pricing early exercise options by monte carlo - an...

Pricing Options with Early Exerciseby Monte Carlo Simulation:

An Introduction

Christian P. [email protected]

December 5, 2005

(Version 0.3)

Abstract

In this paper we review some of the most common methods used in pricing Bermudan (orAmerican) type options by Monte Carlo simulation. We assume basic knowledge of risk neu-tral pricing theory1 and Monte Carlo simulation2.

Note

This version contains only the first chapters onlower bound methods. If you are interested ina discussion ofupper bound methodscheck the papers home page

http://www.christian-fries.de/finmath/earlyexercise

for an updated version.

1 For an introduction to risk neutral pricing see, e.g., [3, 11].2 For an introduction to Monte Carlo simulation see, e.g., [4, 5, 11, 12, 14].

1

mailto:[email protected]


Pricing Early Exercise Options by Monte Carlo Christian P. Fries

Contents

1 Introduction 3

2 Bermudan Options: Notation 32.1 Bermudan Callable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .42.2 Relative Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4

3 Bermudan Option as Optimal Exercise Problem 53.1 Bermudan Option Value as single (unconditioned) Expectation: The Optimal

Exercise Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5

4 Bermudan Option Pricing - The Backward Algorithm 6

5 Re-simulation 7

6 Perfect Foresight 7

7 Conditional Expectation as Functional Dependence 8

8 Binning 98.1 Binning as a Least-Square Regression . . . . . . . . . . . . . . . . . . . . . .10

9 Foresight Bias 11

10 Regression Methods - Least Square Monte Carlo 1210.1 Least Square Approximation of the Conditional Expectation . . . . . . . . . .1210.2 Example: Evaluation of anBermudanOption on a Stock (Backward Algorithm

with Conditional Expectation Estimator) . . . . . . . . . . . . . . . . . . . . .1310.3 Example: Evaluation of an Bermudan Callable . . . . . . . . . . . . . . . . .1410.4 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1710.5 Binning as linear Least-Square Regression . . . . . . . . . . . . . . . . . . . .17

11 Optimization Methods 1911.1 Andersen Algorithm for Bermudan Swaptions . . . . . . . . . . . . . . . . . .1911.2 Review of the Threshold Optimization Method . . . . . . . . . . . . . . . . .20

11.2.1 Fitting the exercise strategy to the product . . . . . . . . . . . . . . . .2011.2.2 Disturbance of the Optimizer through Monte Carlo Error . . . . . . .21

11.3 Optimization of Exercise Strategy: A more general Formulation . . . . . . . .2111.4 Comparisson of Optimization Method and Regression Method . . . . . . . . .22

12 Duality Method: Upper Bound for Bermudan Option Prices 23

References 24

Notes 25

c©2005 Christian Frieshttp://www.christian-fries.de/finmath/earlyexercise

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1 Introduction

Let us first consider the simple case of a Bermudan optionVberm(T1, T2) with two exercisedates only: The option holder has the right to receive an underlyingVunderl,1 in T1 or waitand retain the right to either receive an underlyingVunderl,2 in T2 or receive nothing. Putdifferently, the option holder has the choice or receiving the underlying valueVunderl,1(T1)and the value of an optionVoption(T1) onVunderl,2(T2). The Bermudan may be interpreted asan option on an option. InT1 the optimal exercise is given by choosing the maximum value

max(Vunderl(T1), Voption(T1)), (1)

where (having choosen a NuméraireN )

Voption(T1) = N(T1)EQN (Voption(T2)N(T2)

| FT1

)= N(T1)EQN (max(Vunderl,2(T2), 0)

N(T2)| FT1

). (2)

is the value of the option with exercise inT2, evaluated inT1.

T1 T2T0

Underlying

(Hold)

Underlying

0exercisedate

exercisedate

Figure 1: A simple Bermudan option with two exercise dates.

Thus, to evaluate the exercise criteria (1) it is in general necessary to calculate a conditionalexpectation. The calculation of a conditional expectation within a Monte-Carlo simulation isa non trivial problem. The two main issues arecomplexityandforesight bias, which we willillustrate in Section5and6. In the then following Section we will present methods to efficientlyestimate conditional expectations and/or the Bermudan exercise criteria within Monte-Carlosimulation. The application to the pricing of Bermudan option will always be present, butapart from the description of the backward algorithm in Section4 the methods presented arenot limited to Bermudan option pricing.

2 Bermudan Options: Notation

We now give a faily general definition of an Bermudan option and fix notation. LetTii=1,...,n

denote a set of exercise dates andVunderl,ii=1,...,n a corresponding set of underlyings. TheBermudan option is the right to receive at one and only one timeTi the corresponding under-lying Vunderl,i (with i = 1, . . . , n) or receive nothing.

At each exercise dateTi, the optimal strategy compares the value of the product uponexercise with the value of the product upon non-exercise and chooses the larger one. Thus the


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value of the Bermudan is given recursively

Vberm(Ti, . . . , Tn;Ti) := max(Vberm(Ti+1, . . . , Tn;Ti) , Vunderl,i(Ti)

), (3)

whereVberm(Tn;Tn) := 0 andVunderl,i(Ti) denotes the value of the underlyingVunderl,i atexercise dateTi.

2.1 Bermudan Callable

The most common Bermudan option is the Bermudan Callable. For a Bermudan Calable theunderlyings consist of periodic paymentsXk and differ only in the beginning of the periodicpayments. The value of the underlying then becomes

Vunderl,i(Ti) = Vunderl(Ti, . . . , Tn;Ti) = N(Ti)EQN

(n−1∑k=i

Xk

N(Tk+1)|FTi

)

HereXk denotes a payment fixed inTk (i.e.FTkmeasurable) and paid inTk+1. This is the

common setup for interest rate bermudan callable. Other payment dates are a minor modifica-tion, they simply change the time argument of the Numéraire. For the value upon non exercisewe have as before

Vberm(Ti+1, . . . , Tn;Ti) = N(Ti)EQN

(Vberm(Ti+1, . . . , Tn;Ti+1)

N(Ti+1)|FTi

).

If the value of the underlying may not be expressed by means of an analytical formulatwoconditional expectations have to be evaluated to calculate the exercise strategy (3).

