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2 Relationship between derivative values and simulations: Equities, indices, currencies, commodities

# 2 Relationship between derivative values and simulations: Equities, indices, currencies, commodities

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1264

Part Six numerical methods and programs

The risk-neutral random walk for S is

dS = rS dt + σ S dX.

We can therefore write

option value = e−r(T −t) E payoff(S)

provided that the expectation is with respect to the risk-neutral random walk, not the real one.

This result leads to an estimate for the value of an option by following these simple steps:

1.

Simulate the risk-neutral random walk as discussed below, starting at today’s value of the

asset S0 , 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.

80.3 GENERATING PATHS

The initial part of this algorithm requires ﬁrst of all the generation of random numbers

from a standardized Normal distribution (or some suitable approximation). We discuss

this issue below, but for the moment assume that we can generate such a series in

sufﬁcient quantities. Then one has to update the asset price at each time step using

these random increments. Here we have a choice how to update S.

An obvious choice is to use

δS = rS δt + σ S δt φ,

where φ is drawn from a standardized Normal distribution. This discrete way of simulating

the time series for S is called the Euler method. Simply put the latest value for S into the

right-hand side to calculate δS and hence the next value for S. This method is easy to apply to

any stochastic differential equation. This discretization method has an error of O(δt).1

The above algorithm is illustrated in Figure 80.1. The stock begins at time t = 0 with a value

of 100 and a volatility of 20%. The spreadsheet simultaneously calculates the values of a call

and a put option. They both have an expiry of one year and a strike of 105. The interest rate is

5%. In this spreadsheet we see a small selection of a large number of Monte Carlo simulations

of the random walk for S, using a drift rate of 5%. The time step was chosen to be 0.01. For

each realization the ﬁnal stock price is shown in row 102 (rows 13–93 have been hidden). The

option payoffs are shown in rows 104 and 107. The mean of all these payoffs, over all the

simulations, is shown in rows 105 and 108. In rows 106 and 109 we see the present values

1

There are better approximations, for example the Milstein method which has an error of O(δt 2 ).

Monte Carlo simulation Chapter 80

A

B

C

D

E

F

G

H

I

1

Asset

100

Time

Sim 1

Sim 2

Sim 3 Sim 4

Sim 5

2

Drift

5%

0

100.00

100.00

100.00

100.00

100.00

3 Volatility

20%

0.01

100.15

101.27

100.79

100.16

98.54

0.01

0.02

99.80

100.84

102.72

102.31

101.66

4 Timestep

5

Int. rate

5%

0.03

97.35

103.77

105.27

102.00

105.64

0.04

96.50

103.08

104.72

97.65

105.35

6

7

0.05

101.25

101.61

102.37

102.76

103.63

= D3+\$B\$4

0.06

97.53

100.49

104.47

106.86

99.04

8

9

0 07.

97.41

103.09

107.70

105.73

99.20

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

10

0.08

91.83

102.45

109.22

105.15

98.77

11

0.09

85.74

100.79

109.07

106.01

97.95

12

0.1

81.32

100.99

105.13

105.40

100.32

94

0.92

102.25

105.44

88.51

96.74

96.08

95

0.93

100.68

105.48

90.44

97.04

95.36

96

0.94

102.26

104.01

92.40

99.26

94.67

97

0.95

102.10

103.47

88.99

95.27

97.09

98

0.96

100.11

103.36

88.95

92.74

96.30

99

0.97

101.34

104.06

89.26

93.59

97.19

100

0.98

101.05

104.23

89.29

92.69

95.29

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

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

101

103.71

102.73

90.54

94.43

94.58

=AVERAGE(E104:IV104) 0.99

102

1

104.94

104.47

91.86

95.05

98.79

103

104

Strike

105 CALL

Payoff

0.00

0.00

0.00

0.00

0.00

105 =D105*EXP(Mean

8.43

106 \$B\$5*\$D\$102)

8.02

PV

107

PUTP

ayoff

0.06

0.53

13.14

9.95

6.21

108

Mean

8.31

109

PV

7.9

110

111

Figure 80.1 Spreadsheet showing a Monte Carlo simulation to value a call and a put option.

of the means; these are the option values. For serious option valuation you would not do such

calculations on a spreadsheet. For the present example I took a relatively small number of

sample paths.

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

Figure 80.2 I show how to value an Asian option. This contract pays an amount max(A −

105, 0) where A is the average of the asset price over the one-year life of the contract. The

remaining details of the underlying are as in the previous example. How would the spreadsheet

be modiﬁed if the average were only taken of the last six months of the contract’s life?

