Early exercise and Early exercise and Monte Carlo Monte Carlo

obtaining tight obtaining tight boundsbounds Mark JoshiMark Joshi

Centre for Actuarial SciencesCentre for Actuarial Sciences

University of MelbourneUniversity of Melbourne

www.markjoshi.comwww.markjoshi.com

Bermudan optionalityBermudan optionality

A Bermudan option is an option that be A Bermudan option is an option that be exercised on one of a fixed finite exercised on one of a fixed finite numbers of dates.numbers of dates.

Typically, arises as the right to break a Typically, arises as the right to break a contract. contract. Right to terminate an interest rate swapRight to terminate an interest rate swap Right to redeem note earlyRight to redeem note early

We will focus on equity options here for We will focus on equity options here for simplicity but same arguments hold in simplicity but same arguments hold in IRD land. IRD land.

Why Monte Carlo?Why Monte Carlo? Lattice methods are natural for early Lattice methods are natural for early

exercise problems, we work backwards so exercise problems, we work backwards so continuation value is always known. continuation value is always known.

Lattice methods work well for low-Lattice methods work well for low-dimensional problems but badly for high-dimensional problems but badly for high-dimensional ones.dimensional ones.

Path-dependence is natural for Monte Path-dependence is natural for Monte Carlo Carlo

LIBOR market model difficult on latticesLIBOR market model difficult on lattices Many lower bound methods now exist, e.g. Many lower bound methods now exist, e.g.

Longstaff-SchwartzLongstaff-Schwartz

Buyer’s priceBuyer’s price Holder can choose when to exercise.Holder can choose when to exercise. Can only use information that has already Can only use information that has already

arrived.arrived. Exercise therefore occurs at a stopping Exercise therefore occurs at a stopping

time. time. If D is the derivative and N is numeraire, If D is the derivative and N is numeraire,

value is therefore value is therefore

Expectation taken in martingale measure.Expectation taken in martingale measure.

1 10 0 sup ( )D N E N D

Justifying buyer’s priceJustifying buyer’s price

Buyer chooses stopping time. Buyer chooses stopping time. Once stopping time has been chosen Once stopping time has been chosen

the derivative is effectively an the derivative is effectively an ordinary path-dependent derivative ordinary path-dependent derivative for the buyer.for the buyer.

In a complete market, the buyer can In a complete market, the buyer can dynamically replicate this value. dynamically replicate this value.

Buyer will maximize this value.Buyer will maximize this value. Optimal strategy: exercise whenOptimal strategy: exercise whencontinuation value < continuation value <

exercise value exercise value

Seller’s priceSeller’s price Seller cannot choose the exercise Seller cannot choose the exercise

strategy.strategy. The seller has to have enough cash on The seller has to have enough cash on

hand to cover the exercise value hand to cover the exercise value whenever the buyer exercises. whenever the buyer exercises.

Buyer’s exercise could be random and Buyer’s exercise could be random and would occur at the maximum with non-would occur at the maximum with non-zero probability.zero probability.

So seller must be able to hedge against a So seller must be able to hedge against a buyer exercising with maximal foresight.buyer exercising with maximal foresight.

Seller’s price continuedSeller’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.

1 1sup ( ) (max )tr r r tE N D E N D

Hedging against maximal Hedging against maximal foresightforesight

Suppose we hedge as if buyer using Suppose we hedge as if buyer using optimal stopping time strategy. optimal stopping time strategy.

At each date, either our strategies At each date, either our strategies agree and we are fine agree and we are fine

Or Or 1) buyer exercises and we don’t1) buyer exercises and we don’t 2) buyer doesn’t exercise and we do2) buyer doesn’t exercise and we do

In both of these cases we make In both of these cases we make money!money!

The optimal hedgeThe 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 If optimal strategy says “exercise”. Do so

and buy one unit of option for remaining and buy one unit of option for remaining dates.dates. Pocket cash difference. Pocket cash difference.

As our strategy is optimal at any point As our strategy is optimal at any point where strategy says “do not exercise,” our where strategy says “do not exercise,” our valuation of the option is above the valuation of the option is above the exercise value.exercise value.

Rogers’/Haugh-Kogan Rogers’/Haugh-Kogan methodmethod

Equality of buyer’s and seller’s prices says Equality of buyer’s and seller’s prices says

for correct hedge Pfor correct hedge Pt t with Pwith P0 0 equals zero. equals zero.

If we choose wrong If we choose wrong ττ, price is too low = lower , price is too low = lower boundbound

If we choose wrong PIf we choose wrong Ptt , price is too high= , price is too high= upper bound upper bound

Objective: get them close together. Objective: get them close together.

1 1sup ( ) (max ( ))t t tE N D E N D P

Approximating the Approximating the perfect hedgeperfect hedge

If we know the optimal exercise If we know the optimal exercise strategy, we know the perfect hedge. strategy, we know the perfect hedge.

In practice, we know neither.In practice, we know neither. Anderson-Broadie: pick an exercise Anderson-Broadie: pick an exercise

strategy and use product with this strategy and use product with this strategy as hedge, rolling over as strategy as hedge, rolling over as necessary.necessary.

Main downside: need to run sub-Main downside: need to run sub-simulations to estimate value of hedgesimulations to estimate value of hedge

Main upside: tiny varianceMain upside: tiny variance

Improving Anderson-Improving Anderson-BroadieBroadie

Our upper bound isOur upper bound is

The maximum could occur at a point where The maximum could occur at a point where D=0, which makes no financial sense.D=0, which makes no financial sense.

