Early exercise and Monte
Carlo obtaining tight
bounds
Mark Joshi
Centre for Actuarial Sciences
University of Melbourne
www.markjoshi.com
Bermudan optionality
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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
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We will focus on equity options here for
simplicity but same arguments hold in IRD land.
Why Monte Carlo?
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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
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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
D0 N

1
0
1
 sup E ( N D )
Expectation taken in martingale measure.
Justifying buyer’s price
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Buyer chooses stopping time.
Once stopping time has been chosen the
derivative is effectively an ordinary pathdependent 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
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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
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Maximal foresight price:
1
r
1
supr E( N Dr )  E(max N t Dt )
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Clearly bigger than buyer’s price.
However, seller can hedge.
Hedging against maximal foresight
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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
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In both of these cases we make money!
The optimal hedge
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“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.
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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
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Equality of buyer’s and seller’s prices says
1
1
sup E( N D )  E(max N t ( Dt  Pt ))
for correct hedge Pt with P0 equals zero.
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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
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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
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Our upper bound is
E(max N t1 ( Dt  Pt ))
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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
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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
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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.
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Non-analytic break values
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To apply Rogers/Haugh-Kogan/AndersonBroadie/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
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We can work with the breakable contract
directly.
Rather than thinking of a single cash-flow
arriving at time of exercise, we think of cashflows 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
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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.
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Improving lower bounds
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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.
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Improving Longstaff-Schwartz
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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
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Consider the Asian tail again.
No simple exercise value.
Solution (Amin)
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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.
Example bounds for Asian tail
Asian tail varying jump intensity
1.15
1.13
1.11
1.09
1.07
lower
1.05
upper
1.03
1.01
0.99
0.97
0.95
0
0.1
0.2
0.4
0.8
1.6
3.2
Difference in bounds
Asian tail varying jump intensity: difference in bounds
0.0025
0.002
0.0015
difference
0.001
0.0005
0
0
0.1
0.2
0.4
0.8
1.6
3.2
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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