Using Finite Difference Method to evaluate Option Price Phuong Anh Ha Nguyen Fin 545 - PowerPoint PPT Presentation

Loading...

PPT – Using Finite Difference Method to evaluate Option Price Phuong Anh Ha Nguyen Fin 545 PowerPoint presentation | free to view - id: 97c4c-YzQyM



Loading


The Adobe Flash plugin is needed to view this content

Get the plugin now

View by Category
About This Presentation
Title:

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

Description:

What is Finite Difference Method - A method for solving ordinary differential equations in problem regions with simple boundaries. ... – PowerPoint PPT presentation

Number of Views:245
Avg rating:3.0/5.0
Slides: 28
Provided by: busi305
Category:

less

Write a Comment
User Comments (0)
Transcript and Presenter's Notes

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
About PowerShow.com