Sensitivity Analysis and Hedging in Stochastic String Models

Abstract

We analyze certain results on the stochastic string modeling of the term structure of interest rates and we apply them to study the sensitivities and the hedging of options with payoff functions homogeneous of degree one. Under the same framework, we use an exact multi-factor extension of Jamshidian (1989) to find the sensitivities for swaptions and we prove that it cannot be applied to captions. We present a new approximate result for pricing options on coupon bonds based on the Fenton-Wilkinson method and we show that it generalizes the fast coupon bond option pricing proposed in Munk (1999). This result can be easily applied to the approximate valuation of swaptions and captions.

Appendix of Proofs

Proof of Theorem 2

Using the Euler Theorem for homogeneous functions, the equality

$$ g\left( {x_{1} , \ldots , x_{n} ;M} \right)e^{{\sqrt {\varDelta_{kk} } x_{k} - \frac{1}{2}\varDelta_{kk} }} = g\left( {x_{1} - \frac{{\varDelta_{1k} }}{{\sqrt {\varDelta_{11} } }}, \ldots , x_{n} - \frac{{\varDelta_{nk} }}{{\sqrt {\varDelta_{nn} } }};M} \right) $$

(6)

and $\varDelta_{00 } = 0,$ expression (1) can be rewritten as

$$ \varvec{C}\left[ {t,\varvec{P}_{t} } \right] = \sum\limits_{k = 0}^{n} {\varPhi_{k} P\left( {t,T_{k} } \right)\int\limits_{{x_{1} = - \infty }}^{ + \infty } { \ldots \int\limits_{{x_{j} = \tilde{x}_{j} }}^{ + \infty } { \ldots \int\limits_{{x_{n} = - \infty }}^{ + \infty } {g\left( {x_{1} - \frac{{\varDelta_{1k} }}{{\sqrt {\varDelta_{11} } }}, \ldots , x_{n} - \frac{{\varDelta_{nk} }}{{\sqrt {\varDelta_{nn} } }};M} \right)d\varvec{x}} } } } $$

Taking derivative with respect to $P\left( {t,T_{0} } \right)$ and using $\varDelta_{i0 } = 0$ we obtain the first sensitivity. The second one follows from a similar calculation.

For the next derivative, taking into account that $\tilde{x}_{l}$ is independent of $\varDelta_{ij }$ , we have

$$ \frac{{\partial \varvec{C}\left[ {t,\varvec{P}_{t} } \right]}}{{\partial \varDelta_{ij} }} = \sum\limits_{k = 0}^{n} {\varPhi_{k} P\left( {t,T_{k} } \right)\int\limits_{{x_{1} = - \infty }}^{ + \infty } { \ldots \int\limits_{{x_{l} = \tilde{x}_{l} }}^{ + \infty } { \ldots \int\limits_{{x_{n} = - \infty }}^{ + \infty } {\frac{{\partial g\left( {x_{1} , \ldots , x_{n} ;M} \right)}}{{\partial \varDelta_{ij} }}e^{{\sqrt {\varDelta_{kk} } x_{k} - \frac{1}{2}\varDelta_{kk} }} d\varvec{x}} } } } $$

(7)

The partial derivative of the Gaussian density can be obtained as

$$ \begin{aligned} \frac{{\partial g\left( {x_{1} , \ldots , x_{n} ;M} \right)}}{{\partial \varDelta_{ij} }} & = g\left( {x_{1} , \ldots , x_{n} ;M} \right)\frac{{\partial \,\ln g\left( {x_{1} , \ldots , x_{n} ;M} \right)}}{{\partial \varDelta_{ij} }} \\ & = - \frac{1}{2}g\left( {x_{1} , \ldots , x_{n} ;M} \right)\left[ {\frac{{\partial \,\ln \left| \varvec{M} \right|}}{{\partial \varDelta_{ij} }} + \frac{\partial }{{\partial \varDelta_{ij} }}\left( {\varvec{x}^{T} \varvec{M}^{ - 1} \varvec{x}} \right)} \right] \\ & = g\left( {x_{1} , \ldots , x_{n} ;M} \right)\frac{1}{{\sqrt {\varDelta_{ii} } \sqrt {\varDelta_{jj} } }}\left[ {\left( {\varvec{M}^{ - 1} \varvec{xx}^{T} \varvec{M}^{ - 1} } \right)_{{\varvec{ij}}} - \left( {\varvec{M}^{ - 1} } \right)_{{\varvec{ij}}} } \right] \\ & = g\left( {x_{1} , \ldots , x_{n} ;M} \right)\left[ {\varvec{D} \circ \left( {\varvec{M}^{ - 1} \varvec{xx}^{T} \varvec{M}^{ - 1} - \varvec{M}^{ - 1} } \right)} \right]_{ij} \\ \end{aligned} $$

