# A Fourier-Cosine Method for Pricing Discretely Monitored Barrier Options under Stochastic Volatility and Double Exponential Jump

Authors: Shoude Huang; Xunxiang Guo
Journal: Mathematical Problems in Engineering
 (2020)

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## 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.


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## Body

## 1. Introduction

Since it was proposed in 1973, the well-known Black-Scholes-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. In order to hedge and manage the risk, more options were created, such as American options and exotic options. Barrier option is one of the most actively traded exotic options in international financial derivative markets because of lower cost. For example, 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. And many participants maintain considerable barrier option portfolios, which call for associated valuation and risk management tools for these securities in foreign exchange markets.In view of the important role of barrier options, barrier option pricing is a significant problem in the theoretical researches and applications. Under the BS model framework, closed-form solutions for all kinds of European style barrier options have been obtained [2–4]. But many studies have shown that jump exists in the prices of various financial assets, such as stocks and commodities, especially after financial crises or major natural disasters happen, while the above-mentioned literatures did not take jumps into account. Kou and Wang [5] proposed a jump-diffusion model and obtained analytical solution of the European style barrier options using the Laplace transform method, where the jump size is assumed to follow a double exponential distribution.In financial engineering, the discretely monitored barrier options are becoming a research focus. A growing number of researchers are becoming interested in the valuation of discretely monitored options. Feng and Linetsky [6] proposed the fast Hilbert transform method to obtain the price of the discretely monitored barrier options under Lévy models. Fusai, Germano, and Marazzina [7] utilized Wiener-Hopf factorization to obtain the price of discretely monitored barrier options under exponential Lévy models. Lian, Zhu, Elliott, and Cui [8] proposed a semianalytical solution for the valuation of the discrete barrier options given that the underlying asset price process was driven by a time-dependent Lévy process. Sobhani and Milev [9] presented a Legendre multiwavelets method for pricing discrete double barrier options. Thakoor, Tangman, and Bhuruth [10] proposed a novel spectral method by discretizing the pricing equation of the discrete barrier options whose price process is modelled by the stochastic Alpha Beta Rho (SABR) model. Liu and Zhang [11] proposed a novel jump-diffusion model in present of liquidity risk and derived an approximate solution for the valuation of the discrete barrier options.Unfortunately, these mentioned models are assumed that the volatility is constant. This is different from the well-known facts of empirical researches such as volatility smile or smirk. To overcome these shortcomings, many researchers proposed stochastic volatility models, among which the popular models are raised by Stein and Stein [12], Heston [13]. But a large number of studies have shown that the square-root process setup of the Heston model does not perform well in applications and cannot describe some of the stylized facts observed from the financial time series. For the past few years, the generalized autoregressive conditional heteroskedasticity (GARCH) model proposed by Nelson [14] has attracted extensive attention. Christoffersen P.et al. [15] and Chourdakis and Dotsis [16] further studied the GARCH model and found that the GARCH model describes the features of observations from the option markets much better than other stochastic volatility models, including the Heston model. In addition, Huang et al. [17] reached the same conclusion after their empirical study on Shanghai Stock Exchange (SSE) 50 ETF option.In this paper, we focus on the valuation of discretely monitored barrier options under the GARCH volatility and double exponential jump. Since there is no analytical formula for the characteristic function of the underlying asset log-price, a perturbation method is applied to derive an analytical approximation of the characteristic function for the log-price of the underlying asset. Moreover, we obtain an approximate price for the discretely monitored barrier option using the Fourier-cosine method proposed by Fang and Oosterlee [18]. Fourier-cosine method, as an alternative to the fast Fourier transform, was firstly proposed by Fang and Oosterlee. It can improve the speed of pricing and achieve an exponential convergence rate for European options. This method has been widely used in option pricing, such as Fang and Oosterlee [19, 20], Zhang and Geng [21], and Li and Zhang et al. [11]. Finally, some numerical simulations and sensitivity analysis are proposed to explain our results, and some economic implications of these results are discussed to demonstrate some interesting phenomena.An outline of this paper is as follows. Section2 sets up an option pricing model based on the GARCH volatility and double exponential jump. The approximation of the characteristic function for the underlying asset log-asset price is derived. And an approximate price for discretely monitored barrier option is also derived in Section 3. Some numerical examples and sensitivity analysis are provided to explain our results in Section 4. We conclude the paper in Section 5.


