# An Analysis of Asymptotic Properties and Error Control under the Exponential Jump-Diffusion Model for American Option Pricing

Authors: Mohamed Maidoumi; Boubker Daafi; Mehdi Zahid
Journal: Journal of Applied Mathematics
 (2021)

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## Abstract

Our work is aimed at modeling the American option price by combining the dynamic programming and the optimal stopping time under two asset price models. In doing so, we attempt to control the theoretical error and illustrate the asymptotic characteristics of each model; thus, using a numerical illustration of the convergence of the option price to an equilibrium price, we can notice its behavior when the number of paths tends to be a large number; therefore, we construct a simple estimator on each slice of the number of paths according to an upper and lower bound to control our error. Finally, to highlight our approach, we test it on different asset pricing models, in particular, the exponential Lévy model compared to the simple Black and Scholes model, and we will show how the latter outperforms the former in the real market (Microsoft “MSFT” put option as an example).


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## Body

## 1. Introduction

The first academic work on pricing options, in particular that of Black and Scholes [1], has allowed the development of derivatives markets. But the use of this theory or those which followed for the evaluation of negotiated financial contracts and related hedging strategies deduced could not be done in practice without the choice of a model and its estimate. Today, it is admitted, on the basis of numerous empirical studies, that the Gaussian hypothesis for the log return on financial assets has reduced validity (see Section 2.3.1). The reader will find examples in the article [2] as well as in [3]. The first alternative models to the Gaussian model were the stable model proposed by [4, 5], and within the framework of option valuation, Merton [6] is the first author to have developed a non-Gaussian model; let us note that for a long time, evaluating American option has been considered inadequate with the traditional forward Monte Carlo simulations; hence, the binomial tree model [7] and finite difference [8] methods were used as the only numerical method before the mid-1990s when new methods appeared. The most popular method was proposed by Carriere [9] using optimal stopping times and then developed, improved, and popularized by Longstaff and Schwartz [10] by combining between least square regression and Monte Carlo simulation; for a more comprehensive review of the literature, the reader may profitably consult the works of Tankov et al. [11]. In this paper, we compare between the models of Black and Scholes in 1973 [1] and an exponential diffusion-jump model, but when using a modeling process using discontinuous trajectories, a delicate problem arises, which is the incompleteness of markets, and the risk-neutral measure is not unique. Numerous works such as those of Fujiwara and Miyahara [12] or Tankov et al. [11] have been devoted to this subject. Here, we adopt a more direct point of considering view that the dynamic followed by the price of financial assets is given in a risk-neutral universe.The main advantage of Monte Carlo simulation resides in its convergence rate which does not depend on the number of the underlying, and it is useful thanks to its ease of handling a large range of models. However, Monte Carlo simulations stay slow and their standard form is not suitable for American options; the reason comes from the fact that their trajectories dissociate (date of execution of the option is independent of each trajectory); in fact, the possibility of exercising at any time requires calculating on each trajectory the expected value of the option which is a conditional expectation. [10] proposes an algorithm that requires rejection of certain trajectories (trajectories that are out of the money) and compares the intrinsic value (discounted payoff) to be able to decide whether to execute the option or keep it. This practical term is not feasible even if we limit ourselves to a finite number of possible exercise dates in order to solve a dynamic programming of the optimal exercise policy that requires backward simulations.In Section1, we present the theoretical framework of the Longstaff and Schwartz algorithm and we show the effectiveness of this approach compared to the dynamic algorithm of Tsitsiklis and van Roy. In Section 2, we focus on the numerical implementation of the Longstaff and Schwartz algorithm and compare the B&S model and Lévy jump-diffusion. We also focus on the theoretical and asymptotic convergence of this algorithm, and towards the end, we present our method for controlling the error using a put option on Microsoft “MSFT“ data as an example from “Yahoo Finance” and we implement the validation of these models in a real market.


## 2. Pricing American Option Theoretical Framework

We consider an American option of valueUt,Yt which depends on the time and m factors Yt=Y1t,⋯,Ymt; for example, an option in the B&S model depends on one factor Y1t=St; in the case of exponential Lévy jump-diffusion model, the option depends, respectively, on the underlying price, jump size, and number of jumps in 0,TYt=Y1t=St,Y2t=Jt,Y3t=Nt. We are looking for the empirical distribution of the probability of Ut,Yt or its moments, and we are interested in all values of Ut,Yt in the meantime 0,T (American option).Monte Carlo simulation is a probabilistic method used in the case where the distribution lawUt,Yt or the first moments are not defined analytically. This simulation consists in generating a large number of possible trajectories Stt=t1⋯t=tN; therefore, we build an empirical distribution of St.


### 2.1. Characterization of the Optimal Stopping Time Policy by the Snell Envelope

In this section, we aim to explain how the Snell envelope provides an introduction of optimal stopping time theory for pricing an American option.Definition 1.
LetΩ,F,Ft)t∈0,⋯,T,P be a filtered probability space, which means we discretize the time space 0,T into n pieces and we consider Ω the measure space of the universe of possible values and F⊂Ω is a σ-algebra of the set’s events; P is the probability measure on F; Ft is the natural filtration on Ω,F; i.e., ∀ti,tj∈0,⋯,T, Fti is a σ-algebra and if i<j, Fti⊂Ftj; T is the time horizon, and the underlying Stt∈0,⋯,T is defined on the filtered probability space with a finite time horizon T.
LetOtkk=0⋯n be an adapted and integrable random variable. The Snell envelope of the process Otkk=1⋯n is defined as
(1)Utj=esssupτ∈ΓtjEOτ∣Ftj,j=1⋯n,
whereΓtj is the set of all stopping time in tj,⋯,T.Proposition 2.
The Snell envelopeUtktk=0⋯,tn=T of Otktk=0⋯tn=T fulfills these properties:
(1)
UT=OT(2)
Utk=maxOtk,EUtk+1∣Ftk such that k=0,⋯,n(3)
Utkk=1⋯n is the smallest supermartingale dominating Otkk=1⋯nProof.
See AppendixA.☐☐The optimal time is defined as follows:Definition 3.
τ is an optimal time iffEOτ∣F0=supτ′∈Γ0EOτ∣F0.Using the Snell envelope, we can characterize the optimal stopping time policy to evaluate an American option.Theorem 4.
A stopping timeτ is optimal if and only if
(i)
Uτ=Oτ(ii)
Uτ∧tj is a martingaleProof.
See AppendixB.☐☐Proposition 5.
Letτ=mink≥0∣Uk=Ok; hence, τ is an optimal stopping time.Proof.
See AppendixC.☐☐


