Math6911 S08, HM Zhu
3. Monte Carlo Simulations
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References
1. Chapters 4 and 8, “Numerical Methods in Finance”
2. Chapters 17.6-17.7, “Options, Futures and Other Derivatives”
3. George S. Fishman, Monte Carlo: concepts, algorithms, and applications, Springer, New York, QA 298.F57, 1995
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3.1 Overview
3. Monte Carlo Simulations
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Monte Carlo Simulation
• Monte Carlo simulation, a quite different approach from binomial tree, is based on statistical sampling and analyzing the outputs gives the estimate of a quantity of interest
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Monte Carlo Simulation
• Typically, estimate an expected value with respect to an underlying probability distribution– eg. an option price may be evaluated by computing
the expected payoff w.r.t. risk-neutral probability measure
• Evaluate a portfolio policy by simulating a large number of scenarios
• Interested in not only mean values, but also what happens on the tail of probability distribution, such as Value-at-risk
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Monte Carlo Simulation in Option Pricing
• In option pricing, Monte Carlo simulations uses the risk-neutral valuation result
• More specifically, sample the paths to obtain the expected payoff in a risk-neutral world and then discount this payoff at the risk-neutral rate
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Learning Objectives
• Understanding the stochastic processes of the underlying asset price.
• How to generate random samplings with a given probability distribution
• Choose an optimal number of simulation experiments• Reliability of the estimate, e.g. a confidence interval• Estimate error• Improve quality of the estimates: variance reduction
techniques, quasi-Monte Carlo simulations.
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3.2 Modeling Asset Price Movement
3. Monte Carlo Simulation
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Randomness in stock market
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Outlines
1. Markov process 2. Wiener process3. Generalized Wiener process4. Ito’s process, Ito’s lemma5. Asset price models
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References
1. Chapter 12, “Options, Futures, and Other Derivatives”2. Appendix A for Introduction to MATLAB, Appendix B for
review of probability theory, “Numerical Methods in Finance: A MATLAB Introduction”
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Markov process
• A particular type of stochastic process whose future probability depends only on the present value
• A stochastic process x(t) is called a Markov process if for every n and
• The Markov property implies that the probability distribution of the price at any particular future time is not dependent on the particular path followed by the price in the past
( ) ( )( ) ( ) ( )( )11
21
allfor have we,
−− ≤=≤≤
<<<
nnnnnn
n
txxtxPtttxxtxPttt
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Modeling of asset prices
• Really about modeling the arrival of new info that affects the price.
• Under the assumptions,The past history is fully reflected in the present asset price, which does not hold any further informationMarkets respond immediately to any new information about an asset
• The anticipated changes in the asset price are a Markov process
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Weak-Form Market Efficiency
• This asserts that it is impossible to produce consistently superior returns with a trading rule based on the past history of stock prices. In other words technical analysis does not work.
• A Markov process for stock prices is clearly consistent with weak-form market efficiency
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Example of a Discrete Time Continuous Variable Model
• A stock price is currently at $40. • Over one year, the change in stock price has
a distribution Ν(0,10) where Ν(µ,σ) is a normal distribution with mean µ and standard deviation σ.
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Questions
• What is the probability distribution of the change of the stock price over 2 years?– ½ years?– ¼ years?– ∆t years?
• Taking limits we have defined a continuous variable, continuous time process
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Variances & Standard Deviations
• In Markov processes, changes of the variable in successive periods of time are independent. This means that: – Means are additive– variances are additive– Standard deviations are not additive
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Variances & Standard Deviations (continued)
• In our example it is correct to say that the variance is 100 per year.
• It is strictly speaking not correct to say that the standard deviation is 10 per year.
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A Wiener Process (See pages 265-67, Hull)
• Consider a variable z follows a particular Markov process with a mean change of 0 and a variance rate of 1.0 per year.
• The change in a small interval of time ∆t is ∆z• The variable follows a Wiener process if
1.