## 2. Model Specification

LetΩ,ℱ,ℱt0≤t≤T,ℚ be a complete probability space with a filtration continuous on the right line, where ℚ is a risk-neutral probability. The price process of the underlying asset St and the process of the volatility vt are specified under the risk-neutral measure ℚ as follows:(1)dStSt=r−λmdt+vtdWs,t+eJ−1dNt,(2)dvt=kθ−vtdt+σvtρdWs,t+1−ρ2dWv,t⊥,where CovdWs,t,dWv,t=ρdt. And k,θ,σ mean the speeds of the mean reversion, long-run volatility, and instantaneous volatility of the process vt, respectively. Nt is a Poisson process with constant intensity λ,m=EℚeJ−1. Jump size J is a random variable and follows an asymmetric double exponential distribution with the density(3)fJ=pη1e−η1J1J≥0+qη2eη2J1J<0,η1>1,η2>0,where p,q≤1,p+q=1 are probability of up-move jump and probability of down-move jump, respectively. So we can obtain that m=pη1/η1−1+qη2/η2+1−1.Letxt=lnSt/K. We can transform the equations (1) and (2) into the following equations (4) and (5), respectively:(4)dxt=r−λm−vt2dt+vtdWs,t+JdNt,(5)dvt=kθ−vtdt+σvtρdWs,t+1−ρ2dWv,t⊥.


## 3. Valuation of Discretely Monitored Barrier Options

In this section, we derived the approximate characteristic function of the underlying log-asset price. Then we obtain the approximate prices of discretely monitored barrier options via the Fourier-cosine method based on the proposed model.


### 3.1. Derivation of Approximate Characteristic Function

Following Caldana and Fusai [22], the characteristic function of xT under the risk-neutral ℚ is defined by(6)Φx,v,τ;u=EℚeiuxT|xt=x,vt=v,where T≥t,τ=T−t,i=−1. Then we can obtain the following theorem.Theorem 1.
Given that the underlying asset price follows the dynamics in equations (1) and (2). The approximate characteristic function for xT is given by(7)Φx,v,τ;u=expiux+Au,τv+Bu,τ,where(8)Au,τ=α01−e−ατ−β2+β1e−ατ,Bu,τ=−12θAτ,u−α3α2β1τ+ln−β2+β1e−ατα+r−λmiu+λΛuτ−14θiu+u2τ,α0=−12iu+u2,α1=32θ12σρiu−k,α2=σ2θ,β1=α1+α2,β2=α1−α2,α=α12−4α0α2,α3=12θk+14σρiuθ3/2.Proof.
By Feynman-Kac theorem,Φx,v,τ;u satisfies the following PIDE:(9)−∂Φ∂τ+r−12v−λm∂Φ∂x+12v∂2Φ∂x2+kθ−v∂Φ∂v+12σ2v2∂2Φ∂v2+∂2Φ∂x∂vv3/2σρ+λ∫−∞+∞Φx+J−ΦxfJdJ=0.
The boundary condition for equation (9) is given by(10)Φx,v,0;u=eiuxT.
Notice that the integral term in equation (9),(11)∫−∞+∞Φx+J−ΦxfJdJ=∫−∞+∞Eℚeiux+J−EℚeiuxfJdJ=∫−∞+∞EℚeiuxeiuJ−1fJdJ=∫−∞+∞EℚeiuxEℚeiuJ−1fJdJ=Φx,v,τ;uΛu,where Λu=pη1/η1−iu+qη2/η2+iu−1. Equation (9) is very difficult to solve since it is a nonlinear PDE. So we first linearize it approximately. The idea is to approximate v3/2,v2 in the PIDE using Taylor expansions around the long-run mean of variance as follows:(12)v2≈2θv−θ2,(13)v3/2≈32θ1/2v−12θ3/2.
Substituting equations (11)–(13) into the PDE in equation (9), we have(14)−∂Φ∂τ+r−12v−λm∂Φ∂x+12v∂2Φ∂x2+kθ−v∂Φ∂v+12σ22θv−θ2∂2Φ∂v2+∂2Φ∂x∂v32θ1/2v−12θ3/2σρ+λΦx,v,τ;uΛu=0.
According to Duffie et al. [23], this PDE has an exponential-affine solution of the form(15)Φx,v,τ;u=expiux+Au,τ+Bu,τv,with the boundary conditions(16)Au,0=Bu,0=0.
Substituting equation (15) into equation (14) yields(17)−∂A∂τv+∂B∂τ+r−λm−v2iu+12viu2+kθ−vA+σ222θv−θ2A2+32θ1/2v−12θ3/2σρiuA+λΛu=0.
By matching coefficients, we can derive the following two ordinary differential equations (ODEs):(18)∂A∂τ=σ2θA2+32θ1/2σρiu−kA−12iu+u2,(19)∂B∂τ=r−λmiu+kθA−12σ2θ2A2−12σρθ3/2iuA+λΛu.
Since the ODE (18) is a Riccati equation, its general solution is(20)Au,τ=α01−e−ατ−β2+β1e−ατ,where(21)α0=−12iu+u2,α1=32θ1/2σρiu−k,α2=σ2θ,β1=α1+α2,β2=α1−α2,α=α12−4α0α2.
Multiplyingθ/2 on both sides of equation (18), and substituting it into equation (19), we can obtain:(22)∂B∂τ=r−λmiu+12θk+14σρiuθ3/2A−14θiu+u2−θ2∂A∂τ+λΛu.
Integrating on both sides of the above equation, we have the following result:(23)Bτ,u=−12θAτ,u−α3α2β1τ+ln−β2+β1e−ατα+r−λmiu+λΛuτ−14θiu+u2τ,where(24)α3=12θk+14σρiuθ3/2.
This finished the proof of Theorem1.Remark 1.
Once that the analytic expression of the characteristic function has been obtained, the cumulant oflnST can be computed, which will be used in the truncation of the computational domain of option pricing. Following Fang and Oosterlee [18], the n-th cumulant of lnST is given by(25)cn=1in∂nlnΦu∂un|u=0.