2.2 Relative Prices

Since the conditional expectation of a Numéraire relative price is a Numéraire relative pricethe presentation will be simplified by considering the Numéraire relative quantities. We willtherefore define:

Vunderl,i(Tj) :=Vunderl,i(Tj)

N(Tj)und Vberm,i(Tj) :=

Vberm(Ti, . . . , Tn;Tj)N(Tj)

,

thus we have

Vberm,n ≡ 0,

Vberm,i+1(Ti) = EQN (Vberm,i+1(Ti+1) | FTi

),

Vberm,i(Ti) = max(Vberm,i+1(Ti) , Vunderl,i(Ti)

),

and in the case of a Bermudan callable

Vunderl,i(Tj) = EQN ( n−1∑k=i

Xk(Tk+1) | FTj

)where Xi(Ti+1) :=

Xi

N(Ti).

The relative prices are marked by a tilde.

Remark 1 (Notation): The processest 7→ Vunderl,i(t) andt 7→ Vberm,i(t) areFt conditionalexpectations ofVunderl,i(Ti) andVberm,i(Ti) respectively and thus martingales by definition.The time-discrete processesi 7→ Vunderl,i(Ti), i 7→ Vberm,i(Ti) consist of different productsat different times and are thus not time-discrete martingales in general.


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3 Bermudan Option as Optimal Exercise Problem

A Bermudan option consists of the right to receive one (and only one) of the underlyingsVunderl,i at the corresponding exercise dateTi. The recursive definition (3) represents theoptimal exercise strategy in each exercise time. We formalize this optimal exercise strategy:

For a given pathω ∈ Ω let

T (ω) := minTi : Vberm,i+1(Ti, ω) < Vunderl,i(Ti, ω),

and

η : 1, . . . , n− 1 × Ω → 0, 1, η(i, ω) :=

1 falls T (ω) ≥ Ti

0 sonst.

The definitions ofT andη give equivalent descriptions of the exercise strategy:T (ω) is theoptimal exercise time on a given pathω, η(·, ω) is an indicator function which changes from0 to 1 at the time indexi corresponding toTi = T (ω). The boundary∂η = 1 of the setη = 1 is the so called exercise boundary. It should be noted thatη(k) isFTk

measurable.

3.1 Bermudan Option Value as single (unconditioned) Expectation:The Optimal Exercise Value

With the definition of the optimal exercise strategyT (or η) it is possible to define a randomvariable which allows to express the Bermudan option value as a single (unconditioned) ex-pectation. With

U(Ti) := Vunderl,i(Ti) i = 1, . . . , n

denoting the relative price of thei-th underlying upon its exercise dateTi we have for theBermudan value

Vberm(T0) = EQ(U(T )∣∣ FT0

).

For the Bermudan Callable we may alternatively write

Vberm(T0) = EQ( n−1∑k=1

Xk(Tk+1) · η(k)∣∣ FT0

).

The random variableU(T ) may be calculated directly using thebackward algorithm. We willconsider this in the next section and conclude by givingU(T ) a name:

Definition 2 (Option Value upon Optimal Exercise): qLet U be the stochastic process who’s timet valueU(t) is the (Numéraire relative) option valuereceived upon exercise int. Let T be the optimal exercise strategy. The the random variableU(T ), where

U(T )[ω] := U(T (ω), ω)

is the (Numéraire relative)option value received upon optimal exercise. The (Numéraire rela-tive) Bermudan option value is given byEQ(U(T )

∣∣ FT0

). y

Thus the value ofVberm(T1, . . . , Tn) may be expressed through a single expectation con-ditioned toT0 and does not need any calculation of a conditional expectation at later times,ifwe have the optimal exercise dateT (ω) (and thusη(·, ω)) for any pathω.

Remark 3 (Stopped Prozess): The random variableU is a so calledstopped process. Uis a stochastic process andT is a random variable with the interpretation of a (stochastic)time. FurthermoreT is a stopping time, see Definition??. Here the stochastic processU


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is the family of underlyings received upon exercise, parametrized by exercise time, andT isthe optimal exercise time. ThusU(T ) is the underlying received upon optimal exercise. Allquantities are stochastic, i.e. depend on the path.

4 Bermudan Option Pricing - The Backward Algorithm

The random variableU(T ) may be derived in a Monte-Carlo simulation through the backwardalgorithm,giventhe exercise criteria (3), i.e. the conditional expectation. The algorithm con-sists of the application of the recursive definition of the Bermudan value in (3) with a slightmodification. Let:

Induction start:

Un+1 ≡ 0

Induction step i + 1 → i for i = n, . . . , 1:

Ui =

Ui+1 if Vunderl,i(Ti) < EQ(Ui+1|FTi)Vunderl,i(Ti) else.

From the Tower Law we have by inductionEQ(Ui+1|FTi) = EQ(Vberm,i+1(Ti)|FTi

) andthus

Vberm(T1, . . . , Tn, T0) = EQ(U1|FT0) (4)

andU1 = U(T ) with the notation from the previous section.

Interpretation: The recursive definition ofUi differs from the recursive definition ofVberm,i(Ti). We have

Ui =

Ui+1 if Vunderl,i(Ti) < EQ(Ui+1|FTi)Vunderl,i(Ti) else,

and

Vberm,i(Ti) =

EQ(Vberm,i+1(Ti+1)|FTi

) if Vunderl,i(Ti) < EQ(Ui+1|FTi)

Vunderl,i(Ti) else.

This is a subtle but crucial difference. While both definitions give the Bermudan option value(through application of (4)), we have that the definition ofUi requires the conditional expecta-tion operator only to calculate the exercise criteria.

Since a Monte-Carlo simulation requires advanced methods to obtain an (often not veryaccurate) estimate for the conditional expectation it is important to reduce their use.

Note thatVberm,i(Ti) is FTimeasurable by definition as aFTi

conditional expectation,while all Ui are at mostFTn measurable since they are defined pathwise fromFTk

measurablerandom variablesVunderl,k(Tk) for i ≤ k ≤ n.

The pricing of a Bermudan option may thus be reduced to either the calculation of condi-tional expectations or to the calculation of the optimal exercise strategyT .

As motivation we consider in Sections5 and 6 two methods which are not suitable tocalculate conditional expectations.


