Early exercise and Monte Carlo obtaining tight bounds

1
Early exercise and Monte Carlo obtaining tight
bounds

• Mark Joshi
• Centre for Actuarial Sciences
• University of Melbourne
• www.markjoshi.com

2
Bermudan optionality

• A Bermudan option is an option that be exercised
  on one of a fixed finite numbers of dates.
• Typically, arises as the right to break a
  contract.
• Right to terminate an interest rate swap
• Right to redeem note early
• We will focus on equity options here for
  simplicity but same arguments hold in IRD land.

3
Why Monte Carlo?

• Lattice methods are natural for early exercise
  problems, we work backwards so continuation value
  is always known.
• Lattice methods work well for low-dimensional
  problems but badly for high-dimensional ones.
• Path-dependence is natural for Monte Carlo
• LIBOR market model difficult on lattices
• Many lower bound methods now exist, e.g.
  Longstaff-Schwartz

4
Buyers price

• Holder can choose when to exercise.
• Can only use information that has already
  arrived.
• Exercise therefore occurs at a stopping time.
• If D is the derivative and N is numeraire, value
  is therefore
• Expectation taken in martingale measure.

5
Justifying buyers price

• Buyer chooses stopping time.
• Once stopping time has been chosen the derivative
  is effectively an ordinary path-dependent
  derivative for the buyer.
• In a complete market, the buyer can dynamically
  replicate this value.
• Buyer will maximize this value.
• Optimal strategy exercise when

• continuation value lt exercise value

6
Sellers price

• Seller cannot choose the exercise strategy.
• The seller has to have enough cash on hand to
  cover the exercise value whenever the buyer
  exercises.
• Buyers exercise could be random and would occur
  at the maximum with non-zero probability.
• So seller must be able to hedge against a buyer
  exercising with maximal foresight.

7
Sellers price continued

• Maximal foresight price
• Clearly bigger than buyers price.
• However, seller can hedge.

8
Hedging against maximal foresight

• Suppose we hedge as if buyer using optimal
  stopping time strategy.
• At each date, either our strategies agree and we
  are fine
• Or
• 1) buyer exercises and we dont
• 2) buyer doesnt exercise and we do
• In both of these cases we make money!

9
The optimal hedge

• Buy one unit of the option to be hedged.
• Use optimal exercise strategy.
• If optimal strategy says exercise. Do so and
  buy one unit of option for remaining dates.
• Pocket cash difference.
• As our strategy is optimal at any point where
  strategy says do not exercise, our valuation of
  the option is above the exercise value.

10
Rogers/Haugh-Kogan method

• Equality of buyers and sellers prices says
• for correct hedge Pt with P0 equals zero.
• If we choose wrong t, price is too low lower
  bound
• If we choose wrong Pt , price is too high upper
  bound
• Objective get them close together.

11
Approximating the perfect hedge

• If we know the optimal exercise strategy, we know
  the perfect hedge.
• In practice, we know neither.
• Anderson-Broadie pick an exercise strategy and
  use product with this strategy as hedge, rolling
  over as necessary.
• Main downside need to run sub-simulations to
  estimate value of hedge
• Main upside tiny variance

12
Improving Anderson-Broadie

• Our upper bound is
• The maximum could occur at a point where D0,
  which makes no financial sense.
• Redefine D to equal minus infinity at any point
  out of the money. (except at final time horizon.)
• Buyers price not affected, but upper bound will
  be lower.
• Added bonus fewer points to run sub-simulations
  at.

13
Provable sub-optimality

• Suppose we have a Bermudan put option in a
  Black-Scholes model.
• European put option for each exercise date is
  analytically evaluable.
• Gives quick lower bound on Bermudan price.
• Would never exercise if value lt max European.
• Redefine pay-off again to be minus infinity.
• Similarly, for Bermudan swaption.

14
Breaking structures

• Traditional to change the right to break into the
  right to enter into the opposite contract.
• Asian tail note
• Pays growth in FTSE plus principal after 3 years.
• Growth is measured by taking monthly average in
  3rd year.
• Principal guaranteed.
• Investor can redeem at 0.98 of principal at end
  of years one and two.

15
Non-analytic break values

• To apply Rogers/Haugh-Kogan/Anderson-Broadie/Longs
  taff-Schwartz, we need a derivative that pays a
  cash sum at time of exercise or at least yields
  an analytically evaluable contract.
• Asian-tail note does not satisfy this.
• Neither do many IRD contracts, e.g. callable CMS
  steepener.

16
Working with callability directly

• We can work with the breakable contract directly.
  
• Rather than thinking of a single cash-flow
  arriving at time of exercise, we think of
  cash-flows arriving until the contract is broken.
• Equivalence of buyers and sellers prices still
  holds, with same argument.
• Algorithm model independent and does not require
  analytic break values.

17
Upper bounds for callables

• Fix a break strategy.
• Price product with this strategy.
• Run a Monte Carlo simulation.
• Along each path accumulate discounted cash-flows
  of product and hedge.
• At points where strategy says break. Break the
  hedge and Purchase hedge with one less break
  date, this will typically have a negative cost.
  And pocket cash.
• Take the maximum of the difference of cash-flows.

18
Improving lower bounds

• Most popular lower bounds method is currently
  Longstaff-Schwartz.
• The idea is to regress continuation values along
  paths to get an approximation of the value of the
  unexercised derivative.
• Various tweaks can be made.
• Want to adapt to callable derivatives.

19
The Longstaff-Schwartz algorithm

• Generate a set of model paths
• Work backwards.
• At final time, exercise strategy and value is
  clear.
• At second final time, define continuation value
  to be the value on same path at final time.
• Regress continuation value against a basis.
• Use regressed value to decide exercise strategy.
• Define value at second last time according to
  strategy
• and value at following time.
• Work backwards.

20
Improving Longstaff-Schwartz

• We need an approximation to the unexercise value
  at points where we might exercise.
• By restricting domain, approximation becomes
  easier.
• Exclude points where exercise value is zero.
• Exclude points where exercise value less than
  maximal European value if evaluable.
• Use alternative regression methodology, eg loess

21
Longstaff-Schwartz for breakables

• Consider the Asian tail again.
• No simple exercise value.
• Solution (Amin)
• Redefine continuation value to be cash-flows that
  occur between now and the time of exercise in the
  future for each path.
• Methodology is model-independent.
• Combine with upper bounder to get two-sided
  bounds.

22
Example bounds for Asian tail

23
Difference in bounds
24
References

• A. Amin, Multi-factor cross currency LIBOR market
  model implemntation, calibration and examples,
  preprint, available from http//www.geocities.com/
  anan2999/
• L. Andersen, M. Broadie, A primal-dual simulation
  algorithm for pricing multidimensional American
  options, Management Science, 2004, Vol. 50, No.
  9, pp. 1222-1234.
• P. Glasserman, Monte Carlo Methods in Financial
  Engineering, Springer Verlag, 2003.
• M.Haugh, L. Kogan, Pricing American Options A
  Duality Approach, MIT Sloan Working Paper No.
  4340-01
• M. Joshi, Monte Carlo bounds for callable
  products with non-analytic break costs, preprint
  2006
• F. Longstaff, E. Schwartz, Valuing American
  options by simulation a least squares approach.
  Review of Financial Studies, 14113147, 1998.
• R. Merton, Option pricing when underlying stock
  returns are discontinuous, J. Financial Economics
  3, 125144, 1976
• L.C.G. Rogers Monte Carlo valuation of American
  options, Mathematical Finance,
• Vol. 12, pp. 271-286, 2002