Lecture 7: Simulations

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Lecture 7 Simulations
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http//www.angelfire.com/linux/lecturenotes/
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What will we cover in this lecture

• A introduction to the idea and concept of
  stochastic simulation
• A look at the 2 main methods of stochastic
  simulation Bootstrapping Monte Carlo
  Simulation
• A look at the use of these methods in the
  calculation of risk

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The Idea Behind Simulations

• Simulations are basically the idea of simulating
  a set of possible future outcomes or
  realisations
• Each future realisation is just one possibility
  of what may occur in the future
• By generating many possible future realisations
  we can assess the range of future outcomes
• In an earlier lecture we simulated a possible
  path for stock prices by generating a set of
  possible future returns from a normal
  distribution
• This was an example of a univariate simulation (a
  simulation of one variable)

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The Idea of a Simulation
Possible Future Price Paths
Current Price
Possible Range Of Future Prices
Price
Time
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Useful Simulations

• For a simulation to be useful the paths generated
  must reflect the behaviour of the
  asset/liabilities we are simulating
• Each randomly generated future path will then
  represent a possible future outcome
• The basis for both our Bootstrapping and Monte
  Carlo Simulations will be the Brownian Motion
  Process we have been using so far

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Brownian Motion Process
At each step the proportional change in the value
is a random variable from a distribution
Value
?
Time
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Bootstrapping A simple powerful approach

• Bootstrapping is based on a very simple idea
  that we can use past observations as a bag from
  which we randomly sample to create possible
  future outcomes!
• The premise is a simple one all of our past
  observations were sampled from a given
  distribution and all future observations will
  also be sampled from this same distribution
• Instead of having to estimate the underlying
  distribution and then sampling random variables
  from it we directly sample from our past
  observations which are representative

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Bootstrapping Idea
The Past Outcomes Represent A Random Sample From
The Distribution, We Dont Care About This
Distribution!
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The Simulated Future Outcome Is A Random Sample
Of The PastOutcomes
Simulated Future Outcome
Directly Observed Past Outcome
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Univariate Bootstrap

• Univariate Bootstrap is the generation of a
  possible path for a single simultaneous random
  process
• We generate a pool of observations from the
  processs historic behaviour
• We then generate a future path by randomly
  sampling from this pool of observations

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An Example Bootstrapping a stock price

• If we assume that the continuously compounded
  return of the stock price each day is sampled
  from a constant distribution then we can generate
  future price paths
• Firstly we build up our pool of past daily
  returns by calculating the implied return from
  the price history
• Secondly we randomly sample from this pool to
  generate a possible future path

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Bootstrapping Stock Price
Historic Price Path
Pool of Daily Returns From Historic Observation
Price
Time
Simulated Future Price Path
Price
Randomly Select From Pool To Generate Future Path
Time
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Multivariate Bootstrapping

• It is equally possible to simultaneously generate
  future paths for multiple correlated processes
• Like our univariate bootstrap we build up a pool
  of observations from the past
• We maintain any correlations by grouping
  simtaneous observations together
• Any correlations will be captured in these linked
  historic observations
• When we randomly sample from the pool to generate
  the future paths we preserve these groupings

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Bivariate Bootstrap
Price
Price
Time
Time
The pool is made up of linked pairs which were
observed simultaneously
Price
Price
Time
Time
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What if we do not preserve the grouping of
simultaneous observations?

• If we do not keep the groups then we cannot
  expect our generated paths to reflect any
  correlations
• If we have strong reason to believe that there is
  no correlation between the processes then we
  might break apart the observations and sample
  them completely randomly
• We might do this to increase the size of our
  pool

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Bootstrapping Is Very Powerful

• The power of bootstrapping is that it does not
  rely on any statistical assumptions
• All it says is that the future will be sampled
  from the same distribution as the past
• We need a large sample of historic observations
  to build up a large pool
• The assumption that the future will come from the
  same distribution as the past is crucial and
  requires a stationary series
• This simple bootstrapping technique does not
  capture serial correlation across time.

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Bootstrapping in Excel

• For practical usage bootstrapping is better
  suited to a programming language such as VBA, or
  a specialist tool
• The exercise book for this lecture contains an
  array function called bootstrap
• bootstrap will take an input range and reorder it
  randomly
• bootstrap is an array formula for which an output
  range must be selected and ctrl-shift-enter must
  be used to enter it!

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Monte Carlo Simulation

• The Monte Carlo Simulation generates random
  numbers from probability distribution rather than
  pools of historic observations
• We will focus on Monte Carlo Simulations based
  about the normal distribution
• We will see how we can use the Covariance matrix
  to generate multivariate Monte Carlo Simulations

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Univariate Monte Carlo Simulations

• By generating random variables sampled from a
  normal distribution it is possible to generate a
  brownian motion paths based on this sequence of
  random numbers
• In the example of the stock price if the return
  is normally distributed with mean m and standard
  deviation s then by generating returns from a
  distribution with these properties we can
  generate a possible path for this stock
• The most common method for generating normal
  random numbers is Box-Muller, and we will use
  Excels built in facility

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Stock Price Simulation

• In the case where the stock price is generated
  from the following process

• By generating a sequence of random variables r we
  can generate a future path for P
• This would be an example of a univariate Monte
  Carlo Simulation

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Univariate Stock Price Simulation
Price
Randomly Sampled Returns From Normal Distribution
Randomly generated feasible path
Time
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Multivariate Monte Carlo Simulation

• A multivariate Monte Carlo Simulation is where we
  sample random values from a multivariate
  distribution which captures the various
  correlations
• So if 2 assets have a strong correlation we want
  to see strong correlations in the random numbers
  we generate
• There is a set method which allows us to generate
  a set of correlated normally distributed random
  variables using the covariance matrix

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What we would like

• We have described the statistical properties of
  returns using the expectation vector and the
  covariance matrix.
• We would like to generate sets of random numbers
  that match this expected return vector and
  covariance matrix
• We could then generate simultaneous paths for the
  various assets/liabilities described by these
  statistics

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The Basis of Multivariate Monte Carlo The
Cholesky Decomposition

• The Cholesky decomposition of the covariance
  matrix will allow us to generate sets of random
  variables described by the covariance matrix
• The Cholesky decomposition is sometimes called
  the square root of a matrix
• It takes the form

• Where CV is the covariance matrix and C is the
  cholesky decomposition. C is a lower diagonal
  matrix and of the same dimensions as CV
• The Cholesky decomposition only exists for
  positive definite matrices.