### 2.2. Effectiveness of Longstaff and Schwartz Policy Iteration Approach

We consider the same filtered probability spaceΩ,F,Ft0≤t≤T,ℙ, T≥0, and Wt=Wt1,⋯,Wtd, a Brownian motion of d dimension. We notice Otkk=1⋯N as the payoff process and Utkk=1⋯N as the discounted value of the option (Snell envelope) Utk=esssupτ∈ΓtkEOτ∣Ftk such that Γtk is the set of all Ft-stopping time in t∈tk,⋯,tN=T; we aim to approximate U0.The Longstaff and Schwartz algorithm uses the same trajectories for each step due to its dynamic programming that depends on the stopping time instead of the value function as in the case of Tsitsiklis and van Roy [13]. This approach is very advantageous because it is robust, has low computational burden, and saves time.
(2)τN=tN,τk=tk1Otk≥EOtk+1∣Ftk+τk+11Otk≤EOtk+1∣Ftk,such that τk is the smallest optimal stopping time after tk; the first line of the system (2) highlights the fact that the Snell envelope is equal to the payoff at time T. The second line is a comparison between the payoff at time tk and the conditional expectation of the payoff at time tk+1 on Ftk that we design by continuation value. Now, the main challenge that arises is to approximate conditional expectation.


### 2.3. Longstaff and Schwartz Approach and Monte Carlo Simulation

Theorem 6.
Assume thatXtt=0⋯T is a Markovian process and ϕk is a Ft measurable function; then,
(3)EOtk+1∣Ftk=EOtk+1∣Xtk=ϕkXtk.Remark 7.
Because of the orthogonality of conditional expectation inL2, we can compute ϕk by the least square:
(4)ϕk=infϕ∈ΦEOtk+1−ϕXtk2,where Φ is the set of L2σXtk functions.The classical numerical (LS) algorithm is based on the approximation of conditional expectation by a finitep-dimensional vector (polynomial basis, logistic.) in the L2 space. We notice φkk=1,⋯,p this finite vector such that it is an L2 basis and ϕkpXtk,θk=∑n=1pθnφnXtk.
(i)
LetSt0m,⋯,StNm be the price paths and Ot0m,⋯,OtNm the payoff paths, such that m∈1,⋯,M. We present the LS algorithm with Monte Carlo approximation policy as
(5)τ^Np,m=tN,τ^kp,m=tk1Otkm≥ϕkpStkm,θ^kp,M+τk+11Otkm≤ϕkpStkm,θ^kp,M,such thatτ^kp,m is the smallest optimal stopping time after tk on the mth path for the p-dimensional approximation. Hence, the conditional expectation EOtk+1∣Ftk is computed by the following minimization problem:
(6)θkp=arginfθ∈ΘEOτ^k+1p−ϕkpStk,θ2,or in another way by its Monte Carlo approximation:
(7)θ^kp,M=arginfθ∈Θ1M∑m=1MϕkpStkm,θ−Oτ^k+1p,mm2,such that Θ is a finite set in ℝ. The coefficients θ^kp,Mk=1,⋯,N give us convenient multiplicators of basis vectors in L2 for determining the optimal policy of the system (2). Thus, the option price at time 0 is obtained as U0p=EOτ^1p with τ1=infk∈1,⋯,N∣Ok=Uk. Therefore, the Monte Carlo approximation is
(8)U0p,M=∑m=1MOτ^1p,mm.


#### 2.3.1. The Inadequacy of the Model B&S with the Real Financial Market

We break down the interval0,T in N periods of equal duration Δt=T/N with N large enough which makes it possible to discretize the interval in t0=0,⋯tN=N.
(9)logStjStj−1=r−12σ2Δt+σWΔt.This formula shows us that the log returns of the asset price follow a normal distribution, but as we can see in Figure1, the empirical distribution of Microsoft’s weekly log returns between 14/03/2016 and 17/03/2021 (https://fr.finance.yahoo.com/quote/MSFT/history?p=MSFT) has an excess kurtosis (i.e., density function of logStj/Stj−1 has a bigger pick around mean and heavy tails compared to the normal distribution, which means that logStj/Stj−1 values are more clustered around its mean and the probability that a jump occurs is more considerably presented) and negative skewness (i.e., the tails are shifted to the left compared to a symmetric normal distribution, which means that the decrease in the stock price outweighs the corresponding increase in Microsoft’s log returns).Figure 1
Kernel and normal estimator of Microsoft’s log returns.We can highlight other inconsistencies with the B&S model like stochastic volatility with mean-reverting phenomena; indeed, in the real market volatility, it is not constant and we can also emphasize what is called “volatility clustering” (i.e., an uptick in volatility can be followed by higher volatility) (see [14]). In this paper, we content ourselves with constant volatility models.


