Montecarlo Simulation and Related Methods Option Values Derivatives

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1

Monte Carlo Simulation and RelatedMethods

1 In this lecture. . .

Gthe relationship between option values and expectations for equi-

ties, currencies, commodities and indices

G

the relationship between derivative products and expectations when

interest rates are stochastic

Ghow to do Monte Carlo simulations to calculate derivative prices

and to see the results of speculating with derivatives

Ghow to do simulations in many dimensions using Cholesky factor-

ization

Ghow to do numerical integration in high dimensions to calculate

the price of options on baskets



2

2 Relationship between derivative values

and simulations

Simulations are at the very heart of finance. With simulations you

can explore the unknown future. Simulations can also be used to

price options.

GThe fair value of an option is the present value of the expected pay-

off at expiry under a risk-neutral random walk for the underlying.

The risk-neutral random walk for 6 is

G 6 U 6 G W 6 G ;

We can therefore write

option value H

> 7 % > 9

> payoff 6 @

provided that the expectation is with respect to the risk-neutral ran-

dom walk, not the real one.



3

The algorithm:

(1) Simulate the risk-neutral random walk starting at today’s value of

the asset 6

over the required time horizon. This time period starts

today and continues until the expiry of the option. This gives one

realization of the underlying price path.

(2) For this realization calculate the option payoff.

(3) Perform many more such realizations over the time horizon.

(4) Calculate the average payoff over all realizations.(5) Take the present value of this average, this is the option value.



4

Price paths are simulated using a discrete version of the stochastic

differential equation for 6 An obvious choice is to use

s 6 U 6 s W 6

5

s W

where is from a standardized Normal distribution.

G This way of simulating the time series is called the Euler method.

This method has an error of 2 s W .



5

For the lognormal random walk we are lucky that we can find a

simple, and exact, timestepping algorithm. We can write the risk-neutral stochastic differential equation for 6 in the form

G O R J 6

?

7 >

P

@

/ 9 G ;

This can be integrated exactly to give

6 W 6 H [ S

M

?

7 >

P

@

9 P

=

9

G ;

N

Or, over a timestep I 9

,

$ 9 I 9 $ 9 I $ $ 9 H [ S

K

?

7 >

P

@

I 9 P

5

s W

L

(1)

G Note that s W need not be small, since the expression is exact. Be-

cause this expression is exact and simple it is the best timesteppingalgorithm to use. Because it is exact, if we have a payoff that only

depends on the final asset value, i.e. is European and path indepen-

dent, then we can simulate the final asset price in one giant leap,

using a timestep of 7 .



Spreadsheet showing a Monte Carlo simulation used to value a call and a put option.

1

234

5

67

89

10

1112

94

95

9697

98

99100

101102103

104105106

107

108109

110

111

A B C D E F G H I

Asset 100 Time Sim 1 Sim 2 Sim 3 Sim 4 Sim 5

Drift 5% 0 100.00 100.00 100.00 100.00 100.00Volatility 20% 0.01 100.30 102.34 97.50 98.38 96.14

Timestep 0.01 0.02 103.52 103.38 99.31 98.55 90.84

Int. rate 5% 0.03 106.01 101.98 101.44 96.12 92.94

0.04 101.85 106.67 103.02 94.69 93.170.05 104.75 107.17 105.05 93.25 91.23

0.06 106.57 110.24 109.55 94.50 89.970.07 101.15 107.73 110.77 93.83 91.08

0.08 102.27 107.77 110.01 95.50 93.21

0.09 101.06 111.06 107.49 93.86 93.230.1 103.35 108.60 109.52 92.84 93.82

0.92 109.27 87.76 118.48 69.38 116.43

0.93 110.22 87.59 118.13 72.19 118.66

0.94 105.56 88.18 117.37 73.40 117.150.95 106.53 86.81 118.27 70.86 115.82

0.96 103.14 85.72 116.85 69.36 115.81

0.97 102.20 85.85 112.89 70.24 119.180.98 101.13 84.85 111.49 71.36 115.75

0.99 101.91 87.83 111.69 69.97 116.111 105.99 87.60 111.45 70.67 116.72

Strike 105 CALL Payoff 0.99 0.00 6.45 0.00 11.72

Mean 8.43PV 8.02

PUT Payoff 0.00 17.40 0.00 34.33 0.00

Mean 8.31PV 7.9

= D3+$B$4

=E3*EXP(($B$5-0.5*$B$3*$B$3)*$B$4+$B$3*SQRT($B$4)*NORMSINV(RAND()))