### 3.2. Valuation of Discretely Monitored Barrier Options

It is well known that according to whether the underlying asset price needs to pass or to avoid a certain barrier level to receive a payoff and whether it has the barrier above or below the initial underlying asset price, barrier options can be classified into four cases, i.e., up-and-out option, up-and-in option, down-and-out option, down-and-in option. Without loss of generality, we take knock-out options as an example to illustrate the derivation of pricing formulas of barrier option in this section. The payoff functions are shown in Table1 for different discretely monitored knock-out barrier options with strike K, barrier level H, and maturity T.Table 1
Payoff functions for different types of discretely monitored knock-out barrier options.
Option typePayoffUp-and-out callST−K+−Rb1Sti<H+RbUp-and-out putK−ST+−Rb1Sti<H+RbDown-and-out callST−K+−Rb1Sti>H+RbDown-and-out putK−ST+−Rb1Sti>H+Rbti is the observation date.Letx:=lnStm−1/K and y:=lnStm/K, m=M,M−1,…,2. Then the prices of discretely knock up-and-out and down-and-out barrier option, i.e., vOU and vOD, can be expressed as the following recursive formulas:(26)vOUx,tm−1=cx,tm−1,x<h,e−rT−tm−1Rb,x≥h,(27)vODx,tm−1=e−rT−tm−1Rb,x≤h,cx,tm−1,x>h,where(28)cx,tm−1=e−rΔt∫ℝvy,tmfy|xdy,with(29)vx,t0=e−rΔt∫ℝvy,t1fy|xdy,where h=lnH/K, K is the strike price, T is the maturity, M is the monitoring date and H is the barrier level, Rb is a rebate, r denotes the risk-free interest rate; vx,t and cx,t are the option value and continuation value at time t, respectively; fy|x is the probability density of y given x under the risk-neutral measure ℚ.Usually, the probability density functionfy|x is unknown. Fortunately, Fang and Oosterlee [18] proposed an approximation of the probability density function with a truncated region a,b as follows:(30)fy|x≈2b−a∑k=0N−1′ReΦkπb−aeikπx−a/b−acoskπy−ab−a,where ∑′ means the summation whose first term is multiplied by 1/2, ℜ⋅ denotes taking the real part of a complex number. And Φu is the conditional characteristic function of probability density function fy|x.Replacing thefy|x in equation (26) with its approximation equation (29) and interchanging summation and integration, the approximation of the continuation value cx;tm−1 can be obtained:(31)c^x,tm−1=e−rΔt2b−a∑k=0N−1ℜ′Φkπb−aeikπx−a/b−aVktm,where(32)Vktm=2b−a∫abvy,tmcoskπy−ab−ady.We