### 2.4. Forward Simulation for Pricing Asset Price

Theorem 8 (first fundamental theorem of asset pricing).
We price on a complete market if only if there exists a unique equivalent measureℚ.
In a complete market, the existence and uniqueness are satisfied. For further explanations, see [15]. In the Black and Scholes model under the unique equivalent martingale (risk-neutral measure), the asset price follows the dynamics:
(10)dSt=rStdt+σStdBt,where dBt is the increment of the Brownian motion, r the risk-free rate, and volatility σ.Merton [6] notes that the source of the jumps can generally be caused by information specific to the firm or its industry, and in this case, they have a negligible impact on the rest of the market. The jump component then represents an unsystematic risk and will be uncorrelated with the market. The choice of the risk-neutral measure in the case of incomplete markets remains an open problem, although several lines of research have already been explored in this direction. In this regard, the reader can refer to the different works carried out by [11] or [16]. We will assume in the following that the dynamics of the underlying condition follows a diffusive model with jumps under a given risk-neutral measure Q. This measurement is arbitrary, and we will adjust the parameters; therefore, the evaluation obtained is as close as possible to the prices of market (see Section 3.3).We suppose that the asset follows a Lévy jump-diffusion model:(11)dS=rdt+σdWt+JtdNt,where r is the risk-free rate, σ is a constant volatility, and Jt is the jump amplitude which flows an exponential law with parameter λ1=1, and dNt is the increment of the compound Poisson process, NT is the number of jumps in 0,T, and Ti,i=1,⋯,NT, is the time of jumps between 0 and T. We break down the interval 0,T in N periods of equal duration Δt=T/N to build the discretization t0=0,⋯,tN=T, and therefore, we use the Euler schema to simulate St and Ut,St by discretizing equation (11) as
(12)Stj+1−Stj=rΔt+σWΔt+∑i=0NtjJi,where Wt is calculated from the cumulative of WΔt~N0,Δt; hence, a trajectory Stt=t1⋯t=tN is calculated from the application step-by-step of equation (12) as follows: first of all, we generate the Wiener process with a drift, and we use exponential laws for the duration between jumps wi=Ti+1−Ti with its parameter λ2 (on the average, we have 1/λ2 jumps in the meantime 0,T), and as the case may be, we generate negative exponential law amplitudes for size jumps Ji with its parameter λ1 (i.e., losses which follow exponential laws) (see the example of Figure 2).Theorem 9 (second fundamental theorem of asset pricing).
In a free arbitrary market and under an appropriate equivalent measureℚ, the price at time t is the conditional expected payout at T, i.e.,
(13)Ut=EeT−trUT∣Ft.
Hence,e−rT−tUt is a martingale under ℚ. Such an equivalent martingale measure is also called a risk-neutral measure because the return on the risk-free investment is the same as the expected return on the asset.Figure 2
Modeling asset price paths under Lévy jump-diffusionλ1=1 and λ2=1/4.


### 2.5. Least Square Regression (LS) Algorithm

#### 2.5.1. Methodology

We assume thatπ=t0=0,⋯,tN=T is the set of possible exercise times, p is the number of regressors or the dimension of the L2 basis, and M is the number of paths. We aim to calculate the least square coefficients β=α,β1,β2,⋯,βp. Let φ=φ1,φ2,⋯,φp be a basis of nonzero vectors in the Hilbert space L2, and the orthogonal projection of OStN on L2σStN−1 is the vector: α+β1φ1StN−1+β2φ2StN−1+⋯+βpφpStN−1. At time tN−1, we use an ordinary least square backward to regress OSN on the space generated by regressors φ1StN−1,φ2StN−1,⋯,φpStN−1 by minimizing OStN−α−β1StN−1−β2StN−12; in this case, the continuation value becomes
(14)ϕtN−1p=Ee−rΔtOStNi∣StN−1≅e−rΔtα+β1φ1StN−1+β2φ2StN−1+⋯+βpφpStN−1.On each trajectory fromM′ in the money paths at time tN−1, we obtain the least square coefficients β^tN−1 by the least square estimator:
(15)β^tN−1=αtN−1β1,tN−1⋮βp,tN−1=STS−1STOST1⋯OSTM′,with(16)S=φ1StN−11⋯φ1StN−1M′⋮⋱⋮φpStN−11⋯φpStN−1M′.Thus, we apply the equation (14) backward for ttN−2,tN−3,⋯,t0=0.


### 2.6. Convergence of Monte Carlo Simulation

Theorem 10.
Let0≤j≤N, and we assume that φpStj=φ1Stj,⋯,φpStj be a total basis in L2σStj; then,
(17)limp⟶+∞EOτjp∣Ftj=EOτj∣FtjinL2.Proof.
See [17].
This result can be proven theoretically but cannot be proven numerically, unlike the following convergence result.☐☐Theorem 11.
Assume that for0≤j≤N, PβjφStj=Otj=0 and the simulated M paths are independent; then, limM⟶+∞U0p,M=U0p almost surely.Proof.
First of all, we prove that for0≤j≤N, 1/M∑k=1MOτjk,p,Mk⟶Oτjp when M⟶+∞ almost surely and then conclude our result (see [17]). The numerical illustration of this theorem is shown in Figures 3 and 4.☐☐Figure 3
Numerical convergence of Monte Carlo simulation in the least square algorithm under the B&S asset price model. Confidence level in order of 95% (the estimated value of the American option in blue, upper bound in red, and lower bound in green).
(a)
M from 100 to 1000 by 10 steps(b)
M from 100 to 5000 by 10 steps(c)
M from 100 to 10000 by 10 stepsFigure 4
Numerical convergence of Monte Carlo simulation in the least square algorithm under the Lévy jump-diffusion model. Confidence level in order of 95% (the estimated value of the American option in blue, upper bound in red, and lower bound in green).
(a)
M from 100 to 1000 by 10 steps(b)
M from 100 to 5000 by 10 steps(c)
M from 100 to 10000 by 10 steps


## 2.1. Characterization of the Optimal Stopping Time Policy by the Snell Envelope

In this section, we aim to explain how the Snell envelope provides an introduction of optimal stopping time theory for pricing an American option.Definition 1.
LetΩ,F,Ft)t∈0,⋯,T,P be a filtered probability space, which means we discretize the time space 0,T into n pieces and we consider Ω the measure space of the universe of possible values and F⊂Ω is a σ-algebra of the set’s events; P is the probability measure on F; Ft is the natural filtration on Ω,F; i.e., ∀ti,tj∈0,⋯,T, Fti is a σ-algebra and if i<j, Fti⊂Ftj; T is the time horizon, and the underlying Stt∈0,⋯,T is defined on the filtered probability space with a finite time horizon T.
LetOtkk=0⋯n be an adapted and integrable random variable. The Snell envelope of the process Otkk=1⋯n is defined as
(1)Utj=esssupτ∈ΓtjEOτ∣Ftj,j=1⋯n,
whereΓtj is the set of all stopping time in tj,⋯,T.Proposition 2.
The Snell envelopeUtktk=0⋯,tn=T of Otktk=0⋯tn=T fulfills these properties:
(1)
UT=OT(2)
Utk=maxOtk,EUtk+1∣Ftk such that k=0,⋯,n(3)
Utkk=1⋯n is the smallest supermartingale dominating Otkk=1⋯nProof.
See AppendixA.☐☐The optimal time is defined as follows:Definition 3.
τ is an optimal time iffEOτ∣F0=supτ′∈Γ0EOτ∣F0.Using the Snell envelope, we can characterize the optimal stopping time policy to evaluate an American option.Theorem 4.
A stopping timeτ is optimal if and only if
(i)
Uτ=Oτ(ii)
Uτ∧tj is a martingaleProof.
See AppendixB.☐☐Proposition 5.
Letτ=mink≥0∣Uk=Ok; hence, τ is an optimal stopping time.Proof.
See AppendixC.☐☐