= MAX(G102-$B$104,0)= MAX($B$104-F102,0)

=AVERAGE(E104:IV104)

=D105*EXP(-$B$5*$D$102)



7

The method is particularly suitable for path-dependent options. In

the next spreadsheet is shown the valuation of an Asian option. Thiscontract pays of an amount P D [ $ > where $ is the average

of the asset price over the one-year life of the contract.



Spreadsheet showing a Monte Carlo valuation of an Asian option.

1

234

5

67

89

10

1112

13

93

94

95

96

9798

99100

101102

103104105

106

107108

109

110

A B C D E F G H I

Asset 100 Time Sim 1 Sim 2 Sim 3 Sim 4 Sim 5

Drift 5% 0 100.00 100.00 100.00 100.00 100.00Volatility 20% 0.01 100.88 100.74 99.07 100.73 100.03

Timestep 0.01 0.02 103.01 98.50 100.73 101.71 98.72

Int. rate 5% 0.03 103.47 97.56 98.73 103.43 99.96

0.04 103.92 98.75 97.64 99.90 102.330.05 106.31 97.94 96.76 100.82 101.43

0.06 105.57 99.37 98.38 100.14 97.930.07 107.02 98.46 97.36 102.30 97.19

0.08 107.05 96.74 99.18 100.86 94.89

0.09 104.59 97.27 98.31 101.32 94.260.1 103.80 95.63 100.89 103.75 92.99

0.11 101.61 97.06 99.58 107.24 94.21

0.91 82.50 95.63 105.93 119.52 97.35

0.92 79.69 95.58 105.05 122.86 97.07

0.93 78.91 93.11 105.41 119.11 98.27

0.94 79.10 92.92 106.84 121.56 100.98

0.95 75.42 92.06 107.37 123.28 102.660.96 74.77 92.58 107.40 121.94 102.34

0.97 75.47 91.01 106.90 122.09 101.440.98 76.47 89.98 105.84 122.50 100.14

0.99 77.47 87.15 104.72 121.35 99.691 76.30 86.68 104.48 122.35 100.30

Average 91.27 86.49 95.40 111.87 98.31

Strike 105 ASIAN Payoff 0.00 0.00 0.00 6.87 0.00

Mean 3.18PV 3.02

= D3+$B$4

=E3*EXP(($B$5-0.5*$B$3*$B$3)*$B$4+$B$3*SQRT($B$4)*NORMSINV(RAND()))

= MAX(G104-$B$104,0)=AVERAGE(E104:IV104)

=D105*EXP(-$B$5*$D$102)

=AVERAGE(E2:E102)



9

3 Advantages of Monte Carlo simulation

GThe mathematics that you need to perform a Monte Carlo simula-

tion can be very basic.

GCorrelations can be easily modeled.

GThere is plenty of software available, at the least there are spread-

sheet functions that will suffice for most of the time.

G

To get a better accuracy, just run more simulations.

GThe effort in getting some answer is very low.

G

The models can often be changed without much work.

GComplex path dependency can often be easily incorporated.

G

People accept the technique, and will believe your answers.



10

4 Using random numbers

If the size of the timestep is s W then we may introduce errors of 2 s W

by virtue of the discrete approximation to continuous events. E.g. if

we have a barrier option we may miss the possibility of the barrier

being triggered between steps.Because we are only simulating a finite number of an infinite num-

ber of possible paths, the error due to using 1 realizations of the asset

price paths is 2 1

>

.



11

The error in the price is therefore

2

M

P D [

M

s W

5

1

N N

The total number of calculations required in the estimation of a

derivative price is then 2 1 s W .To minimize the error, while keeping the total computing time fixed

such that 2 1 s W . , we must choose

1 2 .

and s W 2 .

