Abstract
We present an accurate and efficient finite difference method for solving the Black–Scholes (BS) equation without boundary conditions. The BS equation is a backward parabolic partial differential equation for financial option pricing and hedging. When we solve the BS equation numerically, we typically need an artificial far-field boundary condition such as the Dirichlet, Neumann, linearity, or partial differential equation boundary condition. However, in this paper, we propose an explicit finite difference scheme which does not use a far-field boundary condition to solve the BS equation numerically. The main idea of the proposed method is that we reduce one or two computational grid points and only compute the updated numerical solution on that new grid points at each time step. By using this approach, we do not need a boundary condition. This procedure works because option pricing and computation of the Greeks use the values at a couple of grid points neighboring an interesting spot. To demonstrate the efficiency and accuracy of the new algorithm, we perform the numerical experiments such as pricing and computation of the Greeks of the vanilla call, cash-or-nothing, power, and powered options. The computational results show excellent agreement with analytical solutions.
| Original language | English |
|---|---|
| Pages (from-to) | 1-12 |
| Number of pages | 12 |
| Journal | Computational Economics |
| DOIs | |
| Publication status | Accepted/In press - 2017 Jan 25 |
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Keywords
- Black–Scholes equation
- Far field boundary conditions
- Finite difference method
ASJC Scopus subject areas
- Economics, Econometrics and Finance (miscellaneous)
- Computer Science Applications
Cite this
Finite Difference Method for the Black–Scholes Equation Without Boundary Conditions. / Jeong, Darae; Yoo, Minhyun; Kim, Junseok.
In: Computational Economics, 25.01.2017, p. 1-12.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Finite Difference Method for the Black–Scholes Equation Without Boundary Conditions
AU - Jeong, Darae
AU - Yoo, Minhyun
AU - Kim, Junseok
PY - 2017/1/25
Y1 - 2017/1/25
N2 - We present an accurate and efficient finite difference method for solving the Black–Scholes (BS) equation without boundary conditions. The BS equation is a backward parabolic partial differential equation for financial option pricing and hedging. When we solve the BS equation numerically, we typically need an artificial far-field boundary condition such as the Dirichlet, Neumann, linearity, or partial differential equation boundary condition. However, in this paper, we propose an explicit finite difference scheme which does not use a far-field boundary condition to solve the BS equation numerically. The main idea of the proposed method is that we reduce one or two computational grid points and only compute the updated numerical solution on that new grid points at each time step. By using this approach, we do not need a boundary condition. This procedure works because option pricing and computation of the Greeks use the values at a couple of grid points neighboring an interesting spot. To demonstrate the efficiency and accuracy of the new algorithm, we perform the numerical experiments such as pricing and computation of the Greeks of the vanilla call, cash-or-nothing, power, and powered options. The computational results show excellent agreement with analytical solutions.
AB - We present an accurate and efficient finite difference method for solving the Black–Scholes (BS) equation without boundary conditions. The BS equation is a backward parabolic partial differential equation for financial option pricing and hedging. When we solve the BS equation numerically, we typically need an artificial far-field boundary condition such as the Dirichlet, Neumann, linearity, or partial differential equation boundary condition. However, in this paper, we propose an explicit finite difference scheme which does not use a far-field boundary condition to solve the BS equation numerically. The main idea of the proposed method is that we reduce one or two computational grid points and only compute the updated numerical solution on that new grid points at each time step. By using this approach, we do not need a boundary condition. This procedure works because option pricing and computation of the Greeks use the values at a couple of grid points neighboring an interesting spot. To demonstrate the efficiency and accuracy of the new algorithm, we perform the numerical experiments such as pricing and computation of the Greeks of the vanilla call, cash-or-nothing, power, and powered options. The computational results show excellent agreement with analytical solutions.
KW - Black–Scholes equation
KW - Far field boundary conditions
KW - Finite difference method
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UR - http://www.scopus.com/inward/citedby.url?scp=85010739187&partnerID=8YFLogxK
U2 - 10.1007/s10614-017-9653-0
DO - 10.1007/s10614-017-9653-0
M3 - Article
AN - SCOPUS:85010739187
SP - 1
EP - 12
JO - Computational Economics
JF - Computational Economics
SN - 0927-7099
ER -