
Citation: Karolína M. Šišková, Renáta Večeřová, Hana Kubičková, Magdaléna Bryksová, Klára Čépe, Milan Kolář. Revisiting spontaneous silver nanoparticles formation: a factor influencing the determination of minimum inhibitory concentration values?[J]. AIMS Environmental Science, 2015, 2(3): 607-622. doi: 10.3934/environsci.2015.3.607
[1] | Yujie Sheng, Jing-An Cui, Songbai Guo . The modeling and analysis of the COVID-19 pandemic with vaccination and isolation: a case study of Italy. Mathematical Biosciences and Engineering, 2023, 20(3): 5966-5992. doi: 10.3934/mbe.2023258 |
[2] | Shina D. Oloniiju, Olumuyiwa Otegbeye, Absalom E. Ezugwu . Investigating the impact of vaccination and non-pharmaceutical measures in curbing COVID-19 spread: A South Africa perspective. Mathematical Biosciences and Engineering, 2022, 19(1): 1058-1077. doi: 10.3934/mbe.2022049 |
[3] | Minami Ueda, Tetsuro Kobayashi, Hiroshi Nishiura . Basic reproduction number of the COVID-19 Delta variant: Estimation from multiple transmission datasets. Mathematical Biosciences and Engineering, 2022, 19(12): 13137-13151. doi: 10.3934/mbe.2022614 |
[4] | Sha He, Sanyi Tang, Libin Rong . A discrete stochastic model of the COVID-19 outbreak: Forecast and control. Mathematical Biosciences and Engineering, 2020, 17(4): 2792-2804. doi: 10.3934/mbe.2020153 |
[5] | Luyu Zhang, Zhaohua Zhang, Sen Pei, Qing Gao, Wei Chen . Quantifying the presymptomatic transmission of COVID-19 in the USA. Mathematical Biosciences and Engineering, 2024, 21(1): 861-883. doi: 10.3934/mbe.2024036 |
[6] | Akhil Kumar Srivastav, Pankaj Kumar Tiwari, Prashant K Srivastava, Mini Ghosh, Yun Kang . A mathematical model for the impacts of face mask, hospitalization and quarantine on the dynamics of COVID-19 in India: deterministic vs. stochastic. Mathematical Biosciences and Engineering, 2021, 18(1): 182-213. doi: 10.3934/mbe.2021010 |
[7] | Liping Wang, Jing Wang, Hongyong Zhao, Yangyang Shi, Kai Wang, Peng Wu, Lei Shi . Modelling and assessing the effects of medical resources on transmission of novel coronavirus (COVID-19) in Wuhan, China. Mathematical Biosciences and Engineering, 2020, 17(4): 2936-2949. doi: 10.3934/mbe.2020165 |
[8] | Yong Zhou, Minrui Guo . Isolation in the control of epidemic. Mathematical Biosciences and Engineering, 2022, 19(11): 10846-10863. doi: 10.3934/mbe.2022507 |
[9] | Yan-Xia Dang, Zhi-Peng Qiu, Xue-Zhi Li, Maia Martcheva . Global dynamics of a vector-host epidemic model with age of infection. Mathematical Biosciences and Engineering, 2017, 14(5&6): 1159-1186. doi: 10.3934/mbe.2017060 |
[10] | Tingting Zheng, Huaiping Zhu, Zhidong Teng, Linfei Nie, Yantao Luo . Patch model for border reopening and control to prevent new outbreaks of COVID-19. Mathematical Biosciences and Engineering, 2023, 20(4): 7171-7192. doi: 10.3934/mbe.2023310 |
The American option is an important financial tool and is widely used in real market. However, the pricing model of this financial derivative is a free boundary problem so that it is impossible to obtain the analytic pricing formula of this financial derivative. Thus, in the past two decades, the numerical method becomes a mainstream tool to solve the mathematical model.
More and more authors have studied the American option in the case of Black-Scholes model (BS). For example, Geske and Johson [1] investigated the American put option, and an analytic solution to this option was derived. Moreover, based on the analytical solution, the risk hedging coefficient for American put was obtained. Based on this literature, Zhu et al. [2,3] used the integral transformation method to solve the American Contingent Claims pricing mathematical model, and the price and optimal exercise price of this kind of financial derivatives are obtained. Gyulov and Koleva [4] developed a numerical method based on the penalty for American option in the case of the BS model with the regime-switching process. Xiang and Wang [5] proposed an efficient quasi-Monte Carlo method for estimating American option sensitivities. Wang et al. [6] constructed a high-order deferred correction algorithm combined with penalty iteration for solving American option pricing model. Elettra and Rossella [7] use the Recurrent neural network framework for computing prices and deltas of American options in high dimensions. Under the framework of the Cox-Ingersoll-Ross (CIR), Zhang et al. [8] proposed an efficient numerical method for the American options pricing. Additionally for the perpetual American put option, an analytical solution is proposed under the framework of BS in [9].
However, in the case of the classical BS model, the stock price is assumed to follow the Geometric Brownian motion, which cannot reflect the character of the risk asset in the real market. The conclusions in [10,11,12,13] show that the risk asset price should appear to be "phenomenon of jumps" and "asymmetric distribution". Thus, the more complex stochastic differential equation should be used to capture the characters of the risk asset by many scholars. Prominent examples, including the FMLS equation [14], CGMY equation [15], and KoBoL equation [16]. Moreover, as described in [17], both FMLS and CGMY equations are the special cases of the KoBoL equation, so we consider the American call pricing problem in this paper. Under this framework, the function of the option price value is governed by the fractional partial differential equation free boundary problem, which is proved in [18]. Following this work, Chen and Lin [19] used the integral transformation method to obtain the analytical solution of the European option pricing model. For the European double barrier option pricing model, the numerical method is set, and the convergence rate and stability of this numerical method are proved in [17]. Mohapatra et al. [20] considered the numerical solution for the time fractional Black-Scholes model under jump-diffusion involving a Caputo differential operator, and their schemes are investigated for numerous European option pricing jump-diffusion models. Guo et al. [21] proposed a numerical method for European and American option pricing under the time fractional jump-diffusion model in Caputo scene. Fan et al. [22] considered the values and optimal exercise prices of the American option under the CGMY model with the regime-switching process.
Based on the literature, the American call option pricing problem is investigated in the case of the KoBoL model with the jump process (KoBoLJ) in this paper. Under the framework of the KoBoLJ model, the function of the American call value follows a FPIDE, and the pricing model is a free boundary problem. To obtain a FPIDE boundary value problem over a fixed rectangular domain, a nonlinear penalty term is added to the governing equation. However, it is still impossible to achieve the analytical solution of the new mathematical model. Hence, the finite differential method is essentially considered in this paper. Moreover, a dense coefficient matrix resulted from the fractional derivatives in the final linear system, which requires the computational cost in the order of $ \mathcal{O}(M^3) $, where $ M $ is the number of spatial grid nodes. This shows that the computational time of numerical method will increase.
The major contributions of this study can be summarized as follows:
ⅰ) The Poisson jumps is introduced into the KoBoL model due to the need to capture the characters of stock price so that the option pricing models can capture market risk;
ⅱ) The preconditioned conjugate gradient normal residual (PCGNR) method with a Strang's circulant pre-conditioner [23] and the fast Fourier transform (FFT) technique are used, so that the computational cost reduces significantly from $ \mathcal{O}(M^3) $ to $ \mathcal{O}(M\log M) $;
ⅲ) Based on the numerical scheme, we prove that the American call option value generated by the penalty method cannot fall below the value obtained when the American call option is exercised early, i.e., $ V(x, t)\geq \max (e^x - K, 0) $.
The rest of this article is outlined below. In the next section, the pricing mathematical model of American call under the framework of the KoBoLJ model is derived in detail. In Section 3, the numerical scheme is proposed, and we prove that the American call option value obtained by our numerical method is bigger than the exercising value. In Section 4, we prove the PCGNR method with a circulant pre-conditioner and the FFT technique to calculate the final system. Moreover, the numerical experiments are presented with some discussions in Section 5. We conclude this paper in the last section.
