Valuation of certain CMS spreads

Abstract

In this paper, we derive an approximate lognormal process for the swap rate under the multifactor LIBOR market model using a Levy approach. Using the approximate dynamics for the swap rate, the constant maturity swap spread digital range notes with different strike rates are valued in analytic and semi-analytic form. The CMS spread digital range notes are widely traded in the marketplace, or embedded in structure notes.

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Acknowledgements

PW gratefully acknowledges the research supported by an open project of Jiangsu Key Laboratory of Financial Engineering and National Natural Science Foundation of China (71501099). RJE thanks the SSHRC and ARC for continuing support. Both authors thank the referees for helpful suggestions that improved the paper.

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Correspondence to Robert J. Elliott.

Appendices

Appendix A

The terms \(E[\ln L_\alpha (T_\alpha )]\), \(Var[\ln L_\alpha (T_\alpha )]\), \(<\ln S_{\alpha ,m} (T_\alpha ,T_\alpha ,T_m ),\ln S_{\alpha ,n} (T_\alpha ;T_\alpha ,T_n )>\), \(<\ln S_{\alpha ,m} (T_\alpha ,T_\alpha ,T_m ),\ln L_\alpha (T_\alpha ;T_\alpha ,T_{\alpha +1} )>\), and \(<\ln S_{\alpha ,n} (T_\alpha ,T_\alpha ,T_n ),\ln L_\alpha (T_\alpha ;T_\alpha ,T_{\alpha +1} )>\) are now determined.

The approximate dynamics of \(L_\alpha (t)\) under the spot measure are:

$$\begin{aligned} \frac{\mathrm{d}L_\beta (t)}{L_\alpha (t)}=\sum _{j=\eta (t)}^\alpha {\frac{\delta _j L_j (0)\sum _{q=1}^p {\zeta _{j,q} (t)\zeta _{n,q} (t)} }{1+\delta _j L_j (0)}\mathrm{d}t+\sum _{q=1}^p {\zeta _{n,q} (t)\mathrm{d}W_q (t)} } . \end{aligned}$$
(A1)

Therefore,

$$\begin{aligned} L_\alpha (t)= & {} L_\alpha (0)\exp \left( {\int \limits _0^{T_\alpha } {\left( {\sum _{j=\eta (t)}^\alpha {\frac{\delta _j L_j (0)\sum _{q=1}^p {\zeta _{j,q} (t)\zeta } _{n,q} (t)}{1+\delta _j L_j (0)}-\frac{1}{2}\sum _{q=1}^p {\zeta _{n,q}^2 (t)\mathrm{d}W_q (t)} } } \right) } }\right. \nonumber \\&\left. {+\int \limits _0^{T_\alpha } {\sum _{q=1}^p {\zeta _{n,q} (t)\mathrm{d}W_q (t)} } } \right) . \end{aligned}$$
(A2)

Taking the log of both sides:

$$\begin{aligned} \ln L_\alpha (T_\alpha )= & {} \ln L_\alpha (0)+\int \limits _0^{T_\alpha } {\sigma _\alpha } (t)\left( {\sum _{j=\eta (t)}^\alpha {\frac{\delta _j L_j (0)\sum \nolimits _{q=1}^p {\zeta _{j,q} (t)\zeta _{n,q} (t)} }{1+\delta _j L_j (0)}}}\right. \nonumber \\&\left. {{-\frac{1}{2}\sum \limits _{q=1}^p {\zeta { }_{n,q}^2 (t)} \mathrm{d}t} } \right) +\int _0^{T_\alpha } {\sum \limits _{q=1}^p {\zeta _{n,q} (t)} \mathrm{d}W_q } (t). \end{aligned}$$
(A3)

The term \(E[\ln L_\alpha (T_\alpha )]\) is calculated as:

$$\begin{aligned} E[\ln L_\alpha (T_\alpha )]= & {} \ln L_\alpha (0)+\int \limits _0^{T_\alpha } {\sigma _\alpha } (t)\left( {\sum _{j=\eta (t)}^\alpha {\frac{\delta _j L_j (0)\sum \nolimits _{q=1}^p {\zeta _{j,q} (t)\zeta _{n,q} (t)} }{1+\delta _j L_j (0)}}}\right. \nonumber \\&\left. {{-\frac{1}{2}\sum \limits _{q=1}^p {\zeta { }_{n,q}^2 (t)} \mathrm{d}t} } \right) . \end{aligned}$$
(A4)

The term \(Var[\ln L_\alpha (T_\alpha )]\) is calculated as:

$$\begin{aligned} Var[\ln L_\alpha (T_\alpha )]=\int \limits _0^{T_\alpha } {\sum _{q=1}^p {\zeta _{n,q}^2 (t)\mathrm{d}t} } . \end{aligned}$$
(A5)

We wish to find the quadratic covariation \(<\ln S_{\alpha ,m} (T_\alpha ,T_\alpha ,T_m ),\ln S_{\alpha ,n} (T_\alpha ;T_\alpha ,T_n )>\):

