Monte Carlo Methods in Finance

1

• Monte Carlo Methods in Finance
• IIM Ahmedabad, Nov 6, 2005
• Sandeep Juneja
• School of Technology and Computer Science
• Tata Institute of Fundamental Research

2
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

3
European Call Option
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)
BUY CALL
WRITE CALL
Option Payoff
Option Payoff
K
0
profit
0
Underlying price
K
Payoff on the Maturity Date
4
European Put Option
An option to sell an underlying asset at a
specified time for a specified price. Payoff
G(ST) (K-ST)
BUY PUT
WRITE PUT
Option Payoff
Option Payoff
K
0
Underlying price
0
K
Underlying price
Payoff on the Maturity Date
5
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)

6
Examples of Options on Multiple Assets

• Basket Option
• (c1S1(T) c2S2(T) ... cdSd(T) - K)
• Out-performance Option
• (maxc1S1(T), c2S2(T),...,cdSd(T) - K)
• Barrier Option
• I(mini1,..,nS2(ti) ltb(K - S1(T))
• Quantos S2(T)(S1(T) - K)

They all have an associated American version
7
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

8
Simple One Period Binomial Model to Price Options
Two securities exist in this world
S1(H) uS0
1r
1
S0
1r
S1(T) dS0
d lt 1r lt u from no-arbitrage considerations
Consider an option
V1(H), e.g., S1(H)-K
(If S1(T) ltKltS1(H))
V0 ?
V1(T), e.g., 0
9
Create a Replicating Portfolio
10
The Risk Neutral Measure
11
Multi-Period Binomial Model

• The analysis extends to multiple periods to more
  realistic models.

12
Solving for Option Price through Backward
Recursion
13
A Numerical ExamplePricing a Lookback Option
S3(HHH)32
S2(HH)16
S3(HHT,HTH,THH)8
S1(H)8
S2(HT,TH)4
S04
S3(TTH,HTT,THT) 2
S1(T)2
S2(TT)1
S3(TTT)0.5
14
The Discounted Price Process is a Martingale
15
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

16
Fundamental Theorem in Option Pricing
17
Brownian Motion

• A real valued process (W(t)t gt 0), is standard
  Brownian motion if
• For t0 lt t1...lt tn, then W(t1)-W(t0),...,
  W(tn)-W(tn-1) are independent
• W(st)-W(s) is Normally distributed with mean 0
  and variance t
• W(t) is a continuous function of t (with prob
  1).

18
Single Dimension Asset Pricing Model
19
Asset Price an Expectation under Equivalent
Martingale Measure
20
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 Eulers scheme (finer
  discretizations improve accuracy)
• process dSt r Stdt
  s(t) StdW(t)

21
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...Ymgtu) for
  large u.
• Note that P(Y1...Ymgtu) EI(Y1...Ymgtu)
• Loss given default
• E Y1...YmY1...YmgtuEY1...Ym
  I(Y1...Ymgtu)/P(Y1...Ymgtu)

22
Monte Carlo Method

• Motivating the Monte Carlo Approach
• Monte Carlo Method
• Random number generation
• Generating random numbers from general
  distributions
• Popular variance reduction techniques

23
Illustrative Queueing Example

• The inter-arrival times (A1,A2, ) are
  independent identically distributed with
  distribution function
• FA(x) P(A lt x).
• E.g. FA(x) 1 - e-lx
• The service times (S1,S2, ) are independent
  identically distributed with distribution
  function FS(x) P(S lt x).

24
Solve or Run the Model ?
6

• To determine EW we could use deductive
  arguments, e.g.
• Wn1 Wn Sn - An1
• gt ...
• gt
• gt 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

25
Key Statistical Ideas

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

For dice m 11/6 21/6 31/6 41/651/6
61/6 7/2

• Central limit theorem

s2 is the variance of each Xi determines the
convergence rate
26
Pricing Asian Option through Monte Carlo
Asset price
k
k1
27
Constructing Estimators
28
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

29
Generating Uniform (0,1) Pseudo Random Numbers

• Requirement Generate a sequence of numbers U1,
  U2,...so that
• Each Ui is uniformly distributed between 0 and 1
• 2) The Uis are mutually independent

30
Linear Congruential Generators
Popular method A linear congruential
generator Given an initial integer seed x0
between 0 and m, set xi1 a xi mod m ui1
xi1/m a lt m is referred to as multiplier, m
the modulus
31
Properties of a Good Random Number Generator
32
Periodicity of Linear Congruential Generators

• Consider the case where a6, m11.
• Starting from x01, 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-110 values before repeating. Has full
  period
• Consider a3, m11.
• Then x01 yields 1,3,9,5,4,1...
• Then x02 yields 2,6,7,10,8,2...
• In practice we want a generator that produces
  billions and billions of values before repeating

33
Achieving Full Period in an LCG

• Consider LCG xi1 (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
  j1,2,...,m-2
• Example of good LCG a40014, m214748563

34
Random Numbers from LCG lie on a plane
Spectral gap As a discrepancy measure
Ui1
Ui
a6, m11
35
General Random Numbers

• Given i.i.d. sequence of U(0,1) variables,
  generate independent samples from an arbitrary
  distribution F(x) P(X lt x) of X
• Inverse Transform Method
• Suppose X takes values 1,2 and 3 each with prob.
  1/3.

