Cross Asset Model

Files
file	crossassetanalytics.hpp
	analytics for the cross asset model
file	crossassetanalyticsbase.hpp
	basic functions for analytics in the cross asset model
file	crossassetmodel.hpp
	cross asset model
file	crossassetmodelimpliedeqvoltermstructure.hpp
	dynamic black volatility term structure
file	crossassetmodelimpliedfxvoltermstructure.hpp
	dynamic black volatility term structure
file	projectedcrossassetmodel.hpp
	cross asset model projection utils
file	crossassetstateprocess.hpp
	crossasset model state process

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	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	HTtz
struct	rzcrs
struct	rxcrs
struct	rccrs
struct	P2_< E1, E2 >
struct	P3_< E1, E2, E3 >
struct	P4_< E1, E2, E3, E4 >
struct	P5_< E1, E2, E3, E4, E5 >
struct	LC1_< E1 >
struct	LC2_< E1, E2 >
struct	LC3_< E1, E2, E3 >
struct	LC4_< E1, E2, E3, E4 >
class	CrossAssetModel
	Cross Asset Model. More...
class	CrossAssetModelImpliedEqVolTermStructure
	Cross Asset Model Implied EQ Term Structure. More...
class	CrossAssetModelImpliedFxVolTermStructure
	Cross Asset Model Implied FX Term Structure. More...
class	CrossAssetStateProcess
	Cross Asset Model State Process. More...

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	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)

Detailed Description

Grouping of all cross asset models

Function Documentation

ir_expectation_1()

Real QuantExt::CrossAssetAnalytics::ir_expectation_1	(	const CrossAssetModel *	model,
		const Size	i,
		const Time	t0,
		const Real	dt
	)

IR state expectation

This function evaluates part of the expectation \( \mathbb{E}_{t_0}[z_i(t_0+dt)]\).

Using the results above for factor moves \(\Delta z\) over time interval \((s,t)\), we have

\begin{eqnarray} \mathbb{E}_{t_0}[z_i(t_0+\Delta t)] &=& z_i(t_0) + \mathbb{E}_{t_0}[\Delta z_i], \qquad\mbox{with}\quad \Delta z_i = z_i(t_0+\Delta t) - z_i(t_0) \\ &=& z_i(t_0) -\int_{t_0}^{t_0+\Delta t} H^z_i\,(\alpha^z_i)^2\,du + \rho^{zz}_{0i} \int_{t_0}^{t_0+\Delta t} H^z_0\,\alpha^z_0\,\alpha^z_i\,du - \epsilon_i \rho^{zx}_{ii}\int_{t_0}^{t_0+\Delta t} \sigma_i^x\,\alpha^z_i\,du \end{eqnarray}

This function covers the latter three integrals, the state-independent part.

ir_expectation_2()

Real QuantExt::CrossAssetAnalytics::ir_expectation_2	(	const CrossAssetModel *	model,
		const Size	i,
		const Real	zi_0
	)

IR state expectation This function evaluates the state-dependent part of the expectation

\begin{eqnarray} \mathbb{E}_{t_0}[z_i(t_0+\Delta t)] &=& z_i(t_0) -\int_{t_0}^{t_0+\Delta t} H^z_i\,(\alpha^z_i)^2\,du + \rho^{zz}_{0i} \int_{t_0}^{t_0+\Delta t} H^z_0\,\alpha^z_0\,\alpha^z_i\,du - \epsilon_i \rho^{zx}_{ii}\int_{t_0}^{t_0+\Delta t} \sigma_i^x\,\alpha^z_i\,du \end{eqnarray}

i.e. simply the first contribution \( z_i(t_0) \).

inf_jy_expectation_1()

std::pair<QuantLib::Real, QuantLib::Real> QuantExt::CrossAssetAnalytics::inf_jy_expectation_1	(	const CrossAssetModel *	model,
		QuantLib::Size	i,
		QuantLib::Time	t0,
		QuantLib::Real	dt
	)

State independent portion of the JY inflation drift. The first element of the pair relates to the real rate process and the second element of the pair relates to the inflation index process.

inf_jy_expectation_2()

std::pair<QuantLib::Real, QuantLib::Real> QuantExt::CrossAssetAnalytics::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
	)

State dependent portion of the JY inflation drift. The first element of the pair relates to the real rate process and the second element of the pair relates to the inflation index process.

fx_expectation_1()

Real QuantExt::CrossAssetAnalytics::fx_expectation_1	(	const CrossAssetModel *	model,
		const Size	i,
		const Time	t0,
		const Real	dt
	)

FX state expectation

This function evaluates part of the expectation \( \mathbb{E}_{t_0}[\ln x_i(t_0+dt)]\).