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5 Re-simulation

Let us consider the simplified example of a Bermudan option as given in Section1. If noanalytic calculation of the conditional expectation (2) is possible and if Monte-Carlo is thenumerical tool to calculate expectations the straight forward way to calculate the conditionalexpectation is to create inT1 a new Monte-Carlo simulation (conditioned) on each path - seeFigure2. This leads to a much higher number of total simulation paths needed. Especially

0 t

W(t,ω)

T1

Figure 2: Brute-Force Berechnung der bedingten Erwartung in einer Pfad-Simulation durch (pfadweise)Resimulation

if one considers more than one exercise date (option on option on option. . . ) this methodbecomes impractical. The required number of paths, i.e. the complexity of the algorithm andthus the calculation time, grows exponentially with the number of exercise dates. This createsthe need for efficient alternatives.

Interpretation: The calculation of conditional expectation in a path simulation requires fur-ther measures since the path simulation does not offer a suitable discretization of the filtration.

6 Perfect Foresight

If one refuses to use a full resimulation and sticks to the paths generated in the original simu-lation then one effectively estimates the conditional expectation by a single path, namely by

EQ(

V (T2)N(T2)

| FT1

)≈ V (T2)

N(T2).

Put differently this is a limit case of the resimulation where each resimulation consists of asingle path only, namely the one of the original simulation. If this estimate is used in theexercise criteria the exercise will besuper optimalsince it is based on future information thatwould be unkown otherwise.

The exercise criteria at timeT1 may only depend on information available inT1, i.e. onFT1 measureable random variables. The estimateV (T2)

N(T2)is notFT1 measureable.

To illustrate the super optimality consider a simulation consisting of two path, see Figure3.Both path are identical on[0, T1], i.e.FT1 = ∅,Ω = ∅, ω1, ω2. We consider the optionV to receive eitherS(T1) = 2 or S(T2) ∈ 1, 4 at later timeT2. The random variableη : Ω → 0, 1 denotes the exercise strategy forT1: It is 1 on paths that exercise inT1, else0. With a perfect foresightthe super optimal exercise strategy isT (ω1) = T2, T (ω2) = T1,i.e. η(ω1) = 0, η(ω2) = 1 and an average value ofV (T0) = 1

2 (4 + 2) = 62 will be received.

Note that thenη is notFT1 measurable. The exercise decision is made inT1 with knowledge


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t

S(t,ω)4

2

T1

1

ω1

ω2

T2

P(ω1) = 0.5P(ω2) = 0.5

T0

Figure 3: Illustration of perfect foresight.

of the future outcome. If we restrict the exercise strategy to the set ofFT1 measurable randomvariables we either get12 (4 + 1) = 5

2 usingη ≡ 0 or 12 (2 + 2) = 4

2 usingη ≡ 1. Thus theoptimal,FT1 measurable (and thus admissible) exercise strategy isη(ω1) = η(ω2) = 0.

Perfect foresightis not a suitable method to estimate conditional expectation and throughthis the exercise criteria.

7 Conditional Expectation as Functional Dependence

Let us reconsider the calculation of the conditional expectation through brute force re-simulationas described in Section5 and depicted in Figure2. On each path of the original simulation are-simulation has to be created. These re-simulations differ in their initial conditions (e.g. thevalueS(T1) in a simulation of a stock price following a Black-Scholes Modell, or the valuesLi(T1) in a simulation of forward rates following a LIBOR Market Model). The initial condi-tions areFT1 measurable random varialbes (known as ofT1). Thus the conditional expectationis a function of these initial conditions (and possibly other model parameters known inT1. Ifit is known that the conditional expectation is a function of aFT1 measurable random variableZ (we assume here thatZ : Ω → Rd with somed) we have

EQN

(V (T2)N(T2)

| FT1

)= EQN

(V (T2)N(T2)

| Z)

. (5)

Interpretation: If the random variableZ is such thatFT1 is the smallestσ field with respectto whichZ is measurable (i.e. we haveZ−1(B(Rq)) = FT1), then equation (5) is merely thedefinition of an expectation conditioned to an random variable. If however the conditional ex-pectation on the left hand side (i.e.V (T1)

N(T1)) is known to be measurable with respect to a smaller

σ field (e.g. because its functional dependents relies on a smaller set of random variable), thenit might be advantageous to use the right hand side representation. This representation is alsouseful to derive approximation, e.g. if the functional dependents with respect to one componentof Z is known to be weak and thus neglectable.

Example: Consider a LIBOR Market Model with stochastic processes for the forward ratesL1, L2, . . . Ln. In T1 we wish to calculate the conditional expectation of a derivative with aNuméraire relative payoff that depends onL2, . . . , Lk only (e.g. on a swap rate). While thefiltrationFT1 is generated by the full set of forward ratesL1(T1), L2(T1), . . . Ln(T1) it is suffi-cient to knowL2(T1), . . . , Lk(T1) to describe the conditional expectation (i.e. the conditionalvalue of the product).


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We will now describe methods that derive the functional dependance of the conditionalexpectation from a given set of random variables.

8 Binning

In a path simulation the approximation ofEQ(

V (T2)N(T2)

| Z)

will be given by averaging over all

paths for whichZ attains the same value. For the simple example in Figure3 this would removethe perfect foresightsince thereS(T1)−1(2) = Ω. In general the situation will be such thatthere are no two or more paths for whichZ attains the same value - apart from the constructionof the unfeasable resimulation. Thus this approximation will show a perfect foresight.

An improvement is given by abinning, where the averaging will be done over those pathsfor whichZ lies in a neighborhood (bin). If the quantities are continuous we have:

EQ(

V (T2)N(T2)

| Z)

[ω] ≈ EQ(

V (T2)N(T2)

| Z ∈ Uε(Z(ω)))

,

whereUε(Z(ω)) := y | ||Z(ω)− y|| < ε.Instead of defining a binUε(Z(ω)) for each pathω it is more efficient to start with a

partition of Z(Ω) into a finite set of disjoint binsUi ⊂ Z(Ω). The approximation of theconditional expectation

EQ(

V (T2)N(T2)

| Z(ω))

will then be given by

Hi := EQ(

V (T2)N(T2)

| Z ∈ Ui

)whereUi denote the set withZ(ω) ∈ Ui.