2. The values of ∆z for any 2 different (non-overlapping) periods of time are independent
where is N(0,1)z tε ε∆ = ∆
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Properties of a Wiener Process
∆z over a small time interval ∆t:Mean of ∆z is 0Variance of ∆z is ∆tStandard deviation of ∆z is
∆z over a long period of time, T:Mean of [z (T ) – z (0)] is 0Variance of [z (T ) – z (0)] is TStandard deviation of [z (T ) – z (0)] is T
t∆
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Taking Limits . . .
• What does an expression involving dz and dt mean?• It should be interpreted as meaning that the
corresponding expression involving ∆z and ∆t is true in the limit as ∆t tends to zero
• In this respect, stochastic calculus is analogous to ordinary calculus
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Generalized Wiener Processes(See page 267-69, Hull)
• A Wiener process has a drift rate (i.e. average change per unit time) of 0 and a variance rate of 1
• In a generalized Wiener process the drift rate and the variance rate can be set equal to any chosen constants
• The variable x follows a generalized Wiener process with a drift rate of a and a variance rate ofb2 if
dx=a dt + b dz
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Generalized Wiener Processes(continued)
• Mean change in x in time ∆t is a ∆t• Variance of change in x in time ∆t is b2 ∆t• Standard deviation of change in x in time ∆t
is
tbtax ∆ε+∆=∆
b t∆
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Generalized Wiener Processes(continued)
• Mean change in x in time T is aT• Variance of change in x in time T is b2T• Standard deviation of change in x in time T
is
x a T b Tε∆ = +
b T
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Example
• A stock price starts at 40 and at the end of one year, it has a probability distribution of N(40,10)
• If we assume the stochastic process is Markov with no drift then the process is
dS = 10dz • If the stock price were expected to grow by $8 on
average during the year, so that the year-end distribution is N (48,10), the process would be
dS = 8dt + 10dz• What the probability distribution of the stock price at
the end of six months?
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Itô Process (See pages 269, Hull)
• In an Itô process the drift rate and the variance rate are functions of time
dx=a(x,t) dt+b(x,t) dz• The discrete time version, i.e., ∆x over a small time
interval (t, t+∆t)
is only true in the limit as ∆t tends to zerottxbttxax ∆ε+∆=∆ ),(),(
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Why a Generalized Wiener Processis not Appropriate for Stocks
• For a stock price we can conjecture that its expected percentage change in a short period of time remains constant, not its expected absolute change in a short period of time
• We can also conjecture that our uncertainty as to the size of future stock price movements is proportional to the level of the stock price
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where µ is the expected return σ is the volatility.The discrete time equivalent is
A Simple Model for Stock Prices(See pages 269-71, Hull)
dzSdtSdS σ+µ=
tStSS ∆εσ+∆µ=∆
Itô’s process
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• Consider a stock that pays no-dividends, has an expected return 15% per annum with continuous compounding and a volatility of 30% per annum. Observe today’s price $1 per share and with ∆t = 1/250 yr. Simulate a path for the stock price.
Example: Simulate asset prices
-1
1 1
1
G iven , we com pute
Then
i
i i
i i
S
S S t S tS S Sµ σ ε− −
−
∆ = ∆ + ∆
= + ∆
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Example: Simulate stock prices
-0.02490.01190.02290.00080.0057
-0.0025-0.01030.0143
-0.0253-0.0256
-1.34570.61321.19280.01140.2667
-0.1590-0.56520.7189