Using the results above for factor moves \(\Delta \ln x\) over time interval \((s,t)\), we have

\begin{eqnarray} \mathbb{E}_{t_0}[\ln x_i(t_0+\Delta t)] &=& \ln x_i(t_0) + \mathbb{E}_{t_0}[\Delta \ln x_i], \qquad\mbox{with}\quad \Delta \ln x_i = \ln x_i(t_0+\Delta t) - \ln x_i(t_0) \\ &=& \ln x_i(t_0) + \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)\\ &&+ \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 \\ &&+\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)\\ &&-\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)\\ && + \rho^{zx}_{0i} \int_s^t H^z_0\, \alpha^z_0\, \sigma^x_i\,du \\ && - \int_s^t \left(H^z_i(t)-H^z_i\right)\gamma_i \,du, \qquad\mbox{with}\quad s = t_0, \quad t = t_0+\Delta t \end{eqnarray}

where we rearranged terms so that the state-dependent terms are listed on the first line (containing \(\ln x_i(t_0), z_i(t_0), z_0(t_0) \)) and all following terms are state-independent (deterministic, just dependent on initial market data and model parameters).

The last integral above contains \(\gamma_i\) which is (see documentation of the CrossAssetModel)

\[ \gamma_i = -H^z_i\,(\alpha^z_i)^2 + H^z_0\,\alpha^z_0\,\alpha^z_i\,\rho^{zz}_{0i} - \sigma_i^x\,\alpha^z_i\, \rho^{zx}_{ii}. \]

The very last last integral above is therefore broken up into six terms which we do not list here, but which show up in this function's implementation.

This function covers the latter state-independent part of the FX expectation.

fx_expectation_2()

Real QuantExt::CrossAssetAnalytics::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
	)

FX state expectation.

This function covers the state-dependent part of the FX expectation.

eq_expectation_1()

Real QuantExt::CrossAssetAnalytics::eq_expectation_1	(	const CrossAssetModel *	model,
		const Size	i,
		const Time	t0,
		const Real	dt
	)

EQ state expectation

This function covers part of the EQ expectation. The overall expectation formula (taken from the book "Modern Derivatives Pricing and Credit Exposure Analysis" by Lichters, Stamm and Gallagher) is as follows:

\begin{eqnarray} \mathbb{E} \left[\Delta ln[s_k]\right] &=& ln \left[ \frac{P_{\phi(k)}(0,s)}{P_{\phi(k)}(0,t)} \right] - \int_{s}^{t} q_k(u) du - \frac{1}{2} \int_{s}^{t}\sigma_{s_k}^2(u) du\\ && +\rho_{z_0,s_k}\int_{s}^{t}\alpha_0(u) H_0(u) \sigma_{s_k}(u) du - \epsilon_{\phi(k)} \rho_{s_k,x_{\phi(k)}} \int_{s}^{t} \sigma_{s_k}(u)\sigma_{x_{\phi(k)}}(u) du\\ &&+\frac{1}{2} \left( H_{\phi(k)}^2(t) \zeta_{\phi(k)}(t) - H_{\phi(k)}^2(s) \zeta_{\phi(k)}(s) - \int_{s}^{t} H_{\phi(k)}^2(u) \alpha_{\phi(k)}^2(u) du \right)\\ && + (H_{\phi(k)}(t) - H_{\phi(k)}(s)) z_{\phi(k)}(s) +\epsilon_{\phi(k)} \int_{s}^{t} \gamma_{\phi(k)} (u) (H_{\phi(k)}(t) - H_{\phi(k)}(u)) du,\\ && \qquad\mbox{with}\quad s = t_0, \quad t = t_0+\Delta t, \quad \phi(k) = \qquad\mbox{ interest rate for currency of equity k} \end{eqnarray}

This function covers the state-independent part of the EQ expectation

eq_expectation_2()

