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## Using Finite Difference Method to evaluate Option Price Phuong Anh Ha Nguyen Fin 545

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Title: Using Finite Difference Method to evaluate Option Price Phuong Anh Ha Nguyen Fin 545

1
Using Finite Difference Method to evaluate
Option Price Phuong Anh Ha NguyenFin 545
2
Option Valuation _ FDM
• The Black-Scholes PDE
• Option Price Using Finite Difference Method
• - Explicit FDM
• - Implicit FDM
• Matlab Program
• Arbitrage transaction Bond Option

3
Option
• An Option is the right to buy and sell an
asset at a particular date at a particular price
( the strike price X ).
• The purchaser of an option will only exercise
the option if it is profitable.

4
Black-Scholes PDE
• Geometric Brownian Motion
• Ito Lemma
• Black-Scholes FDE for Call Option

5
Why using FDM? What is FDM??
• What is Finite Difference Method
• - A method for solving ordinary differential
equations in problem regions with simple
boundaries.
• - Requires the construction of a mesh defining
local coordinate surfaces. For each node of this
mesh, the unknown function values are found,
replacing the differential equations by
difference equations

6
Call Option Evaluation
• Stable Solution
• y lnS and w(y,t) c(S,t)

7
• Time
to expiration
• lnS t t -?t t - 2?t
2?t ?t 0
• e 0 0 0
0 0 0
0
• ?y

max(0,St X)
• 2?y

max(0,St X)

• max(0,St X)
• y -2?y

max(0,St X)
• y ?y

max(0,St X)
• y

max(0,St X)
• y ?y

max(0,St X)

Explicit
Implicit
8
Implicit FDE for Call Option
w(y,t?t) ?1 w(y-?y,t) ?2 w( y ,t) ?3
w(y?y,t)
• w(y-?y,t)

• w(y,t)
w(y,t?t)( known value )

• w(y?y,t)

?1
?2
?3
9
Implicit FDM
• Using Black-Scholes PDE

w(y,t?t) ?1 w(y-?y,t) ?2 w( y ,t) ?3
w(y?y,t)
10
• Time
to expiration
• lnS t t -?t t - 2?t
2?t ?t 0
• e 0 0 0
0 0 0
0
• ?y

max(0,St X)
• 2?y

max(0,St X)

• max(0,St X)
• y -2?y

max(0,St X)
• y ?y

max(0,St X)
• y

max(0,St X)
• y ?y

max(0,St X)

?1
Implicit
?2
?3
When S ? 8 , ?c(S,t)/?S 1 for all t
11

1 0 0 0 .0
0 0 ?1 ?2 ?3 0 0
0 0 0 ?1 ?2 ?3 0
0 0 0 0
0 ?1 ?2 ?3 0
0 0 ... 0 -1
1 K1000x1000
x
0 m2,1000 m3,1000 . m999,100 B1000-B999 A100010
00x1
m1,999 m2,999 m3,999 . m999,999 m1000,999 a9991
000x1

a K-1 x A
12
Bond Option
• On 11/30/2007
• Bond option
• 1 Year Option . ( Expiration Date 11/30/2008 )
• Bond 5 year Tbond ( maturing on 8/15/2013)
• Yield 3.5, Coupon 5

13
Arbitrage Transaction
• What is the Price of the underlying good in the
option?
• Forward Price
• F ( B I ) x ert
• B Bond Price
• I Present Value of Coupon that will be paid
during the life of the option .

14
QUESTION ??????
• What is B?
• 5 year Tbond or 6 year Tbond price
• Price of 6 year Tbond.
• - Is B quoted price or invoice price ?
• Invoice Price

15
Question
• Spot asset at time t
• Forward price at time t

16
Bond Option
• -Modified the volatility
• -Could use the Finite different method

17
Matlab Program
Inputs X exercise bond price b current
bond price B(1) lowest stock price Y(1)
natural log B(1) Sigma bond volatility R Risk
free rate dt ?t dy ?y
18
Matlab Program
• Input
• t 365 ( time to maturity is 365 days )
• r 1000
• c 1000
• dt ( t/365)/(c/2)
• X 107.91
• B108.8669
• sigma 0.15284
• R .0317

19
Matlab Program
• Matrix
• m zeros(r,c)
• y zeros(r,1)
• B zeros(r,1)
• wr zeros(r,1)
• k zeros(r,r)
• a zeros(r,1)
• T zeros(1,c)
• p zeros(r,c)
• deltay((log(b)/(r/21) 0.001((log(b)/(r/10))

20
Matlab Program
• Change dy
• for n 1length(deltay)
• dy deltay(n)
• Time structure and Price structure matrix
• for ic-12
• T(c)0
• T(i-1)T(i)dt
• end
• for i2r
• B(1) 1
• y(1) log(B(1))
• y(i)y(i-1)dy
• B(i)exp(y(i))
• end

21
Matlab Program
• boundary matrix
• for i1r
• wr(i) B(i)-X
• m(i,c) max(wr(i),0)
• end
• grid equation
• ?1 .5(R-sigma2/2)dt/dy-0.5sigma2
dt/dy2
• ?2 1 sigma2dt/dy2 Rdt
• ?3 -0.5(R-sigma2/2)dt/dy -
.5sigma2dt/dy2

22
Matlab Program
• Matrix BETA
• k(1,1) 1
• k(r,r-1) -1
• k(r,r) 1
• for i 2r-1
• k(i,i-1) ?1
• k(i,i) ?2
• k(i,i1) ?3
• end
• Matrix a
• for i c-12
• a m(1r,i)
• a(1) 0
• a(r) B(r)-B(r-1)
• m(1r,i-1) inv(k) a
• end

23

1 0 0 0 .0
0 0 ?1 ?2 ?3 0 0
0 0 0 ?1 ?2 ?3 0
0 0 0 0
0 ?1 ?2 ?3 0
0 0 ... 0 -1
1 K1000x1000
x
m1,1000 m2,1000 m3,1000 . m999,1000 B1000-B999
A10001000x1
m1,999 m2,999 m3,999 . m999,999 m1000,999 a9991
000x1

a K-1 x A
24
Matlab Program
• Check min(m)
• if (min(m())gt 0)
• disp(sprintf('dy e ',dy ))
• break
• end
• end
• display call put
• for i 1r
• for j 1c
• p(i,j)m(i,j)-B(i)Xexp(R(-T(j)))
• end
• end
• if (dy (log(B)/(r/21))
• disp(sprintf( 'time to maturitye, stock
price e,call price e, put price e'
,(T(500)365),B(500), m(500,500), p(500,500)))
• end

25
Matlab Program
• Black Scholes Modified Fomula
• for i1r
• d1(i)(log(S(i)/X).5sigma2t/365)/(sigm
a(t/365).5)
• d2(i)d1(i)-sigma(t/365).5
• bslcall(i)exp(-Rt/365)(S(i)normcdf(d1(
i))- Xnormcdf(d2(i)))
• bslput(i)bslcall(i)-S(i)Xexp(R(-t/365)
)
• end
• disp(sprintf('blackschole Call optione, Put
option e',bslcall(500),bslput(500)))

26
Matlab Program
• Graph
• for i 1r
• bond(i) B(i)
• option(i) m(i,500)
• end
• plot(bond,option)
• hold on
• plot(bond,bslcall)
• title ('call option maturing at time
11/30/2008')
• xlabel(bond price')
• ylabel('option value')
• gtext('FDM')
• gtext('Black')
• hold off

27
BOND OPTION ARBITRAGE