Asymptotic Convergence Analysis and Error Estimate for Black-Scholes Model of Option Pricing

In this work, we discuss the numerical method for the solution of the Black-Scholes model. First of all, the asymptotic convergence for the solution of Black-Scholes model is proved. Second, we develop a linear, unconditionally stable, and second-order time-accurate numerical scheme for this model. By using the ﬁnite diﬀerence method and Legendre-Galerkin spectral method, we construct a time and space discrete scheme. Finally, we prove that the scheme has second-order accuracy and spectral accuracy in time and space, respectively. Several numerical experiments further verify the convergence rate and eﬀectiveness of the developed scheme.


Introduction
e Black-Scholes (B-S) model is a classical pricing model of European options, which was developed by the economists Black and Scholes [1]. Because of their special contribution to the model, Merton and Scholes were awarded the Nobel Prize in 1997. is model can effectively price financial derivatives such as stocks, currencies, and bonds. Merton [2,3] constructed the theoretical framework of option pricing and gave corresponding examples to verify the theory. Nowadays, B-S model is playing an increasingly important role in derivative financial instruments; it has been widely used in actual trading, which has inspired many options, trading strategies, and hedging strategies [4,5,[5][6][7].
Based on the B-S theoretical framework, Duffie [8] proposed a formula of the market value of securities, in which the arbitrage-free value of derivative securities is obtained by solving partial differential equations. Dennis and Antonio [9] presented a new method to approximately assess price under the background of continuous time model. ey discussed this method under various conditions, including option pricing with random fluctuation, Greek calculation, and term structure of interest rate. He [10] discussed the convergence of contingent claim price model from discrete time to continuous time. ese results show that the contingent claim price and dynamic replication portfolio strategy converge to the continuous time limit.
Kim et al. [11] considered the pricing of a European option on a new multiscale mixed structure of underlying assess price fluctuation, and the result indicates that the option price has an ideal modification to Black-Scholes formula. In addition, this kind of correction can bring significant improvement in fitting the surface of implied volatility through calibration exercises. Lai [12] studied the influence of time discretization on European option pricing. e correction and discrete-time rebalancing strategies caused by discrete transactions are reconsidered, and the higher-order terms are expanded by Taylor series, and the corresponding correction source terms are derived. Yousuf et al. [13] proposed a second-order exponential time differencing scheme to solve nonlinear Black-Scholes model with transaction cost.
In this paper, we will study the asymptotic property of the solution of B-S model. In addition, we develop a time and space discrete scheme, where the time direction has second-order accuracy and the space direction has spectral accuracy.
e stability and convergence of fully discrete scheme are also proved. Finally, numerical examples were shown to verify the accuracy of theoretical analysis.
is paper is organized as follows. In Section 2, the B-S model will be introduced. In Section 3, we will present second-order numerical method. e error estimates will be presented in Section 4. In Section 5, several numerical examples will be used to demonstrate the effectiveness of the fully discrete method. We will show some conclusions in the last section.

Black-Scholes Model
We consider the B-S model as follows: where V is the price of the option, S is the price of the underlying asset, r is the interest rate, t is the time, and σ is the volatility of the stock price.

Remark 1.
e B-S model is actually a stochastic partial differential equation; this model has a closed solution (we should know that most partial differential equations do not have it). at is to say, the option price can be accurately expressed by a function V � f(S, t), and the option price can be directly calculated by substituting the value of the independent variables S, t. In fact, it is difficult to get the desired theoretical and numerical results by directly analyzing the above equations. erefore, we need to make some equivalent transformations to transform the equation into a more general form.
We set t : � T − τ; and V(S, τ) � u(e x , T − τ). us, we can convert model (1) into the following form: where α � 1/2σ 2 , β � r − α, and with the following boundary (barrier) and initial conditions, In order to solve the above model better, we consider constructing a numerical method on the finite interval (a, b), and then the considered model becomes For the solution of models (6)-(8), we have the following asymptotic estimation results.