Take $ (\Omega, \mathcal{F}_t, \mathbb{P}) $ as a filtered probability, where $ t\in[0, T] $. The KoBoL model with the jump process is defined on this probability space. Following the assumptions in [24], under this model, the logarithmic price of the underlying, i.e., $ x_t = \ln(S_t) $, satisfies the following stochastic differential equation
$ dxt=(r−ν−D−ξς)dt+dLKoBoLt+d(Nt∑i=1Yi), $
|
(2.1) |
with solution
$ ST=Ste(r−v)(T−t)+∫TtdLKoBoLu, $
|
where $ x_t $ is the logarithmic form stock price $ x_t = \ln S_t $, $ r $ is the risk-free interest rate, $ D $ is dividend, $ dL_t^{KoBoL} $ is the increment of a $ L\acute{e}vy $ process under the equivalent martingale measure, and $ \nu = \frac{1}{2}\sigma^\alpha[p(\lambda-1)^\alpha+q(\lambda+1)^\alpha-\lambda^\alpha-\alpha\lambda^{\alpha-1}(q-p)] $ is convexity adjustment so that the expectation of $ S_T $ becomes $ \mathbb{E}[S_T] = e^{r(T-t)}S_t $. Parameter $ \alpha\in(1, 2) $ determines whether the KoBoLJ stochastic process has finite or infinite variation. The relative frequency and overall upwind and downwind movements KoBoLJ stochastic process are controlled by $ q > 0, p > 0 (p+q = 1) $. The decay rate of tails of our stochastic process probability density function is controlled by parameter $ \lambda > 0 $. $ \mathcal{N}_t $ is a Poisson process and it is characterized by the jump intensity $ \xi\geq0 $. $ \{Y_i, i = 1, 2, \ldots\} $ is a sequence of independent and identically distributed hyper-exponential random variables with probability density function
$ fY(y)=m1∑i=1ˆpiˆθie−ˆθiy1{y≥0}+m2∑j=1˜pj˜θje−˜θjy1{y≤0}. $
|
Note that $ \hat{p}_i $ ($ i = 1, 2, ..., m_1 $) and $ \tilde{p}_j $ ($ j = 1, 2, ..., m_2 $) denote the probabilities of the ith positive and negative jumps, respectively. They satisfy $ \sum_{i = 1}^{m_1}\hat{p_i}+\sum_{j = 1}^{m_2}\tilde{p_j} = 1 $. $ \hat{\theta}_i > 1 $ ($ i = 1, ..., m_1 $) is the magnitude of the upward jumps and $ \tilde{\theta}_j > 0 $ ($ j = 1, ..., m_2 $) is that of the downward random jumps. The average jump size is given by
$ ς=EP[exp(Y1)−1]=m1∑i=1ˆpiˆθiˆθi−1+m2∑j=1˜pj˜θj˜θj+1−1, $
|
(2.2) |
where $ \mathbb{E}_{\mathbb{P}} $ is the expectation operator under probability measure $ \mathbb{P} $.
Now, we turn to formulate mathematically the pricing of the American option under our model. Financially, the payoff function of American call contract can be written as
$ Π(xT,T)=max(ex−K,0), $
|
(2.3) |
where $ K $ is the strike price. According to the no-arbitrage pricing principle, one can obtain
$ V(x,t)=e−r(T−t)EP[Π(xT,T)]. $
|
(2.4) |
Then, according to the conclusions in [18], it can be obtained that the American call option value $ V(x, t) $ satisfies the following equation
$ ∂V(x,t)∂t+a∂V(x,t)∂x+ξ∫+∞−∞V(x+y,t)fY(y)dy+12σα[peλxxDαxfe−λxV(x,t)+qe−λx−∞DαxeλxV(x,t)]=(b+12σαλα)V(x,t), $
|
(2.5) |
where $ x\in(-\infty, x_f], t\in [0, T] $, $ a = r-\nu-D-\xi\varsigma-\lambda^{\alpha-1}(q-p), b = r+\xi $, and
$ eλxxDαxfe−λxV(x,t)=eλxΓ(2−α)∂2∂x2∫xfxe−λξV(ξ,t)(ξ−x)α−1dξ, $
|
$ e−λx−∞DαxeλxV(x,t)=e−λxΓ(2−α)∂2∂x2∫x−∞eλξV(ξ,t)(x−ξ)α−1dξ. $
|
In fact, the fractional derivative in Eq (2.5) is closely related to the KoBoLJ model. Moreover, the fractional derivatives in the governing Eq (2.5) are non-local in order to describe the American call option value in the holding region $ (-\infty, x_f] $.
In this paper, we take American call as the research object, so the function $ V(x, t) $ satisfies the following boundary conditions:
$ {limx→−∞V(x,t)=0,V(xf,t)=exf−K,∂V(xf,t)∂x=Sf=ef,V(x,T)=max(ex−K,0). $
|
(2.6) |
To sum up, a complete pricing mathematical model of American call under the KoBoLJ process can be obtained as Eqs (2.5) and (2.6). Moreover, we remark that the above FPIDE system is much more difficult to solve than the corresponding BS case with jumps, with the main difficulty resulting from the free boundary and the non-localness of the fractional-integro differential operator. In the following section, a new numerical scheme is proposed to solve it efficiently.
According to the unique characteristics of the American call option, the value function $ V(x, t) $ of this financial derivative should satisfy the following inequality constraint
$ V(x,t)≥max(ex−K,0), $
|
(2.7) |
for all $ t\in[0, T] $ and $ x\leq x_{f} $.
There are two parts in this section. In the first part, the free boundary problem should be changed as one defined on a fixed interval by introducing a nonlinear penalty term. Both the difference scheme and theoretical analysis are displayed in the second part.
Let $ 0 < \epsilon \ll 1 $ and $ C > 0 $ be a fixed constant, and we will determine its specific value. We construct the following nonlinear penalty term
$ εCVε(x,t)+ε−Q(x),andQ(x)=ex−K. $
|
(3.1) |
Then we add it to Eq (2.5)and obtain the following system,
$ ∂Vε(x,t)∂t+a∂Vϵ(x,t)∂x+12σα[peλxxDαxmaxe−λxVε(x,t)+qe−λx−∞DαxeλxVε(x,t)]+ϵCVε(x,t)+ϵ−Q(x)+ξ∫+∞−∞V(x+y,t)fY(y)dy=(b+12σαλα)Vε(x,t), $
|
(3.2) |
where $ x\in(-\infty, x_{\max}], t\in [0, T], 1 < \alpha < 2 $,
$ limx→−∞Vε(x,t)=0, $
|
(3.3) |
$ Vϵ(xmax,t)=exmax−K, $
|
(3.4) |
$ Vϵ(x,T)=max(ex−K,0). $
|
(3.5) |
Moreover, according to the conclusion in [25], the maximum value of risk asset equal to 4 times of $ K $ value. The subscript $ \varepsilon $ of $ V_{\varepsilon}(x, t) $ should be omitted for clarity.