1265

1266

Part Six numerical methods and programs

i

A

B

C

D

E

F

G

H

I

1

Asset

100

Time

Sim 1

Sim 2

Sim 3

Sim 4

Sim 5

2

Drift

5%

0

100.00

100.00

100.00

100.00

100.00

3 Volatility

20%

0.01

98.62

97.68

99.73

99.42

102.98

4 Timestep

0.01

0.02

100.69

96.45

101.13

101.28

101.36

5

Int. rate

5%

0.03

99.60

99.67

102.62

99.37

101.95

6

0.04

99.19

101.15

104.14

98.60

99.51

= D3+\$B\$4

7

0.05

104.10

100.00

105.03

98.97

96.86

8

0.06

104.71

99.11

103.22

96.93

98.89

9

0.07

107.07

95.99

101.60

96.02

96.83

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

10

0.08

108.36

98.96

102.95

97.77

98.23

11

0.09

110.57

100.93

101.59

99.34

98.75

12

0.1

114.50

100.24

99.36

95.67

99.88

13

0.11

114.43

101.32

100.22

94.92

102.90

93

0.91

101.02

111.09

119.38

77.82

85.23

94

0.92

101.54

109.58

118.06

80.13

83.75

95

0.93

101.38

108.20

118.49

79.96

83.54

96

0.94

103.38

107.87

119.79

82.21

83.12

97

0.95

107.58

108.43

116.24

81.58

84.69

98

0.96

108.93

109.39

115.79

81.61

88.50

=AVERAGE(E2:E102)

99

0.97

107.20

112.81

115.56

83.07

90.72

100

0.98

109.18

113.45

116.23

83.59

87.23

= MAX(G104-\$B\$106,0)

101

0.99

110.49

114.04

116.92

85.03

90.41

=AVERAGE(E106:IV106)

102

1

113.23

117.67

120.05

81.49

93.27

103

104

Average

105.98

106.95

109.21

87.43

97.22

105

106

Strike

105 ASIAN

Payoff

0.98

1.95

4.21

0.00

0.00

107 =D107*EXP(Mean

4.79

108 \$B\$5*\$D\$102)

PV

4.55

109

110

Figure 80.2 Spreadsheet showing a Monte Carlo simulation to value an Asian option.

80.4 LOGNORMAL UNDERLYING, NO PATH DEPENDENCY

For the lognormal random walk we are lucky that we can ﬁnd a simple, and exact, time stepping

algorithm. We can write the risk-neutral stochastic differential equation for S in the form

d(log S) = r − 12 σ 2 dt + σ dX.

This can be integrated exactly to give

S(t) = S(0) exp

r − 12 σ 2 t + σ

t

dX .

0

Or, over a time step δt,

S(t + δt) = S(t) + δS = S(t) exp

r − 12 σ 2 δt + σ δt φ .

(80.1)

Note that δt need not be small, since the expression is exact; and because it is exact and simple

it is the best time-stepping algorithm to use. Also, because it is exact, if we have a payoff

Monte Carlo simulation Chapter 80

that only depends on the ﬁnal asset value, i.e. is European and path independent, then we can

simulate the ﬁnal asset price in one giant leap, using a time step of T .

If the option is path-dependent then we have to go back to smaller time increments generally.

80.5

SIMULATION

Now that we have some idea of how Monte Carlo simulations

are related to the pricing of options, I’ll give you some of the

beneﬁts of using such simulations:

• The mathematics that you need to perform a Monte Carlo

simulation can be very basic.

• Correlations can be easily modeled.

• There is plenty of software available, at the least there are spreadsheet functions that will

sufﬁce for most of the time.

• To get a better accuracy, just run more simulations.

• The effort in getting some answer is very low.

• The models can often be changed without much work.

• Complex path dependency can often be easily incorporated.

80.6

USING RANDOM NUMBERS

The Black–Scholes theory as we have seen it has been built on the assumption of either a

simple up-or-down move in the asset price, the binomial model, or a Normally distributed

return. When it comes to simulating a random walk for the asset price it doesn’t matter very

much what distribution we use for the random increments as long as the time step is small and

thus we have a large number of steps from the start to the ﬁnish of the asset price path. All we

need are that the variance of the distribution must be ﬁnite and constant. (The constant must

be such that the annualized volatility, i.e. the annualized standard deviation of returns, is the

correct value. In particular, this means that it must scale with δt 1/2 .) In the limit as the size of

the time step goes to zero the simulations have the same probabilistic properties over a ﬁnite

timescale regardless of the nature of the distribution over the inﬁnitesimal timescale. This is a

result of the central limit theorem.

Nevertheless, the most accurate model is the lognormal model with Normal returns. Since one

has to worry about simulating sufﬁcient paths to get an accurate option price one would ideally

like not to have to worry about the size of the time step too much. As I said above, it is best to

use the exact expression (80.1) and then the choice of time step does not affect the accuracy of

the random walk. In some cases we can take a single time step since the time stepping algorithm

is exact. If we do use such a large time step then we must generate Normally distributed random

variables. I will discuss this below, where I describe the Box–Muller method.

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