## 2.2. Effectiveness of Longstaff and Schwartz Policy Iteration Approach

We consider the same filtered probability spaceΩ,F,Ft0≤t≤T,ℙ, T≥0, and Wt=Wt1,⋯,Wtd, a Brownian motion of d dimension. We notice Otkk=1⋯N as the payoff process and Utkk=1⋯N as the discounted value of the option (Snell envelope) Utk=esssupτ∈ΓtkEOτ∣Ftk such that Γtk is the set of all Ft-stopping time in t∈tk,⋯,tN=T; we aim to approximate U0.The Longstaff and Schwartz algorithm uses the same trajectories for each step due to its dynamic programming that depends on the stopping time instead of the value function as in the case of Tsitsiklis and van Roy [13]. This approach is very advantageous because it is robust, has low computational burden, and saves time.
(2)τN=tN,τk=tk1Otk≥EOtk+1∣Ftk+τk+11Otk≤EOtk+1∣Ftk,such that τk is the smallest optimal stopping time after tk; the first line of the system (2) highlights the fact that the Snell envelope is equal to the payoff at time T. The second line is a comparison between the payoff at time tk and the conditional expectation of the payoff at time tk+1 on Ftk that we design by continuation value. Now, the main challenge that arises is to approximate conditional expectation.


## 2.3. Longstaff and Schwartz Approach and Monte Carlo Simulation

Theorem 6.
Assume thatXtt=0⋯T is a Markovian process and ϕk is a Ft measurable function; then,
(3)EOtk+1∣Ftk=EOtk+1∣Xtk=ϕkXtk.Remark 7.
Because of the orthogonality of conditional expectation inL2, we can compute ϕk by the least square:
(4)ϕk=infϕ∈ΦEOtk+1−ϕXtk2,where Φ is the set of L2σXtk functions.The classical numerical (LS) algorithm is based on the approximation of conditional expectation by a finitep-dimensional vector (polynomial basis, logistic.) in the L2 space. We notice φkk=1,⋯,p this finite vector such that it is an L2 basis and ϕkpXtk,θk=∑n=1pθnφnXtk.
(i)
LetSt0m,⋯,StNm be the price paths and Ot0m,⋯,OtNm the payoff paths, such that m∈1,⋯,M. We present the LS algorithm with Monte Carlo approximation policy as
(5)τ^Np,m=tN,τ^kp,m=tk1Otkm≥ϕkpStkm,θ^kp,M+τk+11Otkm≤ϕkpStkm,θ^kp,M,such thatτ^kp,m is the smallest optimal stopping time after tk on the mth path for the p-dimensional approximation. Hence, the conditional expectation EOtk+1∣Ftk is computed by the following minimization problem:
(6)θkp=arginfθ∈ΘEOτ^k+1p−ϕkpStk,θ2,or in another way by its Monte Carlo approximation:
(7)θ^kp,M=arginfθ∈Θ1M∑m=1MϕkpStkm,θ−Oτ^k+1p,mm2,such that Θ is a finite set in ℝ. The coefficients θ^kp,Mk=1,⋯,N give us convenient multiplicators of basis vectors in L2 for determining the optimal policy of the system (2). Thus, the option price at time 0 is obtained as U0p=EOτ^1p with τ1=infk∈1,⋯,N∣Ok=Uk. Therefore, the Monte Carlo approximation is
(8)U0p,M=∑m=1MOτ^1p,mm.


### 2.3.1. The Inadequacy of the Model B&S with the Real Financial Market

We break down the interval0,T in N periods of equal duration Δt=T/N with N large enough which makes it possible to discretize the interval in t0=0,⋯tN=N.
(9)logStjStj−1=r−12σ2Δt+σWΔt.This formula shows us that the log returns of the asset price follow a normal distribution, but as we can see in Figure1, the empirical distribution of Microsoft’s weekly log returns between 14/03/2016 and 17/03/2021 (https://fr.finance.yahoo.com/quote/MSFT/history?p=MSFT) has an excess kurtosis (i.e., density function of logStj/Stj−1 has a bigger pick around mean and heavy tails compared to the normal distribution, which means that logStj/Stj−1 values are more clustered around its mean and the probability that a jump occurs is more considerably presented) and negative skewness (i.e., the tails are shifted to the left compared to a symmetric normal distribution, which means that the decrease in the stock price outweighs the corresponding increase in Microsoft’s log returns).Figure 1
Kernel and normal estimator of Microsoft’s log returns.We can highlight other inconsistencies with the B&S model like stochastic volatility with mean-reverting phenomena; indeed, in the real market volatility, it is not constant and we can also emphasize what is called “volatility clustering” (i.e., an uptick in volatility can be followed by higher volatility) (see [14]). In this paper, we content ourselves with constant volatility models.


