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. 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. Typically, arises as the right to break a contract. Right to terminate an interest rate swap Right to terminate an interest rate swap Right to redeem note early Right to redeem note early We will focus on equity options here for simplicity but same arguments hold in IRD land. 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 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. Lattice methods work well for low-dimensional problems but badly for high-dimensional ones. Path-dependence is natural for Monte Carlo Path-dependence is natural for Monte Carlo LIBOR market model difficult on lattices LIBOR market model difficult on lattices Many lower bound methods now exist, e.g. Longstaff-Schwartz Many lower bound methods now exist, e.g. Longstaff-Schwartz

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

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

6. Seller’s price Seller cannot choose the exercise strategy. Seller cannot choose the exercise strategy. The seller has to have enough cash on hand to cover the exercise value whenever the buyer exercises. 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. 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. So seller must be able to hedge against a buyer exercising with maximal foresight.

7. Seller’s price continued Maximal foresight price: Maximal foresight price: Clearly bigger than buyer’s price. Clearly bigger than buyer’s price. However, seller can hedge. However, seller can hedge.

8. Hedging against maximal foresight Suppose we hedge as if buyer using optimal stopping time strategy. Suppose we hedge as if buyer using optimal stopping time strategy. At each date, either our strategies agree and we are fine At each date, either our strategies agree and we are fine Or Or 1) buyer exercises and we don’t 1) buyer exercises and we don’t 2) buyer doesn’t exercise and we do 2) buyer doesn’t exercise and we do In both of these cases we make money! In both of these cases we make money!

9. The optimal hedge “Buy” one unit of the option to be hedged. “Buy” one unit of the option to be hedged. Use optimal exercise strategy. Use optimal exercise strategy. If optimal strategy says “exercise”. Do so and buy one unit of option for remaining dates. If optimal strategy says “exercise”. Do so and buy one unit of option for remaining dates. Pocket cash difference. 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. 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 Equality of buyer’s and seller’s prices says for correct hedge P t with P 0 equals zero. If we choose wrong τ, price is too low = lower bound If we choose wrong τ, price is too low = lower bound If we choose wrong P t, price is too high= upper bound If we choose wrong P t, price is too high= upper bound Objective: get them close together. Objective: get them close together.

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

12. Improving Anderson-Broadie Our upper bound is Our upper bound is The maximum could occur at a point where D=0, which makes no financial sense. 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.) 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. Buyer’s price not affected, but upper bound will be lower. Added bonus: fewer points to run sub-simulations at. 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. Suppose we have a Bermudan put option in a Black-Scholes model. European put option for each exercise date is analytically evaluable. European put option for each exercise date is analytically evaluable. Gives quick lower bound on Bermudan price. Gives quick lower bound on Bermudan price. Would never exercise if value < max European. Would never exercise if value < max European. Redefine pay-off again to be minus infinity. Redefine pay-off again to be minus infinity. Similarly, for Bermudan swaption. Similarly, for Bermudan swaption.

14. Breaking structures Traditional to change the right to break into the right to enter into the opposite contract. Traditional to change the right to break into the right to enter into the opposite contract. Asian tail note Asian tail note Pays growth in FTSE plus principal after 3 years. Pays growth in FTSE plus principal after 3 years. Growth is measured by taking monthly average in 3 rd year. Growth is measured by taking monthly average in 3 rd year. Principal guaranteed. Principal guaranteed. Investor can redeem at 0.98 of principal at end of years one and two. 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. 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. Asian-tail note does not satisfy this. Neither do many IRD contracts, e.g. callable CMS steepener. Neither do many IRD contracts, e.g. callable CMS steepener.

16. Working with callability directly We can work with the breakable contract 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. 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. Equivalence of buyer’s and seller’s prices still holds, with same argument. Algorithm model independent and does not require analytic break values. Algorithm model independent and does not require analytic break values.

17. Upper bounds for callables Fix a break strategy. Fix a break strategy. Price product with this strategy. Price product with this strategy. Run a Monte Carlo simulation. Run a Monte Carlo simulation. Along each path accumulate discounted cash-flows of product and hedge. 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. 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. Take the maximum of the difference of cash-flows.

18. Improving lower bounds Most popular lower bounds method is currently Longstaff-Schwartz. 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. 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. Various tweaks can be made. Want to adapt to callable derivatives. Want to adapt to callable derivatives.

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

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

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