Abstract
In this paper, the valuation of the discrete barrier options on the condition that the underlying asset price process follows the GARCH volatility and double exponential jump is studied. We derived an analytical approximation of the characteristic function for the underlying log-asset price. Then, a quasianalytical approximate formula of the price of the discrete barrier option is obtained based the on Fourier-cosine method. Numerical examples show that the Fourier-cosine method is fast and efficient for pricing discrete barrier options compared with the Monte Carlo simulation method. Finally, the influences of some important parameters on the prices of discrete barrier options are studied to further illustrate the rationality of the model.
Highlights
Since it was proposed in 1973, the well-known BlackScholes-Merton (BS) model [1] is often applied in financial industries because the analytic solution of European option under the risk-neutral measure has been obtained by assuming that the underlying price process follows the geometric Brownian motion
The gross market value of the Over-the-Counter (OTC) barrier options accounted for a significant proportion in the US foreign exchange option market in 2018
Many participants maintain considerable barrier option portfolios, which call for associated valuation and risk management tools for these securities in foreign exchange markets
Summary
Since it was proposed in 1973, the well-known BlackScholes-Merton (BS) model [1] is often applied in financial industries because the analytic solution of European option under the risk-neutral measure has been obtained by assuming that the underlying price process follows the geometric Brownian motion. Feng and Linetsky [6] proposed the fast Hilbert transform method to obtain the price of the discretely monitored barrier options under Levy models. Liu and Zhang [11] proposed a novel jumpdiffusion model in present of liquidity risk and derived an approximate solution for the valuation of the discrete barrier options. These mentioned models are assumed that the volatility is constant. We obtain an approximate price for the discretely monitored barrier option using the Fourier-cosine method proposed by Fang and Oosterlee [18].
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