scholarly journals A numerical approach to pricing exchange options under stochastic volatility and jump-diffusion dynamics

2021 ◽  
pp. 1-30
Author(s):  
Len Patrick Dominic M. Garces ◽  
Gerald H. L. Cheang
2009 ◽  
Vol 12 (03) ◽  
pp. 393-425 ◽  
Author(s):  
CARL CHIARELLA ◽  
BODA KANG ◽  
GUNTER H. MEYER ◽  
ANDREW ZIOGAS

This paper considers the problem of numerically evaluating American option prices when the dynamics of the underlying are driven by both stochastic volatility following the square root process of Heston [18], and by a Poisson jump process of the type originally introduced by Merton [25]. We develop a method of lines algorithm to evaluate the price as well as the delta and gamma of the option, thereby extending the method developed by Meyer [26] for the case of jump-diffusion dynamics. The accuracy of the method is tested against two numerical methods that directly solve the integro-partial differential pricing equation. The first is an extension to the jump-diffusion situation of the componentwise splitting method of Ikonen and Toivanen [21]. The second method is a Crank-Nicolson scheme that is solved using projected successive over relaxation and which is taken as the benchmark for the price. The relative efficiency of these methods for computing the American call option price, delta, gamma and free boundary is analysed. If one seeks an algorithm that gives not only the price but also the delta and gamma to the same level of accuracy for a given computational effort then the method of lines seems to perform best amongst the methods considered.


2011 ◽  
Vol 18 (3) ◽  
pp. 245-276 ◽  
Author(s):  
Gerald H. L. Cheang ◽  
Carl Chiarella

Author(s):  
Tumellano Sebehela

The stock jumps of the underlying assets underpinning the Margrabe options have been studied by Cheang and Chiarella [Cheang, GH and Chiarella C (2011). Exchange options under jump-diffusion dynamics. Applied Mathematical Finance, 18(3), 245–276], Cheang and Garces [Cheang, GHL and Garces LPDM (2020). Representation of exchange option prices under stochastic volatility jump-diffusion dynamics. Quantitative Finance, 20(2), 291–310], Cufaro Petroni and Sabino [Cufaro Petroni, N and Sabino P (2020). Pricing exchange options with correlated jump diffusion processes. Quantitate Finance, 20(11), 1811–1823], and Ma et al. [Ma, Y, Pan D and Wang T (2020). Exchange options under clustered jump dynamics. Quantitative Finance, 20(6), 949–967]. Although the authors argue that they explored stock jumps under Hawkes processes, those processes are the Poisson process in their applications. Thus, they studied Hawkes processes in-between two assets while this study explores Hawkes process within any asset. Furthermore, the Poisson process can be flipped into Hawkes process and vice versa. In terms of hedging, this study uses specific Greeks (rho and phi) while some of the mentioned studies used other Greeks (Delta, Theta, Vega, and Gamma). Moreover, hedging is carried out under static and dynamic environments. The results illustrate that the jumpy Margrabe option can be extended to complex barrier option and waiting to invest option. In addition, hedging strategies are robust both under static and dynamic environments.


2015 ◽  
Vol 22 (5) ◽  
pp. 450-462 ◽  
Author(s):  
Gerald H. L. Cheang ◽  
Guanghua Lian

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