>



12

5 Generating Normal variables

GA useful distribution that is easy to implement on a spreadsheet,

and is fast, is the following approximation to the Normal distrib-

ution: U

;

L

U

L

>

where the

L

are independent random variables, drawn from a uni-

form distribution over zero to one.

GTo generate genuinely Normally-distributed random numbers use

the Box–Muller method.

This method takes uniformly-distributed variables and turns them

into Normal. The Box–Muller method takes two uniform randomnumbers

[

and [

between zero and one and combines them to give

two numbers \

and \

that are both Normally distributed:

\

5

> O Q [

F R V ~ [

and \

5

> O Q [

V L Q ~ [



13

0

10

20

30

40

50

60

70

80

90

-3.80 -3.09 -2.37 -1.66 -0.95 -0.24 0.47 1.19 1.90 2.61 3.32

1.The approximation to the Normal distribution using Box–Muller.



14

6 Real versus risk neutral, speculation

versus hedging

In the next figure are shown several realizations of a risk-neutral

asset price random walk. These are the thin lines. The bold lines in

this figure are the corresponding real random walks using the same

random numbers but here with a higher drift instead of the interest

rate drift.

0

20

40

60

80

100

120

140

160

180

0 0.2 0.4 0.6 0.8 1

2.Real and risk-neutral simulations.



15

We can use simulations to estimate the payoff distribution from

holding an unhedged option position. In this situation we are inter-ested in the whole distribution of payoffs (and their present values)

and not just the average or expected value. This is because in holding

an unhedged position we cannot guarantee the return that we (theo-

retically) get from a hedged position.

GIt is therefore valid and important to have the real drift as one of

the parameters; it would be incorrect to estimate the probability

density function for the return from an unhedged position using

the risk-neutral drift.



16

7 Interest rate products

The relationship between expected payoffs and option values when

the interest rate is stochastic is more complicated:

(1) Simulate the random walk for the risk-adjusted spot interest rate 7

over the required time horizon. This gives one realization of the

spot rate path.

(2) For this realization calculate two quantities, the payoff and the

average interest rate realized up until the payoff is received.(3) Perform many more such realizations.

(4) For each realization of the 7

random walk calculate the present

value of the payoff for this realization discounting at the average

rate for this realization.

(5) Calculate the average present value of the payoffs over all realiza-

tions, this is the option value.



17

In other words,

option value

K

0

>

#

%

9

7 Q / Q payoff 7

L

Why is this different from the deterministic interest rate case? Why

discount at the average interest rate? We discount all cashflows at

the average rate because this is the interest rate received by a moneymarket account, and in the risk-neutral world all assets have the same

risk-free growth rate. Recall that cash in the bank grows according

to

G 0

G W

U W 0

The solution of which is

0 W 0 7 H

>

#

%

9

7 Q / Q

This contains the same discount factor as in the option value.



18

The choice of discretization of spot rate models is usually limited

to the Euler method

s U X U W > z U W Z U W G W Z U W

5

s W

In the next spreadsheet the Monte Carlo method is used to price

a contract with payoff P D [ U > . Maturity is in one year.The model used to perform the simulations is Vasicek with constant

parameters. The spot interest rate begins at 6%. The option value is

the average present value in the last row.



Spreadsheet showing the Monte Carlo valuation of an interest rate product.

1

234

5

67

89

10

1112

1314

15

9596

97

9899

100101

102103

104105

106107108

109110

111

112113

114

A B C D E F G H

Spot rate 10% Time Sim 1 Sim 2 Sim 3 Sim 4

Mean rate 8% 0 10.00% 10.00% 10.00% 10.00%Reversion rate 0.2 0.01 10.16% 10.12% 10.02% 10.04%