## 2.3.1. The Inadequacy of the Model B&S with the Real Financial Market

We break down the interval0,T in N periods of equal duration Δt=T/N with N large enough which makes it possible to discretize the interval in t0=0,⋯tN=N.
(9)logStjStj−1=r−12σ2Δt+σWΔt.This formula shows us that the log returns of the asset price follow a normal distribution, but as we can see in Figure1, the empirical distribution of Microsoft’s weekly log returns between 14/03/2016 and 17/03/2021 (https://fr.finance.yahoo.com/quote/MSFT/history?p=MSFT) has an excess kurtosis (i.e., density function of logStj/Stj−1 has a bigger pick around mean and heavy tails compared to the normal distribution, which means that logStj/Stj−1 values are more clustered around its mean and the probability that a jump occurs is more considerably presented) and negative skewness (i.e., the tails are shifted to the left compared to a symmetric normal distribution, which means that the decrease in the stock price outweighs the corresponding increase in Microsoft’s log returns).Figure 1
Kernel and normal estimator of Microsoft’s log returns.We can highlight other inconsistencies with the B&S model like stochastic volatility with mean-reverting phenomena; indeed, in the real market volatility, it is not constant and we can also emphasize what is called “volatility clustering” (i.e., an uptick in volatility can be followed by higher volatility) (see [14]). In this paper, we content ourselves with constant volatility models.


## 2.4. Forward Simulation for Pricing Asset Price

Theorem 8 (first fundamental theorem of asset pricing).
We price on a complete market if only if there exists a unique equivalent measureℚ.
In a complete market, the existence and uniqueness are satisfied. For further explanations, see [15]. In the Black and Scholes model under the unique equivalent martingale (risk-neutral measure), the asset price follows the dynamics:
(10)dSt=rStdt+σStdBt,where dBt is the increment of the Brownian motion, r the risk-free rate, and volatility σ.Merton [6] notes that the source of the jumps can generally be caused by information specific to the firm or its industry, and in this case, they have a negligible impact on the rest of the market. The jump component then represents an unsystematic risk and will be uncorrelated with the market. The choice of the risk-neutral measure in the case of incomplete markets remains an open problem, although several lines of research have already been explored in this direction. In this regard, the reader can refer to the different works carried out by [11] or [16]. We will assume in the following that the dynamics of the underlying condition follows a diffusive model with jumps under a given risk-neutral measure Q. This measurement is arbitrary, and we will adjust the parameters; therefore, the evaluation obtained is as close as possible to the prices of market (see Section 3.3).We suppose that the asset follows a Lévy jump-diffusion model:(11)dS=rdt+σdWt+JtdNt,where r is the risk-free rate, σ is a constant volatility, and Jt is the jump amplitude which flows an exponential law with parameter λ1=1, and dNt is the increment of the compound Poisson process, NT is the number of jumps in 0,T, and Ti,i=1,⋯,NT, is the time of jumps between 0 and T. We break down the interval 0,T in N periods of equal duration Δt=T/N to build the discretization t0=0,⋯,tN=T, and therefore, we use the Euler schema to simulate St and Ut,St by discretizing equation (11) as
(12)Stj+1−Stj=rΔt+σWΔt+∑i=0NtjJi,where Wt is calculated from the cumulative of WΔt~N0,Δt; hence, a trajectory Stt=t1⋯t=tN is calculated from the application step-by-step of equation (12) as follows: first of all, we generate the Wiener process with a drift, and we use exponential laws for the duration between jumps wi=Ti+1−Ti with its parameter λ2 (on the average, we have 1/λ2 jumps in the meantime 0,T), and as the case may be, we generate negative exponential law amplitudes for size jumps Ji with its parameter λ1 (i.e., losses which follow exponential laws) (see the example of Figure 2).Theorem 9 (second fundamental theorem of asset pricing).
In a free arbitrary market and under an appropriate equivalent measureℚ, the price at time t is the conditional expected payout at T, i.e.,
(13)Ut=EeT−trUT∣Ft.
Hence,e−rT−tUt is a martingale under ℚ. Such an equivalent martingale measure is also called a risk-neutral measure because the return on the risk-free investment is the same as the expected return on the asset.Figure 2
Modeling asset price paths under Lévy jump-diffusionλ1=1 and λ2=1/4.


## 2.5. Least Square Regression (LS) Algorithm

### 2.5.1. Methodology

We assume thatπ=t0=0,⋯,tN=T is the set of possible exercise times, p is the number of regressors or the dimension of the L2 basis, and M is the number of paths. We aim to calculate the least square coefficients β=α,β1,β2,⋯,βp. Let φ=φ1,φ2,⋯,φp be a basis of nonzero vectors in the Hilbert space L2, and the orthogonal projection of OStN on L2σStN−1 is the vector: α+β1φ1StN−1+β2φ2StN−1+⋯+βpφpStN−1. At time tN−1, we use an ordinary least square backward to regress OSN on the space generated by regressors φ1StN−1,φ2StN−1,⋯,φpStN−1 by minimizing OStN−α−β1StN−1−β2StN−12; in this case, the continuation value becomes
(14)ϕtN−1p=Ee−rΔtOStNi∣StN−1≅e−rΔtα+β1φ1StN−1+β2φ2StN−1+⋯+βpφpStN−1.On each trajectory fromM′ in the money paths at time tN−1, we obtain the least square coefficients β^tN−1 by the least square estimator:
(15)β^tN−1=αtN−1β1,tN−1⋮βp,tN−1=STS−1STOST1⋯OSTM′,with(16)S=φ1StN−11⋯φ1StN−1M′⋮⋱⋮φpStN−11⋯φpStN−1M′.Thus, we apply the equation (14) backward for ttN−2,tN−3,⋯,t0=0.


## 2.5.1. Methodology

We assume thatπ=t0=0,⋯,tN=T is the set of possible exercise times, p is the number of regressors or the dimension of the L2 basis, and M is the number of paths. We aim to calculate the least square coefficients β=α,β1,β2,⋯,βp. Let φ=φ1,φ2,⋯,φp be a basis of nonzero vectors in the Hilbert space L2, and the orthogonal projection of OStN on L2σStN−1 is the vector: α+β1φ1StN−1+β2φ2StN−1+⋯+βpφpStN−1. At time tN−1, we use an ordinary least square backward to regress OSN on the space generated by regressors φ1StN−1,φ2StN−1,⋯,φpStN−1 by minimizing OStN−α−β1StN−1−β2StN−12; in this case, the continuation value becomes
(14)ϕtN−1p=Ee−rΔtOStNi∣StN−1≅e−rΔtα+β1φ1StN−1+β2φ2StN−1+⋯+βpφpStN−1.On each trajectory fromM′ in the money paths at time tN−1, we obtain the least square coefficients β^tN−1 by the least square estimator:
(15)β^tN−1=αtN−1β1,tN−1⋮βp,tN−1=STS−1STOST1⋯OSTM′,with(16)S=φ1StN−11⋯φ1StN−1M′⋮⋱⋮φpStN−11⋯φpStN−1M′.Thus, we apply the equation (14) backward for ttN−2,tN−3,⋯,t0=0.