(8)

where we have used the definition of $M_{ij }$, the chain rule, the symmetry of M and the equalities $\frac{{\partial \, \ln \left| \varvec{M} \right|}}{{\partial M_{ij} }} = 2(\varvec{M}^{ - 1} )_{{\varvec{ij}}}$ and $\frac{\partial }{{\partial M_{ij} }}\left( {\varvec{x}^{T} \varvec{M}^{ - 1} \varvec{x}} \right) = - 2\left( {\varvec{M}^{ - 1} \varvec{xx}^{T} \varvec{M}^{ - 1} } \right)_{{\varvec{ij}}}$. Replacing (8) in (7) we arrive at the expression for $\frac{{\partial \varvec{C}\left[ {t,\varvec{P}_{t} } \right]}}{{\partial \varDelta_{ij} }}.$

For the last sensitivity we have

$$ \begin{aligned} \frac{{\partial \user2{C}\left[ {t,\user2{P}_{t} } \right]}}{{\partial \Delta _{{ii}} }} & = \frac{\partial }{{\partial \Delta _{{ii}} }}\sum\limits_{{k = 0}}^{n} {\Phi _{k} P\left( {t,T_{k} } \right)\int\limits_{{x_{1} = - \infty }}^{{ + \infty }} { \ldots \int\limits_{{x_{i} = \tilde{x}_{i} }}^{{ + \infty }} \ldots } } \\ & \quad \quad \int\limits_{{x_{n} = - \infty }}^{{ + \infty }} {g\left( {x_{1} - \frac{{\Delta _{{1k}} }}{{\sqrt {\Delta _{{11}} } }}, \ldots ,x_{n} - \frac{{\Delta _{{nk}} }}{{\sqrt {\Delta _{{nn}} } }};M} \right)d\user2{x}} = - \sum\limits_{{k = 0}}^{n} {\Phi _{k} P\left( {t,T_{k} } \right)} \\ & \quad \times \int\limits_{{x_{1} = - \infty }}^{{ + \infty }} { \ldots \int\limits_{{x_{{i - 1}} = - \infty }}^{{ + \infty }} {\int\limits_{{x_{{i + 1}} = - \infty }}^{{ + \infty }} { \ldots \int\limits_{{x_{n} = - \infty }}^{{ + \infty }} {g\left( {x_{1} - \frac{{\Delta _{{1k}} }}{{\sqrt {\Delta _{{11}} } }}, \ldots ,\tilde{x}_{i} } \right.} } } } \\ & \quad \left. { - \frac{{\Delta _{{ik}} }}{{\sqrt {\Delta _{{ii}} } }}, \ldots ,x_{n} - \frac{{\Delta _{{nk}} }}{{\sqrt {\Delta _{{nn}} } }};M} \right) \times \frac{{\partial \tilde{x}_{i} }}{{\partial \Delta _{{ii}} }}d\user2{x}_{i} + \sum\limits_{{k = 0}}^{n} {\Phi _{k} P\left( {t,T_{k} } \right)\int\limits_{{x_{1} = - \infty }}^{{ + \infty }} \ldots } \\ & \quad \quad \int\limits_{{x_{i} = \tilde{x}_{i} }}^{{ + \infty }} { \ldots \int\limits_{{x_{n} = - \infty }}^{{ + \infty }} {\frac{\partial }{{\partial \Delta _{{ii}} }}g\left( {x_{1} - \frac{{\Delta _{{1k}} }}{{\sqrt {\Delta _{{11}} } }}, \ldots ,x_{n} - \frac{{\Delta _{{nk}} }}{{\sqrt {\Delta _{{nn}} } }};M} \right)d\user2{x}} } \\ \end{aligned} $$