Real QuantExt::CrossAssetAnalytics::eq_expectation_2	(	const CrossAssetModel *	model,
		const Size	k,
		const Time	t0,
		const Real	si_0,
		const Real	zi_0,
		const Real	dt
	)

EQ state expectation

This function covers the state-dependent part of the EQ expectation (see overall expression above).

ir_ir_covariance()

Real QuantExt::CrossAssetAnalytics::ir_ir_covariance	(	const CrossAssetModel *	model,
		const Size	i,
		const Size	j,
		const Time	t0,
		const Time	dt
	)

IR-IR Covariance

This function evaluates the covariance term

\f{eqnarray*}{
\mathrm{Cov}[\Delta z_a, \Delta z_b] &=& \rho^{zz}_{ab} \int_s^t \alpha^z_a\,\alpha^z_b\,du
\f}

on the time interval from \f$ s= t_0\f$ to \f$ t=t_0+\Delta t\f$.

ir_fx_covariance()

Real QuantExt::CrossAssetAnalytics::ir_fx_covariance	(	const CrossAssetModel *	model,
		const Size	i,
		const Size	j,
		const Time	t0,
		const Time	dt
	)

IR-FX Covariance

This function evaluates the covariance term

\begin{eqnarray} \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. \end{eqnarray}

on the time interval from \( s= t_0\) to \( t=t_0+\Delta t\).

fx_fx_covariance()

Real QuantExt::CrossAssetAnalytics::fx_fx_covariance	(	const CrossAssetModel *	model,
		const Size	i,
		const Size	j,
		const Time	t0,
		const Time	dt
	)

FX-FX covariance

This function evaluates the covariance term

\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}_{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}_{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^{zx}_{0a}\int_s^t \left(H^z_0(t)-H^z_0\right)\alpha_0^z\,\sigma^x_a\,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}_{ba}\int_s^t \left(H^z_b(t)-H^z_b\right)\alpha_b^z\,\sigma^x_a\, 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^{xx}_{ab}\int_s^t\sigma^x_a\,\sigma^x_b \,du \end{eqnarray}

on the time interval from \( s= t_0\) to \( t=t_0+\Delta t\).

The implementation of this FX-FX covariance in this function further decomposes all terms in order to separate simple and more complex integrations and to allow for tailored efficient numerical integration schemes. Line by line, the covariance above can be written:

\begin{eqnarray} \mathrm{Cov}[\Delta \ln x_a, \Delta \ln x_b] &=& (H^z_0)^2(t)\int_s^t (\alpha_0^z)^2\,du - 2 \,H^z_0(t)\int_s^t H^z_0 (\alpha_0^z)^2\,du + \int_s^t (H^z_0\,\alpha_0^z)^2\,du \\ && - \rho^{zz}_{0a} H^z_0(t) \,H^z_a(t)\int_s^t \alpha_0^z\,\alpha_a^z\,du + \rho^{zz}_{0a} H^z_a(t) \int_s^t H^z_0\,\alpha_0^z\,\alpha_a^z\,du + \rho^{zz}_{0a} H^z_0(t) \int_s^t H^z_a\,\alpha_0^z\,\alpha_a^z\,du - \rho^{zz}_{0a}\int_s^t H^z_0\,H^z_a\,\alpha_0^z\,\alpha_a^z\,du \\ && - \rho^{zz}_{0b} H^z_0(t) \,H^z_b(t)\int_s^t \alpha_0^z\,\alpha_b^z\,du + \rho^{zz}_{0b} H^z_b(t) \int_s^t H^z_0\,\alpha_0^z\,\alpha_b^z\,du + \rho^{zz}_{0b} H^z_0(t) \int_s^t H^z_b\,\alpha_0^z\,\alpha_b^z\,du - \rho^{zz}_{0b}\int_s^t H^z_0\,H^z_b\,\alpha_0^z\,\alpha_b^z\,du \\ && + \rho^{zx}_{0b} H^z_0(t) \int_s^t \alpha_0^z \,\sigma^x_b\,du - \rho^{zx}_{0b} \int_s^t H^z_0 \,\alpha_0^z \,\sigma^x_b\,du \\ && + \rho^{zx}_{0a} H^z_0(t) \int_s^t \alpha_0^z \,\sigma^x_a\,du - \rho^{zx}_{0a} \int_s^t H^z_0 \,\alpha_0^z \,\sigma^x_a\,du \\ && - \rho^{zx}_{ab} H^z_a(t) \int_s^t \alpha_a^z \sigma^x_b,du + \rho^{zx}_{ab} \int_s^t H^z_a\,\alpha_a^z \sigma^x_b,du \\ && - \rho^{zx}_{ba} H^z_b(t) \int_s^t \alpha_b^z \sigma^x_a,du + \rho^{zx}_{ba} \int_s^t H^z_b\,\alpha_b^z \sigma^x_a,du \\ && + \rho^{zz}_{ab} H^z_a(t) H^z_b(t) \int_s^t \alpha_a^z \alpha_b^z \,du - \rho^{zz}_{ab} H^z_b(t) \int_s^t H^z_a \alpha_a^z \alpha_b^z \,du - \rho^{zz}_{ab} H^z_a(t) \int_s^t H^z_b \alpha_a^z \alpha_b^z \,du + \rho^{zz}_{ab} \int_s^t H^z_a H^z_b \alpha_a^z \alpha_b^z \,du \\ && + \rho^{xx}_{ab} \int_s^t\sigma^x_a\,\sigma^x_b \,du \end{eqnarray}

