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The Journal of Derivatives

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Article

It Is Time to Shift Log-Normal

Ren-Raw Chen, Pei-Lin Hsieh and Jeffrey Huang
The Journal of Derivatives Spring 2018, 25 (3) 89-103; DOI: https://doi.org/10.3905/jod.2018.25.3.089
Ren-Raw Chen
is a professor of finance and business economics in the Gabelli School of Business at Fordham University in New York, NY
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Pei-Lin Hsieh
is an assistant professor at the Wang Yanan Institute for Studies in Economics, Xieman University in Xiamen, China
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Jeffrey Huang
is head of global trading at KGI Bank in Taipei, Taiwan
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Abstract

The Libor market model (LMM) and other models for interest rate processes assume the instantaneous short rate is log-normal. It made sense that the rate should be bounded below by zero and that volatility in basis points be proportional to the level of the rate. But there have been some anomalous periods when the model has exhibited substantial problems, such as the 2008 financial crisis and during periods when interest rates fell into negative territory in several major economies, and the log-normal assumption was plainly violated. In this article, the authors show how the assumption that the Libor rate is log-normal can be replaced by assuming 1/(1 + Libor), that is, the price of a zero-coupon bond, is log-normal instead. Under this assumption, the forward short rate follows a shifted log-normal and the drift term in the short rate equation must be modified. With this assumption, the rate is approximately log-normal when rates are high, but it becomes closer to normal when rates are low. The performance of the modified model is illustrated for caps, swaps, and swaptions.

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The Journal of Derivatives: 25 (3)
The Journal of Derivatives
Vol. 25, Issue 3
Spring 2018
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It Is Time to Shift Log-Normal
Ren-Raw Chen, Pei-Lin Hsieh, Jeffrey Huang
The Journal of Derivatives Feb 2018, 25 (3) 89-103; DOI: 10.3905/jod.2018.25.3.089

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It Is Time to Shift Log-Normal
Ren-Raw Chen, Pei-Lin Hsieh, Jeffrey Huang
The Journal of Derivatives Feb 2018, 25 (3) 89-103; DOI: 10.3905/jod.2018.25.3.089
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  • Article
    • Abstract
    • THE EXACT DRIFT ADJUSTMENT
    • CALIBRATION TO CAPS
    • SWAP VALUATION
    • AN EXAMPLE—VASICEK MODEL
    • SUMMARY
    • APPENDIX A
    • APPENDIX B
    • APPENDIX C
    • APPENDIX D
    • ENDNOTES
    • REFERENCES
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