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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.

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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