Example: Pricing of a simple Bermudan Option on a Stock

We illustrate the method in a simpleBlack Scholes modelfor a stockS. In T1 we wish toevaluate the option of receivingN1 ·(S(T1)−K1) in T1 or to receiveN2 ·max(S(T2)−K2, 0)at later timeT2 (whereN1, N2 (notional),K1, K2 (strike) are given). The optimal exercise inT1 compares the exercise value with the value of theT2 option, i.e.

EQ(

N2 ·max(S(T2)−K2, 0)N(T2)

| FT1

).

From the model specification, e.g. here aBlack-Scholes model

dS(t) = r · S(t)dt + σS(t)dW Q(t), N(t) = exp(rt)

it is obvious that the price of theT2 option seen inT1 is a given functionS(T1) and the givenmodel parameters (r, σ). Thus it is sufficient to calculate

EQ(

N2 ·max(S(T2)−K2, 0)N(T2)

| S(T1))

.

In this example the functional dependence is known analytically. It is given by the Black-Scholes formula (??). Nevertheless we use the binning to calculate an approximation to theconditional expectation. If we plot

N2 ·max(S(T2, ωi)−K2, 0)N(T2)︸︷︷︸

Continuation Value

as a function of S(T1, ωi)︸︷︷︸Underlying


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we obtain the scatter plot in Figure4, left. For a givenS(T1) none or very few values of thecontinuation valuesexists. An estimate is not possible or exhibits a foresight bias. For aninterval[S1−ε, S1 +ε] with sufficiently largeε we have enough values to calculate an estimateof

EQ(

N2 ·max(S(T2)−K2, 0)N(T2)

| S(T1) ∈ [S1 − ε, S1 + ε])

which in turn may be used as estimate of

EQ(

N2 ·max(S(T2)−K2, 0)N(T2)

| S(T1) = S1

).

In Figure4, right, we calculate this estimate forS1 = 1 andε = 0.05.

Exercise versus continuation value & Binning

0,0

0,2

0,5

0,8

1,0

Con

tinua

tion

Val

ue

0,0 0,5 1,0 1,5 2,0

Exercise Value0 50 100 150 200

Frequency

Mean

5254 values at 0

Figure 4: The Continuation Value as a function of the Underlying (Spot Value) and the calculation of theconditional expectation through a binning

8.1 Binning as a Least-Square Regression

Consider again the binning: As estimate for the conditional expectationEQ(

V (T2)N(T2)

| Z(ω))

we calculated the conditional expectation

Hi := EQ(

V (T2)N(T2)

| Z ∈ Ui

)(6)

given abin Ui with Z(ω) ∈ Ui.

For the expectation operatorEQ an alternative characterization may be used:

Lemma 7 (Characterization of the Expectation as Least-Square Approximation): Theexpectation of a random variableX is the numberh for whichX−h has the smallest variance(i.e.L2(Ω) norm).

Proof: Let X be a real valued random variable. Then we have for anyh ∈ R

E((X − h)2) = E(X2)− 2 · E(X) · h + h2 =: f(h).


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Sincef ′ = 0 ⇔ h = E(X) andf ′′ = E(X2) ≥ 0 we have thatf attains its minimum inh = E(X). For vector valued random variables this follows componentwise. The same resultsholds for conditional expectations. 2|

Using Lemma7 we may write (6) as minimization problem:

EQ

((V (T2)N(T2)

−Hi

)2

| Z ∈ Ui

)= min

G∈REQ

((V (T2)N(T2)

−G

)2

| Z ∈ Ui

).

For disjoint binsUi this may be written in a single minimization problem for the vector(Hi)i=1,...:

∑i

EQ

((V (T2)N(T2)

−Hi

)2

| Z ∈ Ui

)= min

Gi∈R

∑i

EQ

((V (T2)N(T2)

−G

)2

| Z ∈ Ui

).

(7)This condition admits an alternative interpretation:Hi represents the piecewise constant func-tion (constant onUi) with the minimal distance toV (T2)

N(T2)in the least square sense.

Let H be the space of functionsH : Ω → R being constant on the binsZ−1(Ui).3 LetH ∈ H with H(ω) := Hi for ω ∈ Z−1(Ui). Then (7) is equivalent to

EQ

((V (T2)N(T2)

−H

)2

| Z

)= min

G∈HEQ

((V (T2)N(T2)

−G

)2

| Z

). (8)

Equation (8) is the definition of a regression: Find the functionH from a function spaceHwith minimum distance toV (T2)

N(T2)in theL2 norm. Binning is just a special choice of functional

space:

Lemma 8 (Binning as L2 Regression): Binning is anL2 regression on the space offunctions piecewise constant onUi.

9 Foresight Bias

Definition 9 (Foresight Bias (Definition 1)): qA foresight biasis a super optimal exercise strategy. y

A foresight bias arises due to a violation of the measurability requirements: If the exercisedecision inTi is based on random variable which is notFTi measurable the exercise maybe super optimal, i.e. better than if based on the information theoretically available (FTi

). Ifwe use the same Monte-Carlo simulation to first estimate the exercise criteria and then usethis criteria to price the derivative we most certain generate a foresight bias. In this case theforesight bias is created by the Monte-Carlo error of the estimate, which is in general notFTi

measurable. The existence of this problem becomes obvious if we consider a limit case ofbinning where each bin contains a single path only. Here we would have aperfect foresight.

If our exercise criteria at timeTi uses onlyFTimeasurable random variables then there is -

in theory - no foresight bias. If however the exercise criteria is calculated within a Monte Carlosimulation the Monte-Carlo error of the calculation represents a nonFTi

measurable randomvariable, thus it induces a foresight bias. In this case we may give an alternative definition forthe foresight bias:

Definition 10 (Foresight Bias (Definition 2)): qTheforesight biasis the value of the option on the Monte Carlo error. y

3 Note that the binsUi were defined as subsets ofZ(Ω), here we considerH as a function onΩ.


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With increasing number of paths the foresight bias introduced by binning tends to zerosince the Monte-Carlo error with respect to a bin tends to zero.

A general solution of the problem of a foresight bias is given by using two independantMonte-Carlo simulations: One two estimate the exercise criteria (for binning this is given bytheHi corresponding to theUi’s), the other to apply the criteria in pricing. This is a numericalremoval of the foresigh bias. In [10] an analytic formula for the (Monte Carlo error induced)foresight bias is derivated that may be used to correct the foresight bias analytically.