Define $ \Delta t > 0 $ and $ \Delta x > 0 $ as time and spatial step, respectively. Taking $ N, M $ as the positive $ N * \Delta t = T $ and $ M \Delta x = x_{\max} $. Thus
$ ti=(i−1)Δt,i=1,2,...,N+1,xj=(j−1)Δx,j=1,2,...,M+1. $
|
The forward and backward difference schemes are used for the discrete first-order space. For the time derivative, we use the forward and backward difference schemes, respectively. The approximation of the left-sided and right-sided tempered fractional derivatives given in formula [26] can be used to discretize the left-sided and right-sided tempered fractional derivatives as the following:
$ e−λx−∞Dαx(eλxVji)−λαVji=1(Δx)α∞∑k=0gkVij−k+1+O(Δx),eλx−∞Dαx(e−λxVji)−λαVji=1(Δx)αM−j+1∑k=0gkVij+k−1+O(Δx), $
|
where $ V_j^i $ is the value of function $ V(x, t) $ at grid point $ (x_j, t_i) $. The coefficients $ g_k\; (k = 0, 1, 2, \cdots) $ are used and satisfy the following two equations
$ gk={(−1)k(αk)e−(k−1)λΔx,fork≠1−α−eλΔx(1−e−λΔx)α,fork=1, $
|
(3.6) |
In addition, the integral term contained in the governing equation of Eq (3.2) is approximated by the trapezoidal rules [24], i.e.,
$ ∫+∞−∞V(xj+y,ti)fY(y)dy≈M∑ℓ=0ρMℓ−j[Viℓ+Viℓ+1]+Rj, $
|
where
$ ρMj=12∫(j+1)ΔxjΔxfY(y)dy={12∑m1ℓ=1ˆpℓ(e−ˆθℓjΔx−e−ˆθℓ(j+1)Δx),j≥0,12∑m2ℓ=1˜pℓ(e˜θℓ(j+1)Δx−e˜θℓjΔx),j≤0, $
|
(3.7) |
and
$ Rj=∫+∞xM+1−xj(exj+y−K)fY(y)dy,=exjm1∑ℓ=1ˆpℓˆθℓˆθℓ−1e(1−ˆθℓ)(xmax−xj)−Km1∑ℓ=1ˆpℓe−ˆθℓ(xmax−xj),=(exmax−K)m1∑ℓ=1ˆpℓˆθℓe−ˆθℓ(xmax−xj). $
|
(3.8) |
To sum up, the fully implicit difference scheme for Eq (3.2) can be obtained as follows:
$ Vi+1j−VijΔt+aVij−Vij−1Δx+ξM∑ℓ=0ρMℓ−j[Viℓ+Viℓ+1]+12σα[p(Δx)αM−j+2∑k=0gαk,λVij+k−1+q(Δx)α∞∑k=0gαk,λVij−k+1]+ξRj+ϵCVij+ϵ−qj=bVij, $
|
(3.9) |
and the boundary conditions are approximated as
$ limj→∞Vij=0, $
|
(3.10) |
$ ViM+1=exmax−K, $
|
(3.11) |
$ VN+1j=max(exj−K,0), $
|
(3.12) |
The fact that the values of $ V_{j}^{i} $ for all $ i, j $) must satisfy the constraint condition (2.7) should be strictly proven. In order to ensure completion of proof, we first give two lemmas as follows
Lemma 3.11. ([26]) If $ \alpha \in (1, 2) $, then the coefficients $ g_k $ in Eq (3.6) satisfy
$ {g0=eλΔx,g1=−α−eλΔx(1−e−λΔx)α<0,g2>g3>...>0,∑∞k=0gk=0,∑mk=0gk<0, $
|
where $ m\geq1 $.
Lemma 3.2. ([27]) Both the coefficients $ \rho_j^M $ in Eq (3.7) and $ R_j $ in Eq (3.8) are bounded and satisfy
$ M∑−∞ρMj≤12, $
|
$ Rj≤exmax−K. $
|
Theorem 3.1. If $ \Delta t \leq \frac{1}{\left|b+2\xi \sum^{M}_{\ell = 0 }\rho^M_{\ell-j}\right|} $ and the constant $ C $ satisfies the following inequality
$ C≥|a|exmax−1xmax+σα[(λ+1)α+e(λ+2)xmax]+(b+3ξ)K. $
|
then $ V_{j}^{i} $ obtained by Eq (3.9) satisfies the following inequality $ V_{j}^{i}\ge\max(e^{x_j}-K, 0) $. Here, $ \mathfrak{K} = \exp(x_{\max})-\; K. $
Proof. We are going to complete the proof in two steps: We first prove $ V_{j}^{i}\ge e^{x_j}-K $ and then prove that $ V_{j}^{i}\ge 0 $ for all $ i, \; j $.
Let $ Q_j = e^{x_j}-K, u_{j}^{i} = V_{j}^{i}-Q_j $, then we have
$ ui+1j−aΔtΔxuij−1+12σαΔt[p(Δx)αM−j+2∑k=0gkuij+k−1+q(Δx)α∞∑k=0gkuij−k+1]+ξΔtM∑ℓ=0ρMℓ−j+ξΔtRj+[uiℓ+uiℓ+1]+ϵCΔtuij+ϵ−ΔtF=(1−aΔtΔx+bΔt)uij, $
|
where
$ F=aΔx(qj−qj−1)−bqj−ξM∑ℓ=0ρMℓ−j[ezℓ+ezℓ+1−2K]−ξRj+12σα[p(Δx)αM−j+2∑k=0gkqj+k−1+q(Δx)α∞∑k=0gkqj−k+1]. $
|
Since $ |\frac{e^{\Delta x}-1}{\Delta x}| \leq \frac{e^{x_{\max}}-1}{x_{\max}}\leq 1 $, $ \sum^\infty_{k = 0} g_ke^{x_{j-k+1}} = e^{x_{j+1}}\sum^\infty_{k = 0}g_ke^{-k\Delta x} $, and when $ |z| < 1 $
$ ∞∑k=0(−1)k(αk)zk=(1−z)α. $
|
Hence, we have
$ ∣F∣≤|a|exmax−1xmax+b(exmax−K)+ξ(exmax−K)+12σα[λα+e(λ+1)xmax+(λ+1)α+e(λ+2)xmax]+ξ|M∑ℓ=0ρMℓ−j[exℓ+exℓ+1−2K]|.≤|a|exmax−1xmax+b(exmax−K)+ξ(exmax−K)+σα[(λ+1)α+e(λ+2)xmax]+ξ|M∑ℓ=0ρMℓ−j[exℓ+exℓ+1−2K]|. $
|
Moreover, let $ \mathfrak{K} = [\exp(x_{\max})-K] $, then
$ |M∑ℓ=0ρMℓ−j[exℓ+exℓ+1−2K]|≤2|M∑ℓ=0ρMℓ−jK|2≤KM∑ℓ=0ρMℓ−j≤2K. $
|
Therefore,
$ |F|≤|a|exmax−1xmax+σα[(λ+1)α+e(λ+2)xmax]+(b+3ξ)K. $
|
Let $ u_{J}^{i} = \min_{j}u_{j}^{i} $ and $ u_{L}^{i+1} = \min_{j}u_{j}^{i+1} $, then
$ {1−12σαΔt[p(Δx)αM−j+2∑k=0gαk,λ+q(Δx)α∞∑k=0gαk,λ]}uiJ−bΔtuiJ−ϵCΔtuiJ+ϵ−2ξM∑ℓ=0ρMℓ−juiJΔt+ΔtF≥ui+1L. $
|
Namely,
$ [1−(−b−2ξM∑ℓ=0ρMℓ−j)Δt]ui−ϵCΔtui+ϵ+ΔtF≥ui+1L. $
|
On the other hand, according to Lemma 3.2 and $ \Delta t \leq \frac{1}{\left|b+2\xi \mathop \sum \nolimits_{\ell = 0}^M \rho^M_{\ell-j}\right|} $, we can obtain
$ 1−(−b−2ξM∑ℓ=0ρMℓ−j)Δt≥0. $
|
Let
$ A=1−(−b−2ξM∑ℓ=0ρMℓ−j)Δt, $
|
and define a function $ H(x) $ as
$ H(x)=Ax−ϵCΔtx+ϵ+ΔtF. $
|
(3.13) |
Then, $ H(u^i)\geq0 $ if $ u^{i+1}\geq0 $. Since $ H'(x) = A + \frac{\epsilon C\Delta t}{(x+\epsilon)^2}\geq0 $, $ H(0) = \Delta t(F-C)\leq0 $, and $ u^{N+1}\geq0 $, we obtain $ u^i\geq0 $. Hence, $ u_j^i\geq0 $, and consequently $ V_j^i\geq Q_j $ is satisfied.
