2. Vienna, 18 Oct 08 A perturbative approach to Bermudan Options pricing with applications Roberto Baviera, Rates Derivatives Trader & Structurer, Abaxbank joint work with Lorenzo Giada Vienna, 18 Oct 2008 1

3. Vienna, 18 Oct 08 2 R. Baviera Outline  Problem Formulation & Multifactor models  Bermudan Options Lower Bound: Standard Approach Lower Bound: Perturbative Approach Upper Bound  Examples Model description Example 1: ZC Bermudan Example 2: Step Up Callable Example 3: CMS Spread Bermudan A discussion on accuracy

4. Vienna, 18 Oct 08 3 R. Baviera Callable products: Problem Formulation : class of admissible stopping times with values in Optimal stopping with : Continuation value function Bermudan option: : discount in : payoff in

5. Vienna, 18 Oct 08 4 R. Baviera Rates: Multifactor models MonteCarlo: std approach for Non-Callable products Why MonteCarlo? Lattice methods work poorly for high-dimentional problems.

6. Vienna, 18 Oct 08 5 R. Baviera Callable products: MonteCarlo approach Optimal Stopping h < C Continuation Region Exercise Region Exercise Boundary is a Bermudan option with exercise dates In a MC approach each should come from a new MC simulation starting in !?! Problem: h < C

7. Vienna, 18 Oct 08 6 R. Baviera Approximated Continuation Value Any approximate exercise strategy provides a lower bound using in the exercise decision an approximation where are a set of parameters... Idea: Option value not very sensitive to the exact position of the Exercise Boundary Even a rough approximation of leads to a reasonable approximation of option value Lower Bound Two standard approaches: Longstaff-Schwartz (1998) Andersen (2000)

8. Vienna, 18 Oct 08 7 R. Baviera Standard Approach B: Optimization Lower Bound true Max Optimization exact function flat near true Continuation Value Cont. Value Option Value Max! Optimization with numerical noise (Monte Carlo evaluation) Cont. Value Option Value...then find the best.(Andersen 2000)

9. Vienna, 18 Oct 08 8 R. Baviera New Approach: Approximated continuation value curve true with an arbitrary simple (to compute) function Lower Bound

10. Vienna, 18 Oct 08 curve true 9 R. Baviera New Approach: basic idea approximated Lower Bound

11. Vienna, 18 Oct 08 10 R. Baviera New Approach: Recursive algorithm backwards Starting from the (N-1) Continuation value function, already a simple function, how to getknowing Lower Bound

12. Vienna, 18 Oct 08 11 R. Baviera New Approach: choiceLower Bound a possible choice with the max European option in : where European option valued in with expiry

13. Vienna, 18 Oct 08 12 R. Baviera New Approach: perturbative theoryLower Bound value in Delta in … Gamma in …

14. Vienna, 18 Oct 08 13 R. Baviera Dual MethodUpper Bound Idea: Given a class of martingale processes with values in Lower Bound: L 0 Upper Bound: U 0 (Roger 2001, Andersen Broadie 2004, …)

15. Vienna, 18 Oct 08 14 R. Baviera Dual MethodUpper Bound An approximated continuation value function set martingale process with: …two nested MCs

16. Vienna, 18 Oct 08 15 R. Baviera Examples 1.10y S/A ZC Bermudan option (N = 19) 2.10y S/A Step Up Callable (N = 19) 3.10y A/A Bermudan option on a 10-2 CMS spread (N = 9) : first expiry in Subset of expiries We also consider Lower and Upper bounds for Bermudans with a subset of exercise dates

17. Vienna, 18 Oct 08 16 R. Baviera Model: Notation Forward Libor Rates (in t 0 )Forward ZC Bond (in t 0 ) TodayStartEnd Today StartEnd … and their relation

18. Vienna, 18 Oct 08 17 R. Baviera Model: Bond Market Model BMM Dynamics: spot measure Some BMM Advantages with Fixing Mechanism  Elementary MC: Markov between Reset dates (Gaussian HJM)  Black like formulas for Caps/Floors & Swaptions  Large set of analytical solutions (e.g. CMS & CMS Spread European Options)...

19. Vienna, 18 Oct 08 18 R. Baviera Example 1: ZC Bermudan Option using paths using paths (external MC) & paths (internal MC) Strikes (N=19): Dataset: 14 Jan 05 at 11:15 CET

20. Vienna, 18 Oct 08 19 R. Baviera Example 2: Bermudan Coupon Option, # paths as before... 10y S/A Stepped Up yearly by 0.2% ( 2.9 % - 4.7 % )

21. Vienna, 18 Oct 08 20 R. Baviera Exercise Frequency

22. Vienna, 18 Oct 08 21 R. Baviera New Approach: Accuracy (*) 1 bp = 0.01 % Option value Accuracy in bps (*) standard: new (estim.):

23. Vienna, 18 Oct 08 22 R. Baviera Example 3: CMS Spread Bermudan, # paths as before... Payoff: 5 (CMS 10 – CMS 2 ), floored @ 0.5% capped @ 8%

24. Vienna, 18 Oct 08 23 R. Baviera Example3: Exercise Frequency

25. Vienna, 18 Oct 08 24 R. Baviera Conclusions An elementary new tecnique for pricing Bermudans with Multi-factor models:  Methodology is model independent  “Truly” financial expansion  High precision  Fast (no maximization)  Accuracy control

26. Vienna, 18 Oct 08 25 R. Baviera Bibliography sketch L.B.G. Andersen (2000), A Simple Approach to the Pricing of Bermudan Swaptions in the Multi-Factor Libor Market Model, J. Computational Finance 3, 1-32 L.B.G. Andersen & M. Broadie (2004), A Primal-Dual Simulation Algorithm for Pricing Multi- Dimensional American Options, Management Science 50, 1222-1234 R. Baviera (2006), Bond Market Model, IJTAF 9, 577-596 R. Baviera and L. Giada (2006), A perturbative approach to Bermudan Option pricing, http://ssrn.com/abstract=941318 & http://www.ibleo.it http://ssrn.com/abstract=941318http://www.ibleo.it P. Glasserman (2003), Monte Carlo Methods in Financial Engineering, Springer D. Heath, R. Jarrow and A. Morton (1992), Bond Pricing and the Term Structure of Interest Rates: a New Methodology for Contingent Claims Valuation, Econometrica 60, 77-105 F. Longstaff, E. Schwartz (1998), Valuing American options by simulation: a least squares approach, Rev. Fin. Studies 14,113–147 C. Rogers (2002), Monte Carlo Valuation of American Options, Mathematical Fin. 12, 271-286