-1.3431-1.3913
1.00000.97510.98701.00991.01081.01651.01401.00371.01800.9927
∆S during a time period dt
Random sample for ε
Stock Price Sat start of period
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Example: Simulate stock prices
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Itô’s Lemma (See pages 273-274, Hull)
• If we know the stochastic process followed by x, Itô’s lemma tells us the stochastic process followed by some function G (x, t )
• Since a derivative security is a function of the price of the underlying and time, Itô’s lemma plays an important part in the analysis of derivative securities
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Taylor Series Expansion
A Taylor’s series expansion of G(x, t) gives
…+∆∂∂
+∆∆∂∂
∂+
∆∂∂
+∆∂∂
+∆∂∂
=∆
22
22
22
2
ttGtx
txG
xxGt
tGx
xGG
½
½
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Ignoring Terms of Higher Order Than ∆t and ∆x
t
x
xxGt
tGx
xGG
ttGx
xGG
∆
∆
∆+∆+∆=∆
∆+∆=∆
½
22
2
order of
is whichcomponent a has because
becomes this calculus stochastic In
havewecalculusordinary In
∂∂
∂∂
∂∂
∂∂
∂∂
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Substituting for ∆x
tbxGt
tGx
xGG
ttbtax
dztxbdttxadx
∆ε∂∂
+∆∂∂
+∆∂∂
=∆
∆∆ε∆∆
+=
222
2
½
order thanhigher of termsignoringThen + =
thatso),(),(
Suppose
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The ε2∆t Term
2 2
2
2
2
22
2
Since (0,1), ( ) 0( ) [ ( )] 1( ) 1
It follows that ( )The variance of is proportional to and can be ignored. Hence
12
N EE EE
E t tt t
G G GG x t b tx t x
ε ε
ε εε
ε
∂ ∂ ∂∂ ∂ ∂
≈ =
− =
=
∆ = ∆
∆ ∆
∆ = ∆ + ∆ + ∆
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Taking Limits
Lemma sIto' is This
½ obtain We
ngSubstituti
½ limits Taking
dzbxGdtb
xG
tGa
xGdG
dzbdtadx
dtbxGdt
tGdx
xGdG
∂∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
+∂∂
=
+=∂∂
+∂∂
+∂∂
=
22
2
22
2
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Application of Ito’s Lemmato a Stock Price Process
dzSSGdtS
SG
tGS
SGdG
tSGzdSdtSSd
½
and of function a For
is processprice stockThe
σ∂∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛σ
∂∂
+∂∂
+µ∂∂
=
σ+µ=
222
2
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Example
2
2dG dt dzσµ σ
⎛ ⎞= − +⎜ ⎟
⎝ ⎠
If ln , then ?G S
dG=
=
Generalized Wiener process
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A generalized Wiener process for stock price
( )
( )
( )
2
2
/ 2
The discrete version of this is
ln ( ) ln ( ) / 2
or
( ) ( )
S follows a lognormal distribution. It is often referred to as geometric Brownian motio
t t
S t t S t t t
S t t S t e
t
µ σ σ ε
µ σ σε
− ∆ + ∆
+ ∆ − = − ∆ + ∆
+ ∆ =
n.
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Lognormal Distribution
( )
( )
( )( )( )( ) ( )
( )( )
2
22
0
2
0
22 20
2 20
The corresponding density function for ( ) is
log2
exp2
2
for 0 w ith 0 for 0 .It can be proven that
var
t
t
t
S t
x tS
t
f xx t
x f x x
E S t S e
E S t S e
S t S e e
µ
µ σ
µ σ
σµ
σ
σ π
+
⎛ ⎞⎛ ⎞⎛ ⎞⎛ ⎞⎜ ⎟− − −⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠⎜ ⎟⎝ ⎠⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠=
> = ≤
=
=
= ( )2
1t −
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Example: Simulate stock prices
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Models for stock price in risk neutral world (r: risk-free interest rate)
In a risk neutral world, two process for the underlying asset is:
( ) ( ) ( ) ( )
ˆ (1)
ˆ
dS S dz Sdtor
S t t S t S t t S t t
σ µ
µ σ ε
= +
+ ∆ − = ∆ + ∆
( )
( )
2
2
ln /2
or (2)
ln ( ) ln ( ) / 2
d S r dt dz
S t t S t r t t
σ σ
σ σε
= − +
+ ∆ − = − ∆ + ∆
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Which one is more accurate?
• It is usually more accurate to work with InSbecause it follows a generalized Wiener process and (2) is true for any ∆t while (1) is true only in the limit as ∆t tends to zero.