(9)

Moreover, we have

$$ \frac{{\partial \tilde{x}_{i} }}{{\partial \varDelta_{ii} }} = - \frac{1}{{2\sqrt {\varDelta_{ii} } }}\left( {\frac{1}{{\varDelta_{ii} }}\ln \left( { - \frac{1}{{\varPhi_{i} P\left( {t,T_{i} } \right)}}\sum\limits_{{\begin{array}{*{20}c} {m = 0} \\ {m \ne i} \\ \end{array} }}^{n} {\varPhi_{m} P\left( {t,T_{m} } \right)e^{{\sqrt {\varDelta_{mm} } x_{m} - \frac{1}{2}\varDelta_{mm} }} } } \right) - \frac{1}{2}} \right) $$

(10)

For the second one, by the chain rule we have

$$ \begin{aligned} \frac{{\partial g\left( {x_{1} , \ldots , x_{n} ;M} \right)}}{{\partial \varDelta_{ii} }} & = \sum\limits_{{\begin{array}{*{20}c} {m = 1} \\ {m \ne i} \\ \end{array} }}^{n} {\frac{{\partial M_{im} }}{{\partial \varDelta_{ii} }}\frac{{\partial g\left( {x_{1} , \ldots , x_{n} ;M} \right)}}{{\partial M_{im} }}} \\ & = - \frac{1}{{2\varDelta_{ii} \sqrt {\varDelta_{ii} } }}\sum\limits_{{\begin{array}{*{20}c} {m = 1} \\ {m \ne i} \\ \end{array} }}^{n} {\frac{{\varDelta_{im} }}{{\sqrt {\varDelta_{mm} } }}\frac{{\partial g\left( {x_{1} , \ldots , x_{n} ;M} \right)}}{{\partial M_{im} }}} \\ & = - \frac{1}{{2\varDelta_{ii} }}\sum\limits_{{\begin{array}{*{20}c} {m = 1} \\ {m \ne i} \\ \end{array} }}^{n} {\varDelta_{im} \frac{{\partial g\left( {x_{1} , \ldots , x_{n} ;M} \right)}}{{\partial \varDelta_{im} }}} \\ \end{aligned} $$

Using expressions (6) and (8) and the definition of $M_{ij }$ , we get

$$ \begin{aligned} & \frac{\partial }{{\partial \varDelta_{ii} }}g\left( {x_{1} - \frac{{\varDelta_{1k} }}{{\sqrt {\varDelta_{11} } }}, \ldots , x_{n} - \frac{{\varDelta_{nk} }}{{\sqrt {\varDelta_{nn} } }};M} \right) \\ & \quad = \frac{1}{2}g\left( {x_{1} - \frac{{\varDelta_{1k} }}{{\sqrt {\varDelta_{11} } }}, \ldots , x_{n} - \frac{{\varDelta_{nk} }}{{\sqrt {\varDelta_{nn} } }};M} \right) \\ & \quad\quad \times \left( {\frac{1}{{\varDelta_{ii} }}\sum\limits_{{\begin{array}{*{20}c} {\varvec{m} = 1} \\ {\varvec{m} \ne i} \\ \end{array} }}^{\varvec{n}} {\left[ {\varvec{M} \circ \left( {\varvec{M}^{ - 1} \varvec{xx}^{T} \varvec{M}^{ - 1} - \varvec{M}^{ - 1} } \right)} \right]_{{\varvec{im}}} - \delta_{{\varvec{i}k}} \left( {\frac{{x_{k} }}{{\sqrt {\varDelta_{kk} } }} - 1} \right)} } \right) \\ \end{aligned} $$

(11)

Replacing (10) and (11) in (9) we obtain the last sensitivity.