10 Regression Methods - Least Square Monte Carlo

Motivation (Disadvantage of Binning): The partition of the state spaceZ(Ω) into a finitenumber of bins results in a piecewise constant approximation of the conditional expectation.An obvious improvement would be to give the conditional expectation as a smooth function ofthe state variableZ.

The considerations in Section8.1 suggest a simple yet powerful improvement to the bin-ning: The function giving our estimate for the conditional expectation is defined by a leastsquare approximation (regression).

10.1 Least Square Approximation of the Conditional Expectation

Let us start with a fairly general definition of theleast square approximationof the conditionalexpectation of random variableU .

Definition 12 (Least Square Approximation of the Conditional Expectation): qLet (Ω,F , Q, Ft) be a filtered probability space andV aFT1 measurable random variabledefined as the conditional expectation ofU

V = EQ(U | FT1),

whereU is at leastF measurable. Furthermore letY := (Y1, . . . , Yp) be a givenFT1 measur-able random variable andf : Rp × Rq a given function. LetΩ∗ = ω1, . . . ωN a drawingfrom Ω (e.g. a Monte-Carlo simulation corresponding toQ) andα∗ := (α1, . . . , αq) such that

||U − f(Y, α∗)||L2(Ω∗) = minα||U − f(Y, α)||L2(Ω∗)

where||U − f(Y, α∗)||2L2(Ω∗)=

N∑j=1

(U(ωj)− f(Y (ωj), α∗))2. We set

V LS := f(Y, α∗).

The random variableV LS isFT1 measurable. It is defined overΩ and aleast squareapproxi-mation ofV onΩ∗. y

The method of Carriere, Longstaff and Schwartz4 uses a functionf with q = p and

f(y1, . . . , yp, α1, . . . , αp) :=p∑

i=1

αi · yi,

such thatα∗ may be calculated analytically as a linear regression.

Lemma 13 (Linear Regression): Let Ω∗ = ω1, . . . , ωn be a given sample space,V : Ω∗ → R andY := (Y1, . . . , Yp) : Ω∗ → Rp given random variables. Furthermore let

f(y1, . . . , yp, α1, . . . , αp) :=∑

αiyi.

4 See [6, 15, 16].


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Then we have for anyα∗ with XTXα∗ = XTv

||V − f(Y, α∗)||L2(Ω∗) = minα||V − f(Y, α)||L2(Ω∗),

where

X :=

Y1(ω1) . . . Yp(ω1)...

...Y1(ωn) . . . Yp(ωn)

, v :=

V (ω1)...

V (ωn)

.

If (XTX)−1 exists thenα∗ := (XTX)−1XTv.

Definition 14 (Basis Functions): qThe random variablesY1, . . . , Yp of Lemma13 are calledBasis Functions(explanatory vari-ables). y

10.2 Example: Evaluation of an Bermudan Option on a Stock (Back-ward Algorithm with Conditional Expectation Estimator)

We consider a simple Bermudan option on a Stock. The Bermudan should allow exercise attimesT1 < T2 < . . . Tn. Upon exercise inTi the holder of the option will receive

Ni · (S(Ti)−Ki)

once. If no exercise is made he will receive nothing.We will apply thebackward algorithmto derive the optimal exercise strategy. All payments

will be considered in their Numéraire relative form. Thus the exercise criteria given by acomparison of the conditional expectation of the payments received upon non-exercise withthe payments recieved upon exercise.

Induction start: t > Tn Beyond the last exercise we have:

• The value of the (future) payments isUn+1 = 0

Induction step: t = Ti, i = n, n− 1, n− 2, . . . 1 In Ti we have:

• In the case of exercise is inTi the value is

Vunderl,i(Ti) :=Ni(S(Ti)−Ki)

N(Ti). (9)

• In the case of non-exercise inTi the value isVhold,i(Ti) = EQ(Ui+1 | FTi). This value

is estimated through a regression for given pathsω1, . . . , ωm:

– Let Bj be given (FTi measurable) basis functions.5 Let the matrixX consist of thecolumn vectorsBj(ωk), k = 1, . . . ,m. Then we have Vhold,i(Ti, ω1)

...Vhold,i(Ti, ωm)

≈ X · (XT ·X)−1 ·XT ·

Ui+1(ω1)...

Ui+1(ωm)

. (10)

5 Suitable basis functions for this example are1 (constant),S(Ti), S(Ti)2, S(Ti)

3, etc., such that the regressionfunctionf will be a polynomial inS(Ti).


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• The value of the payments of the product inTi under optimal exercise is given by

Ui :=

Vunderl,i(Ti) if Vhold,i(Ti) < Vunderl,i(Ti)Ui+1 else.

Remark 15: Our example is of course just the backward algorithm with an explicit specifi-cation of an underlying (9) and an explicit specification of an exercise criteria, here given bythe estimator of the conditional expectation (10).

10.3 Example: Evaluation of an Bermudan Callable

We consider a Bermudan Callable. The Bermudan should allow exercise at timesT1 < T2 <. . . Tn. Upon exercise inTi the holder of the option will receive a payment ofXi in Ti+1,i.e. the relative valueXi(Ti+1) := Xi

N(Ti+1).

We will apply thebackward algorithmto derive the optimal exercise strategy. All paymentswill be considered in their Numéraire relative form.




• In the case of exercise is inTi the value is

Vunderl,i(Ti) := EQN

(n−1∑k=i

Xk

N(Tk+1)|FTi

).

This value is estimated through a regression for given pathsω1, . . . , ωm:

Let B1j be given (FTi

measurable) basis functions. Let the matrixX1 consist ofthe column vectorsB1

j (ωk), k = 1, . . . ,m. Then we have Vunderl,i(Ti, ω1)...

Vunderl,i(Ti, ωm)

≈ X1 ·(X1,T ·X1)−1 ·X1,T ·

∑n−1

k=iXk(ω1)

N(Tk+1,ω1)

...∑n−1k=i

Xk(ωm)N(Tk+1,ωm)

.

(11)


is estimated through a regression for given pathsω1, . . . , ωm:

Let B0j be given (FTi measurable) basis functions. Let the matrixX0 consist of

the column vectorsB0j (ωk), k = 1, . . . ,m. Then we have Vhold,i(Ti, ω1)

...Vhold,i(Ti, ωm)

≈ X0 · (X0,T ·X0)−1 ·X0,T ·

Ui+1(ω1)...