## 2.6. Convergence of Monte Carlo Simulation

Theorem 10.
Let0≤j≤N, and we assume that φpStj=φ1Stj,⋯,φpStj be a total basis in L2σStj; then,
(17)limp⟶+∞EOτjp∣Ftj=EOτj∣FtjinL2.Proof.
See [17].
This result can be proven theoretically but cannot be proven numerically, unlike the following convergence result.☐☐Theorem 11.
Assume that for0≤j≤N, PβjφStj=Otj=0 and the simulated M paths are independent; then, limM⟶+∞U0p,M=U0p almost surely.Proof.
First of all, we prove that for0≤j≤N, 1/M∑k=1MOτjk,p,Mk⟶Oτjp when M⟶+∞ almost surely and then conclude our result (see [17]). The numerical illustration of this theorem is shown in Figures 3 and 4.☐☐Figure 3
Numerical convergence of Monte Carlo simulation in the least square algorithm under the B&S asset price model. Confidence level in order of 95% (the estimated value of the American option in blue, upper bound in red, and lower bound in green).
(a)
M from 100 to 1000 by 10 steps(b)
M from 100 to 5000 by 10 steps(c)
M from 100 to 10000 by 10 stepsFigure 4
Numerical convergence of Monte Carlo simulation in the least square algorithm under the Lévy jump-diffusion model. Confidence level in order of 95% (the estimated value of the American option in blue, upper bound in red, and lower bound in green).
(a)
M from 100 to 1000 by 10 steps(b)
M from 100 to 5000 by 10 steps(c)
M from 100 to 10000 by 10 steps


## 3. Numerical Implementation

### 3.1. Specific Example of the Least Square Regression Algorithm

We consider the case ofp=2 to highlight the ordinary least square regression algorithm at time tN−1; for example, we choose the following form of regressor functions: φ1StN−1=StN−1, φ1StN−1=StN−12, and β=αtN−1,β1,tN−1,β2,tN−1, an estimator of the three least square coefficients which minimize the least square error:
(18)∑i=1M′OStNi−α−β1StN−1i−β2StN−1i22.At each trajectory, the terminal valueOStNi equals to the terminal payoff of the option; as of date tN−1, we estimate the continuation value if the trajectory is in the money and we reject if the trajectory is out of the money, so on each trajectory, we have M′ couple StN−1i,OStNi among M which will explain OStN−1i using the following regressors: 1,φ1StN−1=StN−1,φ2StN−1=StN−12. According to equation (14) at tN−1 on trajectory i, the continuation value is
(19)ϕkptN−1,StN−1i=Ee−rΔtOStNi∣StN−1i=StN−1≅e−rΔtαtN−1−β1,tN−1StN−1i−β2,tN−1StN−1i2.We notice that the intrinsic valuee−rΔtOStNi represents the discounted cash flows earned at time tN in the case where the option is not exercised at time tN−1. Thus, we make a comparison between the payoff OStNi and ϕkptN−1,StN−1i on each trajectory if OStNi≥ϕkptN−1,StN−1i; the early exercise of the option has therefore taken place at that time if it has not taken place previously. Hence, the value of option at date tN−1 on trajectory i is
(20)UtN−12,i=maxOStN−1i,ϕkpStN−1i.We do the same for all datestN−2,⋯,t0. Finally, an updated payoff is assigned to each trajectory i: exp−rτiOτi or equals to 0 in the case of no exercise. Note that τ∈t0=0,⋯,tN=T is the first time when OStNi≥ϕkptN−1,StN−1i, and as previously proven, this is an optimal exercising time; in the same logic as Section 2.3, we estimate the final price of the option at time t0 by
(21)U02,M=∑i=1Mexp−rτiOSτii.


### 3.2. Example of Pricing American Put and Validation on the Real Market with the Microsoft Stock Option as an Example

We consider an American put with the following parameters:σ=0.2 the stock volatility, S0=80 the price at time t0, T=1 year of the maturity, K=100 the strike, and r=2% the interest rate. For the Lévy jump-diffusion model, we model the duration between jumps with exponential laws and its parameter which is λ1=1/4, i.e., four jumps per one year. We also model the size of jumps with a negative exponential law with parameter λ2=1; i.e., the size of asset price negatively jumps by 1 dollar.According to the number of simulated paths, we illustrate the numerical convergence of the option price and we notice the asymptotic behavior whenM tends to have a large number. For the same algorithm “least square regression,” we compare asymptotically the convergence between the B&S model and the exponential Lévy jump-diffusion model (see Figure 5). We notice for this example that the B&S model already stabilizes the size of the confidence interval when the M tends to have M=5000, while the Lévy jump-diffusion started to stabilize the confidence interval around M=8000; we come back to this phenomenon of the robustness of B&S in terms of stabilization of the size of the confidence interval in an early number of simulated paths compared to the Lévy jump-diffusion model (see Table 1). Table 2 shows the speed of convergence in terms of time.Figure 5
Asymptotic convergence of the least square algorithmM from 100 to 10000. Confidence level in order of 95% (the estimated value of the American option in blue, upper bound in red, and lower bound in green).
(a)
Equilibrium price option under B&SU02,M=19.8$(b)
Equilibrium price option under the Lévy jump-diffusion processU02,M=21.82$Table 1
Confidence interval according to the number of sampling.
TrajectoriesMean BSMin BSMax BSMean jump-difMin jump-difMax jump-difM=100019.888319.810319.966321.742721.642021.8433M=500019.841419.806119.876821.800021.754621.8454M=800019.851419.801119.872121.800021.7774621.8401M=1000019.852819.807919.871121.827021.795021.8390Table 2
Time consumption (MATLAB R2019a program).
Number of samplingB&SJump-diffusion100 to 1000 by 10 steps24.542 s40.544 s100 to 5000 by 10 steps597.303 s1163.004 s100 to 10000 by 10 steps16305.825 s57092.476 sAccording to the central limit theorem and because of the independence of trajectories, the sum of the option price of each path in the Monte Carlo approximation can be estimated by a normal distributionN0,σ/M (Monte Carlo estimator unbiased); hence, U0M/σ/M~N0,1.We propose an estimator of the theoretical tolerance range with a confidence level of 95% by subtracting the upper and lower bounds of the confidence interval according toM.
(22)I^model=uppermodelM−lowermodelM2=2∗1.96∗σM.For example,M=10000I^levy=21.8390−21.7950/2=0.022 which is a very small range error, so theoretically the price option belongs to 21.82−0.022,21.849+0.022 with a confidence level of 95%.Now, the question that arises is whether B&S or exponential Lévy is more effective in predicting prices in the real market.