know that barrier options are equivalent to European options at timet∈t0,t1. So the price of the discrete barrier option at initial time t0 can be approximated as(33)v^x,t0=e−rΔt∑k=0N−1ℜ′Φkπb−aeikπx−a/b−aVkt1.Now the key point is to obtain the coefficientVktm,m=M,M−1,…,1. For the convenience of later description, we denote:(34)χkx1,x2=∫x1x2excoskπx−ab−adx=11+kπb−a2coskπx2−ab−aex2−coskπx1−ab−aex1+kπb−asinkπx2−ab−aex2−kπb−asinkπx1−ab−aex1,ψkx1,x2=∫x1x2coskπx−ab−adx=b−akπsinkπx2−ab−a−sinkπx1−ab−a,k≠0,x2−x1,k=0.Whenm=M, if h<0, we have the following results:(35)VktM=2b−aRbψkh,b,for up−and−out call,2b−aRbψka,h+Kχk0,b−Kψk0,b,for down−and−out call,2b−aKψka,h−Kχka,h+Rbψkh,b,for up−and−out put,2b−aRbψka,h+Kψkh,0−Kχkh,0,for down−and−out put.Ifh≥0, then we can obtain the following result:(36)VktM=2b−aKχk0,h−Kψk0,h+Rbψkh,b,for up−and−out call,2b−aRbψka,h+Kχkh,b−Kψkh,b,for down−and−out call,2b−aKψka,0−Kχka,0+Rbψkh,b,for up−and−out put,2b−aRbψka,h,for down−and−out put.Form=M−1,M−2,…,1, we have the following theorem.Theorem 2.
For the up-and-out option and down-and-out option, the recurrent coefficientsVktm can be approximately expressed as follows:(37)V^ktm=C^ka,h,tm+e−rT−tmRb2b−aψkh,b,up−and−out,e−rT−tmRb2b−aψka,h+C^kh,b,tm,down−and−out,where C^kx1,x2,tm can be approximately calculated by:(38)C^kx1,x2,tm=e−rΔtRe∑j=0N−1′Φjπb−aV^jtm+1Mk,jx1,x2,with(39)Mkjx1,x2=2b−a∫x1x2eijπx−a/b−acoskπx−ab−adx.Proof.
According to the payoff of discrete barrier options at timeT, we can rewrite vy,T as follows:(40)vy,T=Kαey−1+−Rb1yti<h+Rb,where α=1 for call option and α=−1 for put option. Then we can obtain the above-mentioned results by the following computations.
Form=M−1,M−2,…,1, we have(41)Vktm=2b−a∫abvy,tmcoskπy−ab−ady=2b−a∫ahvy,tmcoskπy−ab−ady+2b−a∫hbvy,tmcoskπy−ab−ady=2b−a∫ahcy,tmcoskπy−ab−ady+2b−a∫hbe−rT−tmRbcoskπy−ab−ady=Cka,h,tm+e−rT−tmRb2b−aψkh,b.
On the other hand,cy,tm can be approximated by c^x,tm, where(42)c^x,tm=e−rΔt∑k=0N−1′ReΦkπb−aeikπx−a/b−aVktm+1.
Substituting the above equation intoCka,h,tm, we can obtain C^ka,h,tm as follows: which is the approximate value of Cka,h,tm:(43)C^ka,h,tm=2b−a∫ahc^y,tmcoskπy−ab−ady=e−rΔtℜ∑j=0N−1Φ′jπb−aVjtm+1ℳk,ja,h,where(44)ℳk,ja,h≔2b−a∫aheijπx−a/b−acoskπx−ab−adx.
Hence,Vktm can be recovered from Vktm+1. Finally, we substitute the approximation V^kt1 into equation (28) to calculate the option price v^x,t0.


