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2. Monte Carlo Methods
Dr P. V. Johnson
Departmentof Mathematics
2020
Dr P. V. Johnson MATH60082
What have we done so far
Introductory lecture discussed errors and how they mightarise.
C++ support classes introduced basic programming.
Today, we discuss Monte Carlo Methods
In the Lab Class this Thursday, “mini task 1” will bedistibuted by email in class
Handin will be Sunday 5pm.
Please check your email over next few days as I check emailthe distribution system.
Dr P. V. Johnson MATH60082
What have we done so far
Introductory lecture discussed errors and how they mightarise.
C++ support classes introduced basic programming.
Today, we discuss Monte Carlo Methods
In the Lab Class this Thursday, “mini task 1” will bedistibuted by email in class
Handin will be Sunday 5pm.
Please check your email over next few days as I check emailthe distribution system.
Dr P. V. Johnson MATH60082
What have we done so far
Introductory lecture discussed errors and how they mightarise.
C++ support classes introduced basic programming.
Today, we discuss Monte Carlo Methods
In the Lab Class this Thursday, “mini task 1” will bedistibuted by email in class
Handin will be Sunday 5pm.
Please check your email over next few days as I check emailthe distribution system.
Dr P. V. Johnson MATH60082
The Random Walk
Many financial models (including the Black-Scholesframework) assume that the underlying asset follows arandom walk
Using random numbers, we can simulate that random walk,
and find the path that the stock price will follow.
The Monte Carlo method takes numerous simulated pathsto estimate the expected option price at expiry,
which can be discounted back to today.
Dr P. V. Johnson MATH60082
The Random Walk
Many financial models (including the Black-Scholesframework) assume that the underlying asset follows arandom walk
Using random numbers, we can simulate that random walk,
and find the path that the stock price will follow.
The Monte Carlo method takes numerous simulated pathsto estimate the expected option price at expiry,
which can be discounted back to today.
Dr P. V. Johnson MATH60082
Quick Reminder:- Option Pricing
Example 2.1:
Plot some example stock price paths on a graph.
Dr P. V. Johnson MATH60082
Monte Carlo techniques
Good for options on more than one underlying asset.
Convergence is slow, errors are random.
Convergence is order N−12 where N is the number of
sample paths.
It is a forward induction technique, which makes valuingpath dependent options easy.
However, it is very difficult to value American style optionsfor exactly the same reason.
Dr P. V. Johnson MATH60082
Monte Carlo techniques
Good for options on more than one underlying asset.
Convergence is slow, errors are random.
Convergence is order N−12 where N is the number of
sample paths.
It is a forward induction technique, which makes valuingpath dependent options easy.
However, it is very difficult to value American style optionsfor exactly the same reason.
Dr P. V. Johnson MATH60082
Note:- The law of large numbers
Large Numbers
Given a sequence of independent, identically distributedrandom variables Y i then
limN→∞
1
N
N∑n=1
Y i = E[Y ]
So the expectation is exactly like taking a long run average.
To decrease errors increase N .
We wish to evaluate E[YT ] = E[V (ST )].
Dr P. V. Johnson MATH60082
Note:- The law of large numbers
Large Numbers
Given a sequence of independent, identically distributedrandom variables Y i then
limN→∞
1
N
N∑n=1
Y i = E[Y ]
So the expectation is exactly like taking a long run average.
To decrease errors increase N .
We wish to evaluate E[YT ] = E[V (ST )].
Dr P. V. Johnson MATH60082
Application to options
Let St be the share price at time t, then the option valueV (St, t) is
V (St, t) = EQt [e−
∫ Tt r(s)dsV (ST , T )]
or if r is constant,
e−r(T−t)EQt [V (ST , T )]
where Q is the risk-neutral measure, Et denotes taking theexpectation at time t, and V (ST , T ) is the payoff at expiry.
Option value is discounted expected price.
Dr P. V. Johnson MATH60082
Application to options
Let St be the share price at time t, then the option valueV (St, t) is
V (St, t) = EQt [e−
∫ Tt r(s)dsV (ST , T )]
or if r is constant,
e−r(T−t)EQt [V (ST , T )]
where Q is the risk-neutral measure, Et denotes taking theexpectation at time t, and V (ST , T ) is the payoff at expiry.
Option value is discounted expected price.
Dr P. V. Johnson MATH60082
Example - GBM
Example 2.2
Outline how to generate random paths of a GeometricBrownian Motion.