Next, we prove that $ V_j^i\geq 0 $. We define $ V^i = \min_{j}V_{j}^{i} $ and let $ J $ satisfy $ V_J^i = V^i $. Hence, according to Eq (3.9), the following inequality can be obtained,
$ {1−12σαΔt[p(Δx)αM−j+2∑k=0gαk,λ+q(Δx)α∞∑k=0gαk,λ]}Vi−bΔtVi−ϵCΔtVi+ϵ−Qj−2ξM∑ℓ=0ρMℓ−jViΔt+ΔtF≥Vi+1J. $
|
Then,
$ [1−(−b−2ξM∑ℓ=0ρMℓ−j)Δt]Vi≥Vi+1J+ϵCΔtVi+ϵ−Qj. $
|
In the first step, $ V_j^i \geq Q_j (\forall i, j) $ is proven, so $ \frac{\epsilon C\Delta t}{V^i+\epsilon-Q_j}\geq 0 $. Thus,
$ [1−(−b−2ξM∑ℓ=0ρMℓ−j)Δt]Vi≥Vi+1J. $
|
Since, $ V_j^{N+1} = \max \left[\exp(x_j)-K, 0\right] \geq 0 $, therefore
$ Vij≥0,∀i,j. $
|
To sum up, we complete the proof.
In fact, the penalty term should result that the discrete system (3.9) is nonlinear; therefore, the Newton iteration method is employed. However, due to the existence of the fractional-integro differential operator, there is a matrix with a dense form in the final system. Thus, we should enhance the computational efficiency while decreasing the storage space.
In order to facilitate the computer to simulate the algorithm (3.9), the original semi-infinite region $ (-\infty, x_{\max}]\times[0, T] $ must be truncated into a limited region $ (x, t)\in(x_{\min}, x_{\max}]\times [0, T], $ where $ x_{\min} = \ln(0.01) $ in the numerical experiments below. Now, the left boundary condition in the original model is changed to $ V(x_{\min}, t) = 0 $.
We should redefine the spatial step size $ \Delta x = (x_{\max}-x_{\min})/M $, then $ x_j = (j-1)\Delta x+x_{\min} $, for $ j = 2, \ldots, M+1 $. Now, we define
$ ϑ=aΔtΔx,β=1−aΔtΔx+Δtb,η=−12Δtσα(Δx)α, $
|
and
$ WMl=ρMl+ρMl−1,l=0,±1,±2,...,±(M−2). $
|
Then, system (3.9) can be rewritten as the following matrix form,
$ [βI+ϑB+η(pA⊤+qA)−ΔtW]Vi−F(Vi)=Vi+1+Zi−ΔtRi, $
|
(4.1) |
where
$ F(Vi)=(F(Vi2),F(Vi3),…,F(ViM−1),F(ViM)),Vi=(Vi2,Vi3,…,ViM−1,ViM), $
|
with
$ F(Vij)=ϵCΔtVij+ϵ−Qj,Ri=(Ri2,Ri2,...,RiM),Zi=(0,0,...,ηqg0+ϑ(ρM0+ρM1+...+ρMM−2))ViM+1. $
|
$ {\bf I} $ is an identity matrix of order $ (M-1) $, and $ {\bf A}^{\top} $ means matrix transpose of $ {\bf A} $. $ {\bf A} $, $ {\bf B} $, and $ {\bf W} $ are Toeplitz matrices, i.e.,
$ W=ξ[WM0WM1WM2⋯WMM−2WM−1WM0WM1⋯WMM−3WM−2WM−1WM0⋯WMM−4⋮⋱⋱⋱⋮WM2−MWM3−MWM4−M⋯WM0], $
|
$ A=[g1g00⋯0g2g10⋯0g3g2g0⋯0⋮⋱⋱⋱⋮gM−2gM−3⋯g1g0gM−1gM−2⋯g2g1]andB=[000⋯00100⋯00010⋯00⋮⋱⋱⋱⋱⋮000⋯00000⋯10]. $
|
In fact, the nonlinear penalty term shows that the system (4.1) cannot be solved directly; therefore, we first use the Newton iteration method to change it as a linear system,
$ [βI+ξB+η(pA⊤+qA)−JF(ωl−1)−ΔtW]Δwl=Vi+1−Zi+1−[βI+ξB+η(pA⊤+qA)−ΔtW]wl−1+F(wl−1)−ΔtRi, $
|
(4.2) |
$ wl=wl−1+κΔwl, $
|
where $ l = 1, 2, 3, \ldots $, $ J_{\mathcal{F}}(\mathfrak{w}^{l-1}) $ is the Jacobian matrix of the vector $ \mathcal{F} (\mathfrak{w}^{l-1}) $, and $ 0 < \kappa < 1 $ is the adjustment factor. During the numerical iteration, it is assumed that for the current time layer $ t_i $, the information of the previous time layer $ t_{i+1} $ is known. Therefore, $ V^{i+1} $ can be taken as the initial value of the iterative sequence $ \mathfrak{w}^{l} $, i.e., $ \mathfrak{w}^{0} = V^{i+1} $. We set $ V^i = \mathfrak{w}^l $ once the stopping criterion $ \parallel \mathfrak{w}^{l}-\mathfrak{w}^{l-1}\parallel\le tol $ for some $ l $ is reached, where $ tol $ is the stopping tolerance of the iterative method. Now, by taking
$ M=βI+ξB+η(pA⊤+qA)−JF(wl−1)−ΔtW,bl=Vi+1+ηZi+1−ΔtRi−[βI+ξB+η(pA⊤+qA)−ΔtW]wl−1+F(wl−1), $
|
Eq (4.2) can be rewritten as
$ [M−JF(wl−1)](δwl)=bl. $
|
(4.3) |
The most challenging part in solving Eq (4.3) is the high computational cost resulting from the fact that both $ {\bf A} $ and $ {\bf W} $ are dense matrices. To overcome this difficulty, we first apply the CGNR method [28], which is to solve $ [\mathfrak{M}-J_{\mathcal{F}}]^{\top}\mathfrak{M}\delta \mathfrak{w}^{l} = [\mathfrak{M}-J_{\mathcal{F}}]^{\top} {\bf b}^{l} $ instead of Eq (4.3).
However, by noticing that the convergence rate of the CGNR method is still quite low due to the fact that the conditional number of the matrix $ \mathfrak{M}^{\top}\mathfrak{M} $ is large, a pre-conditioner technique is applied to accelerate the convergence rate of the CGNR method. It is straightforward to find that the matrix $ J_{\mathcal{F}} $ is not the Toeplitz matrix, and we should approximate this matrix as $ a_0 {\bf I} $, where $ a_0 $ is the average value of main diagonal elements of matrix $ J_{\mathcal{F}} $. Thus, we structure a Toeplitz matrix as follows
$ T=M−a0I. $
|
Next, the Strang's circulant preconditioner $ s(\mathcal{T}) = [s_{j-k}]_{0\leq j, k < M} $ for matrix $ \mathcal{T} $ is structured as
$ s_j = {Tj,0≤j<M/2,0,j=M/2if M is even,andj=(M+1)/2 if M is odd,Tj−M,M/2<j<M,Tj+M,0<−j<M. $
|
Let $ \mathfrak{P} $ denote the Strang's circulant preconditioner $ s(\mathcal{T}) = [s_{j-k}]_{0\leq j, k < M} $ to simplify the expression.
Mathematically, after the PCGNR method with a pre-conditioner $ \mathfrak{P} $ is applied, Eq (4.3) becomes
$ [(P)−1(M−JF)]⊤[(P)−1(M−JF)]δwl=[(P)−1(M−JF)]⊤(P)−1bl. $
|
The pseudo-code of the PCGNR method is displayed in Algorithm 1. The matrix-vector multiplication needs only $ \mathcal{O}(MlogM) $ operations via the fast Fourier transform (FFT) method.
