Monte Carlo Methods in Finance

IIM Ahmedabad, Nov 6, 2005

Sandeep JunejaSchool of Technology and Computer Science

Tata Institute of Fundamental Research

Talk Outline

Motivating Monte Carlo methods in finance through simple Binomial tree models for European options

Monte Carlo Method

Portfolio Credit Risk

Pricing Multi-dimensional American Options

European Call Option

K Underlying price

BUY CALLOptionPayoff

WRITE CALL

K

OptionPayoff

0

0

An option (not an obligation) to purchase an underlying asset at a specified time T (expiration or maturity date) for a specified price K (strike price).

Payoff G(ST)= (ST-K)+

Payoff on the Maturity Date

profit

European Put Option

Underlying price

OptionPayoff

OptionPayoff

K

K0

0Underlying price

BUY PUT WRITE PUT

An option to sell an underlying asset at a specified time for a specified price. Payoff G(ST)= (K-ST)+

Payoff on the Maturity Date

Other Features

American option: Exercise at any time up to the expiration time

Bermudan option: Exercise allowed at a fixed number of times (Intermediate between European and American)

Examples of Options on Multiple Assets

Basket Option ([c1S1(T) + c2S2(T) +...+ cdSd(T)] - K)+

Out-performance Option(max{c1S1(T), c2S2(T),...,cdSd(T)} - K)+

Barrier Option I(mini=1,..,n{S2(ti) <b}(K - S1(T))+

Quantos S2(T)(S1(T) - K)+

They all have an associated American version

Key Problems

The correct price of these options

How to hedge the risk of a portfolio containing options

No arbitrage principle: If 1 dollar = Rs. 40 and 1 pound = Rs. 60, ignoring transaction costs, 1 pound = 1.5 dollar, otherwise by buying low and selling high, an arbitrage may be created

Simple One Period Binomial Model to Price Options

d < 1+r < u from no-arbitrage considerations

S0

S1(H)= uS0

1

1+r

1+rS1(T)= dS0

Two securities exist in this world

V0 ?

V1(H), e.g., S1(H)-K

V1(T), e.g., 0

Consider an option

(If S1(T) <K<S1(H))

Create a Replicating Portfolio

.d-u

r)(1-uq~ and

d-u

d-r1p~ where

, )(~

1

1 ))(~)(~(

1

1 seen that beCan

)())(1()(

and ),())(1()(

thatso valuesseSelect the

shares ofamount purchase wealth,ofamount X Start with

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100010

100010

00

VEr

TVqHVpr

X

TVSXrTS

HVSXrHS

The Risk Neutral Measure

unchanged. is price (.01,0.99)or (0.99,.01) are

hey whether tiesprobabilit physical oft independen is price The

.)(~

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1V price arbitrage No

ies.probabilit neutralrisk thedenote )q~ ,p~( that so

)(~

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d-r1p~ iesprobabilit The

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1110

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X

SEr

TSqHSpr

S

Multi-Period Binomial Model

The analysis extends to multiple periods to more realistic models.

S0

S1(H)

S1(T)

S2(HH)

S2(HT,TH)

S2(TT)

S3(HHH)

S3(HHT,HTH,THH)

S3(TTH,HTT,THT)

S3(TTT)