## 3.1. Derivation of Approximate Characteristic Function

Following Caldana and Fusai [22], the characteristic function of xT under the risk-neutral ℚ is defined by(6)Φx,v,τ;u=EℚeiuxT|xt=x,vt=v,where T≥t,τ=T−t,i=−1. Then we can obtain the following theorem.Theorem 1.
Given that the underlying asset price follows the dynamics in equations (1) and (2). The approximate characteristic function for xT is given by(7)Φx,v,τ;u=expiux+Au,τv+Bu,τ,where(8)Au,τ=α01−e−ατ−β2+β1e−ατ,Bu,τ=−12θAτ,u−α3α2β1τ+ln−β2+β1e−ατα+r−λmiu+λΛuτ−14θiu+u2τ,α0=−12iu+u2,α1=32θ12σρiu−k,α2=σ2θ,β1=α1+α2,β2=α1−α2,α=α12−4α0α2,α3=12θk+14σρiuθ3/2.Proof.
By Feynman-Kac theorem,Φx,v,τ;u satisfies the following PIDE:(9)−∂Φ∂τ+r−12v−λm∂Φ∂x+12v∂2Φ∂x2+kθ−v∂Φ∂v+12σ2v2∂2Φ∂v2+∂2Φ∂x∂vv3/2σρ+λ∫−∞+∞Φx+J−ΦxfJdJ=0.
The boundary condition for equation (9) is given by(10)Φx,v,0;u=eiuxT.
Notice that the integral term in equation (9),(11)∫−∞+∞Φx+J−ΦxfJdJ=∫−∞+∞Eℚeiux+J−EℚeiuxfJdJ=∫−∞+∞EℚeiuxeiuJ−1fJdJ=∫−∞+∞EℚeiuxEℚeiuJ−1fJdJ=Φx,v,τ;uΛu,where Λu=pη1/η1−iu+qη2/η2+iu−1. Equation (9) is very difficult to solve since it is a nonlinear PDE. So we first linearize it approximately. The idea is to approximate v3/2,v2 in the PIDE using Taylor expansions around the long-run mean of variance as follows:(12)v2≈2θv−θ2,(13)v3/2≈32θ1/2v−12θ3/2.
Substituting equations (11)–(13) into the PDE in equation (9), we have(14)−∂Φ∂τ+r−12v−λm∂Φ∂x+12v∂2Φ∂x2+kθ−v∂Φ∂v+12σ22θv−θ2∂2Φ∂v2+∂2Φ∂x∂v32θ1/2v−12θ3/2σρ+λΦx,v,τ;uΛu=0.
According to Duffie et al. [23], this PDE has an exponential-affine solution of the form(15)Φx,v,τ;u=expiux+Au,τ+Bu,τv,with the boundary conditions(16)Au,0=Bu,0=0.
Substituting equation (15) into equation (14) yields(17)−∂A∂τv+∂B∂τ+r−λm−v2iu+12viu2+kθ−vA+σ222θv−θ2A2+32θ1/2v−12θ3/2σρiuA+λΛu=0.
By matching coefficients, we can derive the following two ordinary differential equations (ODEs):(18)∂A∂τ=σ2θA2+32θ1/2σρiu−kA−12iu+u2,(19)∂B∂τ=r−λmiu+kθA−12σ2θ2A2−12σρθ3/2iuA+λΛu.
Since the ODE (18) is a Riccati equation, its general solution is(20)Au,τ=α01−e−ατ−β2+β1e−ατ,where(21)α0=−12iu+u2,α1=32θ1/2σρiu−k,α2=σ2θ,β1=α1+α2,β2=α1−α2,α=α12−4α0α2.
Multiplyingθ/2 on both sides of equation (18), and substituting it into equation (19), we can obtain:(22)∂B∂τ=r−λmiu+12θk+14σρiuθ3/2A−14θiu+u2−θ2∂A∂τ+λΛu.
Integrating on both sides of the above equation, we have the following result:(23)Bτ,u=−12θAτ,u−α3α2β1τ+ln−β2+β1e−ατα+r−λmiu+λΛuτ−14θiu+u2τ,where(24)α3=12θk+14σρiuθ3/2.
This finished the proof of Theorem1.Remark 1.
Once that the analytic expression of the characteristic function has been obtained, the cumulant oflnST can be computed, which will be used in the truncation of the computational domain of option pricing. Following Fang and Oosterlee [18], the n-th cumulant of lnST is given by(25)cn=1in∂nlnΦu∂un|u=0.


