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9 Pricing Derivatives via Monte Carlo Simu-

lation

Suppose, we have a contract at date 0 that

gives us the right but not the obligation to

sell one share of a stock at date T for a

price equal to the maximum during the pe-riod from 0 to T (a lookbackoption).

In such a case the exercise price is random,

depending on the path of St, 0 t T.

Another example is the right to buy, say cur-

rency, at an average rate during the contract

period (an Asian option).

1

In these cases (particularly for the look back

options) the risk neutral approach offers a

considerably simpler alternative to the more

traditional PDE.

Consider the example of the right to sell at

maximum (put option).

The current price (day 0) of the put option

(H) is

H0

= e

rTEP max0tT St ST= erTEP

max0tT St

erTEP [ST]

= erTEP

max0tT

St

S0.

(1)

Remark 9.1: Why is erTEP[ST] =S0?

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In the Monte Carlo simulationm (usually thou-

sands) sample baths are generated under the

risk neutral process, and the average of the

maximums are calculated.

The sample average is used to estimate

EP

max0tT St

.

Denote the average as Smax.

Then

H0=erTSmax S0.(2)

3

Discrete Approximation of a Continuous Pro-

cess

In practice the realizations of continuous pro-

cess are evaluated at discrete time points.

Usually, the solution of the SDE is not known

explicitly, but must be discretely approximated.

The GBM is one rare, though important,

case where the solution of the SDE is known.

That is, if

dS=Sdt + SdW(3)

then

S=S0e(122)t+W,(4)

and we can directly generate sample paths ofS (at discrete time points).

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In the general case, consider the stochastic

process

dSt=a(St, t)dt + b(St, t)dWt.(5)

Using the discrete approximation to calculate

St, divide the sample interval [0, T] into n

equal length subintervals

0 =t0< t1< . . . < tn1< tn=T .(6)

such that tk =tk tk1=T /n h.

In the numerical approximations of St, the

idea is to replace dSt with the discrete differ-

ences St=Stk Stk1.

5

Euler Method

The Euler scheme is:

Stk+1 =Stk+ akh + bkWtk+1,(7)

where ak =a(Stk, tk), bk = (Stk, tk), and

Wtk+1 =Wtk+1 Wtk N(0, h)are independent for k = 0, 1, . . . , n 1.

The procedure is started from a given S0

.

Due to the discretisation of dS, the error of

the Euler method is of order O(h).

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Milstein Method

The Milstein scheme is:

Stk+1 = Stk+ akh + bkWtk+1

+12bkbk

(Wtk+1)

2 h

,(8)

whereb =b(S, t) = b(S, t)/S, evaluated atStk.

The order of error is O(h2).

7

Runge-Kutta Method

Yet another scheme is the Runge-Kutta, which

avoids calculations of the derivatives b(S, t).

The scheme is:

Sk = Stk+ akh + bkhStk+1 = Stk+ akh + bkWtk+1

+ 12

h(bk bk)

(Wtk+1)

2 h

,

(9)

where bk =b(Sk, tk).

Again the order of error is O(h2).

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Which ever method is user, the key is to sim-

ulate the sample paths. In this we can utilize

the property

Wtk+1 =

hZk+1,(10)

where Zk+1 are independent N(0, 1) random

deviates.

The (approximated) sample paths for St are

generated by first generating n independent

N(0, 1) random deviates, Z1, . . . , Z n, and use

them in the equation of the selected method

[(7) (9)] to calculate the (discretely approx-

imated) realization for St.

9

In many software packages N(0, 1) random

number generators are readily available.

An easy, though inefficient, method in Excel

to generate normal random deviates is to use

the inverse function method, such that

Z= 1(U),(11)

where 1(U) (in Excel: = NormSinV(z)) isthe inverse of the standard normal cumula-

tive distribution function, (z), andU U(0, 1)is a uniformly distributed random variable in

the interval (0, 1) (in Excel: = rand()).

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A far more efficient way is to use for exam-

ple the Box-Muller method: Let U1 and U2be two independentU(0, 1) random variables,

then

Z1 =2 log(U1) cos(2U2)

Z2 = 2 log(U1) sin(2U2)(12)

are two independentN(0, 1) random variables.

