Simon Benninga. Financial Modelling 3-rd edition

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608 Chapter 22

To see if this outcome is reasonable, we do some “brute force” simula-

tions. In the next spreadsheet, we examine 100 investment trajectories.

In each cell of the table we consider a function of the following type:

Wealth, end year

Wealth( ) Percent in risky Up

Wealth( )

−∗ ∗

+−∗

1(()1

−∗−

−∗

Percent in risky Drawdown

Wealth( ) Perce

Rand() π

nnt in risky Down

Wealth( ) Percent in risky Drawdown

∗

+−∗− ∗−tR11()

ff Rand() ≤

⎧

⎨

⎪

⎩

⎪

where R = exp(r), Up = exp(μ + σ), Down = exp(μ − σ),

ππ

−

Down

Up Down

and

Up Down

This simulation—like those in Chapter 18—simulates a lognormal

return for the risky asset, taking out the drawdown desired by the inves-

tor in each period. Here’s the output:

DCBA EFGHI

Current wealth 1,000,000

Risk-free rate 8%

Parameters of risky investment

Expected annual return 10%

Standard deviation of return 40%

Proportion invested in risky 40%

Annual drawdown 100,000

Years of investment 10

Runs 1,000

Successful runs 87.60% <-- =successfulruns(B2,B8,B3,B5,B6,B7,B9,B11)

SIMULATION OF 100 PATHS

R 1.0833 <-- =EXP(B3)

Up 1.6487 <-- =EXP(B5+B6)

Down 0.7408 <-- =EXP(B5-B6)

PiUp (risk-neutral prob. of Up) 0.3772 <-- =(B17-B19)/(B18-B19)

PiDown (risk-neutral prob. of Down) 0.6228 <-- =(B18-B17)/(B18-B19)

Number of year 10 wealth > 0 85 <-- =COUNTIF(C40:CX40,">0")

Percentage of successful simulations 85.00% <-- =B25/100

Wealth at end of year

Year

Simulation

0 1,000,000 1,000,000 1,000,000 1,000,000 1,000,000 1,000,000 1,000,000 1,000,000

1 1,209,461 846,300 846,300 1,209,461 846,300 846,300 846,300 846,300

2 1,483,741 700,853 700,853 1,044,512 700,853 700,853 700,853 1,008,196

3 1,842,901 817,739 563,217 888,421 563,217 817,739 817,739 854,055

4 1,643,936 970,798 432,972 1,063,353 432,972 673,826 970,798 708,192

5 1,455,656 818,665 309,721 1,292,419 309,721 537,642 818,665 827,350

6 1,277,487 674,703 305,567 1,123,015 193,089 408,770 674,703 983,382

7 1,108,885 538,471 189,158 962,709 152,842 435,268 538,471 830,574

8 949,338 409,555 79,000 811,011 44,634 311,894 409,555 987,604

9 1,143,120 436,296 3,448 961,987 -41,553 308,413 287,561 834,569

10 981,734 312,866 -96,737 1,159,684 -139,322 191,851 276,550 689,753

Cell C31 contains formula

=IF(RAND()>$B$22,$B$7*C30*$B$18+(1-

$B$7)*C30*$B$17-$B$8,$B$7*C30*$B$19+(1-

$B$7)*C30*$B$17-$B$8)

HOW WELL DO WE DO?

PERCENTAGE OF POSITIVE OUTCOMES

609 An Introduction to Monte Carlo Methods

In this particular simulation (the results change with every press on

F9 to recalculate the spreadsheet), 85 percent of all the simulated paths

ended with positive wealth—substantially in agreement with the results

of the function Successfulruns.

22.5.1 The VBA Code for Successfulruns

We almost forgot! Here it is:

Function SuccessfulRuns(Initial, Drawdown, Interest,

Mean, _

Sigma, PercentRisky, Years, Runs)

Dim PortfolioValue() As Double

ReDim PortfolioValue(Years + 1)

Dim Success As Integer

Up = Exp(Mean + Sigma)

Down = Exp(Mean - Sigma)

PiUp = (Exp(Interest) - Down) / (Up - Down)

