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Early exercise and Monte Carlo obtaining tight bounds Mark Joshi Centre for Actuarial Studies University of Melbourne www.markjoshi.com

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Derivatives pricing We have a contract that pays off according to the movements of some reference rates or asset prices. We have a contract that pays off according to the movements of some reference rates or asset prices. It may also involve some choices on the part of the buyer. It may also involve some choices on the part of the buyer. Simplest example: call option, right but not the obligation to buy a stock on a given date at a given price. Simplest example: call option, right but not the obligation to buy a stock on a given date at a given price.

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Developing a price for path dependents We pick some asset, N, as numeraire and the price of a derivative D is then given by We pick some asset, N, as numeraire and the price of a derivative D is then given by Where T is the final maturity and D(T) includes all cash-flows generated. Where T is the final maturity and D(T) includes all cash-flows generated. Expectation taken in the pricing (martingale) measure. Expectation taken in the pricing (martingale) measure.

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Monte Carlo pricing Develop a path for the underlying. Develop a path for the underlying. Work out pay-off for that path. Work out pay-off for that path. Divide pay-off by the value of the numeraire. Divide pay-off by the value of the numeraire. Average over many paths. Average over many paths. Law of large numbers says that it converges to the expectation. Law of large numbers says that it converges to the expectation. Central limit theorem say error of order Central limit theorem say error of order

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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 the same arguments hold for interest rate derivatives and that is the more important case. We will focus on equity options here for simplicity but the same arguments hold for interest rate derivatives and that is the more important case.

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Exercise strategies Simple options involve no decisions on the part of the holder or the decisions are so simple that they are easily equivalent to derivatives with no decisions. Simple options involve no decisions on the part of the holder or the decisions are so simple that they are easily equivalent to derivatives with no decisions. E.g. a call option, the pay-off is E.g. a call option, the pay-off is No decisions necessary. No decisions necessary.

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Optimal exercise strategies For a Bermudan option, we should exercise if and only if the continuation (i.e. non-exercise) value is greater than the exercise value. For a Bermudan option, we should exercise if and only if the continuation (i.e. non-exercise) value is greater than the exercise value. So an optimal exercise strategy exists and is easily described. So an optimal exercise strategy exists and is easily described. But requires knowledge of the continuation value which may not be available. But requires knowledge of the continuation value which may not be available.

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Why Monte Carlo? Lattice methods are natural for early exercise problems, we work backwards so the continuation value is always known. Lattice methods are natural for early exercise problems, we work backwards so the 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

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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.

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

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Lower bounds The buyer’s price is the supremum over all exercise strategies. The buyer’s price is the supremum over all exercise strategies. So any choice of an exercise strategy will give us a lower bound. So any choice of an exercise strategy will give us a lower bound. Many methods of finding such strategies now exist. Many methods of finding such strategies now exist. Main problem is: how do we know if they are any good? Main problem is: how do we know if they are any good?

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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. The seller’s price is therefore the amount of money required to hedge against any exercise strategy. The seller’s price is therefore the amount of money required to hedge against any exercise strategy.

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Maximal foresight The buyer could choose to exercise at random. The buyer could choose to exercise at random. 1/N chance of exercising at the maximum. 1/N chance of exercising at the maximum. In derivatives pricing we are supposed to cover no matter what happens. In derivatives pricing we are supposed to cover no matter what happens. So we must hedge against someone exercising at the max. So we must hedge against someone exercising at the max. i.e. against someone exercising with maximal foresight. i.e. against someone exercising with maximal foresight.

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Seller’s price continued Maximal foresight price: Maximal foresight price: Clearly bigger than buyer’s price. Clearly bigger than buyer’s price. However, we have neglected the seller’s ability to hedge. However, we have neglected the seller’s ability to hedge.

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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!

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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.

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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.

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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 the product with this strategy as hedge, rolling over as necessary. Anderson-Broadie: pick an exercise strategy and use the 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

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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.

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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.

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Callability Many Bermudan products arise as the right to break a contract. Many Bermudan products arise as the right to break a contract. E.g. early redeem a fixed rate mortgage. E.g. early redeem a fixed rate mortgage. Redeem a fixed coupon bond early. Redeem a fixed coupon bond early. Redeem a bond that pays a complicated coupon early. Redeem a bond that pays a complicated coupon early. Break an interest rate swap. Break an interest rate swap.

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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.

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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.

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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.

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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 along the paths. Take the maximum of the difference of cash-flows along the paths.

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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.

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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.

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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 the domain, approximation becomes easier. By restricting the 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

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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.

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Example bounds for Asian tail

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Difference in bounds

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Jamshidian’s method An alternative approach to upper bounds by Monte Carlo is due to Jamshidian. An alternative approach to upper bounds by Monte Carlo is due to Jamshidian. Suppose a Bermudan option always has positive pay-off at the final time then its value is always positive. Suppose a Bermudan option always has positive pay-off at the final time then its value is always positive. We can therefore take it as numeraire. We can therefore take it as numeraire. The ratio of the value of the pay-off to the value of the derivative is always less than or equal to one at exercise dates (since we can always exercise) The ratio of the value of the pay-off to the value of the derivative is always less than or equal to one at exercise dates (since we can always exercise)

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Jamshidian’s equation This leads to the following equation This leads to the following equation Where H is the hedge consisting of the derivative itself, and on exercise purchasing extra units of itself. Where H is the hedge consisting of the derivative itself, and on exercise purchasing extra units of itself. And N is the numeraire. And N is the numeraire.

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Jamshidian’s method For the correct hedge, we get equality for a general H we get an upper bound. For the correct hedge, we get equality for a general H we get an upper bound. So pick an H close to the derivative and see what you get. So pick an H close to the derivative and see what you get.

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Extending Jamshidian Similarly to Rogers’ method, we can use the derivative itself with a sub-optimal strategy as the hedge and still get an upper bound. Similarly to Rogers’ method, we can use the derivative itself with a sub-optimal strategy as the hedge and still get an upper bound. Involves sub-MC simulations. Involves sub-MC simulations. We can exclude out of the money points as before. We can exclude out of the money points as before. We can make it work even if the final pay-off can be zero: We can make it work even if the final pay-off can be zero:

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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, A simple derivation of and improvements to Jamshidian's and Rogers' upper bound methods for Bermudan options, to appear in Applied Mathematical Finance 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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