Reference manual - version qle_version
Classes | |
| struct | Hz |
| struct | az |
| struct | zetaz |
| struct | sx |
| struct | vx |
| struct | Hy |
| INF H component. May relate to real rate portion of JY model or z component of DK model. More... | |
| struct | ay |
| INF alpha component. May relate to real rate portion of JY model or z component of DK model. More... | |
| struct | zetay |
| INF zeta component. May relate to real rate portion of JY model or z component of DK model. More... | |
| struct | sy |
| JY INF index sigma component. More... | |
| struct | vy |
| JY INF index variance component. More... | |
| struct | Hl |
| struct | al |
| struct | zetal |
| struct | ss |
| struct | vs |
| struct | coms |
| struct | comDiffusionIntegrand |
| struct | rzz |
| struct | rzx |
| struct | rxx |
| struct | ryy |
| struct | rzy |
| struct | rxy |
| struct | rll |
| struct | rzl |
| struct | rxl |
| struct | ryl |
| struct | rss |
| struct | rzs |
| struct | rxs |
| struct | rys |
| struct | rls |
| struct | rcc |
| struct | rzc |
| struct | rxc |
| struct | ryc |
| struct | rsc |
| struct | rlc |
| struct | HTtz |
| struct | rzcrs |
| struct | rxcrs |
| struct | rccrs |
| struct | P2_ |
| struct | P3_ |
| struct | P4_ |
| struct | P5_ |
| struct | LC1_ |
| struct | LC2_ |
| struct | LC3_ |
| struct | LC4_ |
Functions | |
| Real | ir_expectation_1 (const CrossAssetModel &model, const Size i, const Time t0, const Real dt) |
| Real | ir_expectation_2 (const CrossAssetModel &model, const Size i, const Real zi_0) |
| std::pair< QuantLib::Real, QuantLib::Real > | inf_jy_expectation_1 (const CrossAssetModel &model, QuantLib::Size i, QuantLib::Time t0, QuantLib::Real dt) |
| std::pair< QuantLib::Real, QuantLib::Real > | inf_jy_expectation_2 (const CrossAssetModel &model, QuantLib::Size i, QuantLib::Time t0, const std::pair< QuantLib::Real, QuantLib::Real > &state_0, QuantLib::Real zi_i_0, QuantLib::Real dt) |
| Real | fx_expectation_1 (const CrossAssetModel &model, const Size i, const Time t0, const Real dt) |
| Real | fx_expectation_2 (const CrossAssetModel &model, const Size i, const Time t0, const Real xi_0, const Real zi_0, const Real z0_0, const Real dt) |
| Real | eq_expectation_1 (const CrossAssetModel &model, const Size i, const Time t0, const Real dt) |
| Real | eq_expectation_2 (const CrossAssetModel &model, const Size k, const Time t0, const Real si_0, const Real zi_0, const Real dt) |
| Real | com_expectation_1 (const CrossAssetModel &model, const Size i, const Time t0, const Real dt) |
| Real | ir_ir_covariance (const CrossAssetModel &model, const Size i, const Size j, const Time t0, const Time dt) |
| Real | ir_fx_covariance (const CrossAssetModel &model, const Size i, const Size j, const Time t0, const Time dt) |
| Real | fx_fx_covariance (const CrossAssetModel &model, const Size i, const Size j, const Time t0, const Time dt) |
| Real | infz_infz_covariance (const CrossAssetModel &model, const Size i, const Size j, const Time t0, const Time dt) |
| Real | infz_infy_covariance (const CrossAssetModel &model, const Size i, const Size j, const Time t0, const Time dt) |
| Real | infy_infy_covariance (const CrossAssetModel &model, const Size i, const Size j, const Time t0, const Time dt) |
| Real | ir_infz_covariance (const CrossAssetModel &model, const Size i, const Size j, const Time t0, const Time dt) |
| Real | ir_infy_covariance (const CrossAssetModel &model, const Size i, const Size j, const Time t0, const Time dt) |
| Real | fx_infz_covariance (const CrossAssetModel &model, const Size i, const Size j, const Time t0, const Time dt) |
| Real | fx_infy_covariance (const CrossAssetModel &model, const Size i, const Size j, const Time t0, const Time dt) |
| Real | crz_crz_covariance (const CrossAssetModel &model, const Size i, const Size j, const Time t0, const Time dt) |
| Real | crz_cry_covariance (const CrossAssetModel &model, const Size i, const Size j, const Time t0, const Time dt) |
| Real | cry_cry_covariance (const CrossAssetModel &model, const Size i, const Size j, const Time t0, const Time dt) |
| Real | ir_crz_covariance (const CrossAssetModel &model, const Size i, const Size j, const Time t0, const Time dt) |
| Real | ir_cry_covariance (const CrossAssetModel &model, const Size i, const Size j, const Time t0, const Time dt) |
| Real | fx_crz_covariance (const CrossAssetModel &model, const Size i, const Size j, const Time t0, const Time dt) |
| Real | fx_cry_covariance (const CrossAssetModel &model, const Size i, const Size j, const Time t0, const Time dt) |
| Real | infz_crz_covariance (const CrossAssetModel &model, const Size i, const Size j, const Time t0, const Time dt) |
| Real | infz_cry_covariance (const CrossAssetModel &model, const Size i, const Size j, const Time t0, const Time dt) |
| Real | infy_crz_covariance (const CrossAssetModel &model, const Size i, const Size j, const Time t0, const Time dt) |
| Real | infy_cry_covariance (const CrossAssetModel &model, const Size i, const Size j, const Time t0, const Time dt) |
| Real | ir_eq_covariance (const CrossAssetModel &model, const Size irIdx, const Size eqIdx, const Time t0, const Time dt) |
