Mixing
Application of Gram-Charlier expansion for Swaption pricing with drift extension
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I am doing a project for university and I'm stuck at some point.
The aim is to implement the Gram-Charlier expansion into the 2-factor Hull-White model with Drift extension.
I already found out how to implement it without drift extension but I'm not quite sure how to calculate it with. Can anyone help me?
These are the formulas I already have:
- Bond price under the affine term structure model: $P(t,T)=exp {
A(t,T) + B(t,T)^{top}X(t) }$
- moment generating function: $M_m(t)=E^{T_0}[SV(T_0)^m | X_t]=sum limits_{0leq i_1, cdots ,leq i_m leq N} a_{i_1}*cdots * a_{i_m} mu^{T_0}(t,T_0, {T_{i_1},cdots ,T_{i_m} } )$ where $a_i$ are the Cash-flow amounts calculated from $SV(t)=-P(t,T_0)+delta K sum_{I=1}^N P(t,T_i)+P(t,T_N) = sum limits_{i=0}^Na_iP(t,T_i)$
- $mu^T(t, T_0, {T_1, cdots , T_m }) = frac{exp {M(t) + N(t)^{top}X(t) } }{P(t,T)}$
- Swaption Price of Lth-order: $ SOV(t) = C_1 varphi(frac{C_1}{sqrt{C_2}})+sqrt{C_2} theta(frac{C_1}{sqrt{C_2}})+sqrt{C_2}theta(frac{C_1}{sqrt{C_2}})sum limits_{n=3}^L (-1)^nq_nH_{n-2}(frac{C_1}{sqrt{C_2}})$
where $varphi$ is the normal density and $theta$ is the cumulative distribution function - $A(t,T)=-(T-t)(-frac12 sum limits_{i=1}^n sum limits_{j=1}^n frac{rho_{ij} sigma_i sigma_j}{K_iK_j}(1-D[K_i(T-t)]-D[K_j(T-t)]+D[(K_i+K_j)(T-t)])$
- and $B(t,T)=-tau D[K_j(T-t)]$
where $D[x]=(1-e^{-x})/x$ - M(t) and N(t) are solutions of differential equations and X(t) is a Markov-state vector with $dX_t=K(theta-X_t)dt + sum D(X_t)dW_t$ where $sum$ is a Matrix such that $sum sum^{top}$ is positive definite
I know that for a model with drift extension I need $P(t,T)= exp { -int limits_{t}^T Theta(u)du + A(t,T) +B(t,T)^{top}X(t) }$ where $Theta(u)$ is the drift extension
My question now is: How do I integrate the drift extension in the Rest of the formulas respectively which formula changes at all? And how do I then calculate the Drift extension in my program?
Thank you already in advance, I'm really stuck!
finance stochastic-integrals stochastic-approximation
asked Jan 12 at 12:48
cleosercleoser
112
$endgroup$
add a comment |
$begingroup$
I am doing a project for university and I'm stuck at some point.
The aim is to implement the Gram-Charlier expansion into the 2-factor Hull-White model with Drift extension.
I already found out how to implement it without drift extension but I'm not quite sure how to calculate it with. Can anyone help me?
These are the formulas I already have:
- Bond price under the affine term structure model: $P(t,T)=exp {
A(t,T) + B(t,T)^{top}X(t) }$
- moment generating function: $M_m(t)=E^{T_0}[SV(T_0)^m | X_t]=sum limits_{0leq i_1, cdots ,leq i_m leq N} a_{i_1}*cdots * a_{i_m} mu^{T_0}(t,T_0, {T_{i_1},cdots ,T_{i_m} } )$ where $a_i$ are the Cash-flow amounts calculated from $SV(t)=-P(t,T_0)+delta K sum_{I=1}^N P(t,T_i)+P(t,T_N) = sum limits_{i=0}^Na_iP(t,T_i)$
- $mu^T(t, T_0, {T_1, cdots , T_m }) = frac{exp {M(t) + N(t)^{top}X(t) } }{P(t,T)}$
- Swaption Price of Lth-order: $ SOV(t) = C_1 varphi(frac{C_1}{sqrt{C_2}})+sqrt{C_2} theta(frac{C_1}{sqrt{C_2}})+sqrt{C_2}theta(frac{C_1}{sqrt{C_2}})sum limits_{n=3}^L (-1)^nq_nH_{n-2}(frac{C_1}{sqrt{C_2}})$
where $varphi$ is the normal density and $theta$ is the cumulative distribution function - $A(t,T)=-(T-t)(-frac12 sum limits_{i=1}^n sum limits_{j=1}^n frac{rho_{ij} sigma_i sigma_j}{K_iK_j}(1-D[K_i(T-t)]-D[K_j(T-t)]+D[(K_i+K_j)(T-t)])$
- and $B(t,T)=-tau D[K_j(T-t)]$
where $D[x]=(1-e^{-x})/x$ - M(t) and N(t) are solutions of differential equations and X(t) is a Markov-state vector with $dX_t=K(theta-X_t)dt + sum D(X_t)dW_t$ where $sum$ is a Matrix such that $sum sum^{top}$ is positive definite
I know that for a model with drift extension I need $P(t,T)= exp { -int limits_{t}^T Theta(u)du + A(t,T) +B(t,T)^{top}X(t) }$ where $Theta(u)$ is the drift extension
My question now is: How do I integrate the drift extension in the Rest of the formulas respectively which formula changes at all? And how do I then calculate the Drift extension in my program?
Thank you already in advance, I'm really stuck!
finance stochastic-integrals stochastic-approximation
asked Jan 12 at 12:48
cleosercleoser
112
$endgroup$
add a comment |
$begingroup$
I am doing a project for university and I'm stuck at some point.
