The Evaluation Of Discretely Monitored Barrier Option Prices Under Svjd Model Using Fourier Transform Techniques

A derivative is a financial instrument which is constructed from other more basicunderlying assets. With the dramatic growth of the derivatives markets, options becomemore and more complicated, and various exotics have been designed and issued byfinancial engineer, such as package, compound option and barrier option, which arenecessary to traders of derivatives. Also, as the advanced products of market economy,options will be released soon with the development of China, and option pricing is acore issue when options arrive.The traditional Black-Scholes option pricing formula has been widely used foroption pricing and hedging, but this model fails to reflect some empirical phenomena,such as larger random fluctuations, non-normal features of log-return and non-constantvolatility. So numerous option pricing models are derived based on relaxing the BSassumptions about the stochastic process of interest rate and underlying asset price. Themain aspect of improvement for BS model is the stochastic process of underlying assetprice, which can be divided into first-order improvement and second-orderimprovement. The first-order improvement is:(1) adding stochastic volatility into thestochastic process of the underlying asset price, and (2) adding jumps into the stochasticprocess of the underlying asset price; the second-order improvements:(1) addingjumps into the stochastic volatility process,(2)assuming jumps follow doubleexponential distribution instead of normal distribution,(3) assuming a nonzerocorrelation coefficient between the jump parts of the stochastic volatility process and thestochastic process of underlying asset price.As one of exotics, barrier option’s payoff depends on whether or not the asset pricereach specified level during the life of the option, that is to say, it can be activated orextinguished when the price of the underlying asset across certain level. Most modelsfor pricing barrier options assume continuous monitoring of the barrier, under thisassumption, the option can often be priced in closed form. However, in practice, many(if not most) barrier options trades in markets are discretely monitored. There areessentially no formulas for pricing these options, and even numerical pricing is difficult.This paper considers the problem of evaluating barrier option prices when thedynamics of the underlying are driven by stochastic volatility and jump diffusion process. Firstly, we present a model of SVJD, which can reflects larger randomfluctuations, non-normal features of log-return and non-constant volatility, and we canalso present the characteristic function of this model.Secondly, we develop two different methods of pricing formulae by Fouriertransform. The first approach is representing the probability density function of theunderlying’ s process by Fourier transform, and then reducing the value of options bynumerical integration. The second approach is getting the value of options directly byFourier transform. Then we discretize the formula deduced by the second approach andfind out that the expression can be easily applied by computer with the Fast Fouriertransform (FFT) algorithm. Then we compare the result of FFT method with the resultof Monte Carlo method and conclude that their results are very similar, but FFT methodis quicker than Monte Carlo method.Finally, we evaluate discretely monitored barrier options with payoff functions thatare dependent on two different spot values in time using FFT method. Then we comparethe results of the model with different jump diffusion processes and different barriers,and find that the result of the model with higher jump frequency is larger than the resultof the model with lower jump frequency, because the underlying asset price is morefluctuant. Also, we find that the prices of the option with lower barriers is higher thanthe option with higher barriers.