Simon Benninga. Financial Modelling 3-rd edition

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628 Chapter 23

To compute the value of the option, we ﬁ rst compute each price path.

There are 16 such paths. The following spreadsheet shows each path, the

average stock price over the path, the option payoff, and the path’s risk-

neutral probability:

ABCDEFGHI J

Initial stock price 30

Up 1.4

Down 0.8

R, 1+interest rate 1.08

Exercise price 50

State prices

0.4321 <-- =(B5-B4)/(B5*(B3-B4))

0.4938 <-- =(B3-B5)/(B5*(B3-B4))

Risk-neutral probabilities

0.4667 <-- =B9*$B$5

0.5333 <-- =B10*$B$5

115.25 <-- =G18*B3

Stock price

82.32

58.80 65.86 <-- =G18*B4

42.00 47.04

30.00 33.60 37.63 <-- =G20*B4

24.00 26.88

19.20 21.50 <-- =G22*B4

15.36

12.29 <-- =G24*B4

1.3605 <-- =G29*$B$5

Bond price

1.2597

1.1664 1.3605 <-- =G31*$B$5

1.0800 1.2597

1.0000 1.1664 1.3605

1.0800 1.2597

1.1664 1.3605

1.2597

1.3605

ASIAN OPTION PICTURE

629 Using Monte Carlo Methods for Option Pricing

The paths are determined by number and sequence of the up and

down movements of the stock. For explanatory purposes we have high-

lighted two price paths:

•

Along the path {up, down, up, up} the terminal stock price is 65.856,

the average price is 43.699, the option payoff is 13.699, and the dis-

counted expected value using the risk-neutral prices is 0.546:

ABCDEFGHIJKLMNOP

Initial stock price 30

Up 1.40

Down 0.80

Interest 1.08

Option exercise price 30

Formula in cell O16: =$B$11^4

State price, Up: q

0.4321 <-- =(B5-B4)/(B5*(B3-B4)) Formula in cell O18: =$B$11^3*$B$12

State price, Down, q

0.4938 <-- =(B3-B5)/(B5*(B3-B4))

Risk-neutral prob., Up 0.4667 <-- =B8*B5

Risk-neutral prob., Down 0.5333 <-- =B9*B5 Formula in cell N16 =MAX(M16-$B$6,0)

Paths

Period

Average

stock price

Option

payoff

Path

risk-neutral

probability

All up (1 path) up up up up 30.00 42.00 58.80 82.32 115.25 65.67 35.67 0.0474

One down (4 paths) down up up up 30.00 24.00 33.60 47.04 65.86 40.10 10.10 0.0542

up down up up 30.00 42.00 33.60 47.04 65.86 43.70 13.70 0.0542

up up down up 30.00 42.00 58.80 47.04 65.86 48.74 18.74 0.0542

up up up down 30.00 42.00 58.80 82.32 65.86 55.80 25.80 0.0542

Two down (6 paths) down down up up 30.00 24.00 19.20 26.88 37.63 27.54 0.00 0.0619

down up down up 30.00 24.00 33.60 26.88 37.63 30.42 0.42 0.0619

down up up down 30.00 24.00 33.60 47.04 37.63 34.45 4.45 0.0619

up down down up 30.00 42.00 33.60 26.88 37.63 34.02 4.02 0.0619

up up down down 30.00 42.00 58.80 47.04 37.63 43.09 13.09 0.0619

up down up down 30.00 42.00 33.60 47.04 37.63 38.05 8.05 0.0619

Three down (4 paths) up down down down 30.00 42.00 33.60 26.88 21.50 30.80 0.80 0.0708

down up down down 30.00 24.00 33.60 26.88 21.50 27.20 0.00 0.0708

down down up down 30.00 24.00 19.20 26.88 21.50 24.32 0.00 0.0708

down down down up 30.00 24.00 19.20 15.36 21.50 22.01 0.00 0.0708

Four down (1 path) down down down down 30.00 24.00 19.20 15.36 12.29 20.17 0.00 0.0809

Option value

5.3756

Formula in cell N37 =SUMPRODUCT(N16:N35,O16:O35)/B5^4

PRICING AN ASIAN OPTION BY PRICING ALL THE PATHS

Formula in cell M16 =AVERAGE(G16:K16)

STOCK PRICE

630 Chapter 23

AB C

Initial stock price 30

Up 1.40

Down 0.80

Interest 1.08

Option exercise price 30

State price, Up: q

0.4321 <-- =(B5-B4)/(B5*(B3-B4))

State price, Down, q

0.4938 <-- =(B3-B5)/(B5*(B3-B4))