Lemma 1. Suppose u is the solution of (6)-(8), then we find
Proof. Taking the inner product of equation (6) with u, we find Multiplying the above equation by 2e 2rτ , Let H(τ) � e 2rτ ‖u‖ 2 , then we find Integrating with respect to time from 0 to τ, we have Multiplying both sides by e − 2rτ , we find is concludes the proof.

Second-Order Numerical Method
Let K be a positive integer, Δt � T/K be the time step, and τ n � nΔt, n � 0, 1, . . . , K be mesh point. Using Crank-Nicolson formula to time discretization, we can obtain the following time-discrete scheme: Here, u n+1/2 � u n+1 + u n /2, n ≥ 0. We can prove the following unconditional energy stability theorem for scheme (15).

Theorem 1.
e time discretization scheme (15) is unconditionally stable. It satisfies the following properties: Proof. Taking the L 2 inner produce of the (15) with Δt(u n+1 + u n ), we arrive at Giving up the last two terms of the left hand side, we have is yields (16).

Error Estimate
In this part, we consider Legendre-Galerkin spectral method for the time-discrete scheme (12). We will present some error estimates for full-discretization schemes in L 2 norm. First, we denote S N is the Legendre polynomial space, and denote π N : L 2 (Ω) ⟶ S N is the L 2 -projection operator which satisfies as follows: We have the following estimate [14]: Next, we begin to analyze the error estimates of the fulldiscrete scheme (23). We denote the truncation error as follows: We know that R n+1/2 satisfies We denote error functions: e n 1 � π N u τ n − u n N , e n 2 � u τ n − π N u τ n , e n u � e n 1 + e n 2 � u τ n − u n N .
en, we can develop the following full-discrete scheme: We now state the stability results for full-discrete scheme (23).
Proof. Taking ϕ N � Δt(u n+1 N + u n N ), we find that Giving up two positive terms on the right hand side, we have

Mathematical Problems in Engineering 3
Finally, we obtain the desired result (24). □ Theorem 3. For the constructed numerical scheme (24), we have the following error estimate: Proof. For n � 0, equation (23) can be written as Subtracting (28) from (7) at τ 1 , we note that (30) en, we have Setting ϕ N � 2e 1 1 , we have 4 Mathematical Problems in Engineering (32) Dropping some positive terms, we find Subtracting (23) from a reformulation of (7) at τ n+1/2 , we find Letting us, we have Summing up for n � 1, . . . , k, we find By using discrete Gronwall lemma, we can get Note that and using (15), we can get (27).

Verification of the Convergence Order.
In this part, we will test the accuracy and validity of the full-discrete scheme (23). Actually, in the B-S model, researches usually choose α � 0.5σ 2 ; β � r − α. Set T � 1, N � 128 and u 0 � tan(x) + x, σ � 0.1, r � 0.2. In Table 1, we list L 2 error and convergence order for different time step size. We can see that the time direction is obvious of second-order accuracy, while the space direction has a good convergence property.

Effect of Various Parameters.
Next, we will test the asymptotic decay property of the solution. We fix N � 50, Δτ � 0.01, T � 1, Λ � (0, 6), and r � 0.1 and leave the initial value unchanged, and at the same time, let β constantly change. In Figure 1-4, we list the change process of numerical solution with β � 0.2, 0.6, 1.0, 5.0, and we find Mathematical Problems in Engineering  that when β becomes larger, the solution gradually tends to 0.
Fix u 0 � | tan(x) + x|. In the following numerical experiments, we also test the influence of σ and r on the numerical solution. In Figure 5-6, the variation process of numerical solution with σ and r is given. We find that the numerical solution experiences four peaks and valleys; at the same time, with the increase of R, the fluctuation of numerical solution gradually increases.

Concluding Remarks
We construct an effective fully discrete scheme for the B-S model based on Legendre-Galerkin scheme for spatial   discretization.
e scheme is also linear, unconditionally stable, and has second-order accuracy in time, where the Crank-Nicolson method is used for time discretization.
rough the implementations of several numerical examples, we demonstrate the accuracy and effectiveness of the developed scheme, numerically.

Conflicts of Interest
e authors declare that they have no conflicts of interest.