5535 chapter 19

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Basic Numerical Procedure

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Content

1 Binomial Trees

2 Using the binomial tree for options on indices,

currencies, and futures contracts

3 Binomial model for a dividend-paying stock

4 Alternative procedures for constructing trees

5 Time-dependent parameters6 Monte Carlo simulation

7 Variance reduction procedures

8 Finite difference methods

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Binomial Trees

In each small interval of time tthe

stock price is assumed to move up by

a proportional amount u or to move

down by a proportional amount d

Su

Sd

S

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Risk-Neutral Valuation

1. Assume that the expected return from

all traded assets is the risk-free

interest rate.

2. Value payoffs from the derivative by

calculating their expected values and

discounting at the risk-free interestrate.

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Determination ofp, u, and d

Mean: e(r-q)(t =pu + (1p )d

Variance:W

2

(t =pu2 + (1p )d2 e2(r-q)(t

A third condition often imposed is u =1/d

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A solution to the equations, when terms of

higher order than (tare ignored, is

where

)( tqr

t

t

ea

ed

eu

du

dap

(

(

(

!

!

!

!

W

W

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Tree of Asset Prices At time it

S0u2

S0u4

S0d2

S0d4

S0

S0u

S0d

S0 S0

S0u2

S0d2

S0u3

S0u

S0d

S0d3

i0,1,...,j,duSS j-ij0i !!

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Working Backward through the Tree

Example : American put optionS

0=

50; K=

50; r=

10%; W=

40%;T= 5 months = 0.4167;

(t= 1 month = 0.0833

The parameters imply

u =1.1224; d=0.8909;

a =1.0084; p =0.5073

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Example (continued)

9 . 0 7

0 . 0 0

7 9 . 3 5

0 . 0 0

7 0 .7 0 7 0 . 7 0

0 . 0 0 0 . 0 0

6 2 . 9 9 6 2 .9 9

0 . 6 4 0 .0 0

5 6 .1 2 5 6 .1 2 5 6 .1 2

2 .1 6 1 .3 0 0 . 0 0

5 0 . 0 0 5 0 . 0 0 5 0 . 0 0

4 .4 9 3 . 7 7 2 .6 6

4 4 .5 5 4 4 .5 5 4 4 .5 5

6 .9 6 6 .3 8 5 . 4 5

3 9 . 6 9 3 9 .6 9

1 0 . 3 6 1 0 . 3 1

3 5 .3 6 3 5 . 3 6

1 4 .6 4 1 4 . 6 4

3 1 . 5 0

1 8 . 5 0

2 8 . 0 7

2 1 . 9 3

G

F

ED C

B A

39.690.89091.122450du 31310A !vv!! 14.6435.36-50,0)-max(f TG !!! SK 2.665.45)e0.49270(0.5073efp-1pff0.0833-0.1t-r

duE !vv!!v( 9.9014.64)e0.49275.45(0.5073f 0.0833-0.1A !vv!

v4.496.96)e0.49272.16(0.5073f 0.0833-0.1D !vv!

v

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Example (continued) In practice, a smaller value oft, and

many more nodes, would be used.

DerivaGem shows

steps 5 30 50 100 500

f0 4.49 4.263 4.272 4.278 4.283

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Expressing the Approach Algebraically

? A_ aj1,i1j1,it-rj-ij0ji, fp-1pfe,duS-Kmaxf ( !

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Estimating Delta and OtherGreek Letters

deltaat time t

dS-uS

f-f

Sf

00

1,01,1!((!(

S0u2

S0u4

S0d2

S0d4

S0

S0u

S0dS0 S0

S0u2

S0d2

S0u3

S0u

S0d

S0d3

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gamma at time 2t

)dS-u0.5(S

dS-S

f-f

-S-uS

f-f

S

f2

0

2

0

200

2,02,1

02

0

2,12,2

2

2

!x

x!+

S0u2

S0u4

S0d2

S0d4

S0

S0u

S0dS0 S0

S0u2

S0d2

S0u3

S0u

S0d

S0d3

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theta

t2

f-f

t

f 0,02,1

(!x

x!5

S0u

2

S0u4

S0d2

S0d4

S0

S0u

S0d

S0 S0

S0u2

S0d2

S0u3

S0u

S0d

S0d3

f

2

1rS 22 rS !+(5

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Vega

Rho

WR

(!ff -

*

r

ff

(!