Solving for Option Price through Backward Recursion

portfolio greplicatin a construct tostrategy a have weTherefore

2,...0.-N1,-Nnfor

),,...,(),,...,(

),,...,(),,...,(),...,(

)),,...,(~),,...,(~(1

1),...,(V

Set payoff. thisreplicate werecursion backwardThrough

optiondependent -path arbitrary from payoff thedenote ),...,(VLet

1111

11111n

11111n

1N

TSHS

TVHV

TVqHVpr

nnnn

nnnnn

nnnnn

N

A Numerical Example:Pricing a Lookback Option

3/1832

80

)(S)(S

)(V)(V)(

20.3)](V2

1)(V

2

1[

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4)(

8816)()()(V

0)()()(V

Then, .maxV

payoff ah option witlookback a icePr

1/2.q~ and 1/2p~Then 1/4.r 1/2,d

2,u ,4S example. period threeaConsider

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332

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HHTHHHHH

HHTHHHHHV

HHTSHHSHHT

HHHSHHHSHHH

SSnn

S0=4

S1(H)=8

S1(T)=2

S2(HH)=16

S2(HT,TH)=4

S2(TT)=1

S3(HHH)=32

S3(HHT,HTH,THH)=8

S3(TTH,HTT,THT) =2

S3(TTT)=0.5

The Discounted Price Process is a Martingale

)

1

1(

~V priceoption the,particularIn

measure martingaleor neutralrisk

under martingale a is process price discounted The

...)|1

1(

~)|

1

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that Implies

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1),...,(V

0

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nnnn

nnnnn

Vr

E

Vr

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E

TVqHVpr

Binomial Tree Model is Complete

Every security VN can be hedged using a replicating portfolio and hence has a unique price.

If the tree was trinomial, and there were two securities as before not every security could be replicated (incomplete market), only bounds could be developed on prices using the no-

arbitrage condition

Fundamental Theorem in Option Pricing

.generalitygreat in holdsresult This

s.martingale are ):r)(1

V( processes price

option attainable hence and ):r)(1

S( pricesasset

discounted ch theunder whi ,P~

measure martingale equivalent

an of existence theimpliescondition arbitrage no The

nn

nn

Nn

Nn

Brownian Motion

A real valued process (W(t):t > 0), is standard Brownian motion if

For t0 < t1...< tn, then W(t1)-W(t0),..., W(tn)-W(tn-1) are independent

W(s+t)-W(s) is Normally distributed with mean 0 and variance t

W(t) is a continuous function of t (with prob 1).

Single Dimension Asset Pricing Model

)1,0( as ddistribute is W(t)-) W(tHere

W(t))-)W(t)(()()(S(t)-)S(t lly,Heuristica

)()(S)(dS asset freeRisk

)]()()()[(S)(dS asset risky One

N

tttS

dttrtt

tdWtdtttt

Asset Price an Expectation under Equivalent Martingale Measure

.]|)())([exp(~

equals s at time condition, arbitrage-nounder priceoption European The

)]()()()[(S)(dS

isasset risky by the satisfied SDE theh,under whic measure

martingale equivalent unique a exists there,conditions arbitrage-noUnder

T

s

sTSVdttrE

tdWtdttrtt

Generating Sample Paths using Time Discretization

Suppose payoff depends on asset prices at times 0,1,2,...,n

Example: Asian Option

Approximately generate the trajectory of the asset price process using Euler’s scheme (finer discretizations improve accuracy)

process dSt = r Stdt + (t) StdW(t)

n

kk KS

n 1

1

11 )( kkkkk NkSrSSS

Monte Carlo needed in Credit Risk Measurement

Consider a portfolio of loans having m obligors. We wish to manage probability of large losses due to credit defaults

Let Yk denote the loss from obligor k.

Our interest is in estimating P(Y1+...+Ym>u) for large u.

Note that P(Y1+...+Ym>u)= E[I(Y1+...+Ym>u)]

Loss given default E [Y1+...+Ym|Y1+...+Ym>u]=E[Y1+...+Ym I(Y1+...+Ym>u)]/P(Y1+...+Ym>u)

Monte Carlo Method

Motivating the Monte Carlo Approach

Monte Carlo Method Random number generation Generating random numbers from general

distributions

Popular variance reduction techniques

Illustrative Queueing Example

The inter-arrival times (A1,A2, …) are “independent identically distributed” with distribution function

FA(x) =P(A < x).

E.g. FA(x) = 1 - e-x

The service times (S1,S2, …) are independent identically distributed with distribution function FS(x) =P(S < x).