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Cholesky Decomposition Example
C
CT
CV
0.2 0.03
0 0.197
0.04 0.006
0.006 0.04
0.2 0
0.03 0.197


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The Cholesky Transformation

• Image we have 2 random variables A and B each of
  which are sampled from a standard normal
  distribution (mean 0, standard deviation 1). We
  will put these in a vector S (S for stochastic)
• We would like to transform these variables to
  random variables sampled from a distribution with
  mean, variance and covariance described by the
  following

VarA CovA,B
CovA,B VarB
ERA
ERB
A
B
R
CV
S
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• Firstly we perform the Cholesky decomposition on
  the covariance matrix to obtain C
• Then we simply perform the following Cholesky
  transformation

• Where S is a vector of transformed random
  variables sampled from a distribution described
  by R and CV

A
B
S

• Where A will have mean ERA and variance VarA, B
  will have mean ERB and variance VarB, and A and
  B will have Covariance CovA,B

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Cholesky Transformation Graphic
A B Y Z
Input a vector of uncorrelated random variable
with mean 0 and variance 1
Cholesky Transformation
Output a vector of correlated random variable
with mean,variance and covariance described by
the covariance matrix and the expected vector
A B Y Z
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Cholesky Factorisations in Excel

• Cholesky Factorisations are not provided by
  Excels built in matrix support
• The class workbook comes with a VBA macro that
  will perform this factorisation for you, it is an
  array function called cholesky

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Multivariate Monte Carlo Simulation

• Using the Cholesky Transformation we can generate
  sets of correlated random numbers
• Using these sets of correlated random numbers we
  can generate simultaneous future paths for
  correlated processes
• These simulated paths make up one possible future
  outcome in our Monte Carlo Simulation
• Monte Carlo Simulations use 1000s or even
  1000000s of possible future outcomes to build up
  a picture of the range of future possibilities

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Graphic Bivariate Monte Carlo Simulation
Correlated sets of random numbers
Value
Value
Time
Time
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General Multivariate Monte Carlo Algorithm

• 1) Decompose the covariance matrix VC into its
  Cholesky decomposition C
• 2) Generate a vector of N unit variate normal
  random number, where N is the dimensions of the
  simulation
• 3) Multiply the decomposition C by the vector N
  to get a result vector X
• 4) Add the vector X to the expected return vector
  R to get a vector V which contains the fully
  transformed random variables
• 5) Add transformed vector V to the set of
  transformed vectors S
• 6) If S contains less than the desired number of
  random variables then goto step 2

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The montecarlo macro

• Embedded in the exercise spreadsheets there is a
  VBA macro which runs the Monte Carlo algorithm
• It take two parameters first the range
  specifying the covariance matrix, second the
  range specifying the expected return vector
• From these inputs it outputs a set of random
  variables sampled from a distribution described
  by the covariance matrix and expected vector in a
  range specified by the user
• It is important to remember to specify one column
  for every random variate you want to generate and
  press ctrl-shift-enter to tell the computer it is
  an array formula!

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Stochastic Boundaries And Monte Carlo Simulations

• In early lectures we looked at placing boundaries
  on the behaviour of a stochastic process
• These boundaries are linked to Monte Carlo
  Simulations in that we expect only 5 or 2.5 of
  simulations to be outside the respective
  diffusion boundary

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Diffusion Boundary vs Monte Carlo
Only 2.5 of paths will break above the 2.5
upper boundary
Value
Possible Paths
Time
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Uses of Simulation

• Simulations have many uses and allow us to avoid
  a lot of complex mathematics
• People think you are a rocket scientist if you
  know about Monte Carlo Simulations
• One example would be an alternative approach to
  calculating the Value-At-Risk
• We would generate say 100 future paths for a
  portfolios value by generating a set of 100
  simultaneous future paths for each of the assets
  it contains using either a Monte Carlo or
  Bootstrap simulation
• Out of this 100 we would take the worst 5 or 5
  to calculate the 5 VaR

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VaR Using Simulation
Portfolio Value
Take the worst 5 of simulated values as 5 VaR
Time
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Option VaR Using Monte Carlo Simulations

• One particular use of Monte Carlo simulation is
  the accurate calculation of VaR on non-linear
  portfolios containing options
• We generate 100s of possible paths for the assets
  on which the options are written and look at the
  value of the option in each state
• We then take the worst 5 or 2.5 of outcomes to
  calculate the portfolio 5 or 2.5 VaR

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Appendix Cholesky Transformation A Proof

• It is important to note that if we have a vector
  of stochastic variables S with mean zero , then
  the E(S.ST) CV

A
B
A.A A.B
B.A B.B
Var(A) Cov(A,B)
Cov(A,B) Var(B)
A B
E
E

• If we perform the Cholesky transformation on a
  vector of standard normal variables, and assume
  the expectation remains 0

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• Now

• Taking expectations

• Since S is a vector of unit normal variables
  E(S.ST) will be the identity matrix (why?)

• So the transformed random variables will have
  variance and covariance described by the
  covariance matrix C was decomposed from.