( )
( )2
2
/ 2 T
For all T,
ln ( ) ln (0) / 2 T
i.e.,
( ) (0) r T
S T S r T
S T S e σ σε
σ σε
− +
− = − +
=
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3.3 Generate pseudo-random variates
3. Monte Carlo Simulation
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Generate pseudorandom variates
The usual way to generate psedorandom variates, i.e., samples from a given probability distribution:
1. Generate psedurandom numbers, which are variatesfrom the uniform distribution on the interval (0, 1). Denote as U(0, 1)
2. Apply suitable transformation to obtain the desired distribution
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Generate U(0, 1) variables
The standard method is linear congruential generators (LCGs):
( )( )
periodslonger much allows 0) ,state'rand(':function MATLAB Improved
of period maximum a has Usequency The .Periodic
ones real of instead numbers rational generates LCGs each time sequence same thegenerates 0) ,seed'rand(' eg.
sequence. random theof seed thenumber, initial the: Z:Comments
variable1) U(0,a generate which number Return 2.
parameterschosen are and , , wheremod
number Zinteger an Given .1
1-m
0ii
0
1
1,-i
−⎭⎬⎫
⎩⎨⎧ =−
−
−
+=
=
−
mmi
mZmca
mcaZZ
i
ii
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Generate U(0, 1) variables
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Generate U(0, 1) variables
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Generate general distribution F(x) from U(0, 1)
( ) { }
( )( )
{ } ( ){ } ( ){ } ( )xF xFUPxUFP xXP
U FX ,U~U
xX P xF
-
=≤=≤=≤
=
≤=
−1
1
:ddistributeuniformly is Ufact that theand F ofty monotonici the Use:Proof
Return .210number random a Draw 1.
following by the F toaccording variatesrandomgeneratecan we,function on distributi Given the
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• How to generate samples according to standard normal distribution N(0,1) from U(0, 1) ?
Question
- No analytical form of the inverse of the distribution function- Support is not finite
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Generate Samples from Normal Distribution
One simple way to obtain a sample from Ν(0,1) is as follows
( )( )
purposesmost for ry satisfacto ision approximat The-Theorem.Limit Central from isIt -
:Note
0,1N from samples required theis x and,0,1 Ufrom numbers randomt independen are Uwhere
6
i
12
1−= ∑
=iiUx
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Central Limit Theorem
Let X1, X2, X3, ... be a sequence of independent random variables which has the same probability distribution.
Consider the sum :Sn = X1 + ... + Xn. Then the expected value of Snis nμ and its standard deviation is σ n½. Furthermore, informally speaking, the distribution of Sn approaches the normal distribution N(nμ,σn1/2) as n approaches ∞.
Or the standardized Sn, i.e.,
Then the distribution of Zn converges towards the standard normal distribution N(0,1) as n approaches ∞
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A Note
( ) ( )
[ ]
( ) ( )2
Uniform random variable X: PDF:
1 if
0 otherwiseMean:
E X2
Variance:
Var12
, x a,bf x b a
,
b a
b aX
⎧ ∈⎪= −⎨⎪⎩
+=
−=
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Generate Samples from N(0,1): Box-Muller algorithm
( )1 2
21 2
The Box-Muller algorithm can be implemented as follows:1. Generate two independent uniform samples from 0 1
2 Set 2 and =23 Set and where X and Y are indepen
U ,U ~ U ,
. R logU U
. X R cos Y R sinθ π
θ θ= −= =
( )dent samples from 0 1
Note:- Box and Muller (1958), Ann. Math. Statist., 29:610-611- Common and more efficient than the previous method.- But it is costly to evaluate trigonometric functions
N , .
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Generate Samples from N(0,1): Improved Box-Muller algorithm
( )1 2
2 21 1 2 2 1 2
The polar rejection (improved Box-Muller) method:1. Generate two independent uniform samples from 0 1
2 Set 2 -1, 2 -1, and 3 If S>1, return to step 1; otherwise, return
U ,U ~ U ,
. V U V U S V V
.= = = +
( )
1 22 -2lnS and V
Swhere X and Y are required independent samples from 0 1
Note:- Ahrens and Dieter (1988), Comm. ACM 31:1330-1337
- ln SX V YS
N , .
= =
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How to generate samples according to normal distribution N(µ, σ) from N(0, 1)?