11

Example 9.1: Consider the Cox-Ingersoll-Ross process

used in valuation of interest rate derivatives.

drt=( rt)dt + rtdWt,(13)where indicates the long run mean, indicates thespeed of the mean reversion, and is the volatilityparameter.

Thus the drift and volatility parameters are

a(r, t) =( r)and

b(r, t) =

r.

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Let = .50, = 5%, and = 3. Furthermore, let

T = 10 years, h= 1/120 (one month), and supposethat r0 = 5%.

The Euler scheme is

rtk+1 =rtk+ ( rtk)h +

rtk

hZk+1,(14)

whereZk+1 are independent N(0, 1) random variables.

In the Milstein approximation

b =

2

r.(15)

As seen from the figure below the differences between

the approximations are at least in this case immate-

rial.

Cox-Ingersoll-Ross SDE

dr = alpha (mu-r) dt + sigma sqrt(r) dW

alpha = 0.5, mu = 5% (p.a.), sigma = 3%

0

1

2

3

4

5

6

7

8

9

10

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

105

110

115

120

month

Interestrate

Euler

Milstein

Runge-Kutta

Figure 9.1: Euler, Milstein, and Runge-Kutta approx-

imations of the CIR PDE.

13

Simulating the Geometric Brownian Motion

The geometric Brownian motion (GBM) SDE

(3) has the solution (4).

Thus we do not need to use the approx-

imations described above, but can directlysimulate the sample path of St at the time

points tk, k = 0, 1, . . . n 1 with step lengthtk+1 tk =h, for example by the formula

Stk+1 =Stke(122)h+

hZk+1,(16)

where Zk+1 are independent N(0, 1) random

variables and S0 = St0 is given.

The approximation can be made arbitrarily

precise by increasing n (and therefore de-

creasing h).

Unfortunately there are no general rules how

big n should be to make the approximation

adequate.

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Remark 9.2: If we are only interested about the end

values of at the maturity T, as is the case with theEuropean options, we can directly simulate ST, when and are constants.

This is because

ST =S0e(1

22)T+WT,(17)

whereWT N(0, T), so that we can write WT = T Zwith Z N(0, 1).

However, in the case of the path-dependent options

we need to simulate the whole sample paths.

Remark 9.3: If the parameters (usually volatility) istime varying and we are pricing European options, itis usually computationally more efficient to simulatefirst the returns

Rtk = ( 1

2tk )h + tk

hZk,(18)

whereZk N(0, 1) fork = 1, . . . , nand instead of (16)use

ST =S0 exp

nk=1

Rtk

.(19)

Note that tk is non-anticipating.

15

Example 9.2: Consider the look-back option

discussed above.

Suppose that the current price of the stock is

S0 = 90, the risk-free interest rate is r = 5%,

= 25%, and the exercise day is 90 days

from now.

We simulate (16) with replaced by the risk-

free rate, r = 0.05, (see Ch. 8).

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For illustration purposes, let m= 10, then the below

Excel sheet excerpt gives an estimate

Smax= 98.78.(20)

thus the price estimate is

H0=e(0.0590/360)Smax 90 = 7.56.(21)

Figure 9.2 Simulation example of a look-back option.

17

How Many Simulations

The simulation sample size m can be deter-

mined in the usual manner of determining

sample size in a survey research.

H0=erT1

m

m

j=1 Y

jn S0,(22)where Yjn = max0knSjk , and{Sjk}nk=0 isthe jth sample path.

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The Central Limit Theorem (CLT) implies

m

H0 H0 a N0, 2y (n),(23)

where

2y (n) = Var

erTYjn

.(24)

Therefore for large m an asymptotic 95%

confidence interval for H0 is

H0 1.96y(n)

m .(25)

19

Hence, if, for example, we want a H0 that is

within H0 $0.001 with 95% confidence, mmust be chosen so that:

1.96y(n)

m 0.001.(26)

That is,

m 1.96

0.001

22y (n).(27)

Usually Var[Yjn] is not known, but can be

estimated by

Var[Yjn] = 1mm

j=1(Yjn Yn)

2(28)

with

Yn= 1

m

mj=1

Yjn.(29)

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The standard error of the Monte Carlo esti-

mator H0 is

H=y(n)

m(30)

is inversely proportional to the square root of

the number of replications m.

As a result a 50% reduction in standard error

requires four times more replications.

