Early exercise and Early exercise and Monte Carlo Monte Carlo

obtaining tight obtaining tight boundsbounds Mark JoshiMark Joshi

Centre for Actuarial StudiesCentre for Actuarial Studies

University of MelbourneUniversity of Melbourne

www.markjoshi.comwww.markjoshi.com

Derivatives pricingDerivatives pricing

We have a contract that pays off We have a contract that pays off according to the movements of some according to the movements of some reference rates or asset prices. reference rates or asset prices.

It may also involve some choices on It may also involve some choices on the part of the buyer. the part of the buyer.

Simplest example: call option, right Simplest example: call option, right but not the obligation to buy a stock but not the obligation to buy a stock on a given date at a given price.on a given date at a given price.

Developing a price for path Developing a price for path dependentsdependents

We pick some asset, N, as We pick some asset, N, as numeraire numeraire and and the price of a derivative D is then given the price of a derivative D is then given byby

Where T is the final maturity and D(T) Where T is the final maturity and D(T) includes all cash-flows generated. includes all cash-flows generated.

Expectation taken in the pricing Expectation taken in the pricing (martingale) measure.(martingale) measure.

Monte Carlo pricingMonte 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 Divide pay-off by the value of the

numeraire.numeraire. Average over many paths.Average over many paths. Law of large numbers says that it Law of large numbers says that it

converges to the expectation.converges to the expectation. Central limit theorem say error of Central limit theorem say error of

order order

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 numbers of exercised on one of a fixed finite numbers of dates.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 the same arguments hold for simplicity but the same arguments hold for interest rate derivatives and that is the more interest rate derivatives and that is the more important case.important case.

Exercise strategiesExercise strategies

Simple options involve no decisions on Simple options involve no decisions on the part of the holder or the decisions the part of the holder or the decisions are so simple that they are easily are so simple that they are easily equivalent to derivatives with no equivalent to derivatives with no decisions. 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.

Optimal exercise Optimal exercise strategiesstrategies

For a Bermudan option, we should For a Bermudan option, we should exercise if and only if the exercise if and only if the continuation (i.e. non-exercise) value continuation (i.e. non-exercise) value is greater than the exercise value.is greater than the exercise value.

So an optimal exercise strategy So an optimal exercise strategy exists and is easily described.exists and is easily described.

But requires knowledge of the But requires knowledge of the continuation value which may not be continuation value which may not be available.available.

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 the continuation value is always known. the 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 a stopping time has been chosen Once a 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.

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

exercise value exercise value

Lower boundsLower bounds

The buyer’s price is the supremum The buyer’s price is the supremum over all exercise strategies.over all exercise strategies.

So any choice of an exercise strategy So any choice of an exercise strategy will give us a lower bound.will give us a lower bound.

Many methods of finding such Many methods of finding such strategies now exist. strategies now exist.

Main problem is: how do we know if Main problem is: how do we know if they are any good?they are any good?

Seller’s priceSeller’s price

Seller cannot choose the exercise Seller cannot choose the exercise strategy.strategy.

The seller has to have enough cash The seller has to have enough cash on hand to cover the exercise value on hand to cover the exercise value whenever the buyer exercises. whenever the buyer exercises.

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

Maximal foresightMaximal foresight

The buyer could choose to exercise at The buyer could choose to exercise at random.random.

1/N chance of exercising at the maximum.1/N chance of exercising at the maximum. In derivatives pricing we are supposed to In derivatives pricing we are supposed to

cover no matter what happens.cover no matter what happens. So we must hedge against someone So we must hedge against someone

exercising at the max. exercising at the max. i.e. against someone exercising with i.e. against someone exercising with

maximal foresight.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, we have neglected the However, we have neglected the

seller’s ability to hedge.seller’s ability to 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 the product with this strategy and use the 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.

CallabilityCallability

Many Bermudan products arise as Many Bermudan products arise as the right to break a contract.the right to break a contract.

E.g. early redeem a fixed rate E.g. early redeem a fixed rate mortgage.mortgage.

Redeem a fixed coupon bond early.Redeem a fixed coupon bond early. Redeem a bond that pays a Redeem a bond that pays a

complicated coupon early. complicated coupon early. Break an interest rate swap.Break an interest rate swap.

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

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

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

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 the domain, approximation By restricting the 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

Jamshidian’s methodJamshidian’s method

An alternative approach to upper bounds An alternative approach to upper bounds by Monte Carlo is due to Jamshidian.by Monte Carlo is due to Jamshidian.

Suppose a Bermudan option always has Suppose a Bermudan option always has positive pay-off at the final time then its positive pay-off at the final time then its value is always positive.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 The ratio of the value of the pay-off to the

value of the derivative is always less than value of the derivative is always less than or equal to one at exercise dates (since we or equal to one at exercise dates (since we can always exercise) can always exercise)

Jamshidian’s equationJamshidian’s equation

This leads to the following equationThis leads to the following equation

Where H is the hedge consisting of Where H is the hedge consisting of the derivative itself, and on exercise the derivative itself, and on exercise purchasing extra units of itself.purchasing extra units of itself.

And N is the numeraire.And N is the numeraire.

Jamshidian’s methodJamshidian’s method

For the correct hedge, we get For the correct hedge, we get equality for a general H we get an equality for a general H we get an upper bound.upper bound.

So pick an H close to the derivative So pick an H close to the derivative and see what you get.and see what you get.

Extending JamshidianExtending Jamshidian Similarly to Rogers’ method, we can Similarly to Rogers’ method, we can

use the derivative itself with a sub-use the derivative itself with a sub-optimal strategy as the hedge and still optimal strategy as the hedge and still get an upper bound.get an upper bound.

Involves sub-MC simulations.Involves sub-MC simulations. We can exclude out of the money We can exclude out of the money

points as before.points as before. We can make it work even if the final We can make it work even if the final

pay-off can be zero: pay-off can be zero:

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, 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