### 3.3. Validation of the B&S Model Compared to the Exponential Lévy Model on a Real Market with the Microsoft Option as an Example

We notice in Table3 that BS defaults below the spot 230 at time 0, B&S suppose that all paths are out of the money in the meantime 0,T throughout all strikes below 230 80,230, and this comes from the fact that in a complete market, continuous asset prices St with drift are unlikely to fall below 230 from S0=231.02; hence, the American put price is equal to 0. On the other hand, jump-diffusion with negative jumps predicts more efficiently than B&S from S0=80 up to S0=230, and the prediction error remains around 1 dollar. In our case, calibration of the parameters in this spot range is as follows: λ1=−8 which corresponds to the expectation of losses “following the exponential law” of the Microsoft share chosen as the average of share losses between 2020 and 2021” and λ2 which represents the expectation of losses with jump occurrence, chosen equal to 1/3. From strike k=255 to 275 with some calibration (i.e., the assumption of a small positive parameter of jump size “profit jumps” λ1=1) Lévy’s jump-diffusion model outperforms B&S. For k=305,355, B&S is more precise, and it seems that the hypothesis of continuous asset price’s market is more effective for these values. Generally, the mean error of the jump-diffusion model is in the order of 2.8 dollars; on the other hand, B&S mean error is equal to 11.7003, which shows that the Lévy jump-diffusion model outperforms widely B&S to validate our example on the real market.Remark 12.
The use of the improved exponential Lévy jump-diffusion model with hybrid jumps “loss and profit jumps like in Kou model jumps which follows a generalized Laplace nonsymmetric law” [18] can slightly improve our results.Table 3
MSFT put option traded on 19/03/2021 at a price ofS0=231.02, expired on 18/03/2021 (r=4%) (from https://fr.finance.yahoo.com/quote/MSFT?p=MSFT).
StrikeOption priceImplied volatilityB&SErrorJump-diffusionError800.6325%00.630.03380.5921001.112.5%01.10.15570.94431453.5812.5%03.581.28322.29681504.056.25%04.051.83542.21411605.46.25%05.43.02752.37251859.816.25%09.818.52831.281721017.81.56%017.821.79623.996221519.81.56%019.824.8535.05322021.950.78%021.9521.55880.391222524.250.78%024.2525.50891.258923026.60.05%026.627.70621.106225541.480%23.680117.799942.51985.485627051.450%38.656112.793956.93565.485627555.20%43.648111.551957.9354.48530580.530%73.60026.929871.32969.2004355126.680%123.52023.1598121.67775.0023Mean error11.7003Mean error2.8106


## 3.1. Specific Example of the Least Square Regression Algorithm

We consider the case ofp=2 to highlight the ordinary least square regression algorithm at time tN−1; for example, we choose the following form of regressor functions: φ1StN−1=StN−1, φ1StN−1=StN−12, and β=αtN−1,β1,tN−1,β2,tN−1, an estimator of the three least square coefficients which minimize the least square error:
(18)∑i=1M′OStNi−α−β1StN−1i−β2StN−1i22.At each trajectory, the terminal valueOStNi equals to the terminal payoff of the option; as of date tN−1, we estimate the continuation value if the trajectory is in the money and we reject if the trajectory is out of the money, so on each trajectory, we have M′ couple StN−1i,OStNi among M which will explain OStN−1i using the following regressors: 1,φ1StN−1=StN−1,φ2StN−1=StN−12. According to equation (14) at tN−1 on trajectory i, the continuation value is
(19)ϕkptN−1,StN−1i=Ee−rΔtOStNi∣StN−1i=StN−1≅e−rΔtαtN−1−β1,tN−1StN−1i−β2,tN−1StN−1i2.We notice that the intrinsic valuee−rΔtOStNi represents the discounted cash flows earned at time tN in the case where the option is not exercised at time tN−1. Thus, we make a comparison between the payoff OStNi and ϕkptN−1,StN−1i on each trajectory if OStNi≥ϕkptN−1,StN−1i; the early exercise of the option has therefore taken place at that time if it has not taken place previously. Hence, the value of option at date tN−1 on trajectory i is
(20)UtN−12,i=maxOStN−1i,ϕkpStN−1i.We do the same for all datestN−2,⋯,t0. Finally, an updated payoff is assigned to each trajectory i: exp−rτiOτi or equals to 0 in the case of no exercise. Note that τ∈t0=0,⋯,tN=T is the first time when OStNi≥ϕkptN−1,StN−1i, and as previously proven, this is an optimal exercising time; in the same logic as Section 2.3, we estimate the final price of the option at time t0 by
(21)U02,M=∑i=1Mexp−rτiOSτii.


