3. monte carlo simulations - york university · qa 298.f57, 1995. math6911 s08, hm zhu 3.1 overview...

Math6911 S08, HM Zhu

3. Monte Carlo Simulations

2Math6911, S08, HM ZHU

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


3.1 Overview

3. Monte Carlo Simulations

Math6911, S08, HM ZHU

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



• 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


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


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.


3.2 Modeling Asset Price Movement

3. Monte Carlo Simulation


Randomness in stock market


Outlines

1. Markov process 2. Wiener process3. Generalized Wiener process4. Ito’s process, Ito’s lemma5. Asset price models


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”


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


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


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


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.


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


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


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.


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∆ = ∆


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∆


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


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


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∆


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


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?


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+∆=∆ ),(),(


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


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


• 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− −

−

∆ = ∆ + ∆

= + ∆


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


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


Taylor Series Expansion

A Taylor’s series expansion of G(x, t) gives

…+∆∂∂

+∆∆∂∂

∂+

∆∂∂

+∆∂∂

+∆∂∂

=∆

22

22

22

2

ttGtx

txG

xxGt

tGx

xGG

½

½


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

∂∂

∂∂

∂∂

∂∂

∂∂


Substituting for ∆x

tbxGt

tGx

xGG

ttbtax

dztxbdttxadx

∆ε∂∂

+∆∂∂

+∆∂∂

=∆

∆∆ε∆∆

+=

222

2

½

order thanhigher of termsignoringThen + =

thatso),(),(

Suppose


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

∂ ∂ ∂∂ ∂ ∂

≈ =

− =

=

∆ = ∆

∆ ∆

∆ = ∆ + ∆ + ∆


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


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


Example

2

2dG dt dzσµ σ

⎛ ⎞= − +⎜ ⎟

⎝ ⎠

If ln , then ?G S

dG=

=

Generalized Wiener process


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.


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 −


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

= − +

+ ∆ − = − ∆ + ∆


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

− +

− = − +

=


3.3 Generate pseudo-random variates



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


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


Generate U(0, 1) variables


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


• 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


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


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 ∞


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

⎧ ∈⎪= −⎨⎪⎩

+=

−=


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 , .


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 , .

= =


How to generate samples according to normal distribution N(µ, σ) from N(0, 1)?

Question


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

= +

= + + −


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


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?


3.4 Choose the number of simulations



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


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.


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≈


(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− −

−

− < < +


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− < < +


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−

− ≤

≤


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+


3.5 Applications in Option Pricing




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


• 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


• 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


Example: Butterfly Spread


• 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


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−

∆


3.6 Further Comments



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


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∆


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

=

=

= −−

=

≈ ∆ ∴ ≈ =∆

∑

∑

∵


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


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


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


3.7 Variance Reduction Techniques



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


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.



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



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


• 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)



• 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


Example: Butterfly Spread


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

= − +

<

< <


• 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)



• 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=

⎧ ⎫⎛ ⎞⎪ ⎪−⎨ ⎬⎜ ⎟⎝ ⎠⎪ ⎪⎩ ⎭∏


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


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=


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


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


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


Further reading

• www.mcqmc.org• www.mat.sbg.ac.at/~schmidw/links.html


Appendix



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)


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=


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


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


• 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

3. monte carlo simulations - york university · qa 298.f57, 1995. math6911 s08, hm zhu 3.1 overview...

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