Simon Benninga. Financial Modelling 3-rd edition
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638 Chapter 23
It is clear that increasing the number of runs narrows the bounds on
the simulation.
23.7 Pricing Barrier Options with Monte Carlo
5
A barrier option’s payoff depends on whether the price reaches a specifi c
level during the life of the option:
•
A knockin barrier call option has payoff max(S
T
− X, 0) only if at some
time t <T, S
t
> K. A knockin put has the same condition but pays off
max(X − S
T
, 0).
•
A knockout barrier call or put option has these payoffs provided that
at no time before T does the stock price reach the barrier.
Imposing a barrier makes it more diffi cult for an option to be in the
money at expiration; thus barrier options have lower value than regular
options.
23.7.1 A Simple Example of a Barrier Call Option
In this section we show an extended example for a knockout barrier
option that is similar to the example for an Asian option given in section
23.5.
5. Pricing barrier options with Monte Carlo isn’t necessarily a good idea, but it’s a good
exercise. See Broadie, Glaserman, and Kou (1997) for a complete discussion.
639 Using Monte Carlo Methods for Option Pricing
In this example, we model a fi ve-date, four-period barrier call option.
The barrier is 50 (cell B7). A knockout option pays off only if the price
never goes through this barrier, and a knockin option pays off only if the
stock price goes through the barrier. In an equation,
Barrier knockin
call payoff
max[ ]
if Barrier for=
−
⎧
⎨
⎩
><
SX
StT
T
t
,0
0
ootherwise
Barrier knockout
call payoff
max[ ]
if Ba=
−
⎧
⎨
⎩
<
SX
S
T
t
,0
0
rrrier for otherwisetT<
The knockout barrier call illustrated here pays off only when two things
happen simultaneously:
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ABCDEFGHIJKLMNO
P
Initial stock price 30
Up 1.40
Down 0.80
Interest 1.08
Option exercise price 30
Barrier 50.00
q
U
0.4321 <-- =(B5-B4)/(B5*(B3-B4))
q
D
0.4938 <-- =(B3-B5)/(B5*(B3-B4))
Risk-neutral probability, up 0.4667 <-- =B9*B5
Risk-neutral probability, down 0.5333 <-- =B10*B5
Paths Period 1 Period 2 Period 3 Period 4 Period 0 Period 1 Period 2 Period 3 Period 4
Max(S
t
)<
Barrier?
Path
risk-neutral
probability
Knockout
option
payoff
All up (1 path) up up up up 30.00 42.00 58.80 82.32 115.25 FALSE 0.0474 0.00
One down (4 paths) down up up up 30.00 24.00 33.60 47.04 65.86 FALSE 0.0542 0.00
up down up up 30.00 42.00 33.60 47.04 65.86 FALSE 0.0542 0.00
up up down up 30.00 42.00 58.80 47.04 65.86 FALSE 0.0542 0.00
up up up down 30.00 42.00 58.80 82.32 65.86 FALSE 0.0542 0.00
Two down (6 paths) down down up up 30.00 24.00 19.20 26.88 37.63 TRUE 0.0619 7.63
down up down up 30.00 24.00 33.60 26.88 37.63 TRUE 0.0619 7.63
down up up down 30.00 24.00 33.60 47.04 37.63 TRUE 0.0619 7.63
up down down up 30.00 42.00 33.60 26.88 37.63 TRUE 0.0619 7.63
up up down down 30.00 42.00 58.80 47.04 37.63 FALSE 0.0619 0.00
up down up down 30.00 42.00 33.60 47.04 37.63 TRUE 0.0619 7.63
Three down (4 paths) up down down down 30.00 42.00 33.60 26.88 21.50 TRUE 0.0708 0.00
down up down down 30.00 24.00 33.60 26.88 21.50 TRUE 0.0708 0.00
down down up down 30.00 24.00 19.20 26.88 21.50 TRUE 0.0708 0.00
down down down up 30.00 24.00 19.20 15.36 21.50 TRUE 0.0708 0.00
Four down (1 path) down down down down 30.00 24.00 19.20 15.36 12.29 TRUE 0.0809 0.00
Knockout value 1.7375
PRICING A KNOCKOUT BARRIER OPTION
Formula in cell N38:
=SUMPRODUCT(O17:O36,N17:N36)/B5^4
Formula in cell O36:
=M36*MAX(K36-$B$6,0)
STOCK PRICE
Formula in cell M17: =MAX(G17:K17)<$B$7
640 Chapter 23
1. The stock price does not exceed the barrier. This result occurs for all
the paths labeled TRUE in column M. To check this condition in cell
M17 we use the Boolean function (=MAX(G17:K17)<$B$7).