Algorithm 1. PCGNR method for solving $ (\mathfrak{M}-J_{\mathcal{F}})(\delta \mathfrak{w}^{l}) = {\bf b}^{l} $ with a pre-conditioner $ \mathfrak{P} $. |
Given the initial guess $ x_0 $, and a stopping tolerance $ tol $. |
Compute $ r_0 = \left[\mathfrak{P}^{-1}({\bf b}^{l} - (\mathfrak{M}-J_{\mathcal{F}})) x_0 \right] $, |
$ z_0 = \left[(\mathfrak{P})^{-1}(\mathfrak{M}-J_{\mathcal{F}})\right]^{\top}r_0, p_0 = z_0, mr = ||r_0||_2^2 $. |
For $ i = 0, 1, ..., $ |
$ w_i = \left[(\mathfrak{P})^{-1}(\mathfrak{M}-J_{\mathcal{F}})\right]^{\top}p_i $, |
$ \alpha_i = ||z_i||_2^2/||w_i||_2^2 $, |
$ x_{i+1} = x_i + \alpha_i p_i $, |
$ r_{i+1} = r_i - \alpha_i w_i $, |
$ z_{i+1} = \left[(\mathfrak{P})^{-1}(\mathfrak{M}-J_{\mathcal{F}})\right]^{\top} r_{i+1} $, |
$ \beta_{i} = ||z_{i+1}||_2^2/||z_i||_2^2 $, |
$ p_{i+1} = z_{i+1} + \beta_{i} p_i $, |
$ res = ||r_{i+1}||_2^2 $. |
If $ res/mr < tol $, stop; |
otherwise, set $ \delta \mathfrak{w}^{l} = x_{i+1} $. |
End for |
Several numerical examples are given to show the computational efficiency of our numerical method in this part. Moreover, the impacts of the key parameter in our model to the option value and optimal exercise boundary are also discussed. All simulations are implemented using MATLAB2014a on a Lenovo T14 laptop with configuration: Intel(R) Core(TM) i7-1260P 2.10 GHz. The CPU time (in seconds) is estimated by using the timing functions tic/toc.
First, we should examine whether or not the numerical solution preserves the basic properties of American call. This could be viewed as a necessary condition for the reliability of the proposed approach. Mathematically, the current numerical solution must satisfy the inequality $ V_j^i \geq \max(q_j - K, 0) $. Depicted in Figure 1 are the surfaces of $ V_j^i-\max(q_j-K, 0) $ with different parameter settings, which implies that the inequality is preserved.
Figure 2(a), (b) display the American call value surface and option values and payoff function, respectively. First, the curves in the two figures indicate that the American call option price is an increasing function with respect to an underlying asset price, and the 'high contact' condition for American call is also confirmed by such surfaces in Figure 2(a), which shows KoBoLJ model is indeed reasonable. It can be observed from the two figures that the numerical method based on the penalty term produces the smooth and stable approximation solutions. To sum up, both our model and the numerical scheme are reasonable.
To further investigate the performance of the method, we compare the computational efficiency of the Gaussian elimination (GE), the CGNR method, and the PCGNR method, as shown in Table 1. The parameters adopted for computing this table are $ K = 20 $, $ r = 0.05 $, $ \sigma = 0.24 $, $ D = 0.06 $, $ p = 0.6 $, $ \hat{p} = 0.07 $, $ \hat{\theta} = 1.5 $, $ \tilde{\theta} = 0.5 $, , $ \alpha = 1.52 $, $ \xi = 0.2 $, $ \hat{p} = 0.08 $, $ \hat{\theta} = 1.8 $, $ \tilde{\theta} = 0.2 $, $ \xi = 0.1 $. Moreover, in this table, $ Ite-In $ denotes the average iteration number required in each time step. $ OR_{GE} $, $ OR_{CGNR} $ and $ OR_{PCGNR} $ refer the convergence order in $ x $ direction of three different method, respectively. The convergence order is defined as
$ ORi+1=ln(Erri)−ln(Erri+1)ln(Mi+1)−ln(Mi), $
|
where $ M_i $ is the number of spatial grid nodes employed and
$ Err=‖Vj,ik−V(k;x,t)ref‖2, $
|
where $ \parallel \cdot\parallel_2 $ is the $ L_2 $ norm for matrix, and $ V(x, t)_{\mathrm{ref}} $ is the benchmark solution determined directly through matrix operation '$ A\backslash b $' in Matlab with $ (M, N) = (2^{12}, 1000) $.
GE | CGNR | PCGNR | ||||||||||||
$M$ | $Time(s)$ | $Err$ | $OR_{GE}$ | $Ite-In$ | $Time(s)$ | $Err$ | $OR_{CGNR}$ | $Ite-In$ | $Time(s)$ | $Err$ | $OR_{PCGNR}$ | |||
$2^5$ | 37.5201 | 0.0743 | - | 51.2422 | 1.0240 | 0.0793 | - | 5.2846 | 0.6513 | 0.0945 | - | |||
$2^6$ | 150.2406 | 0.0303 | 1.2940 | 52.7299 | 1.9356 | 0.0322 | 1.3003 | 6.4601 | 0.7291 | 0.0402 | 1.2331 | |||
$2^7$ | 628.7839 | 0.0114 | 1.4103 | 50.7812 | 3.5031 | 0.0130 | 1.3085 | 7.0381 | 1.3810 | 0.0164 | 1.2935 | |||
$2^{8}$ | 2494.7124 | 0.0042 | 1.4406 | 52.3963 | 9.5677 | 0.0055 | 1.2410 | 6.8341 | 7.1290 | 0.0068 | 1.2701 | |||
$2^{9}$ | 18290.4211 | 0.0017 | 1.3049 | 52.1256 | 49.9873 | 0.0023 | 1.2578 | 6.8930 | 19.0211 | 0.0026 | 1.3870 | |||
$2^{10}$ | ** | ** | ** | 52.6767 | 250.1262 | 0.0010 | 1.2016 | 7.0025 | 21.4600 | 0.0011 | 1.2410 |
We can observe from Table 1 that for a fixed number of nodal points, the total CPU times required by the CGNR and PCGNR to produce the same level of error are significantly less than that of the GE. Furthermore, the average inner iteration numbers required by the PCGNR method are the lowest. These suggest the superiority of the PCGNR method in computational efficiency over the GE and CGNR methods. Moreover, it is clear that the $ OR_{GE} $, $ OR_{CGNR} $, and $ OR_{PCGNR} $ are close to 1, which indicates that the three schemes are of first-order convergence in the spatial direction.
Similarly, the convergence order and error in the $ t $ direction of the PCGNR method is also examined. First, the $ V(x, t)_{\mathrm{ref}} $ is the benchmark solution that can be determined directly through matrix operation '$ A\backslash b $' in Matlab with $ (M, N) = (2^{12}+1, 1000) $. We increase the grid number in the $ t $ direction from $ 100 $ to $ 800 $. In Table 2, both the $ Err $ and $ OR $ denote the error and convergence order in the $ t $ direction of PCGNR method, respectively. The results are displayed in Table 2. From this table, it is clear that our scheme is first-order convergent.
Number of time steps | Err | OR |
100 | 0.0701 | – |
200 | 0.0334 | 1.0696 |
400 | 0.0156 | 1.0983 |
600 | 0.0103 | 1.0238 |
800 | 0.0046 | 1.1629 |
We first consider the value of parameter $ \alpha $, which affects the optimal exercise boundary of American call. Tow sets of optimal exercise boundary with different $ \alpha $ are computed and displayed in Figure 3. From the curves in this figure, one can find that a bigger value of $ \alpha $ should show a higher optimal exercise boundary. Financially, the $ \alpha $ controls the tail of the distribution of the returns of risk asset, and both tails will be fatter when $ \alpha $ becomes bigger. Thus, as $ \alpha $ becomes bigger, the possibility of smaller stock price increases, and so does the optimal exercise price.