Solve or Run the Model ?

To determine EW we could use deductive arguments, e.g.

Wn+1= [ Wn + Sn - An+1 ]+

==> …...

==> ……

==> EW = …….

Feasible only for simple models

Or we could use the computer to simulate functioning of the queue for a large number of days and do statistical analysis

6

Key Statistical Ideas

Law of large numbers: If X1, X2, … are independent identically distributed random variables with mean m = EX, then

n

XXX n...21

For dice

=1*1/6 +2*1/6 +3*1/6 +4*1/6+5*1/6 +6*1/6 = 7/2

Central limit theorem

),0(... 221 N

n

XXXn n

2 is the variance of each Xi determines the convergence rate

Pricing Asian Option through Monte Carlo

k+1k

Asset

pri

ce

m

n

kkn

n

kk

X,..X

KSn

XS

KSn

., samples m generatetly independen and isRepeat th

1Set ).,...,(Spath sample a Generate

1E~

:optionAsian for Price

1

111

1

Constructing Estimators…

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estimatedy empiricall bemay Variance

96.1ˆ

interval confidence 95% provides oremlimit the Central

1ˆ

equals 1

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for Estimator

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1

mjj

X

mjj

n

kk

Xm

n

Xm

KSn

Now we discuss

Uniformly distributed random number generators: Building blocks for creating randomness

General random number generators

Generating uni-variate and multi-variate normal random variables

Generating Uniform (0,1) Pseudo Random Numbers

Requirement: Generate a sequence of numbers U1, U2,...so that

1) Each Ui is uniformly distributed between 0 and 1

2) The Ui’s are mutually independent

1/2

0

Linear Congruential Generators

Popular method: A linear congruential generator

Given an initial integer seed x0 between 0 and m, setxi+1 = a xi mod mui+1 = xi+1/m

a < m is referred to as multiplier, m the modulus

1/2

0

Properties of a Good Random Number Generator

purposes. reduction varianceand debuggingfor Needed

in this. advantagegreat haveLCG .reproduced

easily becannot numbers random Genuine :ilityReproducib 4.

fast be Should :Speed 3.

large be Should :Length Period 2.

ceindependen of testslstatistica pass shouldThey b)

d. of valuesmoderatefor least at planes ldimensiona-din placed

uniformly be should ),...,,( sequences goverlappin The a)

:Randomness 1.

21 dUUU

Periodicity of Linear Congruential Generators

Consider the case where a=6, m=11.

Starting from x0=1, the next value x1= 6 mod 11 =6, x2= 36 mod 11 =3... The sequence 1,6,3,7,9,10,5,8,4,2,1,6,... is generated

Produces m-1=10 values before repeating. Has full period

Consider a=3, m=11. Then x0=1 yields: 1,3,9,5,4,1... Then x0=2 yields: 2,6,7,10,8,2...

In practice we want a generator that produces billions and billions of values before repeating

Achieving Full Period in an LCG

Consider LCG xi+1 = (a xi) mod m

If m is a prime, full period is obtained if a is a primitive root of m, i.e., am-1 – 1 is a multiple of m aj-1 – 1 is a not a multiple of m for j=1,2,...,m-2

Example of good LCG a=40014, m=214748563

Random Numbers from LCG lie on a plane

Ui

Ui+1

a=6, m=11

Spectral gapAs a discrepancy measure

General Random Numbers

Given i.i.d. sequence of U(0,1) variables, generate independent samples from an arbitrary distribution F(x) = P(X < x) of X

Inverse Transform Method Suppose X takes values 1,2 and 3 each with prob. 1/3.

F-1(U) has distribution function F(x)

1

2/31

1/3

2 3

F(x)1

UF(x)

F-1(U)

x

Inverse Transform Method

Example: F(X) = 1-exp(-a X).

Thus, X is exponentially distributed with rate a.

