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
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Multi-Period Binomial Model
The analysis extends to multiple periods to more realistic models.
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S1(T)
S2(HH)
S2(HT,TH)
S2(TT)
S3(HHH)
S3(HHT,HTH,THH)
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S3(TTT)
Solving for Option Price through Backward Recursion
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A Numerical Example:Pricing a Lookback Option
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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
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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
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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
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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
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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
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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
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2 is the variance of each Xi determines the convergence rate
Pricing Asian Option through Monte Carlo
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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
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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
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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
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1)(1exp((ˆ
))21)(1exp(())2
1exp((
samples Generate
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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
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kk KS
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1
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1exp((0
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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
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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
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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
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• 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
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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
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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)
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American Options and Stopping Times
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Pricing American Options
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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
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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
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The Broad Approach
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Simulation Methodology
Generate n sample paths (sm,j: m=0,...,N and j=1,...,n) of the process (Sm: m=0,1,...,N)
Set
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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
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