PiDown = 1 - PiUp

For Index = 1 To Runs

For j = 1 To Years

Randomize

PortfolioValue(0) = Initial

If Rnd > PiDown Then

PortfolioValue(j) = _

PortfolioValue(j - 1) * PercentRisky * Up + _

PortfolioValue(j - 1) * (1 - PercentRisky) * _

Exp(Interest) - Drawdown

Else

PortfolioValue(j) = PortfolioValue(j - 1) * _

PercentRisky * Down + _

PortfolioValue(j - 1) * (1 - PercentRisky) *

Exp(Interest) _

- Drawdown

End If

Next j

If PortfolioValue(Years) > 0 Then Success = _

Success + 1

Next Index

610 Chapter 22

22.6 Summary

Monte Carlo methods are experimental techniques for determining the

numerical value of a function or a procedure. In general these methods

are to be avoided when there is another, closed-form, way of determining

the value. In cases where no such method exists, however, you can use

Monte Carlo to approximate the value. In this chapter we have used the

example of π to illustrate how Monte Carlo might be applied. In Chapter

23 we use Monte Carlo to price various kinds of options.

Exercises

1. In section 22.3 we designed a macro that calculates the value of π using Monte Carlo

and that updates the screen every iteration. Modify the macro so that it updates the

screen only every 1,000 iterations.

Hints:

•

Use the VBA function Mod. From the VBA help menu, note that this function

has syntax a Mod b. (A similar Excel function has syntax Mod(a,b), but cannot

be used in VBA.)

•

Use the VBA commands Application.ScreenUpdating = True and

Application.ScreenUpdating = False to control the updating of the screen.

2. Put a “switch” on the spreadsheet itself from exercise 1 that controls the updating

of the macro (whether to update, yes or no, and how often to update).

3. Use Monte Carlo to calculate the integral of the function exp(x) for 0 < x < 3. Here’s

the graph of this function:

Graph of exp(

), 0 £

0 0.5 1 1.5 2 2.5 3

exp(x)

611 An Introduction to Monte Carlo Methods

4. One of the messages of this chapter is that while Monte Carlo is a clever method

of calculation, it shouldn’t be used when a better method exists. The MC valuation

of π in section 22.2.1, for example, converges very slowly. It’s a crummy way to

compute π, since well-known methods exist. To see this point, answer the following

questions:

a. Approximately how many runs are required until you get four-decimal accuracy

for π using our MC simulation?

b. Approximately how many runs are required until you get eight-decimal accuracy

for π using our MC simulation?

c. Approximately how many runs are required until you get 16-decimal accuracy

for π using our MC simulation?

Note that in exercises 5 and 6 we show you several alternative ways to compute the

value of π. All are substantially better than Monte Carlo!

5. The ﬁ rst method for computing π is

=+ + +...

Use this series to construct a VBA function to value π, where n is the number of

terms in the series. What value of n gives you four-decimal-place accuracy?

Eight-decimal-place?

6. The great Indian mathematical genius Ramanujan showed that

9 801

4 1 103 26 390

396

∞

∑

,,()!

(!)

()n

The n! indicates the factorial:

n! = n

(n − 1)

(n − 2)

. . .

0! = 1

Excel’s function Fact computes the factorial.

Use this series to construct a VBA function to value π, where n is the number

of terms in the series. Show that two iterations give you more than 15 digits of

accuracy.

612 Chapter 22

Some Comments on the Value of p

Exercise 6 is based on an article by D. H. Bailey, J. M. Borwein, and P. B. Borwein:

“Ramanujan, Modular Equations, and Approximations to π Or How to Compute

One Billion Digits of Pi.” The article originally appeared in the American

Mathematical Monthly, 1987, volume 96, number 3, pages 201–219. It can be down-

loaded at http://www.cecm.sfu.ca/organics/papers/borwein/index.html. Ramanujan’s

method in exercise 6 adds roughly eight digits with each iteration. Ramanujan

developed an even faster method, where only 13 iterations provide more than one

billion digits of π. (Excel’s maximal accuracy is only 15 digits, but there is a nice

Excel add-in that extends the precision to 32,767 digits: http://precisioncalc.com/).

Srinivasan Ramanujan (1887–1920) was one of the great mathematical geniuses

of all time. Read a biography of this unique individual: The Man Who Knew Inﬁ nity:

A Life of the Genius Ramanujan by Robert Kanigel, published by Charles Scribner,

1991 (paperback edition, Washington Square Press, 1992).