Then, X= -log(1-U)/a has the correct distribution

P(F-1(U) < y) = P(U < F(y))=F(y)

Also F(X) has U(0,1) distribution

1

UF(x)

F-1(U)

Acceptance Rejection Method

f(x)

c*g(x)

Need to generate X with pdf f(x)

There exists a pdf g(x) so that f(x) < c g(x) for all x

Algorithm: generate Y using pdf g. Accept the sample if f(Y) < c g(Y). Otherwise, reject and repeat.

Rationale

x

f(x)

Strategy: generate a sample X from f. Spread it uniformlybetween 0 and f(X)

Prob density of being in rectangular strip = f(x)dx * Lx/f(x)= LxdxProb of being in the region= area of the region

This property is retained by the acceptance rejection method

Lx

Generating Normally Distributed Random Numbers

N(0,1) ),N( that Note

invert easy tonot hence form, closedin known not

)dx 2

xexp(-

2

1y)P(N(0,1)F(y)

function on distributi Cumulative

x- ), 2

xexp(-

2

1 (x)

:density N(0,1) Normal univariate standard The

2

y

-

2

2

Algorithm for Normal Distribution

).sin(R Z),cos( set ZThen

U2V , Ulog 2 - RSet ).U,(U Generate : Algorithm

origin at the centered R radius of circle the

on ddistribute uniformly is )Z,(Zpoint the,RGiven 2)

2mean with ddistributelly exponentia is ZZR 1)

thensN(0,1)'t independen are Zand ZIf

RV Normal generate toMethodMuller Box

21

212

21

212

22

21

2

21

VVR

Generating Multivariate Normal Random Numbers

).AA,N( as ddistribute is X then AZ, X

and I)N(0, as ddistribute Zifcheck Easy to

xzero-nonfor ,0

:matrix covariance definite positive symmetric is

x)),-(x)-(x2

1exp(-

||)2(

1 (x)

:density ),N( Normal temultivaria The

T

d

d

d1-T2/12/

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d

Algorithm for Multivariate Normal Random Numbers

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Recap of Monte Carlo Method for Pricing Multi-dimensional European Options

Identify the risk neutral probability measure. Estimate the model from the data Replace drift with the risk free rate

Discretize the state space. Generate sample paths of the assets using the multi-variate Normal random vectors

Collect independent identically distributed samples of option payoffs

Use central limit theorem to develop confidence interval of the price estimate

Ordinary simulation can be computationally expensive

Convergence rate proportional to Slow but for a given variance independent of

problem dimensionGenerating each sample may be expensiveMotivates research in clever variance

reduction techniques to speed up simulations

nX

Common Variance Reduction Techniques

We discuss the following variance reduction techniques

Common random numbers Antithetic variates Control variates Importance sampling

Using Common Random Numbers

Often we need to compare two systems, so we need to estimate

EX - EY = E(X-Y) One way is to

estimate EX by its sample mean Xn

estimate EY by its sample mean Yn

two sample means are independently generated.

Note that Var(X-Y) = Var(X) + Var(Y) - 2Cov(X,Y) Positive correlation between X and Y helps The variations in X-Y cancel

Common Random Numbers to Estimate Sensitivity

difference their of samplest independen of average Take

)ˆ)ˆ2

1)(1exp((ˆ

))21)(1exp(())2

1exp((

samples Generate

1 E

~ Price :y volatilitto

optionAsian of price ofy sensitivit Finding :Example

01

201

01

201

21

1

k

jjk

k

jjkkk

n

kk

NrkSS

NrkSNrSS

KSn

Antithetic Variates

Consider the estimator

Xn = ( X1 + X2 + … + Xn)/n

Var (X1 + X2) = Var (X1) + Var (X2) + 2 Cov (X1, X2)