Question
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To Obtain 2 Correlated Normal Samples
( )1 2
1 2
1 1 1
22 2 1 2
To obtain correlated normal samples and , we first 1. generate two independent stardard normal variates x , x ~N 0,12. and set
1where is the coefficient of correlation.
x
x x
ε ε
ε µ
ε µ ρ ρρ
= +
= + + −
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To Obtain n Correlated Normal Samples
( )1
1 n
T
To obtain correlated normal samples , , , we first 1. generate independent stardard normal variates ~N 0,12. and set = +
Here, is the decomposition
of the
n
ij
nn x , ,x
Cholesky
ε ε
ρ⎡ ⎤= = ⎣ ⎦T
ε µ U X
U U Σ
……
covariance matrix . Σ
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Simulate the correlated asset prices
( )2 2 i i i iˆ / t ti iS ( t t ) S ( t ) e µ σ σ ε− ∆ + ∆
+ ∆ =
To simulate the payoff of a derivative depending on multi-variables, how can we generate samples of each variables?
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3.4 Choose the number of simulations
3. Monte Carlo Simulation
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How many trials is enough?
( )
[ ]
( ) ( ) 22 11
M
i=1
M
i=1
: Sequence of independent samples from the same distribution1 : Sample mean, an unbiased estimator to the M
quantitiy of interest
: Sa
µ
≡
=
⎡ ⎤= −⎣ ⎦−
∑
∑
i
i
i
i
X
X M X
E X
S M X X MM
( )( ) ( )2
2
2
mple variance
E Var : the expected value of square error
as a way to quantify the quality of our estimatorNote that can be estimated by the sample vari
σµ
σ
⎡ ⎤ ⎡ ⎤− = =⎣ ⎦⎢ ⎥⎣ ⎦X M X M
M
( )2ance S M
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Number of Simulations and Accuracy
The number of simulation trials carried out depends on the accuracy required.M: number of independent trials carried outµ: expected value of the derivative,S(M): standard deviation of the discounted payoff
given by the trials.
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Standard Errors in Monte Carlo Simulation
The standard error of the estimate of the option price based on sample size M is:
Clearly as M increases, our estimate is more accurate
( )2S MMM
σ≈
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(1-α) Confidence Interval of Monte Carlo Simulation
In general, the confident interval at level (1- α) may be computed by
Note: Uncertainty is inverse proportional to the square root of the number of trials
( ) ( ) ( ) ( )2 2
1 2 1 2
1 2
where is the critical number from the standard normal distribution.
/ /
/
S M S MX M z X M z
M Mz
α α
α
µ− −
−
− < < +
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95% Confidence Interval of Monte Carlo Simulation
A 95% confidence interval for the price V of the derivative is therefore given by:
( ) ( ) ( ) ( )2 2
1.96 1.96S M S M
X M X MM M
µ− < < +
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Absolute error
( )( )
( )21 2
Suppose we want to controlling the absolute error such that,with probability 1-
where is the maximum acceptable tolerance.Then the number of simulations M must satisfy
/
,
X M ,
z S M M .α
α
µ β
β
β−
− ≤
≤
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Relative error
( )( )
( )( )
21 2
Suppose we want to controlling the relative error such that,with probability 1-
where is the maximum acceptable tolerance.Then the number of simulations M must satisfy
/
,
X M,
z S M MX M
α
α
µγ
µγ
−
−≤
≤1
.γγ+
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3.5 Applications in Option Pricing
3. Monte Carlo Simulation
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Monte Carlo Simulation
When used to value a derivative dependent on a market variable S, this involves the following steps:
1. Simulate 1 path for the stock price in a risk-neutral world
2. Calculate the payoff from the stock option3. Repeat steps 1 and 2 many times to get many sample
payoff4. Calculate mean payoff5. Discount mean payoff at risk-free rate to get an
estimate of the value of the option
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• Consider a European call on a stock with no-dividend payment. S(0) = $50, E = 52, T = 5 months, and annual risk-free interest rate r = 10% and a volatility σ= 40% per annum.
