## Analytical Solution of the Characteristic Function in the Trolle-Schwartz Model

 dc.contributor.advisor McWalter, Thomas dc.contributor.advisor Kienitz, Joerg dc.contributor.author Van Gysen, Richard John dc.date.accessioned 2020-02-25T12:00:49Z dc.date.available 2020-02-25T12:00:49Z dc.date.issued 2019 dc.date.updated 2020-02-25T08:34:58Z dc.description.abstract In 2009, Trolle and Schwartz (2008) produced an instantaneous forward interest rate model with several stylised facts such as stochastic volatility. They derived pricing formulae in order to price bonds and bond options, which can be altered to price interest rate options such as caplets, caps and swaptions. These formulae involve implementing numerical methods for solving an ordinary differential equation (ODE). Schumann (2016) confirmed the accuracy of the pricing formulae in the Trolle and Schwartz (2008) model using Monte-Carlo simulation. Both authors used a numerical ODE solver to estimate the ODE. In this dissertation, a closed-form solution for this ODE is presented. Two solutions were found. However, these solutions rely on a simplification of the instantaneous volatility function originally proposed in the Trolle and Schwartz (2008) model. This case happens to be the stochastic volatility version of the Hull and White (1990) model. The two solutions are compared to an ODE solver for one stochastic volatility term and then extended to three stochastic volatility terms. dc.identifier.apacitation Van Gysen, R. J. (2019). Analytical Solution of the Characteristic Function in the Trolle-Schwartz Model. (). ,Faculty of Commerce ,African Institute of Financial Markets and Risk Management. Retrieved from http://hdl.handle.net/11427/31328 en_ZA dc.identifier.chicagocitation Van Gysen, Richard John. "Analytical Solution of the Characteristic Function in the Trolle-Schwartz Model." ., ,Faculty of Commerce ,African Institute of Financial Markets and Risk Management, 2019. http://hdl.handle.net/11427/31328 en_ZA dc.identifier.citation Van Gysen, R. 2019. Analytical Solution of the Characteristic Function in the Trolle-Schwartz Model. en_ZA dc.identifier.ris TY - Thesis / Dissertation AU - Van Gysen, Richard John AB - In 2009, Trolle and Schwartz (2008) produced an instantaneous forward interest rate model with several stylised facts such as stochastic volatility. They derived pricing formulae in order to price bonds and bond options, which can be altered to price interest rate options such as caplets, caps and swaptions. These formulae involve implementing numerical methods for solving an ordinary differential equation (ODE). Schumann (2016) confirmed the accuracy of the pricing formulae in the Trolle and Schwartz (2008) model using Monte-Carlo simulation. Both authors used a numerical ODE solver to estimate the ODE. In this dissertation, a closed-form solution for this ODE is presented. Two solutions were found. However, these solutions rely on a simplification of the instantaneous volatility function originally proposed in the Trolle and Schwartz (2008) model. This case happens to be the stochastic volatility version of the Hull and White (1990) model. The two solutions are compared to an ODE solver for one stochastic volatility term and then extended to three stochastic volatility terms. DA - 2019 DB - OpenUCT DP - University of Cape Town KW - Mathematical Finance LK - https://open.uct.ac.za PY - 2019 T1 - Analytical Solution of the Characteristic Function in the Trolle-Schwartz Model TI - Analytical Solution of the Characteristic Function in the Trolle-Schwartz Model UR - http://hdl.handle.net/11427/31328 ER - en_ZA dc.identifier.uri http://hdl.handle.net/11427/31328 dc.identifier.vancouvercitation Van Gysen RJ. Analytical Solution of the Characteristic Function in the Trolle-Schwartz Model. []. ,Faculty of Commerce ,African Institute of Financial Markets and Risk Management, 2019 [cited yyyy month dd]. Available from: http://hdl.handle.net/11427/31328 en_ZA dc.language.rfc3066 eng dc.publisher.department African Institute of Financial Markets and Risk Management dc.publisher.faculty Faculty of Commerce dc.subject Mathematical Finance dc.title Analytical Solution of the Characteristic Function in the Trolle-Schwartz Model dc.type Master Thesis dc.type.qualificationlevel Masters dc.type.qualificationname MPhil
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