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Proof of Theorem 4

Bueno-Guerrero et al. (2015b) showed that the exact price of a coupon bond call in the Gaussian stochastic string framework is given by

$$ \begin{aligned} \varvec{Call}_{K} \left[ {t,\varvec{P}_{t} } \right] & = \int\limits_{{{\varOmega }_{t} }} {g\left( {x_{1} , \ldots , x_{n} ;M} \right)\sum\limits_{i = 1}^{n} {C_{i} P\left( {t,T_{i} } \right)e^{{\sqrt {\varDelta_{ii} } x_{i} - \frac{1}{2}\varDelta_{ii} }} d\varvec{x}} } \\ & \quad - KP\left( {t,T_{0} } \right)\int\limits_{{{\varOmega }_{t} }}^{{}} { g\left( {x_{1} , \ldots , x_{n} ;M} \right)d\varvec{x}} \\ \end{aligned} $$

(12)

where

$$ {\varOmega }_{t} = \left\{ {\varvec{x} \in {\mathbb{R}}^{n} :\sum\limits_{i = 1}^{n} {C_{i} P\left( {t,T_{i} } \right)e^{{\sqrt {\varDelta_{ii} } x_{i} - \frac{1}{2}\varDelta_{ii} }} } - KP\left( {t,T_{0} } \right) > 0} \right\} $$

Define ${\text{y}}_{i} = \sqrt {\varDelta_{ii} } x_{i} - \frac{1}{2}\varDelta_{ii}$. As $x_{i}$ has a standard normal distribution, ${\text{y}}_{i} \sim N\left( { - \frac{1}{2}\varDelta_{ii} ,\varDelta_{ii} } \right)$ and then

${\text{w}}_{i} e^{{y_{i} }} \sim {\text{log }}N\left( { - \frac{1}{2}\varDelta_{ii} + \ln {\text{w}}_{i} ,\varDelta_{ii} } \right).$

Define ${\text{l}}_{i} = \log N\left( {m_{li} ,\sigma_{{{\text{l}}_{i} }}^{2} } \right).$The Fenton-Wilkinson method allows to approximate $\sum\nolimits_{i = 1}^{n} {l_{i} }$ by another lognormal variable $Z\sim \, \log N\left( {m_{z} ,\sigma_{z}^{2} } \right)$ with

$$ m_{z} = 2 \, {\text{ln}}\,{\text{u}}_{1} - \frac{1}{2} {\text{ln}}\,{\text{u}}_{2} ,\quad \sigma_{z}^{2} = {\ln} \,{\text{u}}_{2} - 2 \, \ln {\text{u}}_{1} $$

where

$$ \begin{aligned} {\text{u}}_{1} & = \sum\limits_{i = 1}^{n} {e^{{m_{{l_{i} }} + \frac{1}{2}\sigma_{{{\text{l}}_{i} }}^{2} }} } \\ {\text{u}}_{2} & = \sum\limits_{i = 1}^{n} {e^{{2\left( {m_{{l_{i} }} + \sigma_{{{\text{l}}_{i} }}^{2} } \right)}} } + 2\sum\limits_{i = 1}^{n - 1} {\sum\limits_{j = i + 1}^{n} {e^{{m_{{l_{i} }} + m_{lj} }} e^{{\frac{1}{2}\left( {\sigma_{{{\text{l}}_{i} }}^{2} + \sigma_{{{\text{l}}_{j} }}^{2} + 2r_{ij} \sigma_{{{\text{l}}_{i} }} \sigma_{{{\text{l}}_{j} }} } \right)}} } } \\ \end{aligned} $$

and $r_{ij} = {\text{corr}}\left( {l_{i} ,l_{j} } \right)$ (see Pirinen 2003).

In our case, we have ${\text{l}}_{i} = {\text{w}}_{i} e^{{y_{i} }} , r_{ij} = M_{ij}$ , $u_{1} = 1$ and ${\text{u}}_{2} = \sum\nolimits_{i, j = 1}^{n} {w_{i} w_{j} e^{{\varDelta_{ij} }} }$ . Then,

$$ m_{z} = - \frac{1}{2} {\text{ln}}\sum\limits_{i, j = 1}^{n} {w_{i} w_{j} e^{{\varDelta_{ij} }} } ,\quad \sigma_{z}^{2} = - 2\,m_{z} $$

(13)