6
This func-
tion evaluates to TRUE or FALSE, depending on whether the condition
is met. Other cells in column M use a similar condition. When used in a
formula as in item 2, the Boolean function evaluates to 1 if TRUE and
to 0 if FALSE.
2. The terminal stock price S
T
is greater than the option exercise price
of 30. In cell O17 we use the condition M17*MAX(K17-$B$6,0) to evalu-
ate the option payoff.
a. If M17 equals 0 (meaning that S
t
> 50 somewhere along the path and
the option was “knocked out”), then the option doesn’t pay off.
b. If M17 equals 1 (so that S
t
< 50 throughout the path), then the option
has a standard call payoff of max(S
T
− X, 0).
As in all previous cases discussed in this chapter, the barrier call’s
value is the discounted expected payoff of the option, where the proba-
bilities are the risk-neutral probabilities:
Option value
Payoff
All states
==
∑
π
jj
j
R
4
1 7375.
23.7.2 The Knockin Barrier Call
By changing the condition in column O we can price the knockin barrier
call. This time we write (in cell O17, for example) the function =(1-
M17)*MAX(K17-$B$6,0). The value in M17 tests whether the barrier
has never been passed; if this is FALSE (i.e., has a value of zero), then
the option is “knocked in” and the payoff is like that of a regular call. If
M17 is TRUE, then the barrier has not been passed and the option does
not pay off:
6. Boolean functions are discussed in Chapter 34.
641 Using Monte Carlo Methods for Option Pricing
The spreadsheets for the knockout and knockin barriers illustrate
another principle of pricing barrier options: The price of a knockin plus
a knockout call equals the price of a regular plain-vanilla call:
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ABCDEFGHIJKLMNO
P
Initial stock price 30
Up 1.40
Down 0.80
Interest 1.08
Option exercise price 30
Barrier 50.00
q
U
0.4321 <-- =(B5-B4)/(B5*(B3-B4))
q
D
0.4938 <-- =(B3-B5)/(B5*(B3-B4))
Risk-neutral probability, up 0.4667 <-- =B9*B5
Risk-neutral probability, down 0.5333 <-- =B10*B5
Paths Period 1 Period 2 Period 3 Period 4 Period 0 Period 1 Period 2 Period 3 Period 4
Max(S
t
)<
Barrier?
Path
risk-neutral
probability
Option
payoff
All up (1 path) up up up up 30.00 42.00 58.80 82.32 115.25 FALSE 0.0474 85.25
One down (4 paths) down up up up 30.00 24.00 33.60 47.04 65.86 FALSE 0.0542 35.86
up down up up 30.00 42.00 33.60 47.04 65.86 FALSE 0.0542 35.86
up up down up 30.00 42.00 58.80 47.04 65.86 FALSE 0.0542 35.86
up up up down 30.00 42.00 58.80 82.32 65.86 FALSE 0.0542 35.86
Two down (6 paths) down down up up 30.00 24.00 19.20 26.88 37.63 TRUE 0.0619 0.00
down up down up 30.00 24.00 33.60 26.88 37.63 TRUE 0.0619 0.00
down up up down 30.00 24.00 33.60 47.04 37.63 TRUE 0.0619 0.00
up down down up 30.00 42.00 33.60 26.88 37.63 TRUE 0.0619 0.00
up up down down 30.00 42.00 58.80 47.04 37.63 FALSE 0.0619 7.63
up down up down 30.00 42.00 33.60 47.04 37.63 TRUE 0.0619 0.00
Three down (4 paths) up down down down 30.00 42.00 33.60 26.88 21.50 TRUE 0.0708 0.00
down up down down 30.00 24.00 33.60 26.88 21.50 TRUE 0.0708 0.00
down down up down 30.00 24.00 19.20 26.88 21.50 TRUE 0.0708 0.00
down down down up 30.00 24.00 19.20 15.36 21.50 TRUE 0.0708 0.00