Next, we consider how the discrete jumps influence the optimal exercise price of American call. As shown in Figure 4 is the optimal exercise price as a function of the time to expiry with different jump intensity $ \xi $. One can observe from this figure that the optimal exercise price increases with respect to $ \xi $. Financially, a larger jump intensity indicates that the risk asset would change more often so that the American call option contract should be more valuable as it contains more risks. Hence, according to the smooth pasty condition across the free boundary, the monotonicity of $ S_f $ with respect to $ \xi $ holds automatically.
In Figure 5, the optimal exericse price is plotted against the time to expiry with different probabilities of positive jumps $ \hat{p} $. From the curves in this figure, it is straightforward to find that a larger $ \hat{p} $ results in a lower optimal exercise boundary curve. In fact, the logarithmic return of the risk asset is decreasing with respect to $ \hat{p} $, because the return decreases with respect to $ \xi $ from Eq (2.1) and $ \xi $ increases with respect to $ \hat{p} $ from Eq (2.2). Therefore, an increasing $ \hat{p} $ would lower the risk asset value, and thus makes the intermediate American call option less valuable. Therefore, the optimal exercise boundary $ e^{x_f} $ of the intermediate American call decreases with respect to $ \hat{p} $. Similarly, one could explain the monotonicity of the optimal exercise price with respect to $ \hat{\theta} $ and $ \tilde{\theta} $. For the length of the paper, we provide those curves in Figure 6 with no detailed explanations.
Next, we should investigate the impacts of parameters $ p $ and $ q $. In fact, the upward movement frequency of our stochastic process is controlled by parameter $ p $. If the value of parameter $ p $ becomes bigger, which means that our stochastic process should have increased upward movement, then the American call option price should become bigger. As a rational investor, a higher price should be used to exercise the option. Hence, a bigger value of $ p $ should result in a higher optimal exercise price as shown in Figure 7. Similarly, we can analyze the impacts of parameter $ q $ on the optimal exercise boundary.
The optimal exercise boundary curves under different parameter $ \lambda $ are displayed in Figure 8. As described in Section 3.2, the decay rate of tails of our stochastic process probability density function is controlled by parameter $ \lambda > 0 $. Thus, a bigger value of this parameter should result in a thinner tail of the stochastic process density function, and the investor should want to gain a bigger price to exercise the American option.
In this subsection, we consider the stock loans based on the finite moment log-stable process (FMLS). Under this framework of FMLS, the stock loans pricing model is [25].
$ {∂V(x,t)∂t−(r−D−ν)∂V(x,t)∂x+ν−∞DαxV(x,t)=−rV(x,t),limx→−∞V(x,t)=0,V(x,T)=max(ex−KeγT,0),V(xf,t)=exf−Keγt,∂V(xf,t)∂x=exf, $
|
(5.1) |
where $ V(x, t) $ denotes the price of stock loans, $ r $, $ D $, and $ t $ are the risk free interest rate, the dividend and the current time, respectively, $ \sigma $ is a constant, and $ \nu = -\sigma^{\alpha}\sec\frac{\alpha\pi}{2} $ is a convexity adjustment. $ t\in [0, T], x \in(-\infty, x_f), 1 < \alpha < 2, e^{x_f} $ is the optimal redemption price of stock loans. Thus,
$ −∞DαxV(x,t;α)=1Γ(2−α)∂2∂x2∫x−∞V(z,t;α)(x−z)α−1dz. $
|
In fact, the FMLS model is a special case of the KoBoLJ model. Hence, the method is used to solve model (5.1).
We choose the spatial step size $ \Delta x = \frac{x_{\max}-x_{\min}}{2^{11}+1} $ and temporal step size $ \Delta t = \frac{2}{1000} $. Thus, we can obtain the following three figures:
The curved surface in Figure 9 and the curve in Figures 10 and 11 show that our numerical method is effective.
In this paper, we consider the American call option pricing based on the KoBoLJ model. The pricing model is a free boundary problem, and the governing equation is a FPIDE. Thus, a numerical scheme based on the penalty function is set. Both the pricing mathematical model and current scheme are very reliable, which is verified by our numerical results. In order to improve computational efficiency, both the PCGNR and fast Fourier transform technique are used to solve the final linear system. Moreover, the impacts of key parameters $ \alpha, \; p, \; \hat{p}, \; \hat{\theta}, \; \tilde{\theta} $, and $ \lambda $ on optimal exercise price are also analyzed.
At the end of this section, we point out that several issues are not discussed in this paper but the future studies will be implemented for them. First, a risk-free interest rate is a constant in our model. In fact, the constant interest rate cannot describe how the interest rate evolves with respect to the time, especially for the option contracts that have a long time horizon. Second, our numerical results show that the KoBoLJ model is a more comprehensive model than the KoBoJ model; therefore, it should be used to investigate other financial derivative pricing and hedging problems, such as the CDS and Stock Loans. Finally, for the jump processes without the consideration of diffusion processes, we should discuss whether their approaches can be extended to models with jumps and diffusion, such as the stochastic volatility and stochastic liquidity [29,30,31].
The first author is mainly responsible for formula derivation and programming. The second author is primarily responsible for writing the paper and conducting theoretical analysis.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
The work on this paper is partially supported by the Guizhou University of Commerce Natural Science Projects (No. [2022]ZKZD003). Guizhou University of Commerce Social Science Projects (No. 2024XJSDZD07). Guizhou Provincial Department of Education Natural Science Research Project (No. QJJ[2024]179). And the Disaster Remote Sensing Prevention Workstation of the Academician Innovation Team in Guizhou Province (No. KXJZ[2024]006).
The authors declare there is no conflict of interest.
[1] |
Liu HH., Cohen Y (2014) Multimedia environmental distribution of engineered nanomaterials. Environ Sci Technol 48: 3281-3292. doi: 10.1021/es405132z
![]() |
[2] | Lopez-Serrano A, Olivas RM, Landaluze JS, et al. (2014) Nanoparticles: a global vision. Characterization, separation, and quantification methods. Potential environmental and health impact. Anal Methods 6: 38-56. |
[3] |
Li DW, Zhai WL, Li YT, Long YT (2014) Recent progress in surface enhanced Raman spectroscopy for the detection of environmental pollutants. Microchim Acta 181: 23-43. doi: 10.1007/s00604-013-1115-3