Volatility 0.007 0.02 10.25% 9.95% 10.02% 10.03%

Timestep 0.01 0.03 10.27% 9.92% 9.87% 10.00%

0.04 10.18% 9.83% 9.82% 10.00%0.05 10.24% 9.72% 9.91% 10.09%

0.06 10.12% 9.66% 9.93% 10.08%0.07 10.12% 9.69% 9.90% 10.03%

0.08 10.09% 9.76% 9.99% 10.02%

0.09 10.18% 9.71% 10.02% 10.13%0.1 10.23% 9.70% 10.05% 10.15%

0.11 10.22% 9.68% 10.00% 10.19%0.12 10.09% 9.69% 10.11% 10.16%

0.13 10.05% 9.53% 10.09% 10.21%

0.93 9.45% 9.75% 10.11% 10.30%0.94 9.40% 9.74% 10.11% 10.32%

0.95 9.50% 9.56% 10.19% 10.38%

0.96 9.40% 9.58% 10.24% 10.38%0.97 9.30% 9.58% 10.21% 10.33%

0.98 9.38% 9.67% 10.18% 10.27%0.99 9.38% 9.76% 10.17% 10.31%

1 9.40% 9.69% 10.21% 10.32%

Mean rate 9.44% 9.67% 10.23% 10.25%

Strike 10% Payoff 0.0000 0.0000 0.0021 0.0032PV'd 0.0000 0.0000 0.0019 0.0028Mean 0.000688

=D3+$B$5

=F5+$B$3*($B$2-

F5)*$B$5+$B$4*SQRT($B$5)*(RAND()+RAND()+RAND()+RAND()+RAND()+RAND()+RAND()+RAND()+RAND()+RAND()+RAND()+RAND()-6)

=AVERAGE(E107:IV107)

=AVERAGE(E2:E102)

=MAX(E102-$B$106,0)

=E106*EXP(-$D$102*E104)



20

8 Calculating the greeks

The simplest way to calculate the delta using Monte Carlo simulation

is to estimate the option’s value twice. The delta of the option is

d O L P

K

9 6 K W > 9 6 > K W

K

This is an estimate of the first derivative, with an error of 2 K

.

However, the error in the measurement of the two option values at

6 K and 6 > K can be much larger than this for the Monte Carlo sim-

ulation. These Monte Carlo errors are then magnified when dividedby K , resulting in an error of 2 K

.

GTo overcome this problem, estimate the value of the option at

6 K

and 6 > K using the same values for the random numbers. In this

way the errors in the Monte Carlo simulation will cancel each other

out.



21

9 Higher dimensions: Cholesky

factorization

Monte Carlo simulation is a natural method for the pricing of European-

style contracts that depend on many underlying assets. Supposing

that we have an option paying off some function of 6

, 6

, , 6

G

then we could, in theory, write down a partial differential equation

in / variables. Such a problem would be horrendously time con-

suming to compute. The simulation methods discussed above can

easily be extended to cover such a problem. All we need to do is tosimulate

$

L

9 I 9 $

L

9 H [ S

K

?

7 >

P

L

@

I 9 P

L

5

I 9 S

L

L

The catch is that the

L are correlated, ( >

L

M

@

L M

How can we generate correlated random variables? This is where

Cholesky factorization comes in.



22

Let us suppose that we can generate G uncorrelated Normally dis-

tributed variables t

, J

, . . . , J

/ . We can use these variables to getcorrelated variables with the transformation

S M J (2)

where S

and J

are the column vectors with

L

and t

L

in the L

th rows.The matrix M is special and must satisfy

MM 7

E

with E

being the correlation matrix.



23

It is easy to show that this transformation will work. From (2) we

have S S

%

M J J

% M %

(3)

Taking expectations of each entry in this matrix equation gives

(

A

S S

%

B

M

A

J J

%

B

M %

MM %

E

This decomposition of the correlation matrix is not unique. It re-

sults in a matrix M that is lower triangular.



24

10 Speeding up convergence

Monte Carlo simulation is inefficient, compared with finite-difference

methods, in dimensions less than about three. It is natural, therefore,

to ask how can one speed up the convergence. There are several

methods in common use:

GAntithetic variables

GControl variates



25

10.1 Antithetic variables

In this technique one calculates two estimates for an option value

using the one set of random numbers. We do this by

G

using our Normal random numbers to generate one realization of

the asset price path, an option payoff and its present value

Gtaking the same set of random numbers but changing their signs,

thus replacing

with >

and simulating a realization, and calcu-

lating the option payoff and its present value.