Ui+1(ωm)

.

• The value of the payments of the product inTi under optimal exercise is given by

Ui :=

∑n−1k=i

Xk

N(Tk+1)if Vhold,i(Ti) < Vunderl,i(Ti)

Ui+1 else.


14 Version 0.3 (20051205)



Exercise Boundary

Break Even LS Regression Hold Exercise

- 0,2 - 0,1 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9

Value upon Exercise

- 0,2

0,0

0,2

0,5

0,8

1,0

1,2

1,5

1,8

2,0

2,2

2,5V

alue

upo

n H

old

Exercise Boundary


- 0,2 - 0,1 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9

Value upon Exercise

- 0,2

0,0

0,2

0,5

0,8

1,0

1,2

1,5

1,8

2,0

2,2

2,5

Val

ue u

pon

Hol

d

Figure 5: Regression of the conditional expectation estimator with and without restriction of the regres-sion domain: We consider a Bermudan option with two exercise datesT1 = 1.0, T1 = 2.0. Notional andstrike are as follows:N1 = 0.7, N2 = 1.0, K1 = 0.82, K2 = 1.0. The model for the underlyingS isa Black-Scholes model withr = 0.05 andσ = 20%. The plot shows the values received upon exercisedepending on the values received upon non-exercise inT1. Eeach dot corresponds to a path. The regres-sion polynomial gives the estimator for the expectation of the value upon non-exercise. It is optimal toexercise if this estimate lies above the value received upon exercise. In the upper diagram the regressionpolynomial is a second order polynomial inS(T1). In the lower diagram it is a second order polynomialin max(S(T1)−K1, 0). By this values whereS(T1)−K1 ≤ 0 are aggregated into a single point. Forthe product under consideration this is advantageous since forS(T1) − K1 ≤ 0 exercise is not optimalwith probability1. This restriction of the regression domain increases the regression accuracy over theremaining regression domain.


15 Version 0.3 (20051205)



Exercise Boundary


- 0,2 - 0,1 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9

Value upon Exercise

- 0,2

0,0

0,2

0,5

0,8

1,0

1,2

1,5

1,8

2,0

2,2

2,5

Val

ue u

pon

Hol

d

Exercise Boundary


- 0,2 - 0,1 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9

Value upon Exercise

- 0,2

0,0

0,2

0,5

0,8

1,0

1,2

1,5

1,8

2,0

2,2

2,5

Val

ue u

pon

Hol

d

Figure 6: Regression of the conditional expectation estimator - polynomial of fourth (above) and eighth(below) order inmax(S(T1) − K1, 0). Parameters as in Figure5. A polynomial of higher order showswigels at the boundary of the regression domain. However only few paths are affected by the wrongestimate. Resticting the regression domain may reduce the errors (compare the left end of the regressiondomain with the right end).


16 Version 0.3 (20051205)



Remark 16 (Bermudan Callable): The modification to the backward algorithm to price aBermudan callable consists of the inductions of two conditional expectation estimators: onefor the continuation value and (additionally) one for the underlying. As before, the conditionalexpectation estimators are used only for the exercise criteria (and not for the payment).

Remark 17 (Longstaff-Schwartz):

• The estimator of the conditional expectation is used in the estimation of the exercisestrategy only.

• The choice of basis functions is crucial to the quality of the estimate.

Clément, Lamberton and Protter showed convergence of the Longstaff-Schwartz regressionmethod to the exact solution, see [7].

10.4 Implementation

MonteCarloConditionalExpectationLongstaffSchwartz

setBasisFunctionsEstimator(RandomVariable[] basisFunctionsEstimator)setBasisFunctionsPredictor(RandomVariable[] basisFunctionsPredictor)

getConditionalExpectation(RandomVariable randomVariable)

Figure 7: UML Diagram: Conditional Expectation Estimator. The methodsetBasisFunctionsEstimator sets the basis functions which forms the matrixX. ThemethodsetBasisFunctionsPredictor sets the basis functions which form the matrixX∗. Theseare the same basis functions as forX, but possibly evaluated in an independent Monte-Carlo simulation(to avoid foresight bias). The methodgetConditionalExpectation calculates the regressionparameterα∗ = (XTX)−1XT · v from a given vectorv and returns the conditional expectationestimatorX∗ · α∗ of v. See Lemma13.

The Longstraff-Schwartz conditional expectation estimator may easily be implemented ina corresponding class, independent from the given model or Monte-Carlo simulation - seeFigure7. This class contains nothing more than alineare regressionhowever the methodologymay be replaced by more alternative algorithms (e.g. non parametric regressions).

As pointed out in the discussion of the backward algorithm it is in general not necessaryto explicitly calculate the exercise strategy in form ofT to η. It is sufficient to calculate therandom variablesUi in a backward recursion. Since finally onlyU1 is needed to calculatethe price of the Bermudan option theUi’s may be stored in the same vector of Monte-Carlorealizations.

10.5 Binning as linear Least-Square Regression

We return once again to the binning. In Section8.1it turned out thatBinningmay be interpretedas least square regression with a specific set of basis functions: The indicator variables of thebinsUj which we denote by

hj(ω) :=

1 for ω ∈ Uj

0 else.(12)


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Exercise Boundary


- 0,2 - 0,1 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9

Value upon Exercise

- 0,2

0,0

0,2

0,5

0,8

1,0

1,2

1,5

1,8

2,0

2,2

2,5

Val

ue u

pon

Hol

d

Figure 8: Binning using the linear regression algorithm with piecewise constant basis functions: Weuse 20 bins (basis functions). Each bin consist of approximately the same number of paths. Model- undproduct parameters are as in Figure5.

We now give an explicit calculation using the linear regression algorithm with the bin indicatorvariables as basis functions.

Let ωk denote the paths of a Monte-Carlo simulation andX the matrix(hj(ωk)

), j column

index,k row index. Since theUj ’s are disjoint we haveXTX = diag(n1, . . . , nm), wherenj

is the number of paths for whichhj(ωk) = 1. Thus we have for the regression parameter

α∗ = (XTX)−1XT · v = diag(1n1

, . . . ,1

nm) ·XT · v.