Following the approximation of Eq. (13), i.e.,

$$\begin{aligned} S_{\alpha ,m} (t,T_\alpha )\approx \sum _{\mathrm{i}=\alpha +1}^m {w_{m,i} (0)L_i (t,T_i )} ; \end{aligned}$$

and

$$\begin{aligned} S_{\alpha ,n} (t,T_\alpha )\approx \sum _{\mathrm{i}=\alpha +1}^n {w_{n,i} (0)L_i (t,T_i )} . \end{aligned}$$
(A6)

Differentiating both sides:

$$\begin{aligned} \mathrm{d}S_{\alpha ,m} (t,T_\alpha )\approx \sum _{\mathrm{i}=\alpha +1}^m {w_{m,i} (0)\mathrm{d}L_i (t,T_i )} ; \end{aligned}$$

and

$$\begin{aligned} \mathrm{d}S_{\alpha ,n} (t,T_\alpha )\approx \sum _{\mathrm{i}=\alpha +1}^n {w_{n,i} (0)\mathrm{d}L_i (t,T_i )} . \end{aligned}$$
(A7)

Using Lemma 1, the approximate covariance between \(\ln S_{\alpha ,m} (T_\alpha ;T_\alpha ,T_m )\) and \(\ln S_{\alpha ,n} (T_\alpha ;T_\alpha ,T_n )\) is:

$$\begin{aligned} \begin{array}{ll} \left\langle {\ln S_{\alpha ,m} ,\ln S_{\alpha ,n} } \right\rangle &{}=\left\langle {\int {\frac{\mathrm{d}S_{\alpha ,m} }{S_{\alpha ,m} },\int {\frac{\mathrm{d}S_{\alpha ,n} }{S_{\alpha ,n} }} } } \right\rangle \\ &{}\approx \sum _{j=\alpha +1}^m {\sum _{i=\alpha +1}^n \frac{w_{n,j} (0)w_{m,i} (0)L_j (0)L_i (0)}{S_{\alpha ,m} (0)S_{\alpha ,n} (0)}\left\langle {\ln L_j (T_\alpha ),\ln L_i (T_\alpha )} \right\rangle } \\ \end{array}.\nonumber \\ \end{aligned}$$
(A8)

The covariance between \(\ln S_{\alpha ,m} (T_\alpha ;T_\alpha ,T_m )\) and \(\ln S_{\alpha ,n} (T_\alpha ;T_\alpha ,T_n )\) can be approximated as the weighted covariance between \(\ln L_j (T_\alpha )\) and \(\ln L_i (T_\alpha )\). Then,

$$\begin{aligned} \left\langle {\ln L_j (T_\alpha ),\ln L_i (T_\alpha )} \right\rangle =\int \limits _0^{T_\alpha } \sum _{q=1}^p {\zeta _{i,q} (t)\zeta _{j,q} (t)\mathrm{d}t} . \end{aligned}$$
(A9)

Therefore,

$$\begin{aligned} \left\langle {\ln S_{\alpha ,m} ,\ln S_{\alpha ,n} } \right\rangle =\sum _{i=\alpha +1}^m {\sum _{j=\alpha +1}^n \frac{w_{n,j} (0)w_{m,i} (0)L_j (0)L_i (0)}{S_{\alpha ,m} (0)S_{\alpha ,n} (0)}\int \limits _0^{T_\alpha } {\sum _{q=1}^p {\zeta _{i,q} (t)\zeta _{j,q} (t)} } } \mathrm{d}t.\nonumber \\ \end{aligned}$$
(A10)

A further approximation is introduced by freezing all of the \(L_i \), \(S_{\alpha ,m} \), and \(S_{\alpha ,n}\) to their time-zero value. A similar approach can be found in Brigo and Mercurio (2009).

We now wish to calculate: \(<\ln S_{\alpha ,m} (T_\alpha ,T_\alpha ,T_n ),\ln L_\alpha (T_\alpha ;T_\alpha ,T_{\alpha +1} )>\).

The covariance between \(\ln S_{\alpha ,m} (T_\alpha ;T_\alpha ,T_m )\) and \(\ln L_\alpha (T_\alpha ;T_\alpha ,T_{\alpha +1} )\) is not easily determined. Using similar approximations:

$$\begin{aligned}&\begin{array}{ll} \left\langle {\ln S_{\alpha ,m} ,\ln L_\alpha } \right\rangle &{}=\left\langle {\int {\frac{\mathrm{d}S_{\alpha ,m} }{S_{\alpha ,m} },\int {\frac{\mathrm{d}L_\alpha }{L_\alpha }} } } \right\rangle \approx \sum \limits _{i=\alpha +1}^m {w_{m,i} (0)\frac{L_i (0)}{S_{\alpha ,m} (0)}\left\langle {\ln L_i (T_\alpha ),\ln L_\alpha (T_\alpha )} \right\rangle } \\ &{}=\sum \limits _{i=\alpha +1}^m {\frac{w_{m,i} (0)L_i (0)}{S_{\alpha ,m} (0)}\int \limits _0^{T_\alpha } {\sum \limits _{q=1}^p {\zeta _{i,q} (t)} \zeta _{\alpha ,q} (t)\mathrm{d}t} } .\\ \end{array}.\nonumber \\ \end{aligned}$$
(A11)

From Eq. (A11) the covariance between \(\ln S_{\alpha ,m} (T_\alpha ;T_\alpha ,T_m )\) and \(\ln L_\alpha (T_\alpha )\) can be approximated as the weighted covariance between \(\ln L_i (T_\alpha )\) and \(\ln L_\alpha (T_\alpha )\).