To reduce variance we need Cov (X1, X2) < 0

Theorem Given any distribution of rv X and Y (FX

-1(U), FY-1(U)) has the maximum covariance

(FX-1(U), FY

-1(1-U)) has the minimum covariance1

U

Example of Antithetic Technique

Example: Asian Option

n

kk KS

n 1

1

))21)(1exp(())2

1exp((0

12

012

1

k

jjkkk NrkSNrSS

))21)(1exp(())2

1exp((0

12

012

1

k

jjkkk NrkSNrSS

Antithetic

Control Variates

Consider estimating EX via simulation Along with X, suppose that C is also generated and EC is known If C is correlated with X, then knowing C is useful in improving our

estimate

Let Y = X - b ( C - EC) be our new estimate. Note that EY = EX

Best b* = Cov (X,C)/Var(C) Then Var (Y) = (1- 2)Var (X) (: correlation coefficient)

In practice , b = sample covariance(X,C)/sample variance(C) =

and the estimate is Xn + b (Cn - EC)

Pricing Asian Options

KSEKSKS

nn

n

kk

nn

kk

n

kk

/1

1

/1

11

][][ of avg1

of avg

Estimator

ddistributenormally -log is ][ , ][ /1

1

/1

1

nn

kk

nn

kk SKS

Option pay-off

K strike price

T=0

Stock price

Control variate

n

kk KS

n 1

1

Rare Event Simulation Problem

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confidence 95% aget toneeded trials102.75

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Importance Sampling to Rescue

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accuracy 5%for needed samples 7,932 0.8,When

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Portfolio Credit Risk with Extremal Dependence

Credit Risk

Credit Risk: The risk of loss due to obligor defaulting on payments. More generally, due to change in obigor’s credit quality

Market Risk: The risk of losses due to changes in market prices.

In credit risk: Lack of liquidity, time horizons are typically large Relevant model input information: probability of

default, loss given default. Market risk measurement is more concerned with measures such as price volatilities

Credit Risk: Heavier Loss Tails

Portfolio Credit Risk

We focus on measurement of portfolio credit risk The portfolio may comprise loans, defaultable bonds, letters of

credit, credit default swaps (CDS) etc.

Motivation Basel II accord permit the use of internal models for calculating credit

risk The emergence of collateralized debt obligations, where portfolio risk

measurement is crucial

Accurate dependence modeling is critical Literature suggests that extremal dependence may exist among obligor

losses

Section Outline

Describe a commonly used mathematical model for portfolio credit risk

Incorporate extremal dependence in this framework Asymptotic regime to analyze probability of large losses

and expected shortfall Sharp asymptotics for these measures and their

implications Provably efficient importance sampling techniques to

estimate these performance measures

The Portfolio Credit Risk Problem

Consider a portfolio with n obligors The obligor i has exposure ei.

If it defaults, a loss of amount ei is incurredThis amount may be random to incorporate credit

quality changes, recovery variation etc.

The default probability of obligor is pi.

May be measured using historical default data based on its ratings

KMV modifies Merton’s seminal ideas combined with empirical data to come up with Expected Default Frequency

Historical Credit Migration Data to Compute Default Probabilities

AAA AA A BBB BB B CCC D

AAA 93.7% 5.8% 0.4% 0.1% 0.0% 0.0% 0.0% 0.0%AA 0.7% 91.7% 6.9% 0.5% 0.1% 0.1% 0.0% 0.0%A 0.1% 2.3% 91.7% 5.2% 0.5% 0.2% 0.0% 0.0%

BBB 0.0% 0.3% 4.8% 89.2% 4.4% 0.8% 0.2% 0.2%BB 0.0% 0.1% 0.4% 6.7% 83.2% 7.5% 1.0% 1.1%B 0.0% 0.1% 0.3% 0.5% 5.7% 83.6% 3.8% 5.9%

CCC 0.1% 0.0% 0.3% 0.9% 1.9% 10.3% 61.2% 25.3%D 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 100.0%

This data may be adjusted for prevalent conditions.

