A perturbative approach to Bermudan Options pricing

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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
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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

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Callable products Problem Formulation
Bermudan option
class of admissible stopping times with values
in Optimal stopping
with Continuation value function
discount in payoff
in
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Rates Multifactor models
MonteCarlo std approach for Non-Callable products
Why MonteCarlo? Lattice methods work poorly for
high-dimentional problems.
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Callable products MonteCarlo approach
Exercise Region
Exercise Boundary
Continuation Region
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Approximated Continuation Value
Lower Bound
Any approximate exercise strategy provides a
lower bound using in the exercise
decision an approximation
where are a set of
parameters...
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Standard Approach B Optimization
Lower Bound
...then find the best .
(Andersen 2000)
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New Approach Approximated continuation value
Lower Bound
true
curve
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New Approach basic idea
Lower Bound
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New Approach Recursive algorithm backwards
Lower Bound
Starting from the (N-1) Continuation value
function, already a simple function, how to
get knowing
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New Approach choice
Lower Bound
a possible choice
with the max European option in
where European option
valued in with expiry
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New Approach perturbative theory
Lower Bound

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Dual Method
Upper Bound
Idea Given a class of martingale processes
with values in
Lower Bound L0
Upper Bound U0
(Roger 2001, Andersen Broadie 2004, )
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Dual Method
Upper Bound
An approximated continuation value function set

martingale process
with
two nested MCs
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Examples

• 10y S/A ZC Bermudan option (N 19)
• 10y S/A Step Up Callable (N 19)
• 10y A/A Bermudan option on a 10-2 CMS spread (N
  9)

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Model Notation
Forward Libor Rates (in t0)
Forward ZC Bond (in t0)
and their relation
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Model Bond Market Model
BMM Dynamics spot measure
with
Fixing Mechanism
Some BMM Advantages

• 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) ...

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Example 1 ZC Bermudan Option
using paths using paths
(external MC) paths (internal MC)
Strikes (N19)
Dataset 14 Jan 05 at 1115 CET
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Example 2 Bermudan Coupon Option
, paths as before... 10y S/A Stepped Up
yearly by 0.2 ( 2.9 - 4.7 )
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Exercise Frequency
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New Approach Accuracy
Option value
Accuracy in bps() standard new (estim.)
()1 bp 0.01
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Example 3 CMS Spread Bermudan
, paths as before... Payoff 5 (CMS10
CMS 2), floored _at_ 0.5 capped _at_ 8
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Example3 Exercise Frequency
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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

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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/abstract941318
  http//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 14,113147
• C. Rogers (2002), Monte Carlo Valuation of
  American Options, Mathematical Fin. 12, 271-286

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