Defining $\varDelta$ by ${\text{Z}} = \sqrt \varDelta x_{1} - \frac{1}{2}\varDelta$ we have $m_{z} = - \frac{1}{2}\varDelta$ and $\sigma_{z}^{2} = \varDelta$ , which replaced in (13) give

$$ \varDelta = \ln \sum\limits_{i, j = 1}^{n} {w_{i} w_{j} e^{{\varDelta_{ij} }} } $$

Thus we can make $\sum\nolimits_{i, j = 1}^{n} {{\text{w}}_{i} e^{{y_{i} }} } \simeq e^{{\sqrt \varDelta x_{1} - \frac{1}{2}\varDelta }}$ and rewrite (12) approximately as

$$ \begin{aligned} \user2{Call}_{K} \left[ {t,\user2{P}_{t} } \right] & \simeq \sum\limits_{{i = 1}}^{n} {C_{i} P\left( {t,T_{i} } \right)} \int\limits_{{\widehat{\Omega }_{t} }} {g\left( {x_{1} , \ldots ,x_{n} ;M} \right)e^{{\sqrt \Delta x_{1} - \frac{1}{2}\Delta }} d\user2{x}} \\ & \quad \quad - KP\left( {t,T_{0} } \right)\int\limits_{{\widehat{\Omega }_{t} }}^{{}} {g\left( {x_{1} , \ldots ,x_{n} ;M} \right)d\user2{x}} \\ \end{aligned} $$

with

$$ \widehat{\varOmega }_{t} = \left\{ {\varvec{x} \in {\mathbb{R}}^{n} :\sum\limits_{i = 1}^{n} {C_{i} P\left( {t,T_{i} } \right)e^{{\sqrt \varDelta x_{1} - \frac{1}{2}\varDelta }} } - KP\left( {t,T_{0} } \right) > 0} \right\} $$

Integrating in $x_{2} , \ldots , x_{n}$, we get

$$ \begin{aligned} & \user2{Call}_{K} \left[ {t,\user2{P}_{t} } \right] \simeq \sum\limits_{{i = 1}}^{n} {C_{i} P\left( {t,T_{i} } \right)\int\limits_{{\frac{1}{{\sqrt \Delta }}\left[ {\ln \left( {\frac{{\hat{K}P\left( {t,T_{0} } \right)}}{{P\left( {t,t + \hat{D}(t)} \right)}}} \right) + \frac{1}{2}\Delta } \right]}}^{{ + \infty }} {g\left( {x - \sqrt \Delta ;1} \right)dx} - KP\left( {t,T_{0} } \right)} \\ & \quad \int\limits_{{\ln \left( {\frac{{\hat{K}P\left( {t,T_{0} } \right)}}{{P\left( {t,t + \hat{D}(t)} \right)}}} \right) + \frac{1}{2}\Delta }}^{{ + \infty }} {g(x;1)dx} = \frac{{\sum\nolimits_{{i = 1}}^{n} {C_{i} P\left( {t,T_{i} } \right)} }}{{P\left( {t,t + \hat{D}(t)} \right)}}\left[ {P\left( {t,t + \hat{D}(t)} \right)\phi \left( {\frac{{\ln \left( {\frac{{P\left( {t,t + \hat{D}(t)} \right)}}{{\hat{K}P\left( {t,T_{0} } \right)}}} \right) + \frac{1}{2}\Delta }}{{\sqrt \Delta }}} \right)} \right. \\ & \quad \left. { - \hat{K}P\left( {t,T_{0} } \right)\phi \left( {\frac{{\ln \left( {\frac{{P\left( {t,t + \hat{D}(t)} \right)}}{{\hat{K}P\left( {t,T_{0} } \right)}}} \right) - \frac{1}{2}\Delta }}{{\sqrt \Delta }}} \right)} \right] = \frac{{\sum\nolimits_{{i = 1}}^{n} {C_{i} P\left( {t,T_{i} } \right)} }}{{P\left( {t,t + \hat{D}(t)} \right)}}\user2{Call}_{{\hat{K}}} \left( {t,T_{0} ,t + \hat{D}(t)} \right) \\ \end{aligned} $$