When comparing with the code, also note that the integral on line one can be simplified to

\[ \int_s^t (\alpha_0^z)^2\,du = \zeta_0(t) - \zeta_0(s) \]

infz_infz_covariance()

Real QuantExt::CrossAssetAnalytics::infz_infz_covariance	(	const CrossAssetModel *	model,
		const Size	i,
		const Size	j,
		const Time	t0,
		const Time	dt
	)

infz-infz covariance

infz_infy_covariance()

Real QuantExt::CrossAssetAnalytics::infz_infy_covariance	(	const CrossAssetModel *	model,
		const Size	i,
		const Size	j,
		const Time	t0,
		const Time	dt
	)

infz-infy covariance

infy_infy_covariance()

Real QuantExt::CrossAssetAnalytics::infy_infy_covariance	(	const CrossAssetModel *	model,
		const Size	i,
		const Size	j,
		const Time	t0,
		const Time	dt
	)

infy-infy covariance

ir_infz_covariance()

Real QuantExt::CrossAssetAnalytics::ir_infz_covariance	(	const CrossAssetModel *	model,
		const Size	i,
		const Size	j,
		const Time	t0,
		const Time	dt
	)

ir-infz covariance

ir_infy_covariance()

Real QuantExt::CrossAssetAnalytics::ir_infy_covariance	(	const CrossAssetModel *	model,
		const Size	i,
		const Size	j,
		const Time	t0,
		const Time	dt
	)

ir-infy covariance

fx_infz_covariance()

Real QuantExt::CrossAssetAnalytics::fx_infz_covariance	(	const CrossAssetModel *	model,
		const Size	i,
		const Size	j,
		const Time	t0,
		const Time	dt
	)

fx-infz covariance

fx_infy_covariance()

Real QuantExt::CrossAssetAnalytics::fx_infy_covariance	(	const CrossAssetModel *	model,
		const Size	i,
		const Size	j,
		const Time	t0,
		const Time	dt
	)

fx-infy covariance

crz_crz_covariance()

Real QuantExt::CrossAssetAnalytics::crz_crz_covariance	(	const CrossAssetModel *	model,
		const Size	i,
		const Size	j,
		const Time	t0,
		const Time	dt
	)

crz-crz covariance

crz_cry_covariance()

Real QuantExt::CrossAssetAnalytics::crz_cry_covariance	(	const CrossAssetModel *	model,
		const Size	i,
		const Size	j,
		const Time	t0,
		const Time	dt
	)

crz-cry covariance

cry_cry_covariance()

Real QuantExt::CrossAssetAnalytics::cry_cry_covariance	(	const CrossAssetModel *	model,
		const Size	i,
		const Size	j,
		const Time	t0,
		const Time	dt
	)

cry-cry covariance

ir_crz_covariance()

Real QuantExt::CrossAssetAnalytics::ir_crz_covariance	(	const CrossAssetModel *	model,
		const Size	i,
		const Size	j,
		const Time	t0,
		const Time	dt
	)

ir-crz covariance

ir_cry_covariance()