The aim is to implement the Gram-Charlier expansion into the 2-factor Hull-White model with Drift extension.
I already found out how to implement it without drift extension but I'm not quite sure how to calculate it with. Can anyone help me?
These are the formulas I already have:
- Bond price under the affine term structure model: $P(t,T)=exp {
A(t,T) + B(t,T)^{top}X(t) }$
- moment generating function: $M_m(t)=E^{T_0}[SV(T_0)^m | X_t]=sum limits_{0leq i_1, cdots ,leq i_m leq N} a_{i_1}*cdots * a_{i_m} mu^{T_0}(t,T_0, {T_{i_1},cdots ,T_{i_m} } )$ where $a_i$ are the Cash-flow amounts calculated from $SV(t)=-P(t,T_0)+delta K sum_{I=1}^N P(t,T_i)+P(t,T_N) = sum limits_{i=0}^Na_iP(t,T_i)$
- $mu^T(t, T_0, {T_1, cdots , T_m }) = frac{exp {M(t) + N(t)^{top}X(t) } }{P(t,T)}$
- Swaption Price of Lth-order: $ SOV(t) = C_1 varphi(frac{C_1}{sqrt{C_2}})+sqrt{C_2} theta(frac{C_1}{sqrt{C_2}})+sqrt{C_2}theta(frac{C_1}{sqrt{C_2}})sum limits_{n=3}^L (-1)^nq_nH_{n-2}(frac{C_1}{sqrt{C_2}})$
where $varphi$ is the normal density and $theta$ is the cumulative distribution function - $A(t,T)=-(T-t)(-frac12 sum limits_{i=1}^n sum limits_{j=1}^n frac{rho_{ij} sigma_i sigma_j}{K_iK_j}(1-D[K_i(T-t)]-D[K_j(T-t)]+D[(K_i+K_j)(T-t)])$
- and $B(t,T)=-tau D[K_j(T-t)]$
where $D[x]=(1-e^{-x})/x$ - M(t) and N(t) are solutions of differential equations and X(t) is a Markov-state vector with $dX_t=K(theta-X_t)dt + sum D(X_t)dW_t$ where $sum$ is a Matrix such that $sum sum^{top}$ is positive definite
I know that for a model with drift extension I need $P(t,T)= exp { -int limits_{t}^T Theta(u)du + A(t,T) +B(t,T)^{top}X(t) }$ where $Theta(u)$ is the drift extension
My question now is: How do I integrate the drift extension in the Rest of the formulas respectively which formula changes at all? And how do I then calculate the Drift extension in my program?
Thank you already in advance, I'm really stuck!
finance stochastic-integrals stochastic-approximation
asked Jan 12 at 12:48
cleosercleoser
112
$endgroup$
I am doing a project for university and I'm stuck at some point.
The aim is to implement the Gram-Charlier expansion into the 2-factor Hull-White model with Drift extension.
I already found out how to implement it without drift extension but I'm not quite sure how to calculate it with. Can anyone help me?
These are the formulas I already have:
- Bond price under the affine term structure model: $P(t,T)=exp {
A(t,T) + B(t,T)^{top}X(t) }$
- moment generating function: $M_m(t)=E^{T_0}[SV(T_0)^m | X_t]=sum limits_{0leq i_1, cdots ,leq i_m leq N} a_{i_1}*cdots * a_{i_m} mu^{T_0}(t,T_0, {T_{i_1},cdots ,T_{i_m} } )$ where $a_i$ are the Cash-flow amounts calculated from $SV(t)=-P(t,T_0)+delta K sum_{I=1}^N P(t,T_i)+P(t,T_N) = sum limits_{i=0}^Na_iP(t,T_i)$
- $mu^T(t, T_0, {T_1, cdots , T_m }) = frac{exp {M(t) + N(t)^{top}X(t) } }{P(t,T)}$
- Swaption Price of Lth-order: $ SOV(t) = C_1 varphi(frac{C_1}{sqrt{C_2}})+sqrt{C_2} theta(frac{C_1}{sqrt{C_2}})+sqrt{C_2}theta(frac{C_1}{sqrt{C_2}})sum limits_{n=3}^L (-1)^nq_nH_{n-2}(frac{C_1}{sqrt{C_2}})$
where $varphi$ is the normal density and $theta$ is the cumulative distribution function - $A(t,T)=-(T-t)(-frac12 sum limits_{i=1}^n sum limits_{j=1}^n frac{rho_{ij} sigma_i sigma_j}{K_iK_j}(1-D[K_i(T-t)]-D[K_j(T-t)]+D[(K_i+K_j)(T-t)])$
- and $B(t,T)=-tau D[K_j(T-t)]$
where $D[x]=(1-e^{-x})/x$ - M(t) and N(t) are solutions of differential equations and X(t) is a Markov-state vector with $dX_t=K(theta-X_t)dt + sum D(X_t)dW_t$ where $sum$ is a Matrix such that $sum sum^{top}$ is positive definite
I know that for a model with drift extension I need $P(t,T)= exp { -int limits_{t}^T Theta(u)du + A(t,T) +B(t,T)^{top}X(t) }$ where $Theta(u)$ is the drift extension
My question now is: How do I integrate the drift extension in the Rest of the formulas respectively which formula changes at all? And how do I then calculate the Drift extension in my program?
Thank you already in advance, I'm really stuck!
finance stochastic-integrals stochastic-approximation
finance stochastic-integrals stochastic-approximation
asked Jan 12 at 12:48
cleosercleoser
112
asked Jan 12 at 12:48
cleosercleoser
112
edited Jan 12 at 16:53
cleoser
asked Jan 12 at 12:48
cleosercleoser
112
asked Jan 12 at 12:48
cleosercleoser
112
asked Jan 12 at 12:48
cleosercleoser
112
112
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