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Proof of Corollary 3

We write expression (5) under the multifactor HJM stochastic string case (see Bueno-Guerrero et al. 2016) given by $R_{t} (u,y) = \sum\nolimits_{k = 0}^{m} {\sigma_{HJM}^{(k)} (t,u)\sigma_{HJM}^{(k)} (t,y)}$ where $\sigma_{HJM}^{(k)} (t,u),k = 0, 1, \ldots ,m$ are the HJM volatilities in the Musiela parameterization. Expression (5) becomes

$$ \begin{aligned} & \int\limits_{t = s}^{{T_{0} }} {\sum\limits_{k = 0}^{m} {\left[ {\int\limits_{{y = T_{0} - t}}^{{\widehat{D}(t)}} {\sigma_{HJM}^{(k)} (t,y)dy} } \right]^{2} dt} } \\ & \quad = \ln \sum\limits_{i,j = 1}^{n} {w_{i} w_{j} } \exp \left\{ {\int\limits_{t = s}^{{T_{0} }} {\mathop \sum \limits_{k = 0}^{m} \left[ {\mathop \int \limits_{{u = T_{0} - t}}^{{T_{i} - t}} \sigma_{HJM}^{(k)} (t,u)du} \right]\left[ {\int\limits_{{y = T_{0} - t}}^{{T_{j} - t}} {\sigma_{HJM}^{(k)} (t,y)dy} } \right]dt} } \right\} \\ \end{aligned} $$

Taking derivative with respect to s provides

$$ \begin{aligned} & \mathop \sum \limits_{k = 0}^{m} \left[ {\mathop \int \limits_{{y = T_{0} - s}}^{{\widehat{D}(s)}} \sigma_{HJM}^{\left( k \right)} \left( {s,y} \right)dy} \right]^{2} \\ & \quad = \mathop \sum \limits_{k = 0}^{m} \mathop \sum \limits_{i,j = 1}^{n} w_{ij} \left[ {\mathop \int \limits_{{u = T_{0} - s}}^{{T_{i} - s}} \sigma_{HJM}^{\left( k \right)} \left( {s,u} \right)du} \right]\left[ {\mathop \int \limits_{{y = T_{0} - s}}^{{T_{j} - s}} \sigma_{HJM}^{\left( k \right)} \left( {s,y} \right)dy} \right] \\ \end{aligned} $$

(14)

with

$$ \begin{aligned} & w_{ij} \left( {s,T_{0} } \right) \\ & \quad = \frac{{w_{i} w_{j} { \exp }\left\{ {\int_{t = s}^{{T_{0} }} {\sum\nolimits_{k = 0}^{m} {\left[ {\int_{{u = T_{0} - t}}^{{T_{i} - t}} {\sigma_{HJM}^{(k)} (t,u)du} } \right]\left[ {\int_{{y = T_{0} - t}}^{{T_{j} - t}} {\sigma_{HJM}^{(k)} (t,y)dy} } \right]dt} } } \right\}}}{{\sum\nolimits_{i,j = 1}^{n} {w_{i} w_{j} { \exp }\left\{ {\int_{t = s}^{{T_{0} }} {\sum\nolimits_{k = 0}^{m} {\left[ {\int_{{u = T_{0} - t}}^{{T_{i} - t}} {\sigma_{HJM}^{(k)} (t,u)du} } \right]\left[ {\int_{{y = T_{0} - t}}^{{T_{j} - t}} {\sigma_{HJM}^{(k)} (t,y)dy} } \right]dt} } } \right\}} }} \\ \end{aligned} $$

(15)

and $\mathop \sum \limits_{i,j = 1}^{n} w_{ij} \left( {s,T_{0} } \right) = 1.$ Replacing $T_{0}$ by $s$ in expressions (14) and (15), we get

$$ \sum\limits_{k = 0}^{m} {\left[ {\int\limits_{y = 0}^{{\widehat{D}(s)}} {\sigma_{HJM}^{(k)} (s,y)dy} } \right]^{2} } = \sum\limits_{k = 0}^{m} {\left[ {\sum\limits_{i = 1}^{n} {w_{i} \int\limits_{u = 0}^{{T_{i} - s}} {\sigma_{HJM}^{(k)} (s,u)du} } } \right]^{2} } $$

that is exactly the definition of the stochastic duration in Munk (1999) for the HJM case.

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