TY - JOUR
T1 - Chapter 7 Variational Methods in Derivatives Pricing
AU - Feng, Liming
AU - Kovalov, Pavlo
AU - Linetsky, Vadim
AU - Marcozzi, Michael
N1 - Funding Information:
This research was supported by the National Science Foundation under grants DMI-0422937 and DMI-0422985.
PY - 2007
Y1 - 2007
N2 - When underlying financial variables follow a Markov jump-diffusion process, the value function of a derivative security satisfies a partial integro-differential equation (PIDE) for European-style exercise or a partial integro-differential variational inequality (PIDVI) for American-style exercise. Unless the Markov process has a special structure, analytical solutions are generally not available, and it is necessary to solve the PIDE or the PIDVI numerically. In this chapter we briefly survey a computational method for the valuation of options in jump-diffusion models based on: (1) converting the PIDE or PIDVI to a variational (weak) form; (2) discretizing the weak formulation spatially by the Galerkin finite element method to obtain a system of ODEs; and (3) integrating the resulting system of ODEs in time. To introduce the method, we start with the basic examples of European, barrier, and American options in the Black-Scholes-Merton model, then describe the method in the general setting of multi-dimensional jump-diffusion processes, and conclude with a range of examples, including Merton's and Kou's one-dimensional jump-diffusion models, Duffie-Pan-Singleton two-dimensional model with stochastic volatility and jumps in the asset price and its volatility, and multi-asset American options.
AB - When underlying financial variables follow a Markov jump-diffusion process, the value function of a derivative security satisfies a partial integro-differential equation (PIDE) for European-style exercise or a partial integro-differential variational inequality (PIDVI) for American-style exercise. Unless the Markov process has a special structure, analytical solutions are generally not available, and it is necessary to solve the PIDE or the PIDVI numerically. In this chapter we briefly survey a computational method for the valuation of options in jump-diffusion models based on: (1) converting the PIDE or PIDVI to a variational (weak) form; (2) discretizing the weak formulation spatially by the Galerkin finite element method to obtain a system of ODEs; and (3) integrating the resulting system of ODEs in time. To introduce the method, we start with the basic examples of European, barrier, and American options in the Black-Scholes-Merton model, then describe the method in the general setting of multi-dimensional jump-diffusion processes, and conclude with a range of examples, including Merton's and Kou's one-dimensional jump-diffusion models, Duffie-Pan-Singleton two-dimensional model with stochastic volatility and jumps in the asset price and its volatility, and multi-asset American options.
UR - http://www.scopus.com/inward/record.url?scp=77950495349&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=77950495349&partnerID=8YFLogxK
U2 - 10.1016/S0927-0507(07)15007-6
DO - 10.1016/S0927-0507(07)15007-6
M3 - Review article
AN - SCOPUS:77950495349
SN - 0927-0507
VL - 15
SP - 301
EP - 342
JO - Handbooks in Operations Research and Management Science
JF - Handbooks in Operations Research and Management Science
IS - C
ER -