Our estimate for the option value is the average of these two values.

Perform this operation many times to get an accurate estimate for

the option value.

26



26

10.2 Control variate technique

Suppose we have two similar derivatives, the former is the one we

want to value by simulations and the second has a similar (but ‘nicer’)

structure such that we have an explicit formula for its value.

G Use the one set of realizations to value both options.Call the values estimated by the Monte Carlo simulation 9

and '

.

If the accurate value of the second option is '

then a better estimate

than '

for the value of the first option is

'

> '

'

The argument behind this method is that the error in 9

will be the

same as the error in 9

, and the latter is known.

27



27

11 Numerical integration

Often the fair value of an option can be written down analytically

as an integral e.g. non-path-dependent European options contingent

upon G lognormal underlyings, for which we have

9 H

> 7 % > 9

~ 7 > W

> /

Det i

>

P

? ? ? P

/

>

=

? ? ?

=

Payoff $

? ? ? $

/

$

? ? ? $

/

H [ S

M

>

F

%

E

>

F

N

/ $

? ? ? / $

/

where

F

L

P

L

% > 9

M

O R J

M

$

L

$

L

N

M

7 >

L

>

P

L

N

% > 9

N

and E is the correlation matrix for the /

assets and Payoff( ? ? ?

) is the

payoff function.If we do have such a representation of an option’s value then all we

need do to value it is to estimate the value of the multiple integral.

28



28

12 Regular grid

We can do the multiple integration by evaluating the function on a

uniform grid in the /

-dimensional space of assets. There would thus

be

/ grid points in each direction where is the total number of

points used.Supposing we use the trapezium or mid-point rule, the error in the

estimation of the integral will be

> /

and the time taken ap-

proximately since there are function evaluations.

As the dimension / increases this method becomes prohibitivelyslow.

29



29

13 Basic Monte Carlo integration

Suppose that we want to evaluate the integral=

=

I [

[

/

G [

G [

/

over some volume. We can very easily estimate the value of this byMonte Carlo simulation. The idea behind this is that the integral can

be rewritten as=

=

I [

[

/

G [

G [

/

volume of region of integration d average I

where the average of I is taken over the whole of the region of inte-

gration.

30



30

We can sample the average of I by Monte Carlo sampling using

uniformly distributed random numbers in the G -dimensional space.After 1 samples we have

average I {

1

1

;

L

I [

L

(4)

where [

L

is the vector of values of [

[

/

at the L th sampling. As

1 increases, so the approximation improves. Expression (4) is only

an approximation.

31



31

The size of the error can be measured by the standard deviation of

the correct average about the sampled average, this is 7

K

1

> 1

L

where

1

1

;

L

1 [

L

and I

1

1

;

L

1

[

L

The error in the estimation of the value of an integral using a Monte

Carlo simulation is 2 1

>

where is the number of points used,and is independent of the number of dimensions. There are func-

tion evaluations and so the computational time is .

G

The accuracy is much higher than that for a uniform grid if we have

five or more dimensions.

32



32

In financial problems we often have integrals over the range zero to

infinity. The choice of transformation from zero-one to zero-infinityshould be suggested by the problem under consideration. Let us sup-

pose that we have / assets following correlated random walks. The

risk-neutral value of these assets at a time 9 can be written as

$

L

% $

L

9 0

K

7 >

L

>

P

L

L

% > 9 P

L

L

5

7 > W

The random variables

L

are Normally distributed and correlated.

33



33

We can now write the value of our European option as

H

> U % > 9

=

>

? ? ?

=

>

Payoff $

7 6

/

7 S

/

G

G

/

where S

/

is the probability density function for G corre-

lated Normal variables with zero mean and unit standard deviation.

We don’t need to write down S explicitly, we won’t need to know its

functional form

Gas long as we generate numbers from this distribution.

To value the option we must generate suitable Normal variables.

