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. 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.

5. 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

6. 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.

7. Seller’s price continued • Maximal foresight price: • Clearly bigger than buyer’s 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 don’t • 2) buyer doesn’t 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 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.

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 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.

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 < 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/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.

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 buyer’s and seller’s 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, 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