It is more difficult to give Laplace transform directly in a defined form or derive it by Fourier transform in mathematics teaching. The article gives a solution for solving high exponential series sum by using Laplace transform. With the help of Laplace transform, calculus operations can be transformed into complex plane algebra operations. The application of the algorithm to the option hedging strategy verifies the applicability of the algorithm proposed in this article.
Keywords
 Laplace transform method
 Mathematics teaching
 Series summation
 Option hedging
MSC 2010
 44A10
The Laplace transform method is often called arithmetic calculus. It is an effective mathematical method that has developed rapidly in the past century. With the help of Laplace transform, calculus operations can be transformed into complex plane algebra operations. Therefore, it can be used to solve ordinary differential equations and partial differential equations. It greatly simplifies the steps to understand calculus equations, so Laplace transform has become an important tool for solving linear systems. It is widely used in engineering technology, especially in modern automatic control theory. Some scholars apply it to the continuoustime model of options. Some scholars introduced Brownian motion hovering to deal with the pricing problem of options and got a pricing formula with a fourfold integral [1]. Solving this quadruple integral can get the price of the option. Some scholars have given the option inverse Laplace numerical transformation and its calculation error. Some scholars deduced the price of double barrier options based on the Laplace transform. Some scholars have used the FeynmanKac formula to study cumulative options. But this formula cannot price endless options. Some scholars have obtained option price's threedimensional partial differential equation (PDE) and solved it by explicit finitedifference. Some scholars have given the pricing expression and algorithm implementation of moving window options using stoptime simulation. The prices calculated under different parameter conditions are compared by comparing cumulative options, endless options, and moving window options. This experiment verifies the effectiveness of the proposed method. Some scholars have defined the price of nonlinear options based on backward stochastic differential equations. At the same time, scholars gave the option pricing and simulation analysis in uncertain returns and different borrowing rates. Some scholars have given the correct boundary conditions and final value conditions of the PDE of the option and used the directional derivative to reduce the threedimensional PDE to the twodimensional PDE. In this way, the implicit difference method is used to price options [2]. Some scholars solved the problem by decomposing American options into European options with the same parameters and American premium. Option prices can be expressed as a function of implementation boundaries.
The practical application of options is mainly embodied in some structured financial products with the nature of options. For example, the redemption, resale, and downward revision clauses of convertible bonds have optional features. The innovation of this paper is the introduction of Euler's summation method in the Laplace inverse transform to speed up the convergence speed of the original calculation of the truncated series. The article reduces the computational cost and computational complexity and increases the computational speed. The article discusses the difficulty of hedging options and proposes a new relatively stable hedging strategy. This model improves the risk management capabilities of traders.
Here, the option's price based on the Laplace transform is given mainly by knocking down the call option (PDIC) as the entry point.
The 16 symbols mainly used in this article are shown in Table 1. This article is to study option pricing under the framework of noarbitrage equilibrium [3]. Here directly define the price of the underlying asset under the riskneutral measure Q.
List of symbols used in this article.
S  Underlying asset price process  Stock dividend rate or foreign exchange riskfree interest rate  
K  Exercise price  Volatility, assuming it is a normal number  
T  Option expiry date  Laplace parameter, which is a complexvalued variable  
L  The barrier price of the underlying asset price S  m 

D  Option window  b 

D^{′}  The cumulative hovering time of the option unilaterally at the barrier price at time t  k 

x  The initial price of the underlying asset price S 


r  Riskfree interest rate assumed to be constant  d 

Assume that the price of the underlying asset
The new probability measure
The following defines some stopping times related to Brownian motion.
Constant
Use
Both
The present value Π(
From the Fourier analysis theory, it can be known that the Laplace transform
Assume that the ending payment of the option under the underlying asset price
t any moment, the option price is
Its Laplace transform is
Theorem 1 shows that calculating the option value at any time
In the computer realization of numerical calculation, it is necessary to transform the infinite series into a finite sum [5]. The formula is denoted as
The purpose of finding the upper bound of the remainder is to calculate the error of the abovetruncated series.
To achieve the accuracy of
The convergence speed of the censored series is very slow. But the general term of the above series is periodic. So we can define an Euler sum [7]. The approximation of Euler's summation is unbiased. Now consider how fast Euler's sum can converge to the
Then calculate the error:
Also, to obtain the accuracy of
Take
Let's take
Calculation of example parameters.
Barrier price  Option expiration time (years  
9  1  
Euler parameter p  Euler parameter q  
15  23  
Drift rate  Volatility  
0.1  0.2  
Dividend rate  Step size Δ (years)  
0  0.001 
Figure 1 analyzes the changes in the hedge rate
The hedge rate defined in this simulation is
Define the “perfect hedge rate” of a simulation as
The account balance of the hedging strategy with a perfect hedging rate is always zero. However, the perfect hedge rate cannot be known, but only an estimate of the hedge rate can be made [9]. Assuming that a static estimate is used in the estimate of the hedge rate, then
Define the hedging error as
Figure 2–Figure 3 are the results of a simulation. This gives the hedging error and the net value of the hedging account. It can be seen from Figure 3 that the benchmark hedges an underlying asset of 10 yuan without considering transaction friction [10]. This ultimately resulted in account equity. To make the hedging effect more robust, we consider a new hedging strategy. We call it “forward estimation.” Mark it as
We take the price of
It can be seen from Figure 4 that the absolute value of the hedging account equity of the forward method is less than that of the static method for most of the time. And its volatility is also weaker than the static method, which improves the accuracy of hedging.
This paper studies the option pricing and hedging strategies based on the Laplace transform method under the assumption of noarbitrage equilibrium. Discussed the purpose of the numerical method of the inverse Laplace transform is to speed up the convergence speed of the series. This paper presents Euler's summation method as an improved method of inverse transformation. The same accuracy can be obtained by selecting
List of symbols used in this article.
S  Underlying asset price process  Stock dividend rate or foreign exchange riskfree interest rate  
K  Exercise price  Volatility, assuming it is a normal number  
T  Option expiry date  Laplace parameter, which is a complexvalued variable  
L  The barrier price of the underlying asset price S  m 

D  Option window  b 

D^{′}  The cumulative hovering time of the option unilaterally at the barrier price at time t  k 

x  The initial price of the underlying asset price S 


r  Riskfree interest rate assumed to be constant  d 

Calculation of example parameters.
Barrier price  Option expiration time (years  
9  1  
Euler parameter p  Euler parameter q  
15  23  
Drift rate  Volatility  
0.1  0.2  
Dividend rate  Step size Δ (years)  
0  0.001 
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