U.U.D.M. Project Report 2017:13 Pricing of Barrier Options Using a Two-Volatility Model Konstantinos Papakonstantinou Examensarbete i matematik, 30 hp Handledare: Jacob Lundgren, Itiviti Group AB Ämnesgranskare: Erik Ekström Examinator: Denis Gaidashev Juni 2017 Department of Mathematics Uppsala University Abstract The purpose of the current study is to assess and compare performance char- acteristics of a range of trinomial lattice and finite difference solvers for pric- ing barrier options using a two-volatility model. Emphasis is laid upon reduc- ing the problem to a one-dimensional barrier option pricing problem with a time-independent barrier. Additionally, focus is placed upon handling discrete dividend payments and floating interest rates. Improvements such as the use of a non-uniform grid and the local refinement of the adaptive mesh model are also scrutinized. The results of this study reveal that the Crank-Nicolson finite difference scheme almost outperforms among all the considered meth- ods. Only Ritchken’s tree may compete against it when a relatively fast re- sponse time is necessary. Keywords: pricing, valuation, barrier option, two-volatility model, finite dif- ference method, trinomial lattice method, non-uniform grid, adaptive mesh model, discrete dividends. To the memory of my grandmother, Evaggelia Mourti Acknowledgements First and foremost, I wish to thank professor Erik Ekström for his invaluable help in conducting this study. I would like also to thank Itiviti company and especially, Jacob Lundgren and Yuri Shpolyanskiy for their support and guid- ance. For additional helpful comments on this dissertation, I am grateful to Peiyu Wang and Victor Shcherbakov. I would also like to thank my parents Giorgos and Rania, and my sister Maria for their unwavering support throughout my studies. Additionally, I thank Anna, Fragkiski and Vaggelis for their encouragement. Finally, I wish to express my gratitude to my beloved partner Sofia for her wholehearted sup- port. Contents 1. Introduction ............................................................................................ 9 1.1. Barrier Options ............................................................................... 9 1.2. Model and Absence of Arbitrage ................................................. 11 1.3. Numerical Methods ...................................................................... 12 1.4. Objectives ..................................................................................... 13 1.5. Outline .......................................................................................... 13 2. Theory .................................................................................................. 14 2.1. Stochastic Calculus ...................................................................... 14 2.1.1. Stochastic Processes ............................................................. 14 2.1.2. Stochastic Integrals ............................................................... 15 2.1.3. Itô Processes ......................................................................... 15 2.1.4. Itô’s Formula ........................................................................ 16 2.1.5. The Feynman-Kac Formula .................................................. 16 2.1.6. Geometric Brownian Motion ................................................ 16 2.2. The Black-Scholes-Merton Model ............................................... 17 2.2.1. The Black-Scholes Equation ................................................ 18 2.2.2. Risk-Neutral Valuation ......................................................... 19 2.3. Pricing Formulas of Standard Barrier Options ............................. 20 2.4. Trinomial Tree Models ................................................................. 22 2.4.1. Standard Trinomial Tree Model ........................................... 22 2.4.2. Probability-Adjusted Trinomial Tree Model ........................ 24 2.4.3. Ritchken’s Trinomial Tree Model ........................................ 25 2.4.4. Adaptive Mesh Model .......................................................... 27 2.5. Finite Difference Methods ........................................................... 30 2.5.1. Explicit Discretization .......................................................... 31 2.5.2. Fully Implicit Discretization ................................................. 32 2.5.3. Crank-Nicolson Discretization ............................................. 33 2.5.4. Boundary Conditions ............................................................ 34 2.5.5. Non-uniform Grid Points ...................................................... 35 2.5.6. Tridiagonal matrix algorithm ................................................ 38 2.6. Dividends ..................................................................................... 39 2.7. Barycentric Lagrange Interpolation ............................................. 40 3. Methodology ........................................................................................ 42 3.1. Problem ........................................................................................ 42 3.2. Solution ........................................................................................ 42 3.2.1. European Barrier Option ...................................................... 42 3.2.2. Stochastic Process H ............................................................. 43 3.2.3. Pricing of Barrier Options Using a Two-Volatility Model ... 44 3.2.4. Transformation to a Time-independent Barrier Option ........ 45 3.3. Implementation ............................................................................. 46 4. Results .................................................................................................. 47 4.1. Test Specification ......................................................................... 47 4.2. Convergence Test ......................................................................... 48 4.3. Performance Test .......................................................................... 54 4.4. Discussion .................................................................................... 58 5. Conclusion ........................................................................................... 60 6. Bibliography ........................................................................................ 61 List of Figures Figure 1. A Wiener trajectory. ................................................................... 14 Figure 2. A geometric Brownian motion: µ = 1, σ = 0.4. ........................ 17 Figure 3. A trinomial tree model with diffusion paths traversing the barrier between nodes. ......................................... 25 Figure 4. Two trinomial tree models for a down-and-out option with λ = 1 and λ estimated according to Ritchken’s procedure. ....... 27 Figure 5. An illustration of the adaptive mesh model for a down-and-out option. ......................................................... 29 Figure 6. An example of non-uniform grids via an exponential transformation. ................................................... 37 Figure 7. An example of a non-uniform grid via a polynomial transformation. ...................................................... 37 Figure 8. A dividend correction as described by Vellekoop and Nieuwenhuis. ...................................................... 40 Figure 9. An example of a non-uniform grid via a coordinate transformation. ....................................................... 46 Figure 10. Convergence of the finite difference methods for the down-and-out call option with 64-bit precision. ................... 49 Figure 11. Convergence of the finite difference methods for the down-and-out call option with 128-bit precision. ................. 50 Figure 12. Convergence of the finite difference methods for the up-and-out put option with 128-bit precision. ...................... 50 Figure 13. Convergence of the finite difference methods for the down-and-out call option. ..................................................... 51 Figure 14. Convergence of the finite difference methods for the down-and-out call option. ..................................................... 51 Figure 15. Convergence of the finite difference methods for the up-and-out put option. ........................................................... 52 Figure 16. Convergence of the lattice methods for the down-and-out call option. ..................................................... 53 Figure 17. Convergence of the lattice methods for the up-and-out put option. ........................................................... 53 Figure 18. An effective relationship between the number of stock nodes and time points for the down-and-out call option. .. 54 Figure 19. An effective relationship between the number of stock nodes and time points for the up-and-out put option. ........ 55 Figure 20. Performance of the finite difference solvers for the down-and-out call option. ..................................................... 55 Figure 21. Performance of the finite difference solvers for the up-and-out put option. ........................................................... 56 Figure 22. Performance of the lattice solvers for the down-and-out call option. ..................................................... 56 Figure 23. Performance of the lattice solvers for the up-and-out put option. ........................................................... 57 Figure 24. Performance