-*V

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Example

89.0 7

0.00

79.3 5

0.00

70.70 70.70

0.00 0.0062.99 62.99

0.64 0.00

56.12 56.12 56.12

2.16 1.30 0.00

50.00 50.00 50.00

4. 49 3.77 2.6 6

44.55 44.55 44.55

6. 96 6.38 5.45

39.6 9 39.6 9

10.36 10.31

35.36 35.36

14.64 14.6431.50

18.50

28.07

21. 93

G

F

ED C

B A

-0.4144.55-56.12

6.96-2.16

dS-uS

f-f

00

1,01,1 !!!(

0.0311.65

(-0.64)-(-0.24)

39.69)-0.5(62.99

39.69-50

10.36-3.77-

50-62.99

3.77-0.64

)dS-u0.5(S

-2

0

2

0

21

!!

!

((!+

daycalendarper-0.012or

yearper-4.30.08332

4.49-3.77

t2

f-f 0,02,1

!

!v

!(

!5

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Using the binomial tree for options on

indices, currencies, and futures contracts

As with Black-Scholes

For options on stock indices, q equals the

dividend yield on the index

For options on a foreign currency, q

equals the foreign risk-free rate

For options on futures contracts q = r

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Example

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Example

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Binomial model for a dividend-paying

stock

Known Dividend Yield

before

after

Several known dividend yields

i0,1,...,j,duSS j-ij0ji, !!

i0,1,...,j,d)u-(1SS j-ij0ji, !! H

j-ij

i0ji, d)u-(1SS H!

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Known Dollar Dividend

ik

i=k+1

i=k+2

i0,1,...,,duSS j-ij0ji, !!

i0,1,...,jD,-duSS j-ij0ji, !!

1-i0,1,...,jfor

D)d-du(SandD)u-du(SS j-1-ij0j-1-ij

0ji,

!

!

nodes.1mkn

rath

erth

a2)m(kare

th

ere

mkihenandnodes,1inrathertha2iarethere

!

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Simplify the problem

The stock price has two componentsa part that is

uncertain and a part that is the present value of all future

dividends during the life of the option.

Step 1A tree can be structured in the usual way to model .

Step 2By adding to the stock price at each nodes, thepresent value of future dividends, the tree can be converted

into model S.

*S

X

X

X

e(!

"(!( tihen,De-SS

tihen,SS

t)i-r(-*

*

X

X

X(!

"(!

( tihen,DeduSS

tihen,duSS

t)i-r(-j-ij*

0ji,

j-ij*

0ji,

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Example

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Control Variate Technique

1. Using the same tree to calculate both the

value of the American option and the

value of the European option .

2. Calculating the Black-Scholes price of the

European option .

3. This gives the estimate of the value of the

American option asEBSA -fff

Af

Ef

Sf

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Example

B-S model

4.08)(-)(e 12T

BS !! -dNS-dNKf 0-r

4 . 3 2E

f !A 4 . 4 9f !

4.254.32-4.084.49P0 !!

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Alternative procedures for

constructing trees

Instead of setting u =1/dwe can set each of

the 2 probabilities to 0.5 and

ttqr

ttqr

ed

eu

(W(W

(W(W

!

!

)2/(

)2/(

2

2

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Example

0.9703

ed

1.0098

eu

and0.5psetWe

0.25t,0.75T

0.04

0.10r,0.06r

0.795K,0.79S

]0.250.04-.250.0016/2)0-0.1-[(0.06

]0.250.04.250.0016/2)0-0.1-[(0.06

f

0

!

!

!

!

!

!(!

!

!!

!!

