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The Summation of Series Based on the Laplace Transformation Method in Mathematics Teaching

Publicado en línea: 15 Jul 2022
Volumen & Edición: AHEAD OF PRINT
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Recibido: 17 Jan 2022
Aceptado: 25 Mar 2022
Detalles de la revista
License
Formato
Revista
eISSN
2444-8656
Primera edición
01 Jan 2016
Calendario de la edición
2 veces al año
Idiomas
Inglés
Abstract

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

MSC 2010

Introduction

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 continuous-time 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 Feynman-Kac formula to study cumulative options. But this formula cannot price endless options. Some scholars have obtained option price's three-dimensional partial differential equation (PDE) and solved it by explicit finite-difference. Some scholars have given the pricing expression and algorithm implementation of moving window options using stop-time 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 three-dimensional PDE to the two-dimensional 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.

Option pricing equation

Here, the option's price based on the Laplace transform is given mainly by knocking down the call option (PDIC) as the entry point.

Symbolic assumptions and definitions

The 16 symbols mainly used in this article are shown in Table 1. This article is to study option pricing under the framework of no-arbitrage equilibrium [3]. Here directly define the price of the underlying asset under the risk-neutral measure Q.

List of symbols used in this article.

S Underlying asset price process δ Stock dividend rate or foreign exchange risk-free interest rate
K Exercise price σ Volatility, assuming it is a normal number
T Option expiry date λ Laplace parameter, which is a complex-valued variable
L The barrier price of the underlying asset price S m 1σ(1δσ22) {1 \over \sigma}\left({1 - \delta - {{{\sigma ^2}} \over 2}} \right)
D Option window b 1σln(L/x) {1 \over \sigma}\ln (L/x)
D The cumulative hovering time of the option unilaterally at the barrier price at time t k 1σln(K/x) {1 \over \sigma}\ln (K/x)
x The initial price of the underlying asset price S θ 2λ \sqrt {2\lambda}
r Risk-free interest rate assumed to be constant d bkD {{b - k} \over {\sqrt D}}

Assume that the price of the underlying asset S = {St, t ≥ 0} obeys geometric Brownian motion under the risk-neutral measure Q [4]. We use the parameter transformation of Chesney to make the calculation more convenient: m=1σ(rδσ22),b=1σln(Lx) m = {1 \over \sigma}(r - \delta - {{{\sigma ^2}} \over 2}),\,b = {1 \over \sigma}\ln ({L \over x})

The new probability measure P defined by dPdQ|FT=emzm22T {{dP} \over {dQ}}\left| {{F_T} = {e^{mz{{{m^2}} \over 2}T}}} \right. makes Z = {Zt = Wt + mt, 0 ≤ tT} a P Brownian motion. Then St = xeσZt under the P measure.

The following defines some stopping times related to Brownian motion.

Definition 1

Constant b. Define the “last-arrival” process of S. Process gtb g_t^b : gtb=sup{ut|Zu=b} g_t^b = \sup \{u \le t|{Z_u} = b\}

Use gtb g_t^b to define a stop time directly related to the option.

Definition 2

gtb g_t^b and the constant D. Define Tb+ T_b^ + and Tb T_b^ - as Tb+=inf{t>0|(tgtb)1(zt>b)>D}Tb=inf{t>0|(tgtb)1(zt<b)>D} \matrix{{T_b^ + = \inf \{t > 0|(t - g_t^b){1_{({z_t} > b)}} > D\}} \hfill \cr {T_b^ - = \inf \{t > 0|(t - g_t^b){1_{({z_t} < b)}} > D\}} \hfill \cr}

Both Tb+ T_b^ + and Tb T_b^ - are Ft − measurable stop time. For the convenience of calculation, we will not consider the impact of discounting. Therefore, discounting is performed after the entire calculation is over.

