Vietnam Journal of Mathematics

, Volume 44, Issue 3, pp 513–530

# Regularization for the Inverse Problem of Finding the Purely Time-Dependent Volatility

• Dang Duc Trong
• Dinh Ngoc Thanh
• Nguyen Nhu Lan
Article

## Abstract

We consider the inverse problem of finding the volatility σL ρ (0, T) such that $$U_{BS}(X,K,r,t,{{\int }_{0}^{t}}\sigma ^{2}(\tau )d\tau )=u(t)$$, 0≤tT, where U B S is the Black–Scholes formula and u(t) is the observable fair price of an European call option. The problem is ill-posed. Using the residual method, we shall regularize the problem. An explicit error estimate is given.

## Keywords

Calibration Volatility Ill-posed Regularization

## Mathematics Subject Classification (2010)

35R30 65J20 91B24

## Notes

### Acknowledgments

The authors are grateful to three anonymous referees for their precious suggestions leading to the improvement version of our paper.

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© Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2015

## Authors and Affiliations

• Dang Duc Trong
• 1
• Dinh Ngoc Thanh
• 1
• Nguyen Nhu Lan
• 2
Email author
1. 1.Department of Mathematics and Computer ScienceHo Chi Minh City University of Science, Vietnam National UniversityHo Chi Minh CityVietnam
2. 2.Tay Do UniversityCan ThoVietnam