![]() |
[4] |
Lalley J, Dionysiou DD, Varma RS, et al. (2014) Silver-based antibacterial surfaces for drinking water disinfection - an overview. Curr Opin Chem Eng 3: 25-29. doi: 10.1016/j.coche.2013.09.004
![]() |
[5] |
Han Ch, Likodimos V, Khan JA, et al. (2014) UV-visible Light-activated Ag-decorated, monodispersed TiO2 aggregates for treatment of the pharmaceutical oxytetracycline. Environ Sci Pollut Res 21: 11781-11793. doi: 10.1007/s11356-013-2233-5
![]() |
[6] |
Windler I, Height M, Nowack B (2013) Comparative evaluation of antimicrobials for textile applications. Environ Int 53: 62-73. doi: 10.1016/j.envint.2012.12.010
![]() |
[7] |
Geranio L, Heuberger M, Nowack B (2009) The behavior of silver nanotextiles during washing. Environ Sci Technol 43: 8113-8. doi: 10.1021/es9018332
![]() |
[8] |
Adegboyega NF, Sharma VK, Siskova K, et al. (2013) Interactions of Aqueous Ag+ with Fulvic Acids: Mechanisms of Silver Nanoparticle Formation and Investigation of Stability. Environ Sci Technol 47: 757-764. doi: 10.1021/es302305f
![]() |
[9] |
Akaighe N, MacCuspie RI, Navarro DA, et al. (2011) Humic acid-induced silver nanoparticle formation under environmentally relevant conditions. Environ Sci Technol 45: 3895-3901. doi: 10.1021/es103946g
![]() |
[10] |
Nadagouda MN, Iyanna N, Lalley J, et al. (2014) Synthesis of silver and gold nanoparticles using antioxidants from blackberry, blueberry, pomegranate, and turmeric extracts. ACS Sustanaible Chem Eng 2: 1717-1723. doi: 10.1021/sc500237k
![]() |
[11] |
Markova Z, Siskova KM, Filip J, et al. (2013) Air stable magnetic bimetallic Fe-Ag nanoparticles for advanced antimicrobial treatment and phosphorus removal. Environ Sci Technol 47: 5285-5293. doi: 10.1021/es304693g
![]() |
[12] | Sigma-Aldrich. Inc., Product Information, 70192 Mueller Hinton Broth (M-H Broth). Available from: http://www.sigmaaldrich.com/content/dam/sigma-aldrich/docs/Fluka/Datasheet/70192dat.pdf |
[13] |
Sintubin L, Verstraete W, Boon N (2012) Biologically produced nanosilver: current state and future perspectives. Biotechnol Bioeng 109: 2422-2436. doi: 10.1002/bit.24570
![]() |
[14] |
Faramarzi MA, Sadighi A (2013) Insights into biogenic and chemical production of inorganic nanomaterials and nanostructures. Adv Colloid Interface Sci 189-190: 1-20. doi: 10.1016/j.cis.2012.12.001
![]() |
[15] |
Suman TY, Radhika Rajasree SR, Kanchana A, et al. (2013) Biosynthesis, characterization and cytotoxic effect of plant mediated silver nanoparticles using Morinda citrifolia root extract. Colloids Surf B 106: 74-78. doi: 10.1016/j.colsurfb.2013.01.037
![]() |
[16] |
Kharissova OV, Dias HVR, Kharisov BI, et al. (2013) The greener synthesis of nanoparticles. Trends Biotechnol 31: 240-248. doi: 10.1016/j.tibtech.2013.01.003
![]() |
[17] |
Hu B, Wang SB, Wang K, et al. (2008) Microwave-assisted rapid facile “green” synthesis of uniform silver nanoparticles: self-assembly into multilayered films and their optical properties. J Phys Chem C 112: 11169-11174. doi: 10.1021/jp801267j
![]() |
[18] |
Alvarez-Puebla RA, Aroca RF (2009) Synthesis of silver nanoparticles with controllable surface charge and their application to surface-enhanced Raman scattering. Anal Chem 81: 2280-2295. doi: 10.1021/ac8024416
![]() |
[19] |
Khan Z, Talib A (2010) Growth of different morphologies (quantum dots to nanorod) of Ag-nanoparticles: role of cysteine concentration. Coll Surf B 76: 164-169. doi: 10.1016/j.colsurfb.2009.10.029
![]() |
[20] |
Rafey A, Shrivastavaa KBL, Iqbal SA, et al. (2011) Growth of Ag-nanoparticles using aspartic acid in aqueous solutions. J Colloid Interface Sci 354: 190-195. doi: 10.1016/j.jcis.2010.10.046
![]() |
[21] |
Jacob JA, Naumov S, Mukherjee T, et al. (2011) Preparation, characterization, surface modification and redox reactions of silver nanoparticles in the presence of tryptophan. Coll Surf B 87: 498-504. doi: 10.1016/j.colsurfb.2011.06.017
![]() |
[22] |
Sondi I, Goia DV, Matijevic E (2003) Preparation of highly concentrated stable dispersions of uniform silver nanoparticles. J Colloid Interface Sci 260: 75-81. doi: 10.1016/S0021-9797(02)00205-9
![]() |
[23] | Nadagouda MN, Varma RS (2008) Green synthesis of Ag and Pd nanospheres, nanowires, and nanorods using vitamin B2: catalytic polymerisation of aniline and pyrrole. J Nanomat 2008: 1-8. |
[24] |
Tan S, Erol M, Attygalle A, et al. (2007) Synthesis of positively charged silver nanoparticles via photoreduction of AgNO3 in branched polyethyleneimine/HEPES solutions. Langmuir 23: 9836-9843. doi: 10.1021/la701236v
![]() |
[25] |
Frattini A, Pellegri N, Nicastro D, et al. (2005) Effect of amine groups in the synthesis of Ag nanoparticles using aminosilanes. Mater Chem Phys 94: 148-152. doi: 10.1016/j.matchemphys.2005.04.023
![]() |
[26] |
Xie J, Lee JY, Wang DIC, et al. (2007) Silver nanoplates: from biological to biomimetic synthesis. ACS Nano 1: 429-439. doi: 10.1021/nn7000883
![]() |
[27] | Stevanovic M, Savanovic I, Uskokovic V, et al. (2012) A new, simple, green, and one-pot four-component synthesis of bare and poly(alpha,gamma, L-glutamic acid)-capped silver nanoparticles. Colloid Polym Sci 190: 221-231. |
[28] |
Yu DG (2007) Formation of colloidal silver nanoparticles stabilized by Na+-poly(gamma-glutamic acid)-silver nitrate complex via chemical reduction process. Coll Surf B 59: 171-178. doi: 10.1016/j.colsurfb.2007.05.007
![]() |
[29] |
Vigneshwaran N, Nachane RP, Balasubramanya RH (2006) A novel one-pot “green” synthesis of stable silver nanoparticles using soluble starch. Carbohydr Res 341: 2012-2018. doi: 10.1016/j.carres.2006.04.042
![]() |
[30] |
Raveendran P, Fu J, Wallen SL (2003) Completely “green” synthesis and stabilization of metal nanoparticles. J Am Chem Soc 125: 13940-13941. doi: 10.1021/ja029267j
![]() |
[31] |
Tai C, Wang YH, Liu HS (2008) A green process for preparing silver nanoparticles using spinning disk reactor. AIChE J 54: 445-452. doi: 10.1002/aic.11396
![]() |
[32] |
Sun SK, Wang HF, Yan XP (2011) A sensitive and selective resonance light scattering bioassay for homocysteine in biological fluids based on target-involved assembly of polyethyleneimine-capped Ag-nanoclusters. Chem Commun 47: 3817-3819. doi: 10.1039/c0cc04463f
![]() |
[33] |
Lee KJ, Browning LM, Nallathamby PD, et al. (2013) Study of charge-dependent transport and toxicity of peptide-functionalized silver nanoparticles using zebrafish embryos and single nanoparticle plasmonic spectroscopy. Chem Res Toxicol 26: 904-917. doi: 10.1021/tx400087d
![]() |
[34] |
Chernousova S, Epple M (2013) Silver as antibacterial agent: ion, nanoparticle, and metal. Angew Chem Ind Ed 52: 1636-1653. doi: 10.1002/anie.201205923
