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Early exercise and Monte Carlo obtaining tight bounds. Mark Joshi Centre for Actuarial Sciences University of Melbourne www.markjoshi.com. Bermudan optionality. A Bermudan option is an option that be exercised on one of a fixed finite numbers of dates.

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Early exercise and Monte Carlo obtaining tight bounds

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Early exercise and Monte Carlo obtaining tight bounds

Mark Joshi

Centre for Actuarial Sciences

University of Melbourne

www.markjoshi.com


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.


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


Buyer’s 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.


Justifying buyer’s 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 < exercise value


Seller’s 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.

  • Buyer’s 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.


Seller’s price continued

  • Maximal foresight price:

  • Clearly bigger than buyer’s price.

  • However, seller can hedge.


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 don’t

    • 2) buyer doesn’t exercise and we do

  • In both of these cases we make money!


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.


Rogers’/Haugh-Kogan method

  • Equality of buyer’s and seller’s prices says

    for correct hedge Pt with P0 equals zero.

  • If we choose wrong τ, price is too low = lower bound

  • If we choose wrong Pt , price is too high= upper bound

  • Objective: get them close together.


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


Improving Anderson-Broadie

  • Our upper bound is

  • The maximum could occur at a point where D=0, which makes no financial sense.

  • Redefine D to equal minus infinity at any point out of the money. (except at final time horizon.)

  • Buyer’s price not affected, but upper bound will be lower.

  • Added bonus: fewer points to run sub-simulations at.


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 < max European.

  • Redefine pay-off again to be minus infinity.

  • Similarly, for Bermudan swaption.


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.


Non-analytic break values

  • To apply Rogers/Haugh-Kogan/Anderson-Broadie/Longstaff-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.


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 buyer’s and seller’s prices still holds, with same argument.

  • Algorithm model independent and does not require analytic break values.


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.


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.


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.


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


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.


Example bounds for Asian tail


Difference in bounds


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, 14:113–147, 1998.

  • R. Merton, Option pricing when underlying stock returns are discontinuous, J. Financial Economics 3, 125–144, 1976

  • L.C.G. Rogers: Monte Carlo valuation of American options, Mathematical Finance,

    Vol. 12, pp. 271-286, 2002


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