Four down (1 path) down down down down 30.00 24.00 19.20 15.36 12.29 TRUE 0.0809 0.00
Knockin value 9.0334
PRICING A KNOCKIN BARRIER OPTION
Formula in cell N38:
=SUMPRODUCT(O17:O36,N17:N36)/B5^4
Formula in cell O36:
=(1-M36)*MAX(K36-$B$6,0)
STOCK PRICE
Formula in cell M17: =MAX(G17:K17)<$B$7
642 Chapter 23
23.8 Using VBA and Monte Carlo to Price a Barrier Option
We write two VBA functions to price knockin and knockout barrier
options. Here is the function for knockout options:
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ABCDEFGHIJKLMNOP
Q
Initial stock price 30
Up 1.40
Down 0.80
Interest 1.08
Option exercise price 30
Barrier 50.00
q
U
0.4321 <-- =(B5-B4)/(B5*(B3-B4))
q
D
0.4938 <-- =(B3-B5)/(B5*(B3-B4))
Risk-neutral probability, up 0.4667 <-- =B9*B5
Risk-neutral probability, down 0.5333 <-- =B10*B5
Paths Period 1 Period 2 Period 3 Period 4 Period 0 Period 1 Period 2 Period 3 Period 4
Max(S
t
)<
Barrier?
Path
risk-neutral
probability
Knockout
payoff
Knockin
payoff
Plain
vanilla
All up (1 path) up up up up 30.00 42.00 58.80 82.32 115.25 FALSE 0.0474 0 85.2480 85.2480
One down (4 paths) down up up up 30.00 24.00 33.60 47.04 65.86 FALSE 0.0542 0 35.8560 35.8560
up down up up 30.00 42.00 33.60 47.04 65.86 FALSE 0.0542 0 35.8560 35.8560
up up down up 30.00 42.00 58.80 47.04 65.86 FALSE 0.0542 0 35.8560 35.8560
up up up down 30.00 42.00 58.80 82.32 65.86 FALSE 0.0542 0 35.8560 35.8560
Two down (6 paths) down down up up 30.00 24.00 19.20 26.88 37.63 TRUE 0.0619 7.632 0.0000 7.6320
down up down up 30.00 24.00 33.60 26.88 37.63 TRUE 0.0619 7.632 0.0000 7.6320
down up up down 30.00 24.00 33.60 47.04 37.63 TRUE 0.0619 7.632 0.0000 7.6320
up down down up 30.00 42.00 33.60 26.88 37.63 TRUE 0.0619 7.632 0.0000 7.6320
up up down down 30.00 42.00 58.80 47.04 37.63 FALSE 0.0619 0 7.6320 7.6320
up down up down 30.00 42.00 33.60 47.04 37.63 TRUE 0.0619 7.632 0.0000 7.6320
Three down (4 paths) up down down down 30.00 42.00 33.60 26.88 21.50 TRUE 0.0708 0 0.0000 0.0000
down up down down 30.00 24.00 33.60 26.88 21.50 TRUE 0.0708 0 0.0000 0.0000
down down up down 30.00 24.00 19.20 26.88 21.50 TRUE 0.0708 0 0.0000 0.0000
down down down up 30.00 24.00 19.20 15.36 21.50 TRUE 0.0708 0 0.0000 0.0000
Four down (1 path) down down down down 30.00 24.00 19.20 15.36 12.29 TRUE 0.0809 0 0.0000 0.0000
Knockin 9.0334 <-- =SUMPRODUCT(N17:N36,P17:P36)/B5^4
Knockout 1.7375 <-- =SUMPRODUCT(N17:N36,O17:O36)/B5^4
Sum 10.7708 <-- =L38+L39
Plain vanilla 10.7708 <-- =SUMPRODUCT(N17:N36,Q17:Q36)/B5^4
Formula in cell O36:
=M36*MAX(K36-$B$6,0)
KNOCKIN + KNOCKOUT = PLAIN VANILLA
Formula in cell P36:
=(1-M36)*MAX(K36-$B$6,0)
STOCK PRICE
Formula in cell M17: =MAX(G17:K17)<$B$7
Function MCBarrierOut(Initial, Exercise, Barrier, Up, _
Down, Interest, Periods, Runs)
Dim PricePath() As Double
ReDim PricePath(Periods + 1)
‘Risk-neutral probabilities
piup = (Interest - Down) / (Up - Down)
pidown = 1 - piup
643 Using Monte Carlo Methods for Option Pricing
Temp = 0
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
If Application.Max(PricePath) < Barrier Then
Callpayoff = _
Application.Max(PricePath(Periods) - _
Exercise, 0) _
Else Callpayoff = 0
Temp = Temp + Callpayoff
Next Index