Consider now \(<\ln S_{\alpha ,n} (T_\alpha ,T_\alpha ,T_n ),\ln L_\alpha (T_\alpha ;T_\alpha ,T_{\alpha +1} )>\).

As indicated above, the covariance between \(\ln S_{\alpha ,n} (T_\alpha ;T_\alpha ,T_n )\) and \(\ln L_\alpha (T_\alpha ;T_\alpha ,T_{\alpha +1} )\) may be approximated as:

$$\begin{aligned}&\begin{array}{ll} \left\langle {\ln S_{\alpha ,n} ,\ln L_\alpha } \right\rangle &{}=\left\langle {\int {\frac{\mathrm{d}S_{\alpha ,n} }{S_{\alpha ,n} },\int {\frac{\mathrm{d}L_\alpha }{L_\alpha }} } } \right\rangle \approx \sum \limits _{i=\alpha +1}^n {w_{n,i} (0)\frac{L_i (0)}{S_{\alpha ,n} (0)}\left\langle {\ln L_i (T_\alpha ),\ln L_\alpha (T_\alpha )} \right\rangle } \\ &{}=\sum \limits _{i=\alpha +1}^n {\frac{w_{n,i} (0)L_i (0)}{S_{\alpha ,n} (0)}\int \limits _0^{T_\alpha } {\sum \limits _{q=1}^p {\zeta _{i,q} (t)} \zeta _{\alpha ,q} (t)\mathrm{d}t} }. \\ \end{array}. \end{aligned}$$
(A12)

From Eq. (A12) the covariance between \(\ln S_{\alpha ,n} (T_\alpha ;T_\alpha ,T_n )\) and \(\ln L_\alpha (T_\alpha )\) can be approximated as the weighted covariance between \(\ln L_i (T_\alpha )\) and \(\ln L_\alpha (T_\alpha )\).

Appendix B

Proof of Theorem 5

The random variables X and Y are jointly distributed. Denote their density as \(n(x,y;\rho )\). The conditional density of X given \(Y=y\) is \(n(x;\rho y,1-\rho ^{2})\), i.e., a normal density with mean \(\rho y\) and variance \(1-\rho ^{2}\).Thus, we can now compute the approximate price of the digital spread option as:

$$\begin{aligned} \begin{array}{ll} \int \limits _{-\infty }^{+\infty } {\int \limits _{-\infty }^{+\infty } {1_{x\ge \bar{{x}}(y)} n(x,y;\rho )\mathrm{d}x\mathrm{d}y} } &{}=\int \limits _{-\infty }^\infty {n(y)\mathrm{d}y\int \limits _{\bar{{x}}(y)}^\infty {n(x,\rho y,1-\rho ^{2})} } \mathrm{d}x \\ &{}=\int \limits _{-\infty }^{+\infty } {n(y)A(y)\mathrm{d}y.} \\ \end{array} \end{aligned}$$
(B1)

Here,

$$\begin{aligned} A(y)=\frac{\rho y-\bar{{x}}(y)}{\sqrt{1-\rho ^{2}}}. \end{aligned}$$
(B2)

Substitute the following in Eq. (B2):

$$\begin{aligned} \bar{{x}}(y)=\frac{\log (\exp (V_{\alpha ,n} Y+K_\alpha )-M_{\alpha ,n} }{V_{\alpha ,m} }. \end{aligned}$$
(B3)

Then,

$$\begin{aligned} A(y)=\frac{1}{\sqrt{1-\rho (T_\alpha )^{2}}V_{\alpha ,m} }\log \left( {\frac{\exp (\rho (T_\alpha )V_{\alpha ,m} y+M_{\alpha ,m} )}{K_\alpha +\exp (V_{\alpha ,n} y+M_{\alpha ,n} )}} \right) . \end{aligned}$$
(B4)

Using Eqs. (B2) and (B4) the pricing formula can be derived as shown in Theorem 5. \(\square \)

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Wu, P., Elliott, R.J. Valuation of certain CMS spreads. Financ Mark Portf Manag 31, 445–467 (2017). https://doi.org/10.1007/s11408-017-0301-4

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Keywords

  • LIBOR market model
  • Levy approach
  • CMS spread digital range notes

JEL Classification

  • G13
  • C63