W

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Trinomial Trees

6

1

212

3

2

6

1

212

/1

2

2

2

2

3

W

W

(!

!

W

W

(

!

!! (W

rt

p

p

r

t

p

udeu

d

m

u

t

S S

Sd

Su

pu

pm

pd

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Adaptive mesh modelFiglewski and Gao,1999

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Time-dependent parameters

V/Nt)(t then

steps,timeNoftotalaisthereIf

stepth timetheofendtheist

treetheoflifetheisT,T(T)VDefine

maturitytaforyvolatilittheis(t)thatSuppos

:timeoffuctionamakeTo2

ofvalueforwardtheis(t)

rateinterestforwardtheis(t)

esetWe

:timeoffuctiona)(orandmakeTo1

2

2

t(t)]-(t)[

i

i

qg

f

a

rqr

ii

i

gf

f

!

!

! (

W

W

W

W

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Monte Carlo simulation

When used to value an option, Monte Carlo simulationuses the risk-neutral valuation result. It involves thefollowing steps:

1. Simulate a random path forS in a risk neutral world.2. Calculate the payoff from the derivative.

3. Repeat steps 1 and 2 to get many sample values ofthe payoff from the derivative in a risk neutral world.

4. Calculate the mean of the sample payoffs to get anestimate of the expected payoff.

5. Discount this expected payoff at risk-free rate to getan estimate of the value of the derivative.

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Monte Carlo simulation (continued)

In a risk neutral world the process for astock price is

We can simulate a path by choosing timesteps of length t and using the discreteversion of this

where is a random sample from J (0,1)

dS S

dtS

d z! Q W

ttSttStSttS ((!( IWQ )()()(-)(

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Monte Carlo simulation (continued)

TT2

)0(ln)T(ln

thenconstant,areandif

)()(or

2)(ln)(ln

isthisofversiondiscreteThe

2

ln

.nratherthalnestimatetoaccuratemoreisIt

2

2/

2

2

2

WIW

Q

WQ

WIW

Q

WW

Q

IWWQ

!

!(

((

!(

!

((

SS

etSttS

tttSttS

dzdtSd

SS

tt

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Derivatives Dependent on More than

One Market Variable

When a derivative depends on several

underlying variables we can simulate

paths for each of them in a risk-neutralworld to calculate the values for the

derivative

ttSttmttt iiiiiii ((!( IUUUU )()()()(

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Generating the Random Samples from

Normal Distributions

How to get two correlated samples 1

and 2 from univariate standard normal

distributions x1 and x2

n.correlatiooftcoefficientheishere

-1xx

x

2212

11

V

VVI

I

!

!

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Cholesky decomposition

ijfor,

1here

x

j

1k

ijjkik

i

1k

2

ik

i

1k

kiki

!

!

!

!

!

!

VEE

E

EI

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Number of Trials

Denote the mean by and the standard deviation

by .

The standard error of the estimate is

where M is the number of trials.

A 95% confidence interval for the price fof the

derivative is

To double the accuracy of a simulation, we must

quadruple the number of trials.

M

[

Mf

M

[Q[Q 1.961.96-

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Applications

Advantage

1. It tends to be numerically more efficient

increases linearlythan other procedures

increases exponentiallywhen there are

more stochastic variables.

2. It can provide a standard error for the

estimates.3. It is an approach that can accommodate

complex payoffs and complex stocastic

processes.

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Applications (continued)

An estimate for the hedge parameter is

Sampling through a Tree

x

ff*

(

-

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Variance reduction procedures

Antithetic Variable Techniques

standard error of the estimate is

Control Variate Technique

2

21 ff

f

!

BBAA ffff !**

M

[

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Variance reduction procedures

(continued)

Importance Sampling:

Stratified Sampling:

Moment Matching:

Using Quasi-Random Sequences:

)

0.5-

(1-

n

i

N

s

mi*i

-I

I !