Definition 3

The present value Π(T) of the value function at time T and the discount factor in P is exp[(r+m22)T] \exp [- (r + {{{m^2}} \over 2})T] . Assuming * Π(T) is its final value: *Π(T)=e(r+m22)TΠ(T) {}^*\Pi (T) = {e^{(r + {{{m^2}} \over 2})T}}\Pi (T)

From the Fourier analysis theory, it can be known that the Laplace transform Π^(λ) \hat \Pi (\lambda) of Π(T) has Π^(λ)=*Π^(λ+r+m22)) \hat \Pi (\lambda) = {}^*\hat \Pi (\lambda + r + {{{m^2}} \over 2}))

Definition 4

Assume that the ending payment of the option under the underlying asset price S = {St, t ≥ 0} is Φ(ST) = (STK)+. Tb=inf{t>0|(tgtb)1{Zt<b}>D} T_b^ - = \inf \{t > 0|(t - g_t^b){1_{\{{Z_t} < b\}}} > D\} is the stop time defined above, then knock down the call option price (denoted as *PDIC) as *PDIC(x,T;K,L;r,δ)=E(1{Tb<T}(xeσZTK)+emZT) {}^*PDIC(x,T;K,L;r,\delta) = E({1_{\{T_b^ - < T\}}}{(x{e^{\sigma {Z_T}}} - K)^ +}{e^{m{Z^T}}})

Knockdown call options (PDIC)
Theorem 1

t any moment, the option price is PDIC(St,t;x,T;K,L,D,r,δ) PDIC({S_t},t;x,T;K,L,D,r,\delta)

Its Laplace transform is PD^IC(St,t;x,T;K,L,D,r,δ) P\hat DIC({S_t},t;x,T;K,L,D,r,\delta)

Theorem 1 shows that calculating the option value at any time t can be decomposed into three pieces of value and calculated separately. The first block is a standard European option. Its expiration time is T. A second block is an option with an initial price of St. Its expiration time is T. A Laplace transform can represent the third block. We need to invert its Laplace transform.

Numerical methods and comparisons
Numerical approximation of the inverse Laplace transform

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 sp(x). Specifically expressed as sp,h(x)=eax2xf^(a)+eaxxn=1p(1)nRe(f^(a+inh)) {s_{p,h}}(x) = {{{e^{ax}}} \over {2x}}\hat f(a) + {{{e^{ax}}} \over x}\sum\limits_{n = 1}^p {{{(- 1)}^n}{\mathop{\rm Re}\nolimits} (\hat f(a + inh))}

The purpose of finding the upper bound of the remainder is to calculate the error of the above-truncated series. |n=p+1(1)nRe(f^(a+inh))|const1p \left| {\sum\limits_{n = p + 1}^\infty {{{(- 1)}^n}{\mathop{\rm Re}\nolimits} (\hat f(a + inh))}} \right| \le const{1 \over p}

To achieve the accuracy of O(10−7), we need to ask for the sum of the terms. Such calculation cost is very large [6]. We introduce Euler's summation method to accelerate the convergence of the series. The purpose of this is to achieve the above infinite series approximation quickly.

Euler's summation method

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 s true value. We pay attention to the difference E(p+1,q,x) − E(p,q,x) of the continuous term, which can be obtained by scaling |E(p+1,q,x)E(p,q,x)|const12qp2 |E(p + 1,q,x) - E(p,q,x)| \le const{1 \over {{2^q}{p^2}}}

Then calculate the error: |s(x)E(p,q,x)|const12qn=p+1n2const12qp |s(x) - E(p,q,x)| \le const{1 \over {{2^q}}}\sum\limits_{n = p}^{+ \infty} {{1 \over {{n^2}}} \le const{1 \over {{2^q}p}}}

Also, to obtain the accuracy of O(10−7), we can choose p =15 and 23 q = 23. The computational cost required for the iteration is reduced by 1,000 times compared with the previous one. It can be seen that Euler's summation method greatly reduces the computational complexity.

Take PUOC as an example to calculate the calculation time and relative error under different precision parameter selection [8]. The price of PUOC calculated by p = 108 is taken as the true value by Poisson's summation method. Fix the Euler parameter p in the parameter selection to observe the improvement of accuracy by different q. In calculating time, we adopt the method of calculating the average price of 100 times to reduce the systematic error. Euler's summation method only needs to take p = 12, q = 16 to obtain the accuracy of O(10−9) and the average calculation time for each option is less than 2 milliseconds. To obtain the accuracy of O(10−5), the Poisson summation method already requires p = 122880 calculations. The average time is 1184 milliseconds. It can be seen that Euler's summation method greatly improves the algorithmic efficiency of options, which is better than Poisson's summation method.

Brief analysis

Let's take PDIC as an example to analyze the sensitivity of option prices and hedging strategies. The calculation example parameters are shown in Table 2.

Calculation of example parameters.