MCBarrierOut = (Temp / Interest ^ Periods) / Runs
End Function
Since this function is very similar to the function MCAsian of section
23.6, we will not discuss it, except to point out that the operative part for
the “knockin” option is contained in the following lines (note the use of
Excel’s Max function—in the form of Application.Max; VBA does not
have its own maximum function):
If Application.Max(PricePath) < Barrier Then
Callpayoff = _
Application.Max(PricePath(Periods) - Exercise, 0) _
Else Callpayoff = 0
644 Chapter 23
In the next spreadsheet we use this function and its associated function
MCBarrierOut to price the options previously priced in our extensive
example:
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ABCDE
F
Up 1.4
Down 0.8
Interest 1.08
Initial price 30
Periods 4
Exercise 30
Barrier 50
Runs 100
Knockin option value 7.0810 <-- =mcbarrierin(B6,B8,B9,B2,B3,B4,B7,B11)
Actual value 9.0334 <-- ='Initial knockin'!N38
Knockout option value 1.6829 <-- =mcbarrierout(B6,B8,B9,B2,B3,B4,B7,B11)
Actual value 1.7375 <-- Determined from fully-worked out example
7.7944 7.2408 8.4210 9.2159 6.8481 13.1353
11.6143 8.3860 9.5916 9.1598 7.6430 11.3200
9.3154 6.7095 7.2673 9.2593 9.7948 8.9174
11.0269 10.1018 7.4959 7.5435 9.3069 9.3799
11.7265 8.4210 6.5369 10.6289 8.8613 8.6327
9.5016 9.6350 9.1037 7.9066 8.2262 8.9565
10.9358 7.4440 9.2550 12.1806 8.1056 8.9216
6.3771 7.4440 9.4318 9.3323 9.5831 9.1428
Average of simulations 9.0162 <-- =AVERAGE(A20:F27)
True value 9.0334 <-- =B14
6.3771 <-- =MIN(A20:F27)
13.1353 <-- =MAX(A20:F27)
1.5066 <-- =STDEV(A20:F27)
PRICING BARRIER OPTIONS
BY MONTE CARLO
48 iterations of MCBarrierIn
Finally, we can also show the implementation of the functions MCBar-
rierIn and MCBarrierOut for the case where the unit period is divided
into n subperiods:
645 Using Monte Carlo Methods for Option Pricing
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BACDEFG H
S
0
, current stock price
50
X, exercise price 45
Barrier 50
T, time to option exercise 0.4
r, interest rate 8%
μ
, mean stock return
15%
σ
, standard deviation of stock return
22%
n, number of subintervals of 1 period 80
Delta t 0.0125 <-- =1/B10
Up over 1 subinterval 1.0268 <-- =EXP(B7*B11+B8*SQRT(B11))
Down over 1 subinterval 0.9775 <-- =EXP(B7*B11-B8*SQRT(B11))
Interest over 1 subinterval 1.0010 <-- =EXP(B6*B11)
Runs 100
6.9666 7.8251 6.0301 7.5631 6.2022 6.0205 <-- =mcbarrierin($B$2,$B$3,$B$4,$B$13,$B$14,$B$15,INT($B$10*$B$5),$B$17)
6.2149 7.2956 7.0748 7.7456 6.4665 7.3994
7.2747 7.8567 7.4531 7.4395 7.5946 6.8954
6.6258 6.8220 6.2379 6.7264 6.9056 6.4062
6.8421 8.4597 7.2974 7.6032 6.3056 6.5011
8.4227 6.6866 7.3832 6.6680 7.6663 6.8732
7.5956 7.6050 6.9231 6.7689 7.1278 8.7135
7.5495 7.1480 6.4259 7.5199 6.1428 6.7183
Average of above 7.0831 <-- =AVERAGE(A19:F26)
Minimum 6.0205 <-- =MIN(A19:F26)
Maxmimum 8.7135 <-- =MAX(A19:F26)
Standard deviation 0.6412 <-- =STDEV(A19:F26)
PRICING BARRIER OPTIONS--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
As for the case discussed in section 23.6 for Asian options, as the
number of runs (cell B17) gets larger, the approximations become better,
although the improvement is not dramatic:
