Download presentation

Presentation is loading. Please wait.

Published byJoel Lindsey Modified over 2 years ago

1
1 Richardson Extrapolation Techniques for the Pricing of American-style Options Chuang-Chang Chang, San-Lin Chung 1 and Richard C. Stapleton 2 This Draft: March, 2002 1 Department of Finance, The Management School, National Central University, Chung-Li 320, Taiwan. Tel: 886-3-4227424, Fax: 886-3- 4252961, Email: chungs@mgt.ncu.edu.tw 2 Department of Accounting and Finance, Strathclyde University, United Kingdom. Email: rcs@staplet.demon.co.uk 1 Department of Finance, The Management School, National Central University, Chung-Li 320, Taiwan. Tel: 886-3-4227424, Fax: 886-3- 4252961, Email: chungs@mgt.ncu.edu.tw 2 Department of Accounting and Finance, Strathclyde University, United Kingdom. Email: rcs@staplet.demon.co.uk

2
2 Introduction Among many analytical approximation methods, Geske and Johnson (1984) method is widely used in the literature. Geske and Johnson (1984) showed that it was possible to value an American-style option by using a series of options exercisable at one of a finite number of exercise points (known as Bermudan- style options). They employed Richardson extrapolation techniques to derive an efficient computational formula using the values of Bermudan options.

3
3 Two problems in the Geske and Johnson (1984) method: the problem of non-uniform convergence it is difficult to determine the accuracy of the approximation

4
4 Literature Review Geske and Johnson (1984) Geske and Johnson (1984) show that an American put option can be estimated with a high degree of accuracy using a Richardson approximation. If P(n) is the price of a Bermudan-style option exercisable at one of n equally-spaced exercise dates, then, for example, using P(1), P(2) and P(3), the price of the American put is estimated by

5
5 where P(1,2,3) denotes the approximated value of the American option using the values of Bermudan options with 1, 2 and 3 possible exercise points. Omberg (1987) Omberg (1987) shows that there may be a problem of non-uniform convergence since P(2) in equation (1) is computed using exercise points at time T and T/2, where T is the time to maturity of option, and P(3) is computed using exercise points at time T/3, 2T/3, and T.

6
6 Bunch and Johnson (1992) Bunch and Johnson (1992) suggest a modification of the Geske-Johnson method based on the use of an approximation: where P max (2) is the value of an option exercisable at two points at time, when the exercise points are chosen so as to maximize the option’s value.

7
7 Broadie and Detemple (1996) Due to the uniform convergence property of the BBS method, Broadie and Detemple (1996) also suggest a method termed the Binomial Black and Scholes model with Richardson extrapolation (BBSR). In particular, the BBSR method with n steps computes the BBS prices corresponding to m=n/2 steps (say P m ) and n steps (say P n ) and then sets the BBSR price to P=2P n -P m.

8
8 The Repeated-Richardson Extrapolation Algorithm Assume that with known exponents and, but unknown a 1, a 2, a 3, etc., where denotes a quantity whose size is proportional to, or possibly smaller. According to Schmidt (1968), we can establish the following algorithm when,.

9
9 Algorithm: For i=1,2,3..., set A i,0 =F(h i ), and compute for m=1, 2, 3, …, k-1. where A i,m is an approximate value of a 0 obtained from an m times Repeated-Richardson extrapolation using step sizes of, and.

10
10 The computations can be conveniently set up in the following scheme: The Geske-Johnson Formulae

11
11 The Modified Geske-Johnson Formulae The Error Bounds and Predictive Intervals of American Option Values Schmidt’s Inequality Schmidt (1968) shows that it always holds that when i is sufficiently large

12
12 Numerical Results

13
13

14
14

15
15

16
16

17
17

18
18

19
19

20
20

21
21

22
22

23
23

Similar presentations

Presentation is loading. Please wait....

OK

Numerical Computation

Numerical Computation

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on power grid india Ppt on media revolution help Ppt on wind power plant Ppt on panel discussion rubric Ppt on cse related topics about information Ppt on global marketing management Upload and view ppt online training Ppt on statistics in maths what is the factor Ppt on standing order mandate Ppt on chapter 3 atoms and molecules activities