## 3.2. Valuation of Discretely Monitored Barrier Options

It is well known that according to whether the underlying asset price needs to pass or to avoid a certain barrier level to receive a payoff and whether it has the barrier above or below the initial underlying asset price, barrier options can be classified into four cases, i.e., up-and-out option, up-and-in option, down-and-out option, down-and-in option. Without loss of generality, we take knock-out options as an example to illustrate the derivation of pricing formulas of barrier option in this section. The payoff functions are shown in Table1 for different discretely monitored knock-out barrier options with strike K, barrier level H, and maturity T.Table 1
Payoff functions for different types of discretely monitored knock-out barrier options.
Option typePayoffUp-and-out callST−K+−Rb1Sti<H+RbUp-and-out putK−ST+−Rb1Sti<H+RbDown-and-out callST−K+−Rb1Sti>H+RbDown-and-out putK−ST+−Rb1Sti>H+Rbti is the observation date.Letx:=lnStm−1/K and y:=lnStm/K, m=M,M−1,…,2. Then the prices of discretely knock up-and-out and down-and-out barrier option, i.e., vOU and vOD, can be expressed as the following recursive formulas:(26)vOUx,tm−1=cx,tm−1,x<h,e−rT−tm−1Rb,x≥h,(27)vODx,tm−1=e−rT−tm−1Rb,x≤h,cx,tm−1,x>h,where(28)cx,tm−1=e−rΔt∫ℝvy,tmfy|xdy,with(29)vx,t0=e−rΔt∫ℝvy,t1fy|xdy,where h=lnH/K, K is the strike price, T is the maturity, M is the monitoring date and H is the barrier level, Rb is a rebate, r denotes the risk-free interest rate; vx,t and cx,t are the option value and continuation value at time t, respectively; fy|x is the probability density of y given x under the risk-neutral measure ℚ.Usually, the probability density functionfy|x is unknown. Fortunately, Fang and Oosterlee [18] proposed an approximation of the probability density function with a truncated region a,b as follows:(30)fy|x≈2b−a∑k=0N−1′ReΦkπb−aeikπx−a/b−acoskπy−ab−a,where ∑′ means the summation whose first term is multiplied by 1/2, ℜ⋅ denotes taking the real part of a complex number. And Φu is the conditional characteristic function of probability density function fy|x.Replacing thefy|x in equation (26) with its approximation equation (29) and interchanging summation and integration, the approximation of the continuation value cx;tm−1 can be obtained:(31)c^x,tm−1=e−rΔt2b−a∑k=0N−1ℜ′Φkπb−aeikπx−a/b−aVktm,where(32)Vktm=2b−a∫abvy,tmcoskπy−ab−ady.We know that barrier options are equivalent to European options at timet∈t0,t1. So the price of the discrete barrier option at initial time t0 can be approximated as(33)v^x,t0=e−rΔt∑k=0N−1ℜ′Φkπb−aeikπx−a/b−aVkt1.Now the key point is to obtain the coefficientVktm,m=M,M−1,…,1. For the convenience of later description, we denote:(34)χkx1,x2=∫x1x2excoskπx−ab−adx=11+kπb−a2coskπx2−ab−aex2−coskπx1−ab−aex1+kπb−asinkπx2−ab−aex2−kπb−asinkπx1−ab−aex1,ψkx1,x2=∫x1x2coskπx−ab−adx=b−akπsinkπx2−ab−a−sinkπx1−ab−a,k≠0,x2−x1,k=0.Whenm=M, if h<0, we have the following results:(35)VktM=2b−aRbψkh,b,for up−and−out call,2b−aRbψka,h+Kχk0,b−Kψk0,b,for down−and−out call,2b−aKψka,h−Kχka,h+Rbψkh,b,for up−and−out put,2b−aRbψka,h+Kψkh,0−Kχkh,0,for down−and−out put.Ifh≥0, then we can obtain the following result:(36)VktM=2b−aKχk0,h−Kψk0,h+Rbψkh,b,for up−and−out call,2b−aRbψka,h+Kχkh,b−Kψkh,b,for down−and−out call,2b−aKψka,0−Kχka,0+Rbψkh,b,for up−and−out put,2b−aRbψka,h,for down−and−out put.Form=M−1,M−2,…,1, we have the following theorem.Theorem 2.
For the up-and-out option and down-and-out option, the recurrent coefficientsVktm can be approximately expressed as follows:(37)V^ktm=C^ka,h,tm+e−rT−tmRb2b−aψkh,b,up−and−out,e−rT−tmRb2b−aψka,h+C^kh,b,tm,down−and−out,where C^kx1,x2,tm can be approximately calculated by:(38)C^kx1,x2,tm=e−rΔtRe∑j=0N−1′Φjπb−aV^jtm+1Mk,jx1,x2,with(39)Mkjx1,x2=2b−a∫x1x2eijπx−a/b−acoskπx−ab−adx.Proof.
According to the payoff of discrete barrier options at timeT, we can rewrite vy,T as follows:(40)vy,T=Kαey−1+−Rb1yti<h+Rb,where α=1 for call option and α=−1 for put option. Then we can obtain the above-mentioned results by the following computations.
Form=M−1,M−2,…,1, we have(41)Vktm=2b−a∫abvy,tmcoskπy−ab−ady=2b−a∫ahvy,tmcoskπy−ab−ady+2b−a∫hbvy,tmcoskπy−ab−ady=2b−a∫ahcy,tmcoskπy−ab−ady+2b−a∫hbe−rT−tmRbcoskπy−ab−ady=Cka,h,tm+e−rT−tmRb2b−aψkh,b.
On the other hand,cy,tm can be approximated by c^x,tm, where(42)c^x,tm=e−rΔt∑k=0N−1′ReΦkπb−aeikπx−a/b−aVktm+1.
Substituting the above equation intoCka,h,tm, we can obtain C^ka,h,tm as follows: which is the approximate value of Cka,h,tm:(43)C^ka,h,tm=2b−a∫ahc^y,tmcoskπy−ab−ady=e−rΔtℜ∑j=0N−1Φ′jπb−aVjtm+1ℳk,ja,h,where(44)ℳk,ja,h≔2b−a∫aheijπx−a/b−acoskπx−ab−adx.
Hence,Vktm can be recovered from Vktm+1. Finally, we substitute the approximation V^kt1 into equation (28) to calculate the option price v^x,t0.