## 3.2. Example of Pricing American Put and Validation on the Real Market with the Microsoft Stock Option as an Example

We consider an American put with the following parameters:σ=0.2 the stock volatility, S0=80 the price at time t0, T=1 year of the maturity, K=100 the strike, and r=2% the interest rate. For the Lévy jump-diffusion model, we model the duration between jumps with exponential laws and its parameter which is λ1=1/4, i.e., four jumps per one year. We also model the size of jumps with a negative exponential law with parameter λ2=1; i.e., the size of asset price negatively jumps by 1 dollar.According to the number of simulated paths, we illustrate the numerical convergence of the option price and we notice the asymptotic behavior whenM tends to have a large number. For the same algorithm “least square regression,” we compare asymptotically the convergence between the B&S model and the exponential Lévy jump-diffusion model (see Figure 5). We notice for this example that the B&S model already stabilizes the size of the confidence interval when the M tends to have M=5000, while the Lévy jump-diffusion started to stabilize the confidence interval around M=8000; we come back to this phenomenon of the robustness of B&S in terms of stabilization of the size of the confidence interval in an early number of simulated paths compared to the Lévy jump-diffusion model (see Table 1). Table 2 shows the speed of convergence in terms of time.Figure 5
Asymptotic convergence of the least square algorithmM from 100 to 10000. Confidence level in order of 95% (the estimated value of the American option in blue, upper bound in red, and lower bound in green).
(a)
Equilibrium price option under B&SU02,M=19.8$(b)
Equilibrium price option under the Lévy jump-diffusion processU02,M=21.82$Table 1
Confidence interval according to the number of sampling.
TrajectoriesMean BSMin BSMax BSMean jump-difMin jump-difMax jump-difM=100019.888319.810319.966321.742721.642021.8433M=500019.841419.806119.876821.800021.754621.8454M=800019.851419.801119.872121.800021.7774621.8401M=1000019.852819.807919.871121.827021.795021.8390Table 2
Time consumption (MATLAB R2019a program).
Number of samplingB&SJump-diffusion100 to 1000 by 10 steps24.542 s40.544 s100 to 5000 by 10 steps597.303 s1163.004 s100 to 10000 by 10 steps16305.825 s57092.476 sAccording to the central limit theorem and because of the independence of trajectories, the sum of the option price of each path in the Monte Carlo approximation can be estimated by a normal distributionN0,σ/M (Monte Carlo estimator unbiased); hence, U0M/σ/M~N0,1.We propose an estimator of the theoretical tolerance range with a confidence level of 95% by subtracting the upper and lower bounds of the confidence interval according toM.
(22)I^model=uppermodelM−lowermodelM2=2∗1.96∗σM.For example,M=10000I^levy=21.8390−21.7950/2=0.022 which is a very small range error, so theoretically the price option belongs to 21.82−0.022,21.849+0.022 with a confidence level of 95%.Now, the question that arises is whether B&S or exponential Lévy is more effective in predicting prices in the real market.


## 3.3. Validation of the B&S Model Compared to the Exponential Lévy Model on a Real Market with the Microsoft Option as an Example

We notice in Table3 that BS defaults below the spot 230 at time 0, B&S suppose that all paths are out of the money in the meantime 0,T throughout all strikes below 230 80,230, and this comes from the fact that in a complete market, continuous asset prices St with drift are unlikely to fall below 230 from S0=231.02; hence, the American put price is equal to 0. On the other hand, jump-diffusion with negative jumps predicts more efficiently than B&S from S0=80 up to S0=230, and the prediction error remains around 1 dollar. In our case, calibration of the parameters in this spot range is as follows: λ1=−8 which corresponds to the expectation of losses “following the exponential law” of the Microsoft share chosen as the average of share losses between 2020 and 2021” and λ2 which represents the expectation of losses with jump occurrence, chosen equal to 1/3. From strike k=255 to 275 with some calibration (i.e., the assumption of a small positive parameter of jump size “profit jumps” λ1=1) Lévy’s jump-diffusion model outperforms B&S. For k=305,355, B&S is more precise, and it seems that the hypothesis of continuous asset price’s market is more effective for these values. Generally, the mean error of the jump-diffusion model is in the order of 2.8 dollars; on the other hand, B&S mean error is equal to 11.7003, which shows that the Lévy jump-diffusion model outperforms widely B&S to validate our example on the real market.Remark 12.
The use of the improved exponential Lévy jump-diffusion model with hybrid jumps “loss and profit jumps like in Kou model jumps which follows a generalized Laplace nonsymmetric law” [18] can slightly improve our results.Table 3
MSFT put option traded on 19/03/2021 at a price ofS0=231.02, expired on 18/03/2021 (r=4%) (from https://fr.finance.yahoo.com/quote/MSFT?p=MSFT).
StrikeOption priceImplied volatilityB&SErrorJump-diffusionError800.6325%00.630.03380.5921001.112.5%01.10.15570.94431453.5812.5%03.581.28322.29681504.056.25%04.051.83542.21411605.46.25%05.43.02752.37251859.816.25%09.818.52831.281721017.81.56%017.821.79623.996221519.81.56%019.824.8535.05322021.950.78%021.9521.55880.391222524.250.78%024.2525.50891.258923026.60.05%026.627.70621.106225541.480%23.680117.799942.51985.485627051.450%38.656112.793956.93565.485627555.20%43.648111.551957.9354.48530580.530%73.60026.929871.32969.2004355126.680%123.52023.1598121.67775.0023Mean error11.7003Mean error2.8106


## 4. Conclusion

In this paper, we have presented a comparison between the least square algorithm under B&S and exponential Lévy jump-diffusion processes in terms of time-consuming and asymptotic convergence properties depending on the number of simulated trajectories, and we pointed out that B&S converges faster and its option price variance is less volatile than that of Lévy’s jump-diffusion model which makes the calculation more easier and faster, which is useful in the case of multidimensional underlying assets. However, as we have pointed out in Sections2.3.1 and 3.3, the Lévy model stays more adequate than B&S in our validation example on Microsoft’s put option. We have also proposed a simple method to estimate theoretical tolerated range according to the confidence level in Section 3.2. We also noted that by varying the number of paths, we numerically highlighted the equilibrium option price.More calibrations can perform better our exponential Lévy jump-diffusion model as done in the context European option of the S&P100 index in this paper [16].Recent work on machine learning and the parallelism method for solving BSDEs opens a horizon as in [19] where the author tried to cluster the computational method using machine learning instead of using a brute force approximation of the true solution, which is time-consuming as in [20, 21].



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