| Real | fx_eq_covariance (const CrossAssetModel &model, const Size i, const Size j, const Time t0, const Time dt) |
| Real | infz_eq_covariance (const CrossAssetModel &model, const Size i, const Size j, const Time t0, const Time dt) |
| Real | infy_eq_covariance (const CrossAssetModel &model, const Size i, const Size j, const Time t0, const Time dt) |
| Real | crz_eq_covariance (const CrossAssetModel &model, const Size i, const Size j, const Time t0, const Time dt) |
| Real | cry_eq_covariance (const CrossAssetModel &model, const Size i, const Size j, const Time t0, const Time dt) |
| Real | eq_eq_covariance (const CrossAssetModel &model, const Size i, const Size j, const Time t0, const Time dt) |
| Real | aux_aux_covariance (const CrossAssetModel &model, const Time t0, const Time dt) |
| Real | aux_ir_covariance (const CrossAssetModel &model, const Size j, const Time t0, const Time dt) |
| Real | aux_fx_covariance (const CrossAssetModel &model, const Size j, const Time t0, const Time dt) |
| Real | com_com_covariance (const CrossAssetModel &model, const Size i, const Size j, const Time t0, const Time dt) |
| Real | ir_com_covariance (const CrossAssetModel &model, const Size i, const Size j, const Time t0, const Time dt) |
| Real | fx_com_covariance (const CrossAssetModel &model, const Size i, const Size j, const Time t0, const Time dt) |
| Real | infz_com_covariance (const CrossAssetModel &model, const Size i, const Size j, const Time t0, const Time dt) |
| Real | infy_com_covariance (const CrossAssetModel &model, const Size i, const Size j, const Time t0, const Time dt) |
| Real | cry_com_covariance (const CrossAssetModel &model, const Size i, const Size j, const Time t0, const Time dt) |
| Real | crz_com_covariance (const CrossAssetModel &model, const Size i, const Size j, const Time t0, const Time dt) |
| Real | eq_com_covariance (const CrossAssetModel &model, const Size i, const Size j, const Time t0, const Time dt) |
| Real | ir_crstate_covariance (const CrossAssetModel &model, const Size i, const Size j, const Time t0, const Time dt) |
| Real | fx_crstate_covariance (const CrossAssetModel &model, const Size i, const Size j, const Time t0, const Time dt) |
| Real | crstate_crstate_covariance (const CrossAssetModel &model, const Size i, const Size j, const Time t0, const Time dt) |
| template<class E> | |
| Real | integral_helper (const CrossAssetModel &x, const E &e, const Real t) |
| template<typename E> | |
| Real | integral (const CrossAssetModel &model, const E &e, const Real a, const Real b) |
| template<class E1, class E2> | |
| const P2_< E1, E2 > | P (const E1 &e1, const E2 &e2) |
| template<class E1, class E2, class E3> | |
| const P3_< E1, E2, E3 > | P (const E1 &e1, const E2 &e2, const E3 &e3) |
| template<class E1, class E2, class E3, class E4> | |
| const P4_< E1, E2, E3, E4 > | P (const E1 &e1, const E2 &e2, const E3 &e3, const E4 &e4) |
| template<class E1, class E2, class E3, class E4, class E5> | |
| const P5_< E1, E2, E3, E4, E5 > | P (const E1 &e1, const E2 &e2, const E3 &e3, const E4 &e4, const E5 &e5) |
| template<class E1> | |
| const LC1_< E1 > | LC (QuantLib::Real c, QuantLib::Real c1, const E1 &e1) |
| template<class E1, class E2> | |
| const LC2_< E1, E2 > | LC (QuantLib::Real c, QuantLib::Real c1, const E1 &e1, QuantLib::Real c2, const E2 &e2) |
| template<class E1, class E2, class E3> | |
| const LC3_< E1, E2, E3 > | LC (QuantLib::Real c, QuantLib::Real c1, const E1 &e1, QuantLib::Real c2, const E2 &e2, QuantLib::Real c3, const E3 &e3) |
| template<class E1, class E2, class E3, class E4> | |
| const LC4_< E1, E2, E3, E4 > | LC (QuantLib::Real c, QuantLib::Real c1, const E1 &e1, QuantLib::Real c2, const E2 &e2, QuantLib::Real c3, const E3 &e3, QuantLib::Real c4, const E4 &e4) |
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 \((s,t)\). We aim at computing conditional expectations of factors at time \(t\) given their state at time \(s\), likewise covariances of factor moves \(\Delta z\) and \(\Delta x\) over time interval \((s,t)\).
Starting with the interest rate processes
\begin{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. \end{eqnarray*}
we get the factor move by integration
\begin{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. \\ \end{eqnarray*}
Thus, conditional expectation and covariances are
\begin{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 \end{eqnarray*}
Proceeding similarly with the foreign exchange rate processes
\[dx_i / x_i = \mu^x_i \, dt +\sigma_i^x\,dW^x_i, \]
we get the following log-moves by integration
\begin{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 \end{eqnarray*}
Integration by parts yields
\begin{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) \end{eqnarray*}
so that the expectation is
\begin{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} \end{eqnarray}
and IR-FX and FX-FX covariances are
\begin{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} \end{eqnarray}
Based on these expectations of factor moves and log-moves, respectively, we can work out the conditonal expectations of the factor levels at time \(t\). These expectations have state-dependent parts (levels at time \(s\)) 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.