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BC DEFG H IJ
Data table: Sensitivity of results on number of runs
Runs
Average
of 48
MCBarrierIn
Minimum Maximum
Standard
deviation
<-- =B31 , data table header (hidden)
50 7.1829 5.7672 8.9774 0.7927
100 6.9170 5.9491 8.3480 0.5323
150 6.8797 5.9086 8.0689 0.4978
200 6.8515 5.8530 7.6812 0.3939
250 6.9466 5.7513 7.7856 0.3819
300 6.9271 5.9635 7.8805 0.3943
350 6.9796 6.2143 7.7238 0.3588
400 6.9940 6.1547 7.7151 0.3365
450 6.9165 6.1976 7.5807 0.2864
500 6.9398 5.7926 7.5645 0.3337
550 6.8902 6.1393 7.4763 0.2693
48 MCBarrierIn Simulations for Various Numbers of
Runs
0.000
1.000
2.000
3.000
4.000
5.000
6.000
7.000
8.000
9.000
10.000
50 100 150 200 250 300 350 400 450 500 550
Runs per simulation
Average
of 48 MCBarrierIn
Minimum
Maximum
Standard
deviation
646 Chapter 23
Finally, we can show that the sum of the knockin plus knockout is
approximately equal to the Black-Scholes call when n, the number of
divisions of the unit time interval, is very large:
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BAC
S
0
, current stock price
30
X, exercise price 30
Barrier 40
T, time to option exercise 0.4
r, interest rate 8%
μ
, mean stock return
15%
σ
, standard deviation of stock return
22%
n, number of subintervals of 1 period 200
Delta t 0.0050 <-- =1/B10
Up over 1 subinterval 1.0164 <-- =EXP(B7*B11+B8*SQRT(B11))
Down over 1 subinterval 0.9853 <-- =EXP(B7*B11-B8*SQRT(B11))
Interest over 1 subinterval 1.0004 <-- =EXP(B6*B11)
Runs 700
Knockout barrier 1.653148
<-- =mcbarrierout
(B2,B3,B4,B13,B14,B15,INT(B10*B5),B17)
Knockin barrier 0.504321
<-- =mcbarrierin(B2,B3,B4,B13,B14,B15,INT(B10*B5),B17)
Sum of knockout + knockin 2.157468
<-- =B19+B20
Black-Scholes call price 2.153173 <-- =BSCall(B2,B3,B5,B6,B8)
KNOCKIN + KNOCKOUT = CALL
The almost-continuous case
23.9 Summary
Monte Carlo methods—simulations of option pricing by tracing out
many paths of the stock price—are at best a “second best” method of
pricing. But in cases where no analytical formulas are available, Monte
Carlo is easy to program in VBA and easy to see in Excel. In this chapter
we have illustrated Monte Carlo methods for plain-vanilla options, Asian
options, and barrier options. Other variations of path-dependent options
and their Monte Carlo solutions are considered in the exercises.
Exercises
1. Create a VBA subroutine [call it Exercise1( )] that generates a random number and
prints it on the screen in a message box that looks like this:
647 Using Monte Carlo Methods for Option Pricing
Note Use the VBA keyword Rnd.
2. Create a VBA subroutine [call it Exercise2( )] that generates fi ve random numbers
and prints them out on the screen in a message box like the following:
Note Use FormatNumber(Expression,NumDigitsAfterDecimal) to print out only
four digits.
3. Create a VBA subroutine [call it Exercise3( )] that generates fi ve random digits of
either 1 or 0 and prints them out on the screen in a message box like the
following:
4. Suppose a stock price follows a binomial distribution. We want to create random
price path for the stock in a VBA macro. The picture shows the input with some
sample output. Write an appropriate VBA subroutine.