Finally: So what’s the value of π? Mathematica—a very sophisticated mathemati-

cal programming language (http://www.wolfram.com)—gives the following values

for π:

Number of

Signiﬁ cant Digits Value of π

25 3.141592653589793238462643

50 3.1415926535897932384626433832795028841971693993751

75 3.1415926535897932384626433832795028841971693993751058

2097494459230781640629

500 3.1415926535897932384626433832795028841971693993751058

209749445923078164062862089986280348253421170679821480

865132823066470938446095505822317253594081284811174502

841027019385211055596446229489549303819644288109756659

334461284756482337867831652712019091456485669234603486

104543266482133936072602491412737245870066063155881748

815209209628292540917153643678925903600113305305488204

665213841469519415116094330572703657595919530921861173

819326117931051185480744623799627495673518857527248912

279381830119491

Using Monte Carlo Methods for Option Pricing

23.1 Overview

This chapter continues the discussion of the previous chapter and shows

how to implement Monte Carlo methods for pricing options. The main

object of the chapter is to show how to price Asian options and barrier

options. An Asian option has a payoff that depends on the average price

of the underlying asset for some period before option maturity, and a

barrier option’s payoff depends on the underlying price reaching a

particular level at some point before maturity. In sections 23.5–23.7 we

will make these general statements more explicit.

Both Asian and barrier options are path-dependent options—options

whose price depends not only on the terminal price of the asset, but also

on the path of the prices by which the terminal price was reached. In

general a path-dependent option does not have an analytic price solu-

tion. Monte Carlo provides us with a handy numerical tool for pricing

such options.

Monte Carlo pricing of options depends on a simulation of the price

path of the underlying asset. In order to clarify the Monte Carlo pricing

algorithms, we start in section 23.2 with a brief review of state prices and

risk neutrality. We then show, in section 23.3, how to price a plain-vanilla

option with a Monte Carlo algorithm. Since plain-vanilla options—jargon

by which we simply mean European calls and puts on a stock whose price

process is lognormal—are accurately priced using the Black-Scholes

formula, this exercise allows us to check our pricing method against a

known result and also allows us to develop the proper intuitions about

the Monte Carlo pricing of more complicated options.

23.2 State Prices, Probabilities, and Risk Neutrality

We start with a brief recapitulation of the basic facts about state prices

and risk-neutral pricing discussed in Chapter 17.

Suppose we have binomial framework with stock price S that grows

at U or D in every period. Suppose the interest rate is R.

Risk neutral-

ity is a property of every set of state prices: Given state prices

1. Properly speaking, U, D, and R are one plus the growth and interest rates. For the sake

of linguistic parsimony, we will use “up growth,” “down growth,” and “interest rate”

even though we mean something slightly different.

614 Chapter 23

RU D

−

⎧

⎨

⎩

⎫

⎬

⎭

()

, the risk-neutral probabilities are

deﬁ ned by {π

= Rq

}. The risk-neutral probabilities can be

used to price assets by taking the discounted expected value of the asset

payoffs.

An asset with payoffs one period hence of {Payoff

, Payoff

} in

states U and D, respectively, has value today of

Asset value today Payoff Payoff

Payoff Payoff

=+=

↑

UUDD

ππ

hhe risk-neutral expected payoff

(expectation computed

with the rissk-neutral probabilities)

discounted at the risk-free interest

raate

 

The {Payoff

, Payoff

} are usually functions of the underlying asset price.

For an ordinary call, for example, {Payoff

= max(S*U − X,0), Payoff

max(S*D − X,0)}.

We can extend the risk-neutral pricing scheme to multiperiod frame-

works. Consider a multiperiod binomial setting where U and D do not

change over time, and indicate the date-n state payoffs by Payoff

n,j

j = 0, . . . , n. The notation Payoff

n,j

indicates the date-n payoff of the asset

in a state where there are j up moves on the binomial tree; for a call in

a binomial framework, Payoff

n,j

= max(S*U

n−j

− X,0) Then the value of

this asset is given by

Asset value today Payoff=

⎛

⎝

⎜

⎞

⎠

⎟

⎛

⎝

⎜

⎞

⎠

⎟

−

∑

πππ

−

↑

∑

Payoff

The risk-neutral expected

discounted v

aalue



This particular notation assumes that the tree is recombining. It assumes,

in other words, that the date-n payoffs are path independent—the option

payoff is a function only of the terminal stock price and does not depend

on the path by which this price is reached (Fig. 23.1). We will return to

this topic later in this chapter.

2. As discussed in Chapter 17, the risk-neutral probabilities are not the actual proba-

bilities of the state occurrences. They are in fact “pseudoprobabilities” that derive

from the state prices.

615 Using Monte Carlo Methods for Option Pricing

23.3 Pricing a Plain-Vanilla Call Using Monte Carlo Methods

In this section we explore the pricing of a European call using Monte

Carlo. In one sense, of course, this is an immense waste of time—the

Black-Scholes formula (Chapter 19) gives a wonderful pricing solution

for European calls and puts. However, like the example of estimating the

value of π discussed in Chapter 22, the exercise of pricing a plain-vanilla

call using Monte Carlo methods gives us considerable insight into the

application of Monte Carlo methods.

