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# Early exercise and Monte Carlo obtaining tight bounds - PowerPoint PPT Presentation

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

Mark Joshi

Centre for Actuarial Sciences

University of Melbourne

www.markjoshi.com

• 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.

• 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

• 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.

• 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 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.

• Maximal foresight price:

• Clearly bigger than buyer’s price.

• However, seller can hedge.

• 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!

• “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.

• 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.

• 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

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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

• 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.

• 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