Risk-neutral prob., Up 0.4667 <-- =B8*B5

Risk-neutral prob., Down 0.5333 <-- =B9*B5

Date

Price at

beginning of

period

Price movement:

Up or Down

030.000

142.000 Up

2 33.600 Down

347.040 Up

465.856 Up

Average price along

path

43.699 <-- =AVERAGE(B15:B19)

Option payoff at path end 13.699 <-- =MAX(B20-$B$6,0)

Path risk-neutral price 0.0542 <-- =B11^COUNTIF(C16:C19,"Up")*B12^COUNTIF(C16:C19,"Down")

Value of path: Payoff *

Risk-neutral price *

Discount factor

0.546 <-- =B21*B22/B5^4

PATH PRICE EXAMPLE: {Up, Down, Up, Up}

The average stock price along this path is 43.699, so that the option pays

off max[43.699 − 30, 0] = 13.699.

•

Along the path {up, up, down, up} the terminal stock price is the same

as before: 65.856. However, the average price and thus the option payoff

and value are different:

631 Using Monte Carlo Methods for Option Pricing

These two paths illustrate what we mean when we say that an Asian

option price is path dependent: Two paths—both starting at the initial

stock price of 30 and ending at 65.856—have different option payoffs

because the stock-price average along the path is different.

This example, which we’ve not yet concluded, also illustrates the difﬁ -

culty of pricing Asian options: Each single path must be dealt with—16

separate paths. This fact distinguishes Asian options from the case of

ordinary options (“plain vanilla,” in the jargon of option pricers); for the

particular example we’re considering here, a plain-vanilla option requires

dealing only with ﬁ ve ending prices.

To price the Asian option, we attach a risk-neutral probability to each

price path:

AB C

Initial stock price 30

Up 1.40

Down 0.80

Interest 1.08

Option exercise price 30

State price, Up: q

0.4321 <-- =(B5-B4)/(B5*(B3-B4))

State price, Down, q

0.4938 <-- =(B3-B5)/(B5*(B3-B4))

Risk-neutral prob., Up 0.4667 <-- =B8*B5

Risk-neutral prob., Down 0.5333 <-- =B9*B5

Date

Price at

beginning of

period

Price movement:

Up or Down

030.000

142.000 Up

258.800 Up

3 47.040 Down

465.856 Up

Average price along

path

48.739 <-- =AVERAGE(B15:B19)

Option payoff at path end 18.739 <-- =MAX(B20-$B$6,0)

Path risk-neutral price 0.0542 <-- =B11^COUNTIF(C16:C19,"Up")*B12^COUNTIF(C16:C19,"Down")

Value of path: Payoff *

Risk-neutral price *

Discount factor

0.747 <-- =B21*B22/B5^4

PATH PRICE EXAMPLE: {Up, Up, Down, Up}

632 Chapter 23

The option price is the discounted expected payoff value, where the

expectations are computed with the risk-neutral probabilities:

Path

All paths

Option payoff on path∗

∑

5 3756.

MNO P

Average

stock price

Option

payoff

Path

risk-neutral

probability

65.67 35.67 0.0474 <-- =B11^4

40.10 10.10 0.0542 <-- =$B$11^3*$B$12

43.70 13.70 0.0542

48.74 18.74 0.0542

55.80 25.80 0.0542

27.54 0.00 0.0619 <-- =$B$11^2*$B$12^2

30.42 0.42 0.0619

34.45 4.45 0.0619

34.02 4.02 0.0619

43.09 13.09 0.0619

38.05 8.05 0.0619

30.80 0.80 0.0708 <-- =$B$11*$B$12^3

27.20 0.00 0.0708

24.32 0.00 0.0708

22.01 0.00 0.0708

20.17 0.00 0.0809 <-- =$B$12^4

Option value

5.3756

<-- =SUMPRODUCT(N16:N35,O16:O35)/B5^4

Risk-Neutral Probabilities—Again

We repeat our earlier comments (p. 617) about the role of risk-

neutral probabilities: Each path is priced by its discounted risk-

neutral probability, which is a function of the Up, Down, and R.

The actual state probabilities are not relevant.

633 Using Monte Carlo Methods for Option Pricing

23.6 Pricing Asian Options with a VBA Program

Our spreadsheet example in the previous section illustrated both the

principle of Monte Carlo pricing of options and the problematics of

pricing the options directly in a spreadsheet. For four periods, we require

the computation of 2

= 16 paths. For a more general problem with n

periods, there are 2

paths to consider; this number rapidly becomes too

large for even a very powerful computer. In order to price the options

accurately, you would need hundreds or thousands of simulations. Doing

these in a spreadsheet directly would be cumbersome. The obvious

answer is to write some VBA code to automate the process and allow us

to run an arbitrarily large number of simulations.