• Black-Schole Price: $5.1911• MC Price with 1000 simulations: $5.4445
Confidence Interval [4.8776, 6.0115]• MC Price with 200,000 simulations: $5.1780
Confidence Interval [5.1393, 5.2167]
Example: European call
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• A butterfly spread consists of buying a European call option with exercise price K1 and another with exercise price K3 (K1< K3) and by selling two calls with exercise price K2 = (K1 + K3)/2, for the same asset and expiry date
• Consider a butterfly spread with S(0) = $50, K1 = $55, K2= $60, K3 = 65, T = 5 months, and annual risk-free interest rate r = 10% and a volatility σ= 40% per annum.
• Exact price: $0.6124• MC price with 100,000 simulations: $0.6095
Confidence Interval [0.6017, 0.6173]
Example: Butterfly Spread
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• Consider pricing an Asian average rate call option with discrete arithmetic averaging. The option payoff is
• Consider the option on a stock with no-dividend payment. S(0) = $50, K = 50, T = 5 months, and annual risk-free interest rate r = 10% and a volatility σ= 40% per annum.
Example: Arithmetic Average Asian Option
( )1
1 0
where and =
⎧ ⎫−⎨ ⎬⎩ ⎭
= ∆ ∆ =
∑N
ii
i
max S t K ,N
Tt i t t .N
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Determining Greek Letters
For ∆:1. Make a small change to asset price2. Carry out the simulation again using the same random number streams3. Estimate ∆ as the change in the option price divided by the change in
the asset price
Proceed in a similar manner for other Greek letters
*MC MCV V
S−
∆
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3.6 Further Comments
3. Monte Carlo Simulation
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The parameters in the model
• Our analysis so far is useless unless we know the parameters µ and σ
• The price of an option or derivative is in general independent of µ
• However, the asset price volatility σ is critically important to option price. Typically, 20%< σ < 50%
• Volatility of stock price can be defined as the standard deviation of the return provided by the stock in one year when the return is expressed using continuous compounding
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Estimate volatility σ from historical data
i
i1
Assume:n 1: number of observation at fixed time interval
Stock price at end of ith (i 0, 1, ..., n) interval: Length of observing time interval in years
n number of approximatioi
i
S :t
Su ln :S −
+=
∆
⎛ ⎞= ⎜ ⎟
⎝ ⎠n for the change
of over ln S t∆
78Math6911, S08, HM ZHU
Estimate volatility σ from historical data
( )
i
2
1
i
1
Compute the standard deviation of the u
11
where the mean of the u is1un
s
The standard error of estimate is about 2n
nii
nii
' s :
s u un
' s
u
sˆtt
ˆ
σ σ σ
σ
=
=
= −−
=
≈ ∆ ∴ ≈ =∆
∑
∑
∵
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Advantages and Limitations
• More efficient for options dependent on multiple underlying stochastic variables. As the number of variables increase, Monte Carlo simulations increases linearly while the others increases exponentially
• Easily deal with path dependent options and options with complex payoffs / complex stochastic processes
• Has an error estimate
• It cannot easily deal with American options• Very time-consuming
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Extensions to multiple variables
When a derivative depends on several underlying variables we can simulate paths for each of them in a risk-neutral world to calculate the values for the derivative
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Sampling Through the Tree
• Instead of sampling from the stochastic process we can sample paths randomly through a binomial or trinomial tree to value a derivative
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3.7 Variance Reduction Techniques
3. Monte Carlo Simulation
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Variance Reduction Procedures
• Usually, a very large value of M is needed to estimate V with reasonable accuracy.
• Variance reduction techniques lead to dramatic savings in computational time
• Antithetic variable technique• Control variate technique• Importance sampling• Stratified sampling• Moment matching• Using quasi-random sequences
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Antithetic Variable Technique
( )( ) ( )
( ) ( )
( ) ( )
1
0 1
1 0 1
12
M
MC i ii
i iAV
V E f U U N , .
V f U U N , .M
f U f UV
M
=
=
=
−=
∑
∼
∼
Consider to estimate where
The standard Monte Carlo estimate:
with i.i.d.