23.3.1 A Really Simple Example

We start with a ridiculously simple example. We use Monte Carlo to price

a European call in a two-period setting in which the stock price goes

either up or down each period. In the following spreadsheet we use

Monte Carlo methods to price an at-the-money option on a stock whose

price today is S

= 50. There are two periods, and in each period the stock

= S

Date 0 Date 1 Date 2 Date 0 Date 1 Date 2

Payoff

Value today

Payoff

= Payoff

Value today

Payoff

Date 0 Date 1 Date 2 Date 0 Date 1 Date 2

Underlying Assets and Path-Independent

versus Path-Dependent Payoffs

Stock prices Bond value

Path-independent asset with payoffs at date 2 Path-dependent asset with payoffs at date 2

Figure 23.1

Recombining versus nonrecombining binomial models.

616 Chapter 23

price goes either Up = 1.4 or Down = 0.9; the interest rate is R = 1.05.

Given Up and Down, the stock-price tree looks like Figure 23.2.

The following spreadsheet shows two random price paths and their

pricing:

Down

1.4

0.9

= 98 = 50 * 1.4

= 50 S

= S

= 63 = 50 * 1.4 * 0.9

= 40.5 = 50 * 0.9

Date 0 Date 1 Date 2

= 70

= 45

Figure 23.2

Stock-price tree in a standard recombining binomial model.

BADC

Initial stock price 50

5X0

Up 1.4

Down 0.9

1R.05

State prices

0.2857 <-- =(B7-B6)/(B7*(B5-B6))

0.6667 <-- =(B7-B6)/(B7*(B5-B6))

Risk-neutral probabilities

0.3000 <-- =(B7-B6)/(B7*(B5-B6))

0.7000 <-- =(B7-B6)/(B7*(B5-B6))

First period, up (1) or down (0)? 0 0 <-- =IF(RAND()>$B$15,1,0)

Second period, up (1) or down (0)? 0 1 <-- =IF(RAND()>$B$15,1,0)

Total ups

0 1 <-- =SUM(C18:C19)

Terminal stock price

40.5 63 <-- =$B$2*up^C21*down^(2-C21)

Option payoff

0 13 <-- =MAX(C22-$B$3,0)

Average discounted payoff

5.8957 <-- =AVERAGE(23:23)/R_^2

Computing the actual option price with state prices

Payoffs

top 48 <-- =MAX(B2*up^2-B3,0)

middle 13 <-- =MAX(B2*up*down-B3,0)

bottom 0 <-- =MAX(B2*down^2-B3,0)

Actual option price 8.8707 <-- =q_up^2*B29+2*q_up*q_down*B30+q_down^2*B31

Random paths and the Monte Carlo price

SIMPLE SIMULATION:

TWO PATHS IN A TWO-DATE MODEL

617 Using Monte Carlo Methods for Option Pricing

The state prices and risk-neutral probabilities are computed in cells

B10:B11 and B14:B15. Cells B18:B29 and C18:C19 show two random

price paths. In each period we use a random number from Excel gener-

ated by the function Rand. If Rand() > π

, then the stock price goes up,

and if Rand() £ π

then the stock price goes down.

•

In the ﬁ rst price path (cells B18:B19), the stock price goes down in

both periods. The terminal stock price is 40.5, and the option’s payoff is

0 (cell B23).

•

In the second price path (cells C18:C19), the stock price goes up in the

ﬁ rst period and down in the second period. The terminal stock price is

63 and the option’s payoff is 13 (cell C23).

If these were the only two random price paths, the Monte Carlo option

price would be the discounted average 5.8957 (cell B25). Notice that we

have also computed the actual call price using the state prices; cell B32

shows this price to be 8.8707.

Risk-Neutral Probabilities in Monte Carlo

Notice the role of the risk-neutral probabilities in the Monte Carlo

simulation: The price path is determined not by the actual proba-

bilities, but by the risk-neutral probabilities π

and π

. There is no

role for the actual probabilities in Monte Carlo pricing.

23.3.2 Extending the Two-Period Model

You can’t, of course, run a Monte Carlo simulation by using only two

price paths. In the next spreadsheet we’ve extended our price paths to

all the columns in the spreadsheet. Excel’s spreadsheet is 256 columns

wide; therefore, the computation in cell B25 is the average of the option

value over 255 simulated price paths.