Real QuantExt::CrossAssetAnalytics::ir_cry_covariance	(	const CrossAssetModel *	model,
		const Size	i,
		const Size	j,
		const Time	t0,
		const Time	dt
	)

ir-cry covariance

fx_crz_covariance()

Real QuantExt::CrossAssetAnalytics::fx_crz_covariance	(	const CrossAssetModel *	model,
		const Size	i,
		const Size	j,
		const Time	t0,
		const Time	dt
	)

fx-crz covariance

fx_cry_covariance()

Real QuantExt::CrossAssetAnalytics::fx_cry_covariance	(	const CrossAssetModel *	model,
		const Size	i,
		const Size	j,
		const Time	t0,
		const Time	dt
	)

fx-cry covariance

infz_crz_covariance()

Real QuantExt::CrossAssetAnalytics::infz_crz_covariance	(	const CrossAssetModel *	model,
		const Size	i,
		const Size	j,
		const Time	t0,
		const Time	dt
	)

infz-crz covariance

infz_cry_covariance()

Real QuantExt::CrossAssetAnalytics::infz_cry_covariance	(	const CrossAssetModel *	model,
		const Size	i,
		const Size	j,
		const Time	t0,
		const Time	dt
	)

infz-cry covariance

infy_crz_covariance()

Real QuantExt::CrossAssetAnalytics::infy_crz_covariance	(	const CrossAssetModel *	model,
		const Size	i,
		const Size	j,
		const Time	t0,
		const Time	dt
	)

infy-crz covariance

infy_cry_covariance()

Real QuantExt::CrossAssetAnalytics::infy_cry_covariance	(	const CrossAssetModel *	model,
		const Size	i,
		const Size	j,
		const Time	t0,
		const Time	dt
	)

infy-cry covariance

ir_eq_covariance()

Real QuantExt::CrossAssetAnalytics::ir_eq_covariance	(	const CrossAssetModel *	model,
		const Size	irIdx,
		const Size	eqIdx,
		const Time	t0,
		const Time	dt
	)

IR-EQ Covariance

\begin{eqnarray} Cov \left[\Delta ln[s_i], \Delta z_j \right] &=& \rho_{z_{\phi(i)},z_j} \int_{s}^{t} (H_{\phi(i)} (t) - H_{\phi(i)} (u)) \alpha_{\phi(i)} (u) \alpha_j (u) du\\ &&+ \rho_{s_i,z_j} \int_{s}^{t} \sigma_{s_i} (u) \alpha_j (u) du\\ \end{eqnarray}

fx_eq_covariance()

Real QuantExt::CrossAssetAnalytics::fx_eq_covariance	(	const CrossAssetModel *	model,
		const Size	i,
		const Size	j,
		const Time	t0,
		const Time	dt
	)

FX-EQ Covariance

\begin{eqnarray} Cov \left[\Delta ln[s_i], \Delta ln[x_j] \right] &=& \rho_{z_{\phi(i)},z_0} \int_{s}^{t} (H_{\phi(i)} (t) - H_{\phi(i)} (u)) (H_0 (t) - H_0 (u)) \alpha_{\phi(i)}(u) \alpha_0(u) du\\ && - \rho_{z_{\phi(i)},z_j} \int_{s}^{t} (H_{\phi(i)} (t) - H_{\phi(i)} (u)) (H_j (t) - H_j (u)) \alpha_{\phi(i)} (u)\alpha_j(u) du\\ && + \rho_{z_{\phi(i)},x_j} \int_{s}^{t} (H_{\phi(i)} (t) - H_{\phi(i)} (u)) \alpha_{\phi(i)} (u) \sigma_{x_j}(u) du\\ &&+ \rho_{s_i,z_0} \int_{s}^{t} (H_0 (t) - H_0 (u)) \alpha_0 (u) \sigma_{s_i}(u) du\\ &&- \rho_{s_i,z_j} \int_{s}^{t} (H_j (t) - H_j (u)) \alpha_j (u) \sigma_{s_i}(u) du\\ &&+ \rho_{s_i,x_j} \int_{s}^{t} \sigma_{s_i}(u) \sigma_{x_j}(u) du\\ \end{eqnarray}

infz_eq_covariance()

