Valuation of certain CMS spreads


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.

This is a preview of subscription content, log in to check access.


  1. Andersen, L., Piterbarg, W.: Interest Rate Modeling: Volumes I, II, and III. Atlantic Financial Press, London (2010)

    Google Scholar 

  2. Brace, A., Dun, T. A., Barton, G.: Towards a central interest rate model. Paper presented at the Global Derivatives Conference (1998)

  3. Brace, A., Gatarek, D., Musiela, M.: The market model of interest rate dynamics. Math. Finance 7, 127–155 (1997)

    Article  Google Scholar 

  4. Brace, A., Womersley, R. S.: Exact fit to the swaption volatility matrix using semi-definite programming. Paper presented at the ICBI Global Derivatives Conference (2000)

  5. Brigo, D., Mercurio, F.: Interest Rate Models: Theory and Practice. Springer, Heidelberg (2009)

    Google Scholar 

  6. Crow, E.L., Shimizu, K.: Lognormal Distribution: Theory and Application. Dekker, New York (1988)

    Google Scholar 

  7. Deng, S., Li, M., Zhou, J.: Closed-form approximations for spread option prices and Greeks. Working paper (2008)

  8. Eydeland, A., Wolyniec, K.: Energy and Power Risk Management: New Development in Modeling, Pricing and Hedging. Wiley, Hoboken (2002)

    Google Scholar 

  9. Heath, D., Jarrow, R., Morton, A.: Bond pricing and the term structure of interest rates: a new methodology for contingent claim valuations. Econometrica 60, 645–653 (1992)

    Article  Google Scholar 

  10. Jamshidian, F.: LIBOR and swap market models and measures. Finance Stoch. 1, 293–330 (1997)

    Article  Google Scholar 

  11. Kirk, E.: Correlations in the energy markets. In: Kaminsk, V. (ed.) Managing Energy Price Risk. Risk Publications, pp 71–78. London (1995)

  12. Levy, E.: Pricing European average rate currency options. J. Int. Money Finance 11, 471–491 (1992)

    Article  Google Scholar 

  13. Margrabe, W.: The value of an option to exchange one asset for another. J. Finance 33, 177–186 (1978)

    Article  Google Scholar 

  14. Mitchell, R.L.: Permanence of the lognormal distribution. J. Opt. Soc. Am. 58, 1267–1272 (1968)

    Article  Google Scholar 

  15. Navatte, P., Quittard-Pinon, F.: The valuation of interest rate digital options and range notes revisited. Eur. Financ. Manag 5, 425–440 (1999)

    Article  Google Scholar 

  16. Rebonato, R.: Volatility and Correlation: The Perfect Hedger and the Fox. Wiley, New York (2004)

    Google Scholar 

  17. Turnbull, S.: Interest rate digital options and range notes. J. Deriv. 3, 92–101 (1995)

    Article  Google Scholar 

  18. Wu, P., Elliott, R.J.: Valuation of CMS digital range notes in a multifactor LIBOR market model. Int. J. Financ. Eng. 3, 2424–2433 (2016)

    Article  Google Scholar 

  19. Wu, T.P., Chen, S.N.: Valuation of floating range notes in a LIBOR market model. J. Futures Mark. 28, 697–710 (2008)

    Article  Google Scholar 

  20. Wu, T.P., Chen, S.N.: Valuation of CMS spread options with nonzero strike rate. J. Deriv. 1, 41–55 (2011)

    Article  Google Scholar 

Download references


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.

Author information



Corresponding author

Correspondence to Robert J. Elliott.


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:

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


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

Taking the log of both sides:

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

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

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

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

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

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.,

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


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

Differentiating both sides:

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


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

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:

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

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,

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


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

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:

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

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:

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

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:

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


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

Substitute the following in Eq. (B2):

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


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

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

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Wu, P., Elliott, R.J. Valuation of certain CMS spreads. Financ Mark Portf Manag 31, 445–467 (2017).

Download citation


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

JEL Classification

  • G13
  • C63