It may be used to compute losses due to change in credit quality

Latent Variable Approach based on Merton’s Model

inferred be alsocan debt of value

theanddefault ofy probabilit The y. volatilitits and ueasset val

implied find toused becan This debt. sfirm' theof valueface

toequal price strike with valuefirm on theoption an isEquity

equation thefollows process ueAsset val dWrdtX

dX

t

t

1 Yr

Distribution of asset value at horizon

AssetValue

Today Time

Value

Debt face value

Default Path

KMV’s Approach to Finding Expected Default Frequency

1 Yr

Distribution of asset value at horizon

AssetValue

Today

EDF

Time

Value

Default PointDistance-to-Default =3 Standard deviations

Asset Volatility(1 Std Dev)

Courtesy: KMV website

Modeling Dependence through Multi-Variate Latent Variables

Latent random variable Xi models the value of obligor i If Xi goes below a threshold xi the obligor i defaults resulting

in loss ei

Total Loss L= e1I(X1<x1) + e2I(X2<x2) …. + enI(Xn<xn)

We focus on developing sharp asymptotics and Monte Carlo importance sampling techniques to estimate P(L>x) and E(L-x|L>x) for large x

when latent variables (X1, X2,…, Xn) have extremal dependence

Typically Latent Variables are assumed to have Normal Distributions

J. P. Morgan’s CreditMetrics and Moody’s KMV system assume that the latent variables (X1, X2,…, Xn) follow a multi-variate normal distribution. Correlations captured through dependence on factors

risk. ticidiosyncra

captures that variablenormal a is factors. loading are ),...,,(

obligors.on effectsother andindustry country, global, ofimpact

measure that variablesrandom normal standardt independen are ,...,, Here

...

21

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ccc

ZZZ

cZcZcZcX

Normal Variables often Inadequate to Capture Extremal Dependence

• Empirical evidence suggests that financial variables often exhibit stronger dependence than that captured by correlation based multi-variate normal model.

• Example: P(X1>x | X2>x) 0 as x infinity, in normal setting

• If instead latent variables have a multivariate t-distribution, extremal dependence is captured, i.e., random variables may take large values together with non-negligible probability

• T-distributions often show better fit to financial data

Modeling Extremal Dependence

upset in this case special a ison distributi t variate-multi The

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).1:( oft independen variable,randomt independenanother is Z

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Asymptotic Regime to Analyze Loss Distribution

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Comparison with Normal Copula

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21/n

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nb

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Z

i

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Monte Carlo Simulation

Accurate estimation via Monte Carlo Simulation Naïve implementation

Generate samples of Z, W and the Bernoulli variables with probability of success P(Xi< -ai n | Z, W) for each i.

Then a sample of I{ Ln>nb } is seen.

Average of many samples provides an estimator for P(Ln>nb)

Central limit theorem may be used to construct confidence intervals

Computational problem of estimating rare event probabilities

Importance Sampling in Our Setting

j. populationin successes thedenotes Y

)()(p~-1

1

p~)(f~

)(f~

)(f)(f

isestimator sampling importance Then the

).:p~( and )(f~

),(f~

notationy with probabilit new aunder samples generate weSuppose

)}.({X of iesprobabilit Bernoulli lconditiona denote

):(pLet W.and Zof pdf denote )(f),(fLet

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Evaluating Importance Sampling Estimator

error relative bounded with scheme sampling importancean propose We

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nbLPnbLIERnbLIR

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Performance of Importance Sampling Algorithms

In the range of practical importance, P(Ln>nb) approximately 1 in 1000, algorithm 1 reduces variance by about 150 times.