Real QuantExt::CrossAssetAnalytics::infz_eq_covariance	(	const CrossAssetModel *	model,
		const Size	i,
		const Size	j,
		const Time	t0,
		const Time	dt
	)

infz-eq covariance

infy_eq_covariance()

Real QuantExt::CrossAssetAnalytics::infy_eq_covariance	(	const CrossAssetModel *	model,
		const Size	i,
		const Size	j,
		const Time	t0,
		const Time	dt
	)

infy-eq covariance

crz_eq_covariance()

Real QuantExt::CrossAssetAnalytics::crz_eq_covariance	(	const CrossAssetModel *	model,
		const Size	i,
		const Size	j,
		const Time	t0,
		const Time	dt
	)

crz-eq covariance

cry_eq_covariance()

Real QuantExt::CrossAssetAnalytics::cry_eq_covariance	(	const CrossAssetModel *	model,
		const Size	i,
		const Size	j,
		const Time	t0,
		const Time	dt
	)

cry-eq covariance

eq_eq_covariance()

Real QuantExt::CrossAssetAnalytics::eq_eq_covariance	(	const CrossAssetModel *	model,
		const Size	i,
		const Size	j,
		const Time	t0,
		const Time	dt
	)

EQ-EQ Covariance

\begin{eqnarray} Cov \left[\Delta ln[s_i], \Delta ln[s_j] \right] &=& \rho_{z_{\phi(i)},z_{\phi(j)}} \int_{s}^{t} (H_{\phi(i)} (t) - H_{\phi(i)} (u)) (H_{\phi(j)} (t)\\ && - H_{\phi(j)} (u)) \alpha_{\phi(i)}(u) \alpha_{\phi(j)}(u) du\\ &&+ \rho_{z_{\phi(i)},s_j} \int_{s}^{t} (H_{\phi(i)} (t) - H_{\phi(i)} (u)) \alpha_{\phi(i)}(u) \sigma_{s_j}(u) du\\ &&+ \rho_{z_{\phi(j)},s_i} \int_{s}^{t} (H_{\phi(j)} (t) - H_{\phi(j)} (u)) \alpha_{\phi(j)}(u) \sigma_{s_i}(u) du\\ &&+ \rho_{s_i,s_j} \int_{s}^{t} \sigma_{s_i}(u) \sigma_{s_j}(u) du\\ \end{eqnarray}

aux_aux_covariance()

Real QuantExt::CrossAssetAnalytics::aux_aux_covariance	(	const CrossAssetModel *	model,
		const Time	t0,
		const Time	dt
	)

AUX-AUX Covariance

This function evaluates the covariance term for the domestic auxiliary state variable in the bank account measure

\f{eqnarray*}{
\mathrm{Cov}[\Delta y_0, \Delta y_0] &=& \int_s^t (\alpha^z_0)^2 \,(H^z_0)^2\,du
\f}

on the time interval from \f$ s= t_0\f$ to \f$ t=t_0+\Delta t\f$.

aux_ir_covariance()

Real QuantExt::CrossAssetAnalytics::aux_ir_covariance	(	const CrossAssetModel *	model,
		const Size	j,
		const Time	t0,
		const Time	dt
	)

AUX-IR Covariance

This function evaluates the covariance term for the domestic auxiliary state variable in the bank account measure with any other LGM state variable

\f{eqnarray*}{
\mathrm{Cov}[\Delta y_0, \Delta z_b] &=& \rho^{zz}_{0b} \int_s^t H^z_0\,\alpha^z_0\,\alpha^z_b\,du
\f}

on the time interval from \f$ s= t_0\f$ to \f$ t=t_0+\Delta t\f$.

aux_fx_covariance()

Real QuantExt::CrossAssetAnalytics::aux_fx_covariance	(	const CrossAssetModel *	model,
		const Size	j,
		const Time	t0,
		const Time	dt
	)

AUX-FX Covariance

This function evaluates the covariance term

\begin{eqnarray} \mathrm{Cov} [\Delta y_0, \Delta \ln x_b]) &=& \int_s^t \left(H^z_0(t)-H^z_0\right) (\alpha^z_0)^2\,H^z_0\,du \nonumber\\ &&- \rho^{zz}_{0b}\int_s^t \alpha^z_0\,H^z_0 \left(H^z_b(t)-H^z_b\right) \alpha^z_b \,du \nonumber\\ &&+\rho^{zx}_{0b}\int_s^t \alpha^z_0\,H^z_0 \, \sigma^x_b \,du. \end{eqnarray}

on the time interval from \( s= t_0\) to \( t=t_0+\Delta t\).