Redefine D to equal minus infinity at any point Redefine D to equal minus infinity at any point out of the money. (except at final time out of the money. (except at final time horizon.)horizon.)

Buyer’s price not affected, but upper bound Buyer’s price not affected, but upper bound will be lower.will be lower.

Added bonus: fewer points to run sub-Added bonus: fewer points to run sub-simulations at.simulations at.

1(max ( ))t t tE N D P

Provable sub-optimalityProvable sub-optimality

Suppose we have a Bermudan put option in Suppose we have a Bermudan put option in a Black-Scholes model. a Black-Scholes model.

European put option for each exercise date European put option for each exercise date is analytically evaluable. is analytically evaluable.

Gives quick lower bound on Bermudan Gives quick lower bound on Bermudan price. price.

Would never exercise if value < max Would never exercise if value < max European.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.

Breaking structuresBreaking structures

Traditional to change the right to break Traditional to change the right to break into the right to enter into the opposite into the right to enter into the opposite contract. contract.

Asian tail noteAsian tail note Pays growth in FTSE plus principal after 3 Pays growth in FTSE plus principal after 3

years.years. Growth is measured by taking monthly Growth is measured by taking monthly

average in 3average in 3rdrd year. year. Principal guaranteed.Principal guaranteed. Investor can redeem at 0.98 of principal at Investor can redeem at 0.98 of principal at

end of years one and two. end of years one and two.

Non-analytic break Non-analytic break valuesvalues

To apply To apply Rogers/Haugh-Kogan/Anderson-Rogers/Haugh-Kogan/Anderson-Broadie/Longstaff-Schwartz, we need Broadie/Longstaff-Schwartz, we need a derivative that pays a cash sum at a derivative that pays a cash sum at time of exercise or at least yields an time of exercise or at least yields an analytically evaluable contract. analytically evaluable contract.

Asian-tail note does not satisfy this. Asian-tail note does not satisfy this. Neither do many IRD contracts, e.g. Neither do many IRD contracts, e.g.

callable CMS steepener. callable CMS steepener.

Working with callability Working with callability directlydirectly

We can work with the breakable We can work with the breakable contract directly. contract directly.

Rather than thinking of a single cash-Rather than thinking of a single cash-flow arriving at time of exercise, we flow arriving at time of exercise, we think of cash-flows arriving until the think of cash-flows arriving until the contract is broken.contract is broken.

Equivalence of buyer’s and seller’s Equivalence of buyer’s and seller’s prices still holds, with same argument.prices still holds, with same argument.

Algorithm model independent and does Algorithm model independent and does not require analytic break values.not require analytic break values.

Upper bounds for Upper bounds for callablescallables

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 Along each path accumulate discounted cash-flows of product and hedge.cash-flows of product and hedge.

At points where strategy says break. Break At points where strategy says break. Break the hedge and “Purchase” hedge with one the hedge and “Purchase” hedge with one less break date, this will typically have a less break date, this will typically have a negative cost. And pocket cash.negative cost. And pocket cash.

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

Improving lower boundsImproving lower bounds Most popular lower bounds method is Most popular lower bounds method is

currently Longstaff-Schwartz. currently Longstaff-Schwartz. The idea is to regress continuation The idea is to regress continuation

values along paths to get an values along paths to get an approximation of the value of the approximation of the value of the unexercised derivative.unexercised derivative.

Various tweaks can be made. Various tweaks can be made. Want to adapt to callable derivatives.Want to adapt to callable derivatives.

The Longstaff-Schwartz The Longstaff-Schwartz algorithmalgorithm

Generate a set of model pathsGenerate a set of model paths Work backwards.Work backwards. At final time, exercise strategy and value is At final time, exercise strategy and value is

clear.clear. At second final time, define continuation value At second final time, define continuation value

to be the value on same path at final time.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 Use regressed value to decide exercise

strategy.strategy. Define value at second last time according to Define value at second last time according to

strategystrategyand value at following time.and value at following time. Work backwards.Work backwards.

Improving Longstaff-Improving Longstaff-SchwartzSchwartz

We need an approximation to the We need an approximation to the unexercise value at points where we might unexercise value at points where we might exercise.exercise.

By restricting domain, approximation By restricting domain, approximation becomes easier.becomes easier.

Exclude points where exercise value is zero.Exclude points where exercise value is zero. Exclude points where exercise value less Exclude points where exercise value less

than maximal European value if evaluable.than maximal European value if evaluable. Use alternative regression methodology, eg Use alternative regression methodology, eg

loessloess

Longstaff-Schwartz for Longstaff-Schwartz for breakablesbreakables

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-Redefine continuation value to be cash-flows that occur between now and the flows that occur between now and the time of exercise in the future for each time of exercise in the future for each path.path.

Methodology is model-independent.Methodology is model-independent. Combine with upper bounder to get Combine with upper bounder to get

two-sided bounds.two-sided bounds.

Example bounds for Example bounds for Asian tailAsian tail

Asian tail varying jump intensity

0.95

0.97

0.99

1.01

1.03

1.05

1.07

1.09

1.11

1.13

1.15

0 0.1 0.2 0.4 0.8 1.6 3.2

lower

upper

Difference in boundsDifference in bounds

Asian tail varying jump intensity: difference in bounds

0

0.0005

0.001

0.0015

0.002

0.0025

0 0.1 0.2 0.4 0.8 1.6 3.2

difference

ReferencesReferences 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