It follows that the regression parameter gives the expectation on the corresponding bin

α∗j =1nj

∑vk∈Uj

vk.


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11 Optimization Methods

Motivation: In the discussion of thebackward algorithmit has become obvious that theconditional expectation estimator is needed to derive the optimal exercise strategy only. Since asub-optimal exercise will lead to a lower Bermudan price the optimal exercise has an alternativecharacterization: it maximizes the Bermudan value. A solution to the pricing problem of theBermudan thus consists of maximize the Bermudan value over a suitable, sufficiently largespace of admissible6 exercise strategy.

11.1 Andersen Algorithm for Bermudan Swaptions

The following method was proposed for the valuation of Bermudan swaptions by L. Ander-sen in [1]. We thus restrict our presentation to the evaluation of the Bermudan Callable anduse the notation of Section10.3. In [1] the method appears less generic than the Longstraff& Schwartz regression. However one might reformulate the optimization method in a fairlygeneric way. Since then the optimization is a high dimensional one the method becomes lessuseful in practice.

The exercise strategy is given by a parametrized function of the underlyings

Ii(λ) := f(Vunderl,i(Ti, ω), . . . , Vunderl,n−1(Ti, ω), λ)

where we replace the optimal exercise

Vunderl,i(Ti) < EQ(Ui+1 | FTi)

byIi(λ) > 0.

Here the functionf may represent a varity of exercise criterias, e.g.

Ii(ω, λ) = Vunderl,i(Ti, ω)− λ. (13)

We assume thatIi is such that it may be calculated without resimulation, i.e. we assume that theunderlyingsVunderl,j(Ti) are either given by an analytic formula or a suitable approximation.E.g. this is the case for a swap within a LIBOR Market Model. Using the optimization methodwithin the backward algorithm now locks as follows:




• In the case of exercise is inTi the value isVunderl,i(Ti)


is estimated through an optimization for given pathsω1, . . . , ωm:

– Ii(λ, ω) = f(Vunderl,i(Ti, ω), . . . , Vunderl,n−1(Ti, ω), λ).

6 By an admissible excercise strategy we denote one that respects the measurability requirements. As we noted, aviolation of measurability requirements, i.e. a foresight bias or even perfect foresight, will result in a super optimalstrategy. The Bermudan value with a super optimal strategy is hight than the Bermudan value with the optimalstrategy, however super optimal exercise is impossible.


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– Ui(λ, ω) :=

Vunderl,i(Ti) if Ii(λ, ω) > 0Ui+1 else.

– Vberm,i(T0, λ) = EQ(Ui(λ) | FT0

)≈ 1

m

∑k Ui(λ, ωk)

– λ∗ = arg maxλ

(1m

∑k Ui(λ, ωk)

)• The value of the payments of the product inTi under optimal exercise is given by

Ui(ω) :=

Vunderl,i(Ti) if Ii(λ∗, ω) > 0Ui+1 else.

= Ui(λ∗, ω)

The exercise strategy is estimated inTi by choosing theλ∗ for whichIi gives the maximalBermudan option value. This is done by going backward in time, from exercise date to exercisedate.

11.2 Review of the Threshold Optimization Method

11.2.1 Fitting the exercise strategy to the product

We apply the optimization method to the pricing of a simple Bermudan option on a stockfollowing a Black-Scholes model. We show that choosing the exercise strategy too simple willgive surprisingly unreliable results.

1. option 2. option bermudan exercise nach schwellenwert

0,00

0,25

0,50

0,75

1,00

1,25

1,50

Aus

zahl

ung

/ Opt

ions

wer

t

0,0 1,0 2,0

Spot0,0 1,0 2,0 3,0 4,0 5,0

Schwellenwert

0,455

0,458

0,460

0,462

0,465

0,468

0,470

0,473

Optionsw

ert

Figure 9: Example of the successful optimization of the exercise criteria (intersection of the two pricecurves (left). The graph on the right shows the Bermudan option value as a function of the exercisethresholdλ.

The simple strategy (13) fails for the most simple type of a Bermudan option. We considerthe option to receive

N1 · (S(T1)−K1)

in T1 or receivemax(N2 · (S(T2)−K2), 0) in T2

in T2, where - as before -Ni andKi denote notional and strike andS follows a Black-Scholesmodel. This gives us an analytic formula for the option inT2 and thus the true optimal exercise.


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Figure9 shows an example where the optimization of the simple strategy (13) gives the valueof the Bermudan option.

1. option 2. option bermudan exercise nach schwellenwert

0,00

0,25

0,50

0,75

1,00

1,25

1,50A

usza

hlun

g / O

ptio

nsw

ert

0,0 1,0 2,0

Spot0,0 1,0 2,0 3,0 4,0 5,0

Schwellenwert

0,325

0,350

0,375

0,400

0,425

0,450

0,475

0,500

0,525

0,550

Optionsw

ert

Figure 10: Example of a failing optimization of the exercise criteria (intersection of the two price curves(left). The graph on the right shows the Bermudan option value as a function of the exercise thresholdλ.

A small change of notianalN1 and strikeK1 changes the picture. If both are smaller thanN2 andK2 respectively we may optain two intersection points of the exercise and continuationvalue. InT1 it is optimal to exercise in between these two intersection point. Our simpleexercise criteria cannot render this case. Optimizing the threshold parameterλ shows twomaxima: the value of the two european options "exercise never" and "exercise always". Bothvalues are below the true Bermudan option value, see Figure fig:andersenBad, right.

The conclusion of this is example is that the choice of the exercise strategy had to be madecarefully in accordance with the product. But this remark applies to any method in varyingextends.

11.2.2 Disturbance of the Optimizer through Monte Carlo Error

We are maximizing a Monte-Carlo price. Since the Monte-Carlo price is an average of a finitenumber of path values it is (in general) a discontinuous function of the parameterλ. Theprice (as a function ofλ) will not only exhibit discontinuities but also small local minima, seeFigure9 (right). These may prevent the minimizer algorithm from finding the global minima.Note that for sufficiently many path the local minima appear only on a small scale. Thus witha robust minimizer one would rarely encounter this problem.

11.3 Optimization of Exercise Strategy: A more general Formulation

There is a trivial generalization of the optimization method considered in Section11.1:

• The exercise criteria will be given as a function of arbitraryFTi measurable randomvariables.