GThe first step is to generate uncorrelated variables and then

G

to transform them into correlated variables.

Both of these steps have been explained above; use Box–Mullerand then Cholesky. The option value is then estimated by the average

of the payoff over all the randomly generated numbers.

34



34

14 Low-discrepancy sequences

With the basic Monte Carlo method we cannot be certain that the

generated points in the /

-dimensional space are ‘nicely’ distributed.

There is inevitably a good deal of clumping.

Suppose that we want to calculate the value of an integral in twodimensions and we use a Monte Carlo simulation to generate a large

number of points in two dimensions at which to sample the integrand.

The choice of points may look something like the following.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0 .2 0 .3 0 .4 0.5 0 .6 0 .7 0 .8 0.9 1

3.A Monte Carlo sample in two dimensions.

35



Now suppose we want to add a few hundred more points to im-

prove the accuracy. Where should we put the new points? If we putthe new points in the gaps between others then we increase the ac-

curacy of the integral. If we put the points close to where there are

already many points then we could make matters worse.

We want a way of choosing points such that

G

they are not too bunched, but nicely spread out

Gwe can add more points later without spoiling our distribution.

Monte Carlo is bad for evenness of distribution, but a uniform griddoes not stay uniform if we add an arbitrary number of extra points.

Low discrepancy sequences or quasi-random sequences have the

properties we require.

36



There are many such sequences with names such as Sobol’, Faure,

Haselgrove and Halton. Halton sequences are by far the easiest todescribe.

The Halton sequence is a sequence of numbers K L E for

L

? ? ?. The integer

Eis the base. The numbers all lie between zero

and one. The numbers are constructed as follows. First choose yourbase. Let us choose 2. Now write the positive integers in ascending

order in base 2, i.e. 1, 10, 11, 100, 101, 110, 111 etc. The Halton se-

quence base 2 is the reflection of the positive integers in the decimal

point i.e.

Integers Integers Halton sequence Halton number

base 10 base 2 base 2 base 10

1 1 @

0.52 10 @

@

0.25

3 11 @

@

0.75

4 100 @

@

@

0.125

? ? ? ? ? ? ? ? ? ? ? ?

37



Generally, the integer Q

can be written as

L

2

;

M

D

M

E

M

in base E , where T D

M

E . The Halton numbers are then given by

K L E

2

;

M

D

M

E

> M >

The resulting sequence is nice because as we add more and more

numbers, more and more ‘dots,’ we fill in the range zero to one atfiner and finer levels.

38



The next figure shows the approximation to the Normal distribu-

tion using 500 points from a Halton sequence and the Box–Mullermethod.

0

10

20

30

40

50

60

70

-2.65 -1.88 -1.11 -0.34 0.43 1.19 1.96 2.73

4.The approximation to Normal using Halton numbers.

39



When distributing numbers in two dimensions choose, for exam-

ple, Halton sequence of bases 2 and 3 so that the integrand is calcu-lated at the points

K L K L for L ? ? ? 1 . The bases in

the two sequences should be prime numbers.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

5.Halton points in two dimensions.

40



The estimate of the G -dimensional integral=

? ? ?

=

1 [

? ? ? [

/

/ [

? ? ? G [

/

is then

1

;

L

1 K L E

? ? ? K L E

Q

where E

M

are distinct prime numbers.

The error in these quasi-random methods is

2 O R J 1

G

1

>

and is even better than Monte Carlo at all dimensions. The time

taken is .

41



The error is clearly sensitive to the number of dimensions / . To

fully appreciate the inverse relationship to can require a lot of points. However, even with fewer points, in practice the method at

its worst has the same error as Monte Carlo.

In the next figure is shown the relative error in the estimate of value

of a five-dimensional contract as a function of the number of pointsused. The inverse relationship with the Halton sequence is obvious.

-6

-5.5

-5

-4.5

-4

-3.5

-3

-2.5

-2

2 2.5 3 3.5 4

MC

Halton

6.Error estimates.

Montecarlo Simulation and Related Methods Option Values Derivatives

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Transcript of Montecarlo Simulation and Related Methods Option Values Derivatives