Pricing Exotic Option Under Jump-Diffusion Models by the Quadrature Method
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Pricing Exotic Option Under Jump‑Diffusion Models by the Quadrature Method Jin‑Yu Zhang1 · Wen‑Bo Wu2 · Yong Li3 · Zhu‑Sheng Lou2 Accepted: 22 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract This paper extends the quadrature method to price exotic options under jumpdiffusion models. We compute the transition density of jump-extended models using convolution integrals. Furthermore, a simpler and more efficient lattice grid is introduced to implement the recursion more directly in matrix form. It can be shown that a lot of running time can be saved. At last, we apply the developed approach to the different jump-extended models to demonstrate its universality and provide a detailed comparison for the discrete path-dependent options to demonstrate its advantages in terms of speed and accuracy. Keywords Finance · Discrete path-dependent options · Quadrature · Jump-diffusion model · Option hedging JEL Classification G13 · C63
1 Introduction In the empirical finance, it is well documented that the jump phenomenon has appeared in the returns of various assets. For example, Bates (1996) found that jumps in the exchange rate process are able to explain the volatility smile in the Deutsche Mark option. Johannes (2004) provided the evidence of jumps in shortterm interest rate models and showed that they played an important role in option pricing. The pricing and hedging of exotic options are of great importance to modern derivative markets. In the literature, numerous methods have been developed for this * Yong Li [email protected] 1
School of Finance, Nanjing Audit University, Nanjing 210000, China
2
Hanqing Advanced Institute of Economics and Finance, Renmin University of China, Beijing 100872, China
3
School of Economics, Renmin University of China, Beijing 100872, China
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purpose. Typical examples are including Figlewski and Gao (1999), Boyle and Tian (1999), Fusai and Meucci (2008), Cai et al. (2015), and Roul (2020). Hence, how to price options with incorporating jump components aroused a lot of interest, see Kadalbajoo et al. (2016), Fu et al. (2016), Mollapourasl et al. (2018), Yousuf et al. (2018), etc. Most numerical methods for pricing options, especially the tree methods, are based on diffusion models. Unfortunately, it seems somewhat difficult to apply the tree methods directly to the jump-diffusion process because the transformation of Nelson and Ramaswamy (1990) required by tree methods is often complex when jumps are included. Amin (1993) established a tractable discrete time model for option pricing by constructing multivariate jumps superimposed on the binomial model. Hilliard and Schwartz (2005) developed a robust bivariate tree approach for option pricing under the jump-diffusion process. However, these methods are difficult to be extended beyond the Black–Scholes framework into different models with jump components. Recently, Beliaeva and Nawalkha (2012) proposed the mixed jump-diffusion trees approach fo
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