectively.
    Moreover, the implementation splits up the integrals further in order to separate simple and more 
    complex integrations and to allow for tailored efficient numerical integration schemes. 
 
    In the following we rearrange the expectations and covariances above such that the expressions 
    correspond 1:1 to their implementations below.
 
    \todo Rearrange integrals to achieve 1:1 correspondence with code
*/
namespace CrossAssetAnalytics{}
 
/*! 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:
 
    - provide an already calibrated parametrization for a component
      extracted from some external model
    - do the calibration within the CrossAssetModel using one
      of the calibration procedures
 
    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 \f$n\f$ currencies is specified by the following SDE
 
    \f{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\} 
    \f}
    Factors \f$z_i\f$ drive the LGM interest rate processes (index \f$i=0\f$ denotes the domestic currency),
    and factors \f$x_i\f$ the foreign exchange rate processes. 
 
    The no-arbitrage drift terms are 
    \f{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}
    \f}
    where we have dropped time-dependencies to lighten notation.  
 
    The short rate \f$r_i(t)\f$ in currency \f$i\f$ is connected with the instantaneous forward curve \f$f_i(0,t)\f$ 
    and model parameters \f$H_i(t)\f$ and \f$\alpha_i(t)\f$ via
    \f{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  \\ \\ 
    \f}
 
    Parameters \f$H_i(t)\f$ and \f$\alpha_i(t)\f$ (or alternatively \f$\zeta_i(t)\f$) are LGM model parameters 
    which determine, together with the stochastic factor \f$z_i(t)\f$, the evolution of numeraire and 
    zero bond prices in the LGM model:
    \f{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\}. 
    \f}
 
*/
namespace CrossAssetModelTypes{}
 
/*!
*/
namespace tag{}
 
}