The antithetic variate technique:+
with i ( )
( ) ( ) ( )( ) ( ) ( )( )
( )( )
10 1
12 2
12
M
ii
i ii i i
i
U N ,
f U f Uvar var f U cov f U , f U
var f U
f
=
⎛ ⎞−⎡ ⎤= + −⎜ ⎟ ⎣ ⎦
⎝ ⎠
≤
∑ ∼.i.d.
We can prove that
+
if is monotonic.
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Antithetic Variable Technique
It involves calculating two values of the derivative.V1: calculated in the usual wayV2: calculated by the changing the sign of all the random standard normal samples used for V1
V is average of the V1 and V2 , the final estimate of the value of the derivative
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Antithetic Variable Technique
1Note:
Monte-Carlo works when the simulated variables "spread out" as closely as possible to the true distribution. 2. Antithetic variates relies upon finding samplings that are anticorrelated
.
with the original random variable.3. Works well when the payoff is monotonic w.r.t. 4. Further reading: --P.P. Boyle, Option: a Monte Carlo approach, J. of Finanical Economics, 4:323-338 (1977)
S
--Boyle, Broadie and Glasserman, Monte Carlo methods for security pricing J. of Economic Dynamics and Control, 21: 1267-1321 (1997) --N. Madras, Lectures on Monte Carlo Methods, 2002
87Math6911, S08, HM ZHU
• Consider a European call on a stock with no-dividend payment. S(0) = $50, K = 52, T = 5 months, and annual risk-free interest rate r = 10% and a volatility σ= 40% per annum.
• Black-Schole Price: $5.1911• MC Price with 200,000 simulations: $5.1780
Confidence Interval [5.1393, 5.2167]• MCAV Price with 200,000 simulations: $5.1837
Confidence Interval [5.1615, 5.2058] (ratio: 1.75)
Example: European call
88Math6911, S08, HM ZHU
• Consider a butterfly spread with S(0) = $50, K1 = $55, K2= $60, K3 = 65, T = 5 months, and annual risk-free interest rate r = 10% and a volatility σ= 40% per annum.
• Exact price: $0.6124• MC price with 100,000 simulations: $0.6095
Confidence Interval [0.6017, 0.6173]• MCAV price with 50,000 simulations: $0.6090
Confidence Interval [0.5982, 0.6198]
Example: Butterfly Spread
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Control Variate Technique
( )( )( )
( ) ( )
.
var var
We can generalize the previous technique to
for any Here, is "close" to with known mean .In this case,
A B B
B A B
A B
Z V E V V
V V E V
Z V V
θ
θ
θ
θ
θ
= + −
∈
= −
( ) ( ) ( )( ) ( )( )
( )
2var 2 cov , var
var var
cov ,0 2
var
We can prove that if and only if
.
A A B B
A
A B
B
V V V V
Z V
V VV
θ
θ θ
θ
= − +
<
< <
90Math6911, S08, HM ZHU
• Consider a European call on a stock with no-dividend payment. S(0) = $50, K = 52, T = 5 months, and annual risk-free interest rate r = 10% and a volatility σ= 40% per annum.
• Black-Schole Price: $5.1911• MC Price with 200,000 simulations: $5.1780
Confidence Interval [5.1393, 5.2167]• MCCV Price with 200,000 simulations: $5.1883
Confidence Interval [5.1712, 5.2054] (ratio: 2.2645)
Example: European call
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• Consider the option on a stock with no-dividend payment. S(0) = $50, K = 50, T = 5 months, and annual risk-free interest rate r = 10% and a volatility σ= 40% per annum.
• We could use the sum of the asset prices as a control variate as we know its expected value and Y is correlated to the option itself
• Another choice of the control variate is the payoff of a geometric average option as this is known analytically
Example: Arithmetic Average Asian Option
( ) ( )( )1
0 00 0 0
11
r N tN N Nri t
i r ti i i
eE S t E S i t S e Se
+ ∆∆
∆= = =
−⎡ ⎤ ⎡ ⎤= ∆ = =⎢ ⎥ ⎣ ⎦ −⎣ ⎦∑ ∑ ∑
( )1
1
max , 0N N
ii
S t K=
⎧ ⎫⎛ ⎞⎪ ⎪−⎨ ⎬⎜ ⎟⎝ ⎠⎪ ⎪⎩ ⎭∏
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Importance Sampling Technique
It is a way to distort the probability measure in order to sample from critical region. For example, in evaluating European call option:
: the unconditional probability distribution function for the asset price at time T: the probability of the asset price K at maturity, known analytically : the probability distribution
F
qG F / q
≥= of the asset conditional
on the asset price K, i.e., importance functionInstead of sampling from F, we sample from G. Then the value of the option is the average discounted payoff
≥
multiplied by . q
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Stratified Sampling Technique
Sampling representative values rather than random values from a probability distribution is usually more accurate.