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Finite difference methods

Define i,j as the

value of at time

i(twhen the stockprice is j(S

T=T/N;

S=Smax /M

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Implicit Finite Difference Method

Forward difference approximation

backward difference approximation

S

ff

S

f jiji

(

!xx ,1,

S

ff

S

f jiji

(!

x

x 1,,

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Implicit Finite Difference Methods(continued)

2

or

2

setwe

2

1In

2

,1,1,

2

2

1,,,1,

2

2

1,1,

2

222

SS

SSSS

SS

rS

SS

rSt

jijiji

jijijiji

jiji

(

!

(

(

(

!

(

!

!

xx

xx

xx

xx

Wxx

xx

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tj2

1-(r-q)j

2

1-c

rtj1b

tj2

1-(r-q)j

2

1a

ffcfbfa

jSt

ff

t

f

22

j

22

j

22

j

jijijjijjij

jiji

t

t

there

:obtaine

tandsetalsoeIf

,11,,1,

,,1

!

!

!

!

!(

!

xx

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Explicit Finite Difference Methods

:obtainwe

point)(atthearetheyaspoint)1(atthe

samethebetoassumedareandIf22

i,j,ji

SfSf

xxxx

2

1,11,11,

2

2

11,11,

2

2

SS

SS

jijiji

jiji

(

!

(

!

xx

xx

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Explicit Finite Difference Methods(continued)

1,1

*

,1

*

1,1

*

,

isequationdifferenceThe

! jijjijjijji cbaf

)t(1

1

)-(1

1

)t(-1

1where *

tj2

1(r-q)j

2

1

trc

tj1tr

b

tj2

1(r-q)j

2

1

tra

22*

j

22*

j

22

j

(

!(

!

(

!

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Explicit Finite Difference Methods(continued)

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Difference between implicit and

explicit finite difference methods

i+1,j+1

i,j i+1,j

i+1,j1

i+1,ji,j

i,j 1

i,j+1

Implicit Method Explicit Method

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Change of Variable

Z

2

1

Z

)

2-q-(

,lnZDefine

2

22

2

rrt

S

!

!

xx

WxxW

xx

2

j

2

j

2

j

jijijjijjij

2

)-(r-q2

r1

2

)-(r-q2

ffff

2

2

2

2

2

,11,,1,

Z

t

2

-

Z

t-

tZ

t

Z

t

2

-

Z

there

becomesmethodimplicitfortheequationdifferenceThe

(

(

(

(!

(

(!

(

(

(

(!

!

WK

F

WE

KFE

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Change of Variable (continued)

1,1

*

,1

*

1,1

*

,

becomesmethodexplicitfortheequationdifferenceThe

! jijjijjijjif KFE

)Z

t)

2-(

Z

t(

1

1

)

Z

t-(

1

1

)Z

t)

2-(

Z

t(-

1

1where

2

2

2

2

2*

2*

j

2*

j

2

j

2

r-q2tr

1

tr

2

r-q2tr

((

((

(!

(

(

(!

((

((

(!

WK

WE

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Relation to Trinomial Tree Approaches

The three probabilities sum to unity.

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Relation to Trinomial Tree Approaches(continued)

2 2

2

2

2

2

2

1- j tisnegative henj 13.

Wecanusechange-of-variableapproch:

t t - ( - )

Z 2 Z

t -

Zt t

( - )Z 2 Z

2

u

2

m

2

d

p r - q 2 2

p 1

p r - q 2 2

W

W

W

( u

( (! ( (

(!

(( (

! ( (

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Other Finite Difference Methods

Hopscotch method

Crank-Nicolson scheme

Quadratic approximation

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Summary

We have three different numerical procedures for valuingderivatives when no analytic solution: trees, Monte Carlosimulation, and finite difference methods.

Trees: derivative price are calculated by starting at theend of the tree and working backwards.

Monte Carlo simulation: works forward from thebeginning, and becomes relatively more efficient as thenumber of underlying variables increases.

Finite difference method: similar to tree approaches. Theimplicit finite difference method is more complicated buthas the advantage that does not have to take any specialprecautions to ensure convergence.

5535 chapter 19

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