21

Variance Reduction Techniques

A number of variance-reduction techniques

have been developed to improve the efficiency

of the simulation estimators.

A simple technique is to replace estimatorswith their population counterparts whenever

possible.

For example, when simulating risk-neutral as-

set returns, the sampling variation can becorrected by adding the difference between

the riskless rate and the sample mean to each

observation in the replication.

That is, suppose

{rjk

}nk=1 be the jth sample

with the sample mean rj = 1n nk=1 rjk, thenthe corrected path is rjk =rjk (r rj) withrj =r, which has no sampling error.

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The efficiency gain depends on the extend

how much the estimation error in the sam-

ple mean contributes to the overall sampling

variation.

23

Antithetic Variate Methods

Symmetry, like population distribution with

respect to mean (risk-free return), is another

population information that can be exploited.

Producing m/2 sample paths the other m/2sample paths can be produced by mirroring

the first ones through the mean. This is a

simple example of more general antithetic-

variates methods.

Other methods are stratified sampling, im-

portance sampling, etc.

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As an illustration consider the antithetic-variate

estimation.

For each simulated price path{Sjk}nk=0 an-other path can be obtained by reverting the

sign of each randomly generated IID standard

normal variates on which the price series isbased, yielding a second path {Sjk}nk=0whichis negatively correlated with the first.

25

The resulting antithetic-variates estimate of

H0 is

Ha0 =erT 1

2m

mj=1

Yjn +

mj=1

Yjn

S0(31)or

Ha0 =e2rT1

m

mj=1

Yjn + Yjn2

S0,(32)

where

Yjn = max0kn

Sjk and Y

jn = max0kn

Sjk .(33)

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Then

Var[Ha0] =e2rT1

mVar

Yjn + Y

jn

,(34)

which reduces to

Var[Ha0] = 1

2m2y (n)(1 + ),(35)

where(36)

2y (n) = Var[erTYjn] = Var[erTYjn],

and

= Corr erTYjn, erTYjn.(37)The reduction of variance comes from two

sources:

(a) The number of replications is doubled

from m to 2m

(b) The factor 1 +, which should be less

than one if the correlation of the antithetic

variates is negative.

27

Path Dependent Options: PDE Method

Simulation approach is an easy procedure for

European type path dependent option. How-

ever, it is time consuming and hard to imple-

ment if Early exercise is allowed.

Several path dependent options can be easily

implemented in the PDE frame work.

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Assume generally

dS=a(S, t)dt + (S, t)dW .(38)

Suppose that the path dependent quantity

can be represented by an integral of some

function of the asset

I(t) = t

0f(S, u)du,(39)

such that

dI=f(S, t)dt.(40)

Typical examples are averages: A(T) =I(T)/T

Arithmetic Average: f(S, t) =S.

Geometric Average: f(S, t) = log(S).

Remark 9.4: f(S, t) is assumed to be measurable w.r.t

the information set and non-anticipating.

29

The Pricing Equation

LetF(S , I , t) denote the value of the contact.

The hedge portfolio is

P =F(S , I , t)

S,(41)

where is the position making the portfolio

risk-free (delta hedge).

Using Ito

dP =F

t +

1

22(S, t)

2F

S2

dt +

F

IdI+

FS

dS,(42)

where 2(S, t) = ((S, t))2.

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Choosing

=F

S(43)

the portfolio becomes risk-free.

No-arbitrage implies,

dP =rP dt.(44)

Thus, because dI=f(S, t)dt, we get finally

F

t +

1

22(S, t)

2F

S2 +

F

If(S, t)

dt = r(F SF

S)dt(45)

or

rF =F

t + rS

F

S + f(S, t)

F

I +

1

22(S, t)

2F

S2(46)

This is solved subject to the payoff condition

at the expiry

F(S, I) =G(S, I).(47)

31

Example 9.3: SupposeSt follows the geometric Brow-

nian motion. Asian option has aPayoff that dependson the average of the asset price of some period. Ifthe period is form zero to expiry and the average isarithmetic, then f(S, t) =S and

I=

t0

S du.(48)

The payoff function at the expiry for the call option

is

G(S, I) = max[S I/T, 0](49)and the PDE to be solved is

rF =F

t + rS

F

S + S

F

I +

1

22S2

2F

S2.(50)

32