[ ]
{ }1
It involves dividing the distribution into intervals andsampling from each interval (stratum) according to the distribution:
E X
where P for j 1, ,m, is known.
Note: 1. For eac
m
j jj
j j
p E X Y y
Y y p=
⎡ ⎤= =⎣ ⎦
= = =
∑
h stratum, we sample conditioned on the even
2. The representative values are typically mean or median for each interval jX Y y=
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Moment Matching
Moment matching involves adjusting the samples taken from a standard normal distribution so that the first, second, or possibly higher moments are matched.
i
ii
For example, we want to sample from N(0,1). Suppose that the samples are (1 i n). To match the first two moments, we adjustify the samples by
where m and s are the mean and standard deviati
* ms
ε
εε
≤ ≤
−=
i
on of samples. The adjusted samples has the correct mean 0 and standard deviation 1.
We then use the adjusted samples for calculation.
*ε
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Quasi-Random Sequences
• Also called a low-discrepancy sequence is a sequence of representative samples from a probability distribution
• At each stage of the simulation, the sampled points are roughly evenly distributed throughout the probability space
• The standard error of the estimate is proportional to 1M
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Further reading
• R. Brotherton-Ratcliffe, “Monte Carlo Motoring”, Risk, 1994:53-58.
• W.H.Press, et al, “Numerical Reciptes in C”, Cambridge Univ. Press, 1992
• I. M. Sobol, “USSR Computational Mathematics and Mathematical Physics”, 7, 4, (1967): 86-112;P. Jaeckel, “Monte Carlo Methods in finance”, Wiley, Chichester, 2002
• P. Glasserman, “Monte Carlo Methods in Financial Engineering”, Springer-Verlag, New York, 2004
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Further reading
• www.mcqmc.org• www.mat.sbg.ac.at/~schmidw/links.html
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Appendix
3. Monte Carlo Simulation
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Asian Options (Chapter 22)
• Payoff related to average stock price• Average Price options pay:
– Call: max(Save – K, 0)– Put: max(K – Save , 0)
• Average Strike options pay:– Call: max(ST – Save , 0)– Put: max(Save – ST , 0)
100Math6911, S08, HM ZHU
Asian Options (Chapter 22)
• For arithmetic averaging:
• For geometric averaging:
( )1
1 where N
ave ii
S S t , T N tN =
= = ∆∑
( ) ( ) ( )1 2N
ave NS S t S t S t=
101Math6911, S08, HM ZHU
Asian Options (geometric averaging)
• Closed form (Kemna & Vorst,1990, J. of Banking and Finance, 14:113-129) because the geometric average of the underlying prices follows a lognormal distribution as well.
where N(x) is the cumulative normal distribution function and
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Asian Options (arithmetic averaging)
• No analytic solution• Can be valued by assuming (as an
approximation) that the average stock price is lognormally distributed
• Turnbull and Wakeman, 1991, J. of Financial and Quantitative Finance, 26:377-389
• Levy,1992, J. of International Money and Finance, 14:474-491
103Math6911, S08, HM ZHU
• Consider an Asian Put on a stock with non-dividend payment. S(0) = $50, X = 50, T = 5 months, and annual risk-free interest rate r = 10% and a volatility σ= 40% per annum.
• MC Price with 100,000 simulations: $3.9806 Confidence Interval [3.9195, 4.0227]
• MCAV Price with 100,000 simulations: $3.9581Confidence Interval [3.9854, 3.9309]
Example: Asian Put with arithmetricaverage
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