In this section we write VBA code to do a Monte Carlo simulation of

Asian pricing options. We generate price paths by simulating a sequence

of Up and Down movements of the underlying stock price; the probabil-

ity of Up or Down depends on the risk-neutral probabilities—in this

sense our Monte Carlo simulation for Asian options is similar to that

illustrated in section 23.4 for plain-vanilla options. For each price path

generated, we calculate the option payoff, and after generating a large

number of price paths, we compute the option price by discounting and

averaging these payoffs. The VBA function MCAsian, which prices the

Asian options, follows:

VBA Asian-Option-Pricing Function

Function FCAsian(initial, Exercise, Up, _

Down, Interest, Periods, Runs)

Dim PricePath() As Double

ReDim PricePath (Periods + 1)

‘Risk-neutral probabilities

piup = (Interest - Down) - (Up - Down)

pidown = 1 - piup

Temp = 0

634 Chapter 23

Here’s the implementation of the function in a spreadsheet:

For Index = 1 To Runs

‘Generate path

For i = 1 To Periods

Pricepath(0) = initial

pathprob = 1

If Rnd > pidown Then

PricePath(i) = PricePath(i - 1) * Up

Else:

PricePath(i) = PricePath(i - 1) * Down

End If

Next i

PriceAverage = Application.Sum _

(PricePath) / (Periods + 1)

callpayoff = Application.Max _

(PriceAverage - Exercise, 0)

Temp = Temp + callpayoff

Next Index

MCAsian = (Temp / Interests ^ Periods) _

/ Runs

End Function

CBA

Up 1.4

Down 0.8

Interest 1.08

Initial price 30

Periods 20

Exercise 30

Runs 500

Asian call value 9.7253 <-- =MCAsian(B5,B7,B2,B3,B4,B6,B8)

PRICING AN ASIAN OPTION BY MONTE CARLO

635 Using Monte Carlo Methods for Option Pricing

The function in cell B9 is our Monte Carlo valuation of the option—the

simulated value. Any recalculation of the spreadsheet will cause the

function to rerun and recompute the option value.

In the next spreadsheet we show a block of replications of the function.

Each of the cells A10:F17 contains =MCAsian(Initialprice,exercise,Up,

Down,Interest,Periods,Runs), so that we calculate 48 simulations of the

option value.

ABCDEF G

Up 1.4

Down 0.8

Interest 1.08

Initial price 30

Periods 4

Exercise 30

Runs 100

6.1553 4.2242 6.5765 4.8856 5.0011 5.3158 <-- =MCAsian($B$5,$B$7,$B$2,$B$3,$B$4,$B$6,$B$8)

5.6501 6.0547 5.3120 5.1381 5.0030 3.9862

4.3910 5.5588 5.3586 5.4741 5.4985 6.3849

6.2332 4.9783 4.2952 4.7955 5.1351 3.8102

5.6826 5.3159 5.5378 6.3488 6.3157 5.3347

5.5535 5.6785 5.0737 6.2634 5.8400 5.1982

5.6582 6.0569 6.1157 5.3885 5.0708 5.7090

5.5615 5.4652 5.1437 5.4799 5.7300 4.9654

Average of MC simulations 5.4105 <-- =AVERAGE(A10:F17)

True value 5.3756 <-- From Section 23.5

3.8102 <-- =MIN(A10:F17)

6.5765 <-- =MAX(A10:F17)

0.6227 <-- =STDEV(A10:F17)

PRICING AN ASIAN OPTION--VBA FUNCTION

Prices an Asian option with four periods and 100 runs for each simulation

We have deliberately priced the Asian option for which we know the

true value—as we showed in section 23.5, the value of an Asian option

in a four-period model with the parameters for Up, Down, and Interest

illustrated here is 5.3756. The average of our Monte Carlo simulations is

5.4105, with a standard deviation of 0.6227.

When we increase the number of runs (cell B8), we will usually

decrease the standard deviation of our estimates (cell B24); this decrease

is equivalent to increasing the accuracy of the simulation. In the next

example, we have run the 48 simulations 500 runs each:

636 Chapter 23

As you can see, the standard deviation is much reduced—about half

of the standard deviation with 100 runs.

23.6.1 Asian Options with More Periods

In the following spreadsheet we divide the unit time interval into n sub-

periods. We follow the procedure of section 23.4 for deﬁ ning the returns,

state prices, and risk-neutral probabilities over the subperiod Δt. Follow-

ing is the resulting spreadsheet.