All else being equal, greater the impact of W in causing the rare event, better the performance

The results extend easily to multi-factor models

Monte Carlo Methods for Pricing American Options

Multi-Period Binomial Model: American Options

The decision to exercise can be made at any point in the lattice

S0

S1(H)

S1(T)

S2(HH)

S2(HT,TH)

S2(TT)

S3(HHH)

S3(HHT,HTH,THH)

S3(TTH,HTT,THT)

S3(TTT)

American Options and Stopping Times

.|r)(1

GE~

by given is n

at timestrategy h thisoption wit theof valuearbitrage-no The

. timesstopping called are variablesrandomSuch ).,...,(

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n. at time exercised isoption theif payoff thedenote 0),...,(GLet

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Pricing American Options

with time valueloses process price theaverageOn

)1(

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priceoption theThen,

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11

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Dynamic Programming Recursion satisfied by the Price Process

),,...,(~),,...,(~()1(

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and V

recursions programing dynamic thesatisfiesV process price The :Result

1111

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General Models

We assume that the option can be exercised at times 0,1,2,...,N (Bermudan option)

The discounted value of the option at time m if exercised at time m equals Gm(Sm) > 0

Let Tm denote the set of stopping times taking values in (m, m+1, ...,N)

Then

Where the expectation is under risk neutral measure

If s0 denotes the initial price then our interest is in finding J0(s0)

)|)((sup)( sSSGEsV mT

mm

Dynamic Programming Formulation

Let Cm(s) = E(Vm+1(Sm+1)|Sm=s)=Vm+1(y) fm(s,y)dy = Pm(Vm+1)(s)

denote the continuation value.

VN(s) = GN(s)

Vm(s) = max(Gm(s), Cm(s)) for m=0,1,...,N-1

Alternatively, Cn-1(s) = Pn(Gn)(s)

Cm(s) =Pm(max(Gm+1,Cm+1))(s) for m=0,1,2,…N-2

Even if the state space is discretized, the DP formulation suffers fromthe curse of dimensionality

Monte Carlo Methods for American Options

Random Tree Method

Regression based Function Approximation method

The Random Tree Method(Broadie and Glasserman 1997)

t=0 t=1

.b as )( Ushown that beCan

]1

),(max[

)(

000

1

...1

......

......

212121

2121

sJ

Ub

sGU

sGUb

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iiimm

iiim

iiiNN

iiiN

tmm

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Generate a sample treeWith b branchesFrom every statevisited

s0

s11

s12

s13

s211

s231

Random Tree Method

Does not depend upon the number of underlying securities

The effort increases exponentially with the number of exercise opportunities.

Regression Based Function Approximations

meach for

))(r),...,1((r tscoefficienK finding of problem

the toreduced is )(C finding of problem theThus

s and meach for )(C )r(s,C

thatso r,...,r,r vectorsparameter find

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are where)()( r)(s,CLet

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s

s

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The Broad Approach

policy. exercise eapproximat good a determine toused becan two theHowever,

remains. )(*)( and )(between error The .m as *rr that see Easy to

))J(s(s)(s)(sr

:as estimated bemay thisy,Empiricall

)]()([~

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1

1

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1T

1

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n

i

n

i

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kk

krssJ

SJSE

dssfkrssJ

Simulation Methodology

Generate n sample paths (sm,j: m=0,...,N and j=1,...,n) of the process (Sm: m=0,1,...,N)

Set

n

j

K

kjmkjrm

n

j

K

kjNkjNNrN

krssGr

krssGr

jmjm1 1

2,,

*

jm,

1 1

2,,

*1

)]()()([minarg

m. timeit till exercisednot have wegiven that

j,path alongoption theexercise to time the

know weSuppose m. imeconsider t y,Recursivel

)]()()([minarg

,,

Nm

Regression based Methodology

Using this methodology the optimal exercise policy * is learnt quickly

The expectation corresponding to this stopping policy is evaluated using the usual Monte-Carlo to generate samples of G*(S*)

The first phase is empirically seen to be quick. Mistakes here are not crucial.

The second phase requires significant effort...hence a need to speed-up through variance reduction