![]() |
[35] |
Sharma VK, Siskova KM, Zboril R, et al. (2014) Organic-coated silver nanoparticles in biological and environmental conditions: Fate, stability and toxicity. Adv Colloid Interface Sci 204: 15-34. doi: 10.1016/j.cis.2013.12.002
![]() |
[36] |
Rai MK, Deshmukh SD, Ingle AP, et al. (2012) Silver nanoparticles: the powerful nanoweapon against multidrug-resistant bacteria. J Appl Microbiol 112: 841-852. doi: 10.1111/j.1365-2672.2012.05253.x
![]() |
[37] |
Li WR, Xie XB, Shi QS, et al. (2010) Antibacterial activity and mechanism of silver nanoparticles on Escherichia coli. Appl Microbiol Biotechnol 85: 1115-1122. doi: 10.1007/s00253-009-2159-5
![]() |
[38] |
Li WR, Xie XB, Shi QS, et al. (2011) Antibacterial effect of silver nanoparticles on Staphylococcus aureus. Biometals 24: 135-141. doi: 10.1007/s10534-010-9381-6
![]() |
[39] |
Lara HH, Ayala-Nunez NV, Turrent LCI, et al. (2010) Bactericidal effect of silver nanoparticles against multidrug-resistant bacteria. World J Microbiol Biotechnol 26: 615-621. doi: 10.1007/s11274-009-0211-3
![]() |
[40] |
Lok CN, Ho CM, Chen R, et. al. (2006) Proteomic analysis of the mode of antibacterial action of silver nanoparticles. J Proteome Res 5: 916-924. doi: 10.1021/pr0504079
![]() |
[41] |
Yang XY, Gondikas AP, Marinakos SM, et al. (2012) Mechanism of silver nanoparticle toxicity is dependent on dissolved silver and surface coating in Caenorhabditis elegans. Environ Sci Technol 46: 1119-1127. doi: 10.1021/es202417t
![]() |
[42] |
Morones JR, Elechiguerra, JL, Camacho A, et al. (2005) The bactericidal effect of silver nanoparticles. Nanotechnology 16: 2346. doi: 10.1088/0957-4484/16/10/059
![]() |
[43] | Sondi I, Salopek-Sondi B (2004) Silver nanoparticles as antimicrobial agent: a case study on E. coli as a model for gram-negative bacteria. J Colloid Interface Sci 275: 177-182. |
[44] |
Xu H, Qu F, Xu H, et al. (2012) Role of reactive oxygen species in the antibacterial mechanism of silver nanoparticles on Escherichia coli O157:H7. Biometals 25: 45-53. doi: 10.1007/s10534-011-9482-x
![]() |
[45] |
Choi O, Hu Z (2008) Size dependent and reactive oxygen species related nanosilver toxicity to nitrifying bacteria. Environ Sci Technol 42: 4583-4588. doi: 10.1021/es703238h
![]() |
[46] |
Xiu ZM, Zhang QB, Puppala HL, et al. (2012) Negligible particle-specific antibacterial activity of silver nanoparticles. Nano Lett 12: 4271-4275. doi: 10.1021/nl301934w
![]() |
[47] |
El Badawy AM, Luxton TP, Silva RG, et al. (2010) Impact of environmental conditions (pH, ionic strength, and electrolyte type) on the surface charge and aggregation of silver nanoparticles suspensions. Environ Sci Technol 44: 1260-1266. doi: 10.1021/es902240k
![]() |
[48] |
Siskova KM, Machala L, Tucek J, et al. (2013) Mixtures of L-amino acids as reaction medium for iron nanoparticles formation: the order of addition into ferrous salt solution matters. Int J Mol Sci 14: 19452-19473. doi: 10.3390/ijms141019452
![]() |
[49] |
Sloufova I, Siskova K, Vlckova B, et al. (2008) SERS-activating effect of chlorides on borate-stabilized silver nanoparticles: formation of new reduced adsorption sites and induced nanoparticle fusion. Phys Chem Chem Phys 10: 2233-2242. doi: 10.1039/b718178g
![]() |
[50] |
Siskova K, Becicka O, Safarova K, et al. (2013) HCl effect on two types of Ag nanoparticles utilizable in detection of low concentrations of organic species. Sustainable nanotechnology and the environment: advances and achievements 1124: 151-163. doi: 10.1021/bk-2013-1124.ch009
![]() |
[51] |
Sriramulu DD, Lunsdorf H, Lam JS, et al. (2005) Microcolony formation: a novel biofilm model of Pseudomonas aeruginosa for the cystic fibrosis lung. J Med Microbiol 54: 667-676. doi: 10.1099/jmm.0.45969-0
![]() |
[52] | Nakamoto K (2009) Infrared and Raman spectra of inorganic and coordination compounds, part B, Hoboken, New Jersey, USA, John Wiley & Sons, Inc., 26-84 and 388-392. |
[53] |
Moskovits M, Suh JS (1985) Confirmation of mono- and dicarboxylic acids adsorbed on silver surfaces. J Am Chem Soc 107: 6826-6829. doi: 10.1021/ja00310a014
![]() |
[54] |
Munro CH, Smith WE, Garner M, et al. (1995) Characterization of the surface of a citrate-reduced colloid optimized for use as a substrate for surface-enhanced resonance Raman scattering. Langmuir 11: 3712-3720. doi: 10.1021/la00010a021
![]() |
[55] | Lindon JC (2000) Encyclopedia of spectroscopy and spectrometry, part II, San Diego, Academic press, 1035-1057. |
[56] | Schaumann GE, Philippe A, Bundschuh M, et al. (2014) Understanding the fate and biological effects of Ag- and TiO2- nanoparticles in the environment: The quest for advanced analytics and interdisciplinary concepts. Sci Total Environ: S0048-9697: 1473-1479. |
GE | CGNR | PCGNR | ||||||||||||
$M$ | $Time(s)$ | $Err$ | $OR_{GE}$ | $Ite-In$ | $Time(s)$ | $Err$ | $OR_{CGNR}$ | $Ite-In$ | $Time(s)$ | $Err$ | $OR_{PCGNR}$ | |||
$2^5$ | 37.5201 | 0.0743 | - | 51.2422 | 1.0240 | 0.0793 | - | 5.2846 | 0.6513 | 0.0945 | - | |||
$2^6$ | 150.2406 | 0.0303 | 1.2940 | 52.7299 | 1.9356 | 0.0322 | 1.3003 | 6.4601 | 0.7291 | 0.0402 | 1.2331 | |||
$2^7$ | 628.7839 | 0.0114 | 1.4103 | 50.7812 | 3.5031 | 0.0130 | 1.3085 | 7.0381 | 1.3810 | 0.0164 | 1.2935 | |||
$2^{8}$ | 2494.7124 | 0.0042 | 1.4406 | 52.3963 | 9.5677 | 0.0055 | 1.2410 | 6.8341 | 7.1290 | 0.0068 | 1.2701 | |||
$2^{9}$ | 18290.4211 | 0.0017 | 1.3049 | 52.1256 | 49.9873 | 0.0023 | 1.2578 | 6.8930 | 19.0211 | 0.0026 | 1.3870 | |||
$2^{10}$ | ** | ** | ** | 52.6767 | 250.1262 | 0.0010 | 1.2016 | 7.0025 | 21.4600 | 0.0011 | 1.2410 |
Number of time steps | Err | OR |
100 | 0.0701 | – |
200 | 0.0334 | 1.0696 |
400 | 0.0156 | 1.0983 |
600 | 0.0103 | 1.0238 |
800 | 0.0046 | 1.1629 |
GE | CGNR | PCGNR | ||||||||||||
$M$ | $Time(s)$ | $Err$ | $OR_{GE}$ | $Ite-In$ | $Time(s)$ | $Err$ | $OR_{CGNR}$ | $Ite-In$ | $Time(s)$ | $Err$ | $OR_{PCGNR}$ | |||
$2^5$ | 37.5201 | 0.0743 | - | 51.2422 | 1.0240 | 0.0793 | - | 5.2846 | 0.6513 | 0.0945 | - | |||
$2^6$ | 150.2406 | 0.0303 | 1.2940 | 52.7299 | 1.9356 | 0.0322 | 1.3003 | 6.4601 | 0.7291 | 0.0402 | 1.2331 | |||
$2^7$ | 628.7839 | 0.0114 | 1.4103 | 50.7812 | 3.5031 | 0.0130 | 1.3085 | 7.0381 | 1.3810 | 0.0164 | 1.2935 | |||
$2^{8}$ | 2494.7124 | 0.0042 | 1.4406 | 52.3963 | 9.5677 | 0.0055 | 1.2410 | 6.8341 | 7.1290 | 0.0068 | 1.2701 | |||
$2^{9}$ | 18290.4211 | 0.0017 | 1.3049 | 52.1256 | 49.9873 | 0.0023 | 1.2578 | 6.8930 | 19.0211 | 0.0026 | 1.3870 | |||
$2^{10}$ | ** | ** | ** | 52.6767 | 250.1262 | 0.0010 | 1.2016 | 7.0025 | 21.4600 | 0.0011 | 1.2410 |
Number of time steps | Err | OR |
100 | 0.0701 | – |
200 | 0.0334 | 1.0696 |
400 | 0.0156 | 1.0983 |
600 | 0.0103 | 1.0238 |
800 | 0.0046 | 1.1629 |