ABCDEF G

Up 1.4

Down 0.8

Interest 1.08

Initial price 30

Periods 4

Exercise 30

Runs 500

5.2446 5.3915 5.5127 4.7900 5.7154 5.4045 <-- =MCAsian($B$5,$B$7,$B$2,$B$3,$B$4,$B$6,$B$8)

5.5467 5.8816 5.1299 5.4879 5.5207 5.1372

5.7815 5.0557 5.3597 5.6537 4.9744 5.1238

5.0642 4.9823 5.4147 5.5262 5.3900 5.3530

4.9365 5.2247 5.3370 5.2451 4.9379 5.6638

5.5245 5.1445 5.1902 5.9107 5.9324 5.3908

5.6054 5.1735 5.6206 4.9663 5.8997 5.0975

4.9088 4.9482 5.3003 4.9104 4.9713 5.5325

Average of MC simulations 5.3295 <-- =AVERAGE(A10:F17)

True value 5.3756 <-- From Section 23.5

4.7900 <-- =MIN(A10:F17)

5.9324 <-- =MAX(A10:F17)

0.3051 <-- =STDEV(A10:F17)

PRICING AN ASIAN OPTION--VBA FUNCTION

Pricing an Asian option with four periods and 500 runs for each simulation

4. The bad news, of course, is that we have to increase the number of runs by a factor of

5 to reduce the standard deviation by one-half.

637 Using Monte Carlo Methods for Option Pricing

The block of results in cells A18:F25 gives 48 results for running the

function MCAsian. Following this block we give the statistics for these

simulations.

These simulations use 100 price paths per iteration of the function

MCAsian; this is the number contained in cell B16. We can use Data|Table

to see the effect of changing the number of runs:

BACDEF HG

, current stock price

X, exercise price 45

T, time to option exercise 0.4

r, interest rate 8%

, mean stock return

15%

, standard deviation of stock return

22%

n, number of sub-intervals of T 80

Delta t 0.0125 <-- =1/B9

Up over 1 sub-interval 1.0268 <-- =EXP(B6*B10+B7*SQRT(B10))

Down over 1 sub-interval 0.9775 <-- =EXP(B6*B10-B7*SQRT(B10))

Interest over 1 sub-interval 1.0010 <-- =EXP(B5*B10)

Runs 100

6.2475 5.9146 6.0242 5.5575 6.5308 5.0464 <-- =MCAsian($B$2,$B$3,$B$12,$B$13,$B$14,$B$4*$B$9,$B$16)

5.3977 5.6106 5.6373 6.7208 5.4349 5.2232

4.8632 5.7098 5.3437 5.6366 5.1776 5.1993

5.7539 5.4739 6.3559 5.9198 5.9364 5.2982

6.1369 5.4303 4.9872 5.3160 5.4158 5.3885

5.9276 5.5548 5.7701 5.8524 5.4137 5.6423

5.4583 5.4490 5.4570 5.3789 5.6247 5.2845

5.5649 5.6722 5.9362 5.4512 5.5657 5.4095

Average of above 5.6063 <-- =AVERAGE(A18:F25)

Minimum 4.8632 <-- =MIN(A18:F25)

Maxmimum 6.7208 <-- =MAX(A18:F25)

Standard deviation 0.3775 <-- =STDEV(A18:F25)

PRICING AN ASIAN OPTION--VBA FUNCTION

Each time interval is divided into n subintervals. In this simulation the initial stock price = 50.00, the exercise

price = 45.00, the time to maturity = 0.40, and the unit time interval is divided into 80 subintervals. The stock

price process has mean return = 15.00% and standard deviation = 1.25%, and the interest rate = 8.00%.

There are 100 runs in each Monte Carlo simulation

BC D EFG H I J

Data table: Sensitivity of results on number of runs

Runs

Average

of 48

MCAsian

Minimum Maximum

Standard

deviation

<-- =B30 , data table header (hidden)

50 5.7270 4.7990 6.6145 0.4571

100 5.7683 5.0842 6.7896 0.3842

150 5.6694 5.1515 6.1435 0.2473

200 5.6556 5.1862 6.0654 0.2321

250 5.7597 5.2828 6.4856 0.2460

300 5.7175 5.0849 6.3364 0.2568

350 5.7279 5.3174 6.3799 0.2214

400 5.7266 5.2333 6.3176 0.2318

450 5.7281 5.3281 6.0717 0.1739

500 5.7133 5.3376 6.1541 0.1944

550 5.7138 5.4471 6.0796 0.1515

48 MCAsian Simulations for Various Numbers of

Runs

0.000

1.000

2.000

3.000

4.000

5.000

6.000

7.000

8.000

50 100 150 200 250 300 350 400 450 500 550

Runs per simulation

Average

of 48 MCAsian

Minimum

Maximum

Standard

deviation