## 4. Numerical Analysis

### 4.1. Comparison of the Approximate Solutions against Monte Carlo Simulations

In this section, some numerical examples are performed to show the performance of the Fourier-cosine method for pricing discrete barrier options against its alternative competitor, the Monte Carlo simulation, as a benchmark. We perform numerical examples on the discretely monitored up-and-out call and up-and-out put barrier options. Following Fang and Oosterlee [19], the integration interval [a, b] is chosen as follows:(45)a,b=c1+a0−Lc2+c4,c1+a0+Lc2+c4,with a0=lnS0,L=10 and cn is the n-th cumulant of lnST.For Monte Carlo method, we use100,000 numbers of simulations, 200 numbers of time steps. Let N=210 for Fourier-cosine method. The parameter values are listed in Table 2 for all our numerical examples. The computer used in the experiments equips an Intel Core i3 CPU with a 2.53 GHZ processor. And all of our numerical examples were performed using Matlab 2016a.Table 2
Parameter values for the numerical experiments.
Parameterrλkθσpη1TS0ρv0HRbη2Value0.05350.20.70.3101100−0.50.212055Our numerical results show that the Fourier-cosine method is fast and accurate for pricing discrete up-and-out barrier options. The absolute relative error (Abs R.E.) and CPU time information, comparing the Fourier-cosine method and the Monte Carlo simulation, are presented in Tables3 and 4 for pricing discrete up-and-out call and put barrier options. Tables 3 and 4 also compares relative errors between the Fourier-cosine method and the Monte Carlo simulation for pricing discrete up-and-out call and put barrier options and they show that the Fourier-cosine method is significantly faster than Monte Carlo simulation. Furthermore, Tables 3 and 4 compare the pricing accuracy between the two methods across a range of strike prices, and the relative percentage of price differences of Fourier-cosine method are all less than 1.6%. Therefore, numerical results show that the Fourier-cosine method is accurate and effective.Table 3
Comparisons of the CPU time and accuracy for Fourier-cosine method(FC method) and the Monte Carlo simulation (MC Simulation) for pricing discrete up-and-out call barrier options.
MKFC methodMC simulationAbs R.E.4903.62313.66811.231002.86402.90861.531102.47092.50831.49cpu time (sec)1.142110.95212903.57033.60030.831003.06303.09090.901102.82212.84890.94cpu time (sec)1.2721144.13626903.57093.61281.161003.15843.18820.931102.97402.99650.75cpu time (sec)1.429136.54652903.58153.62161.101003.22393.25110.831103.07133.08920.58cpu time (sec)1.492197.555252903.60753.63730.821003.31683.33430.521103.20213.21090.27cpu time(sec)1.013324.363Table 4
Comparisons of the CPU time and accuracy for Fourier-cosine method (FC method) and the Monte Carlo simulation(MC Simulation) for pricing discrete up-and-out put barrier options.
MKFC methodMC simulationAbs R.E. (%)49015.305115.29120.0910019.289819.23490.2911023.640523.54680.04cpu time (sec)1.219237.383129014.335514.29950.2510017.799017.71570.4711021.529121.40190.59cpu time (sec)1.066289.999269013.865113.82810.2710017.093517.00500.5211020.549920.41380.67cpu time (sec)1.099136.546529013.538713.50970.2110016.611516.53500.4611019.889319.76950.61cpu time (sec)1.383155.3112529013.069613.04700.1710015.927615.86410.4011018.961418.86270.52cpu time (sec)1.170113.495


### 4.2. Sensitivity Analysis of the Model Parameters

In this section, we will evaluate the impact of parameters on pricing discretely monitored up-and-out call barrier options. We mainly study the effects of the following parameters on the price of discretely monitored up-and-out call barrier options: (i) the jump intensity λ, (ii) the long-run mean level θ, (iii) the reciprocal of the mean for the positive jumps η1, and (iv) the reciprocal of the mean for the negative jumps η2.Figures1 and 2 show the variation trend of the price of the discrete knock up-out call barrier option with respect to the jump intensity λ and the long-run mean level θ, respectively. As shown in Figure 1, a higher jump intensity λ induces a lower price of discrete knock up-out call barrier option. The explanation for this phenomenon is that it makes the underlying asset price process more variable as the jump intensity λ increases. So it makes the underlying asset price more likely to cross the barrier level H at the monitoring dates. Therefore, the price of discrete up-and-out call barrier option is a decreasing function of the jump intensity λ. The implication of Figure 2 can be expressed in the similar way.Figure 1
Prices of discrete up-and-out call barrier options with respect to the parameters ofλ.Figure 2
Prices of discrete up-and-out call barrier options with respect toθ.Figures3 and 4 illustrate the effects of the reciprocal of the mean for the positive jumps η1, and the negative jumps η2 on the price of discrete knock up-out call barrier option. We can see that the price of discrete knock up-out call barrier option increases with respect to the parameters η1 and η2, respectively. The reason is that the probability density function of the underlying log-asset price becomes more leptokurtosis and heavy tails when the parameters η1 and η2 increase.Figure 3
Prices of discrete up-and-out call barrier options with respect to the parameters ofη1.Figure 4
Prices of discrete up-and-out call barrier options with respect to the parameters ofη2.