com_com_covariance()

Real QuantExt::CrossAssetAnalytics::com_com_covariance	(	const CrossAssetModel *	model,
		const Size	i,
		const Size	j,
		const Time	t0,
		const Time	dt
	)

COM-COM state variable covariance, mean-reverting single-factor case

\begin{eqnarray} Cov \left[\Delta Y_i, \Delta Y_j] \right] &=& \rho_{Y_i,Y_j} \int_{s}^{t} \sigma_{Y_i}(u) \exp(\kappa_i u) \sigma_{Y_j}(u) \exp(\kappa_j u) du\\ \end{eqnarray}

ir_com_covariance()

Real QuantExt::CrossAssetAnalytics::ir_com_covariance	(	const CrossAssetModel *	model,
		const Size	i,
		const Size	j,
		const Time	t0,
		const Time	dt
	)

TODO: COM covariance with all other risk factors

ir_crstate_covariance()

Real QuantExt::CrossAssetAnalytics::ir_crstate_covariance	(	const CrossAssetModel *	model,
		const Size	i,
		const Size	j,
		const Time	t0,
		const Time	dt
	)

IR_CrState Covariance

fx_crstate_covariance()

Real QuantExt::CrossAssetAnalytics::fx_crstate_covariance	(	const CrossAssetModel *	model,
		const Size	i,
		const Size	j,
		const Time	t0,
		const Time	dt
	)

FX_CrState Covariance

crstate_crstate_covariance()

Real QuantExt::CrossAssetAnalytics::crstate_crstate_covariance	(	const CrossAssetModel *	model,
		const Size	i,
		const Size	j,
		const Time	t0,
		const Time	dt
	)

CrState_CrState Covariance

integral_helper()

Real integral_helper	(	const CrossAssetModel *	x,
		const E &	e,
		const Real	t
	)

generic integrand

integral()

Real integral	(	const CrossAssetModel *	model,
		const E &	e,
		const Real	a,
		const Real	b
	)

generic integral calculation

P() [1/4]

const P2_<E1, E2> QuantExt::CrossAssetAnalytics::P	(	const E1 &	e1,
		const E2 &	e2
	)

creator function for product expression, 2 factors

P() [2/4]

const P3_<E1, E2, E3> QuantExt::CrossAssetAnalytics::P	(	const E1 &	e1,
		const E2 &	e2,
		const E3 &	e3
	)

creator function for product expression, 3 factors

P() [3/4]

const P4_<E1, E2, E3, E4> QuantExt::CrossAssetAnalytics::P	(	const E1 &	e1,
		const E2 &	e2,
		const E3 &	e3,
		const E4 &	e4
	)

creator function for product expression, 4 factors

P() [4/4]

const P5_<E1, E2, E3, E4, E5> QuantExt::CrossAssetAnalytics::P	(	const E1 &	e1,
		const E2 &	e2,
		const E3 &	e3,
		const E4 &	e4,
		const E5 &	e5
	)

creator function for product expression, 5 factors

LC() [1/4]

const LC1_<E1> QuantExt::CrossAssetAnalytics::LC	(	QuantLib::Real	c,
		QuantLib::Real	c1,
		const E1 &	e1
	)

creator function for linear combination, 1 factor

LC() [2/4]

const LC2_<E1, E2> QuantExt::CrossAssetAnalytics::LC	(	QuantLib::Real	c,
		QuantLib::Real	c1,
		const E1 &	e1,
		QuantLib::Real	c2,
		const E2 &	e2
	)

creator function for linear combination, 2 factors

LC() [3/4]

const LC3_<E1, E2, E3> QuantExt::CrossAssetAnalytics::LC	(	QuantLib::Real	c,
		QuantLib::Real	c1,
		const E1 &	e1,
		QuantLib::Real	c2,
		const E2 &	e2,
		QuantLib::Real	c3,
		const E3 &	e3
	)

creator function for linear combination, 3 factors

LC() [4/4]

const LC4_<E1, E2, E3, E4> QuantExt::CrossAssetAnalytics::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
	)

creator function for linear combination, 4 factors