1
F(x)
2/3
1/3
1
2
3
x
F-1(U) has distribution function F(x)
36
Inverse Transform Method
P(F-1(U) lt y) P(U lt F(y))F(y) Also F(X) has
U(0,1) distribution
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
37
Acceptance Rejection Method
cg(x)
f(x)
Need to generate X with pdf f(x) There exists a
pdf g(x) so that f(x) lt c g(x) for all
x Algorithm generate Y using pdf g. Accept the
sample if f(Y) lt c g(Y). Otherwise, reject and
repeat.
38
Rationale
Strategy generate a sample X from f. Spread it
uniformly between 0 and f(X)
f(x)
Lx
x
Prob density of being in rectangular strip
f(x)dx Lx/f(x) Lxdx Prob of being in the
region area of the region This property is
retained by the acceptance rejection method
39
Generating Normally Distributed Random Numbers
40
Algorithm for Normal Distribution
41
Generating Multivariate Normal Random Numbers
42
Algorithm for Multivariate Normal Random Numbers
43
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

44
Ordinary simulation can be computationally
expensive

• Convergence rate proportional to
• Slow but for a given variance independent of
  problem dimension
• Generating each sample may be expensive
• Motivates research in clever variance reduction
  techniques to speed up simulations

45
Common Variance Reduction Techniques

• We discuss the following variance reduction
  techniques
• Common random numbers
• Antithetic variates
• Control variates
• Importance sampling

46
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

47
Common Random Numbers to Estimate Sensitivity
48
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) lt 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 covariance

49
Example of Antithetic Technique

• Example Asian Option

Antithetic
50
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- r2)Var (X) (r correlation
  coefficient)
• In practice , b sample covariance(X,C)/sample
  variance(C)
• and the estimate is Xn b (Cn - EC)

51
Pricing Asian Options
Option pay-off
Control variate
52
Rare Event Simulation Problem
53
Importance Sampling to Rescue
54
Importance Sampling in Abstract Setting
55
Importance Sampling (contd.)
56
Importance Sampling for Sums of continuous RV
taking Large Values
57
Portfolio Credit Risk with Extremal Dependence
58
Credit Risk

• Credit Risk The risk of loss due to obligor
  defaulting on payments. More generally, due to
  change in obigors 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

59
Credit Risk Heavier Loss Tails
60
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

61
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

62
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 incurred
• This 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 Mertons seminal ideas combined with
  empirical data to come up with Expected Default
  Frequency

63
Historical Credit Migration Data to Compute
Default Probabilities
This data may be adjusted for prevalent
conditions. It may be used to compute losses due
to change in credit quality
64
Latent Variable Approach based on Mertons Model
65
KMVs Approach to Finding Expected Default
Frequency
Courtesy KMV website
66
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(X1ltx1) e2I(X2ltx2) . enI(Xnltxn)
• We focus on developing sharp asymptotics and
  Monte Carlo importance sampling techniques to
  estimate
• P(Lgtx) and E(L-xLgtx) for large x
• when latent variables (X1, X2,, Xn) have
  extremal dependence

67
Typically Latent Variables are assumed to have
Normal Distributions

• J. P. Morgans CreditMetrics and Moodys KMV
  system assume that the latent variables (X1,
  X2,, Xn) follow a multi-variate normal
  distribution. Correlations captured through
  dependence on factors

68
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(X1gtx X2gtx) ? 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

69
Modeling Extremal Dependence
70
Asymptotic Regime to Analyze Loss Distribution
71
Sharp Asymptotic for Loss Probability
72
Comparison with Normal Copula
73
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(Xilt -ai
  n Z, W) for each i.
• Then a sample of I Lngtnb is seen.
• Average of many samples provides an estimator for
  P(Lngtnb)
• Central limit theorem may be used to construct
  confidence intervals
• Computational problem of estimating rare event
  probabilities

74
Importance Sampling in Our Setting
75
Evaluating Importance Sampling Estimator
76
Performance of Importance Sampling Algorithms

• In the range of practical importance, P(Lngtnb)
  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

77

• Monte Carlo Methods for Pricing American Options

78
Multi-Period Binomial Model American Options

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

79
American Options and Stopping Times
80
Pricing American Options
81
Dynamic Programming Recursion satisfied by the
Price Process
82
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) gt 0
• Let Tm denote the set of stopping times taking
  values in (m, m1, ...,N)
• Then
• Where the expectation is under risk neutral
  measure
• If s0 denotes the initial price then our interest
  is in finding J0(s0)

83
Dynamic Programming Formulation

• Let Cm(s) E(Vm1(Sm1)Sms) ò Vm1(y)
  fm(s,y)dy Pm(Vm1)(s)
• denote the continuation value.
• VN(s) GN(s)
• Vm(s) max(Gm(s), Cm(s)) for m0,1,...,N-1
• Alternatively, Cn-1(s) Pn(Gn)(s)
• Cm(s) Pm(max(Gm1,Cm1))(s) for m0,1,2,N-2

Even if the state space is discretized, the DP
formulation suffers from the curse of
dimensionality
84
Monte Carlo Methods for American Options

• Random Tree Method
• Regression based Function Approximation method

85
The Random Tree Method(Broadie and Glasserman
1997)
s211
s11
Generate a sample tree With b branches From every
state visited
s12
s0
s13
s231
t0
t1
86
Random Tree Method

• Does not depend upon the number of underlying
  securities
• The effort increases exponentially with the
  number of exercise opportunities.

87
Regression Based Function Approximations
88
The Broad Approach
89
Simulation Methodology

• Generate n sample paths (sm,j m0,...,N and
  j1,...,n) of the process (Sm m0,1,...,N)
• Set

90
Regression based Methodology

• Using this methodology the optimal exercise
  policy t is learnt quickly
• The expectation corresponding to this stopping
  policy is evaluated using the usual Monte-Carlo
  to generate samples of Gt(St)
• 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