• The exercise criteria will be given as a function of a parameter vectorλ ∈ Rk.

Thus we replace the ”truet’t’ exercise criteria used in the backward algorithm

Vunderl,i(Ti) < EQ(Ui+1 | FTi)


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by a functionIi(λ, ω) := f(Bi,1(ω), . . . , Bi,m(ω), λ),

whereBi,j is a set ofFTi measurable random variables andλ ∈ Rk.

11.4 Comparisson of Optimization Method and Regression Method

The difference between the optimization method and the regression method becomes appearendin Figur 5-6. While the regression method requires that the regression functions give a goodfit to the conditional expectations across the whole domain of the independent variable, theoptimization method only requires that the functionalIi(λ) captures the exercise boundary. InFigures5-6 the conditional expectation estimator is a curve, but the exercise boundary is givenby two points only (the intersection of the conditional expectation estimator with the bisector).

Thus the optimization method can cope with far less parameters than the regression. Onthe other hand, as noted in the example above, it is far more important that the functional isadapted to the Bermudan product under consideration.

It is trivial to choose the mapIi(λ) such that the optimalλ∗ give the same or a better valuethat the least-square regression. IfBi,j denote the basis functions used in the least-squareregression for exercise dateTi we set

Ii(λ1, . . . , λn) = (λ1Bi,1 + . . . + λ1Bi,m)− Vunderl,i(Ti).

Then we have that forλ = α∗, whereα∗ is the regression parameter from the least-squareregression, the exercise criteria agrees with the one of the least-square regression. This resulthowever is rarely an advantage of the optimization method since in practice a high dimensionaloptimization does not represent an alternative.


22 Version 0.3 (20051205)



12 Duality Method: Upper Bound for Bermudan OptionPrices

This section is under revision.

It will reappear on or before February 30th, 2006. Please checkhttp://www.christian-fries.de/finmath/earlyexercise

for updates.


23 Version 0.3 (20051205)




References

[1] ANDERSEN, LEIF: A simple approach to the pricing of Bermudan swaptions inthe multi-factor Libor market model. Working paper. General Re Financial Products,1999.

[2] ANDERSEN, LEIF; BROADIE, MARK: A Primal-Dual Simulation Algorithm for Pric-ing Multi-Dimensional American Options. Working paper. General Re Financial Prod-ucts, 1999.

[3] BINGHAM , NICHOLAS H.; K IESEL, RÜDIGER: Risk-neutral valuation: pricing andhedging of financial derivatives. (Springer Finance). Springer Verlag, London, 1998.ISBN 1-852-33458-4. 296 pages.

[4] BOYLE, PHELIM ; BOADIE, MARK ; GLASSERMAN, PAUL : Monte Carlo methods forsecurity pricing. Journal of Economic Dynamics and Control, 21, 1267-1321 (1997).

[5] BROADIE, MARK ; GLASSERMAN, PAUL : Pricing American-Style Securities by Sim-ulation.J. Econom. Dynam. Control, 1997, Vol. 21, 1323-1352.

[6] CARRIERE, JACQUES F.: Valuation of Early-Exercise Price of Options Using Sim-ulations and Nonparametric Regression.Insurance: Mathematics and Economics19,19-30, 1996.

[7] CLÉMENT, EMMANUELLE ; LAMBERTON, DAMIEN ; PROTTER, PHILIP: An anal-ysis of a least squares regression method for American option pricing.Finance andStochastics6, 449-471, 2002.

[8] DAVIS , MARK ; KARATZAS, IOANNIS: A Deterministic Approach to Optimal Stop-ping, with Applications. In: Whittle, Peter (Ed.): Probability, Statistics and Optimiza-tion: A Tribute to Peter Whittle, 455-466, 1994. John Wiley & Sons, New York andChichester, 1994.

[9] DRAPER, NORMAN R.; SMITH , HARRY: Applied Regression Analysis. Third Edi-tion. Wiley-Interscience, 1998.ISBN 0-471-02995-5.

[10] FRIES, CHRISTIAN P.: Foresight Bias and Suboptimality Correction in Monte-CarloPricing of Options with Early Exercise: Classification, Calculation and Removal.2005.http://www.christian-fries.de/finmath/foresightbias .

[11] FRIES, CHRISTIAN: Finanzmathematik. Theorie, Modellierung, Implementierung.Lecture Notes. Frankfurt am Main, 2005,http://www.christian-fries.de/finmath/script .

[12] GLASSERMAN, PAUL : Monte Carlo Methods in Financial Engineering. 596 pages.Springer, 2003.ISBN 0-387-00451-3.

[13] HAUGH, MARTIN ; KOGAN, LEONIS: Pricing American Options: A Duality Ap-proach, MIT Sloan Working Paper No. 4340-01, 2001.

[14] JÄCKEL , PETER: Monte Carlo Methods in Finance. 238 Seiten. John Wiley and SonsLtd., 2002.ISBN 0-471-49741-X.

[15] LONGSTAFF, FRANCIS A.; SCHWARTZ EDUARDO S.: Valuing American Optionsby Simulation: A Simple Least-Square Approach.Review of Financial Studies14:1,113-147, 2001.


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http://www.amazon.de/exec/obidos/ASIN/1852334584/space-21


http://www.christian-fries.de/finmath/foresightbias

http://www.christian-fries.de/finmath/script

http://www.christian-fries.de/finmath/script


http://www.amazon.de/exec/obidos/ASIN/047149741X/space-21



[16] PITERBARG, VLADIMIR V.: A practitioner’s guide to pricing and hedging callableLIBOR exotics in forward LIBOR models,Preprint. 2003.

[17] ROGERS, L. C. G.: Monte Carlo valuation of American options,Preprint. 2001.

Notes

Suggested Citation

Fries, Christian P.: Pricing Options with Early Exercise by Monte Carlo Simulation: AnIntroduction. (2005).http://www.christian-fries.de/finmath/earlyexercise .

MSC-class: 65C05 (Primary), 68U20, 60H35 (Secondary).ACM-class: G.3; I.6.8.JEL-class: C15, G13.Keywords: Early-Exercise Options, Monte-Carlo, Foresight

25 pages. 10 figures. 0 tables.


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