Strike price Barrier price Option expiration time (years
9.5 9 1
Options window (years) Euler parameter p Euler parameter q
0.1 15 23
Initial price Drift rate Volatility
10 0.1 0.2
Risk-free rate Dividend rate Step size Δ (years)
4.50% 0 0.001
Option sensitivity analysis

Figure 1 analyzes the changes in the hedge rate Δ=ps \Delta = {{\partial p} \over {\partial s}} of options. The X axis is the price of the underlying asset, the Y axis is the cumulative hovering time, and the Z axis is Δ. The bottom of Figure 1 shows the contours about the Z axis. It can be seen that the contour lines at the lower limit of the barrier level L and the option window D are getting denser and converge to −∞ quickly. So this is a singularity of Δ. At this point, hedging will become very difficult, and the hedging error will diverge to infinity.

Figure 1

The combined effect of the cumulative hovering time and the underlying asset price process on the hedging rate of the option

Numerical simulation and hedging strategy analysis

The hedge rate defined in this simulation is Δt=ptst=pt(st+Δs)pt(stΔs)2Δs {\Delta _t} = {{\partial {p_t}} \over {\partial {s_t}}} = {{{p_t}({s_t} + \Delta s) - {p_t}({s_t} - \Delta s)} \over {2\Delta s}}

Define the “perfect hedge rate” of a simulation as Δ˜ t=pt+1(st+1)+pt(st)st+1st {\tilde \Delta _t} = {{{p_{t + 1}}({s_{t + 1}}) + {p_t}({s_t})} \over {{s_{t + 1}} - {s_t}}}

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 Δ˜ t=Δt=pt(st+Δs)+pt(stΔs)2Δs {\tilde \Delta _t} = {\Delta _t} = {{{p_t}({s_t} + \Delta s) + {p_t}({s_t} - \Delta s)} \over {2\Delta s}}

Define the hedging error as Δ˜ tΔ^ t {\tilde \Delta _t} - {\hat \Delta _t}

Figure 2Figure 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 Δ t+1 {\vec \Delta _{t + 1}} . Use the following hedge rate estimate: Δ t+1=pt+1(st+Δs)pt+1(stΔs)2Δs {\vec \Delta _{t + 1}} = {{{p_{t + 1}}({s_t} + \Delta s) - {p_{t + 1}}({s_t} - \Delta s)} \over {2\Delta s}}

Figure 2

Hedging error

Figure 3

Hedging account equity

We take the price of pt+1 ( ) instead of pt ( ) in our calculations. The forward method considers the actual change direction of the option price when the option is close to the options window at the time of accumulated hovering [11]. In this way, the option price can be more accurately judged in the sense of probability.

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.

Figure 4

Comparison of two hedging strategies

Conclusion

This paper studies the option pricing and hedging strategies based on the Laplace transform method under the assumption of no-arbitrage 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 p = 15 and q = 23 as parameters. The computational complexity of this algorithm is very small. Research shows that the forward hedging method can reduce the hedging error by more than half in most cases. Forward hedging is a relatively robust hedging strategy.

Figure 1

The combined effect of the cumulative hovering time and the underlying asset price process on the hedging rate of the option
The combined effect of the cumulative hovering time and the underlying asset price process on the hedging rate of the option

Figure 2

Hedging error
Hedging error

Figure 3

Hedging account equity
Hedging account equity

Figure 4

Comparison of two hedging strategies
Comparison of two hedging strategies

List of symbols used in this article.

S Underlying asset price process δ Stock dividend rate or foreign exchange risk-free interest rate
K Exercise price σ Volatility, assuming it is a normal number
T Option expiry date λ Laplace parameter, which is a complex-valued variable
L The barrier price of the underlying asset price S m 1σ(1δσ22) {1 \over \sigma}\left({1 - \delta - {{{\sigma ^2}} \over 2}} \right)
D Option window b 1σln(L/x) {1 \over \sigma}\ln (L/x)
D The cumulative hovering time of the option unilaterally at the barrier price at time t k 1σln(K/x) {1 \over \sigma}\ln (K/x)
x The initial price of the underlying asset price S θ 2λ \sqrt {2\lambda}
r Risk-free interest rate assumed to be constant d bkD {{b - k} \over {\sqrt D}}

Calculation of example parameters.

Strike price Barrier price Option expiration time (years
9.5 9 1
Options window (years) Euler parameter p Euler parameter q
0.1 15 23
Initial price Drift rate Volatility
10 0.1 0.2
Risk-free rate Dividend rate Step size Δ (years)
4.50% 0 0.001

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