Option Pricing In Markov-modulated Exponential Lévy Model

Options are indispensable risk management tools in the capital market,and option pricing is one of the core issues in financial mathematics and one of the most challenging tasks.Recently,the classical Black-Scholes model has been maturely developed.Owing to the inherent shortcomings of the Black-Scholes model,a large number of studies tried to promote it to the models based on Lévy process.Furthermore,in order to relax the independent and stationary increment of Lévy process and make it more joint at the real market,the Markov regime-switching Lévy model has gradually aroused increasingly attention.This paper mainly discusses the problem of pricing options whose underlying assets prices dynamics follow the Markov-modulated exponential Lévy models.Then the pricing of several options is discussed concretely.Firstly,we consider the problem of pricing European options,namely vanilla option,binary option and exchange option,whose underlying assets prices dynamics follow Markov-modulated exponential Lévy models with stochastic interest rates.We assume that the interest rates are driven by Markov-modulated Hull-White process.The integral representations for the option prices are derived via Fourier transform(FT)technique.Then we chiefly concentrate on the 3-state case,by using the built-in quadgk function in MATLAB,some corresponding numerical results for Merton jump-diffusion model are achieved.The numerical results demonstrate that the option prices for state 1 and state 2 are also relatively close to MMLP,whereas the state 3 yields much higher prices,which can be explained by the assumption of more sharply fluctuation when the economic condition in state 3.The effect of interest rate variability over short time horizons is negligible,but it is significant over longer time horizons for vanilla option and binary option.The influence for exchange option whether the interest rate is stochastic or not is slighter than the other two options.Moreover,the higher proportion of price differentiation lead to much lower prices for exchange option.Secondly,we consider the valuation of single and double barrier knock-out call options in a Markov-modulated Black-Scholes model with specific rebates.The integral representations for the option prices are derived via Fourier transform andmatrix Wiener-Hopf factorizations.We concentrate on the 2-state case where the matrix Wiener-Hopf factorization is available analytically,then some corresponding numerical results are achieved,which demonstrate that the option price with rebate is higher than that without rebate.Besides,when the initial price 0is close to the barrier,it will bring about higher option price with rebate,which can be explained by the closer to barrier the initial price is,the more risk the investor will face,therefore the more amount of rebates should be paid.Furthermore,from a theoretical viewpoint,extending the model used in Chapter 4 to jump-diffusion models is rather straightforward and can be achieved by the same techniques.Nevertheless,from a numerical viewpoint,the decomposition and transformation for the barrier options,to some extent,lead to more tedious numerical calculations.At last,we consider the problem of pricing European-style discretely monitored geometric Asian options under Markov-modulated Lévy model.We first derive the characteristic function of the geometric average of the underlying asset prices based on a matrix exponentiation.It is worth noting that there exists an exact formula for computing the matrix exponentiation for a two-state case,however,for more than two states,we introduce the Carathéodory–Fejér approximations.Then we develop an efficient pricing method,namely modified Fourier cosine series(M-COS)method,based on Fourier cosine expansions(COS)to evaluate the option prices.As discussed of the error analysis,the M-COS method converges exponentially.We employ the two states Markov-modulated Merton jump-diffusion model,Markovmodulated CGMY model and Markov-modulated VG model to accomplish numerical illustrations.The various numerical examples demonstrate that the M-COS method and COS method are faster than Carr-Madan method and the CPU time does not increase significantly with the increase of monitoring date's quantity.Although COS method is also faster than M-COS method,the accuracy is lower and it is more significant for MM-CGMY model.