Example 2.3
What is the error if we calculate an option value with a singlepath?
Dr P. V. Johnson MATH60082
Example - GBM
Example 2.2
Outline how to generate random paths of a GeometricBrownian Motion.
Example 2.3
What is the error if we calculate an option value with a singlepath?
Dr P. V. Johnson MATH60082
Errors
Central Limit Theorem
If V (SiT ) is a sequence of independent and identically
distributed random variables with mean EQt [V (ST )] and
variance η2, then we can say that
√n
(1
n
n∑i=1
V (SiT )− EQ
t [V (ST )]
)d−→ N(0, η2).
Example 2.4
What does this mean we can say about our calculation errors?
Dr P. V. Johnson MATH60082
Errors
Central Limit Theorem
If V (SiT ) is a sequence of independent and identically
distributed random variables with mean EQt [V (ST )] and
variance η2, then we can say that
√n
(1
n
n∑i=1
V (SiT )− EQ
t [V (ST )]
)d−→ N(0, η2).
Example 2.4
What does this mean we can say about our calculation errors?
Dr P. V. Johnson MATH60082
European Call Option
For a European call option the payoff at maturity V (ST ) isgiven by
V (ST ) = max(ST −X, 0)
Simulate n possible paths for ST with n independent drawsfrom N(0, 1) (φi)
Then
for: 1 ≤ i ≤ n
SiT = S0 exp[(r − 1
2σ2)T + σφi
√T ]
V (SiT ) = max(Si
T −X, 0)
V (S0, t = 0) = e−rT1
n
n∑i=1
V (SiT )
Dr P. V. Johnson MATH60082
European Call Option
For a European call option the payoff at maturity V (ST ) isgiven by
V (ST ) = max(ST −X, 0)
Simulate n possible paths for ST with n independent drawsfrom N(0, 1) (φi)
Then
for: 1 ≤ i ≤ n
SiT = S0 exp[(r − 1
2σ2)T + σφi
√T ]
V (SiT ) = max(Si
T −X, 0)
V (S0, t = 0) = e−rT1
n
n∑i=1
V (SiT )
Dr P. V. Johnson MATH60082
Psuedo code
Example 2.5
Write down a simple Monte Carlo option algorithm in pseudocode.
Dr P. V. Johnson MATH60082
Path Dependent Options
A path dependent option is one whose payoff depends onthe path followed by the underlying asset
The terminal condition may depend on the path (Asian,lookback, etc.)
or there may exist a condition at time t (barrier options).
Assume the share price is observed at K + 1 points in time
S(t0), S(t1), . . . , S(tK)
If the terminal boundary condition depends on the paththe payoff becomes
V (S(t0), S(t1), . . . , S(tK), T )
Dr P. V. Johnson MATH60082
Path Dependent Options
A path dependent option is one whose payoff depends onthe path followed by the underlying asset
The terminal condition may depend on the path (Asian,lookback, etc.)
or there may exist a condition at time t (barrier options).
Assume the share price is observed at K + 1 points in time
S(t0), S(t1), . . . , S(tK)
If the terminal boundary condition depends on the paththe payoff becomes
V (S(t0), S(t1), . . . , S(tK), T )
Dr P. V. Johnson MATH60082
Creating a Path
Example 2.6
Outline the procedure for valuing a Path Dependent option.
Dr P. V. Johnson MATH60082
Is it random enough?
A computer can never generate a set of random numbers
only one that ‘appears’ random, subject to some statisticaltests.
One of the most important tests is that of zero correlation.
In some circumstances, generating a correlated randomsequence will perform better than one with close to zerocorrelation
Dr P. V. Johnson MATH60082
What is random
Example 2.7
Draw a unit square and place 10 crosses inside, assuming thatthe x and y coordinates are drawn from uniform distributions.What does this look like if a computer does it?
Dr P. V. Johnson MATH60082
Monte-Carlo Methods
Simple to program and to understand
Convergence is slow, extrapolation impossible.
Forward looking method ideal for path dependentderivatives
Good for derivatives where there are multiple sources ofuncertainty, as the computational effort only increaseslinearly.
To simulate the paths we typically use the solution to theSDE or the Euler approximation, along with a decentgenerator of Normally distributed random variables.
Check the notes for how to generate multi-asset paths.
Dr P. V. Johnson MATH60082
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