## 4.1. Comparison of the Approximate Solutions against Monte Carlo Simulations

In this section, some numerical examples are performed to show the performance of the Fourier-cosine method for pricing discrete barrier options against its alternative competitor, the Monte Carlo simulation, as a benchmark. We perform numerical examples on the discretely monitored up-and-out call and up-and-out put barrier options. Following Fang and Oosterlee [19], the integration interval [a, b] is chosen as follows:(45)a,b=c1+a0−Lc2+c4,c1+a0+Lc2+c4,with a0=lnS0,L=10 and cn is the n-th cumulant of lnST.For Monte Carlo method, we use100,000 numbers of simulations, 200 numbers of time steps. Let N=210 for Fourier-cosine method. The parameter values are listed in Table 2 for all our numerical examples. The computer used in the experiments equips an Intel Core i3 CPU with a 2.53 GHZ processor. And all of our numerical examples were performed using Matlab 2016a.Table 2
Parameter values for the numerical experiments.
Parameterrλkθσpη1TS0ρv0HRbη2Value0.05350.20.70.3101100−0.50.212055Our numerical results show that the Fourier-cosine method is fast and accurate for pricing discrete up-and-out barrier options. The absolute relative error (Abs R.E.) and CPU time information, comparing the Fourier-cosine method and the Monte Carlo simulation, are presented in Tables3 and 4 for pricing discrete up-and-out call and put barrier options. Tables 3 and 4 also compares relative errors between the Fourier-cosine method and the Monte Carlo simulation for pricing discrete up-and-out call and put barrier options and they show that the Fourier-cosine method is significantly faster than Monte Carlo simulation. Furthermore, Tables 3 and 4 compare the pricing accuracy between the two methods across a range of strike prices, and the relative percentage of price differences of Fourier-cosine method are all less than 1.6%. Therefore, numerical results show that the Fourier-cosine method is accurate and effective.Table 3
Comparisons of the CPU time and accuracy for Fourier-cosine method(FC method) and the Monte Carlo simulation (MC Simulation) for pricing discrete up-and-out call barrier options.
MKFC methodMC simulationAbs R.E.4903.62313.66811.231002.86402.90861.531102.47092.50831.49cpu time (sec)1.142110.95212903.57033.60030.831003.06303.09090.901102.82212.84890.94cpu time (sec)1.2721144.13626903.57093.61281.161003.15843.18820.931102.97402.99650.75cpu time (sec)1.429136.54652903.58153.62161.101003.22393.25110.831103.07133.08920.58cpu time (sec)1.492197.555252903.60753.63730.821003.31683.33430.521103.20213.21090.27cpu time(sec)1.013324.363Table 4
Comparisons of the CPU time and accuracy for Fourier-cosine method (FC method) and the Monte Carlo simulation(MC Simulation) for pricing discrete up-and-out put barrier options.
MKFC methodMC simulationAbs R.E. (%)49015.305115.29120.0910019.289819.23490.2911023.640523.54680.04cpu time (sec)1.219237.383129014.335514.29950.2510017.799017.71570.4711021.529121.40190.59cpu time (sec)1.066289.999269013.865113.82810.2710017.093517.00500.5211020.549920.41380.67cpu time (sec)1.099136.546529013.538713.50970.2110016.611516.53500.4611019.889319.76950.61cpu time (sec)1.383155.3112529013.069613.04700.1710015.927615.86410.4011018.961418.86270.52cpu time (sec)1.170113.495


## 4.2. Sensitivity Analysis of the Model Parameters

In this section, we will evaluate the impact of parameters on pricing discretely monitored up-and-out call barrier options. We mainly study the effects of the following parameters on the price of discretely monitored up-and-out call barrier options: (i) the jump intensity λ, (ii) the long-run mean level θ, (iii) the reciprocal of the mean for the positive jumps η1, and (iv) the reciprocal of the mean for the negative jumps η2.Figures1 and 2 show the variation trend of the price of the discrete knock up-out call barrier option with respect to the jump intensity λ and the long-run mean level θ, respectively. As shown in Figure 1, a higher jump intensity λ induces a lower price of discrete knock up-out call barrier option. The explanation for this phenomenon is that it makes the underlying asset price process more variable as the jump intensity λ increases. So it makes the underlying asset price more likely to cross the barrier level H at the monitoring dates. Therefore, the price of discrete up-and-out call barrier option is a decreasing function of the jump intensity λ. The implication of Figure 2 can be expressed in the similar way.Figure 1
Prices of discrete up-and-out call barrier options with respect to the parameters ofλ.Figure 2
Prices of discrete up-and-out call barrier options with respect toθ.Figures3 and 4 illustrate the effects of the reciprocal of the mean for the positive jumps η1, and the negative jumps η2 on the price of discrete knock up-out call barrier option. We can see that the price of discrete knock up-out call barrier option increases with respect to the parameters η1 and η2, respectively. The reason is that the probability density function of the underlying log-asset price becomes more leptokurtosis and heavy tails when the parameters η1 and η2 increase.Figure 3
Prices of discrete up-and-out call barrier options with respect to the parameters ofη1.Figure 4
Prices of discrete up-and-out call barrier options with respect to the parameters ofη2.


## 5. Conclusion

This paper proposed an efficient method for the valuating the discrete barrier options when the underlying asset price process is governed by the GARCH volatility and double exponential jump. We utilized Taylor expansion method to derive an analytical approximation of the characteristic function for the underlying log-asset price. Based on Fourier-cosine method, a quasianalytical approximate formula of the price of the discrete barrier is obtained. The Fourier-cosine method is fast and efficient for pricing discrete barrier options compared with Monte Carlo simulation. Finally, the sensitivities of some parameters in the model are provided to explain our results.



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